iii Remerciements J'ai eu la grande chance (tout au long de ma thèse) d'être soutenue par des chercheurs de renommée internationale. J'ai donc toutes les raisons pour me considérer privilegiée. Maintenant j'ai une belle occasion d'adresser un grand "Merci " aux personnes qui ont contribué à ma formation professionnelle et humaine dans l'étape doctorale.

Je tiens à remercier Bruno Bouchard, parce qu'il a dirigé mes activités de recherche dans le cadre du groupe de travail dans la seconde année de master et (ensuite) ma thèse. Il m'a profondément influencée par sa personnalité scientifique et par sa manière exigeante et amicale en même temps de travailler avec les jeunes chercheurs. Ses nombreux conseils en mathématiques, toujours très judicieux, ses encouragements, la confiance qu'il m'a manifestée et sa constante gentillesse m'ont donné l'envie de me perfectionner, de progresser chaque jour. Je tiens également à le remercier de m'avoir offert la très honorante opportunité de donner les travaux dirigés pour son cours de M2 (finance mathématique du Master Masef), ainsi que de m'avoir impliquée dans l'organisation du Séminaire Bachelier.

Je remercie également Romuald Elie pour son investissement dans mon travail, pour le temps qu'il m'a accordé avec grande générosité. Il a contribué pleinement, par ses idées toujours stimulantes, au developpement de mon goût et de mes connaissances en finance mathématique.

Une très profonde gratitude se dirige vers Agnès Sulem que j'ai eu la grande chance de rencontrer comme Professeur pendant le M2. Elle a accepté de diriger mon stage de recherche (du master 2) et, ulterieurement, elle m'a adressé une proposition inesperée : celle de poursuivre des travaux de recherche dans l'équipe Mathrisk, ce qui fut pour moi une réelle opportunité. En tant que directrice de Mathrisk, elle a suivi très attentivement mon évolution professionnelle, elle m'a proposé des sujets variés (sur le plan théorique et numérique) et m'a offert la très honorante possibilité de travailler avec elle et d'intéragir avec d'autres chercheurs. Je la remercie pour son inconditionnelle disponibilité à mon égard et pour toute sa confiance: délicate, très chaleureuse et ferme en même temps, Agnès Sulem a toujours su me donner les meilleurs conseils afin que je puisse m'intégrer dans le monde (si competitif) de la recheche francaise en mathématiques financières.

A son tour, Marie-Claire Quenez a toujours été près de moi, en me soutenant avec un enthousiame qui m'a profondément touchée. Le travail avec Marie-Claire Quenez m'a beaucoup apporté du point de vue scientifique: ses connaissances, son esprit pédagogique et sa rigueur m'ont influencé d'une manière substantielle dans ma formation de jeune mathématicien. C'est un réel plaisir et une grande chance de travailler avec elle. Je lui serai toujours reconnaissante pour sa gentillesse, sa confiance et pour tout le temps qu'elle m'a accordé sans jamais le compter.

Pendant ma thèse, j'ai aussi eu la chance de travailler avec des chercheurs plus jeunes: je tiens les remercier pour leur soutien. Je pense en particulier à Céline Labart: la collaboration avec elle a été réellement très enrichissante. Je remercie aussi Dylan Possamai d'avoir accepté de travailler avec moi.

J'adresse mes plus sincères remerciements à Nizar Touzi qui m'a fait l'honneur d'être examinateur dans mon jury de thèse, ainsi qu'à Dirk Becherer, Philippe Briand et Rainer Buckdahn, pour tout le temps investi à rapporter ma thèse et pour leurs précieux conseils.

Je ne peux pas oublier d'autres personnes qui ont représenté des repères importants dans mon activité, leur permanente disponiblité et leur constant appui étant très importants pour moi: Massimiliano Gubinelli (directeur du programme doctoral en mathématiques), Jean-Yves Chemin et Jean Dolbeault (directeurs de la Fondation de Sciences Mathématiques), Vlad Bally (professeur à Marne-la-Vallée et membre de l'équipe Mathrisk) et Gabriel Turinici (professeur à Dauphine).

Je suis aussi très reconnaissante envers Dan Crisan et Mete Soner de m'avoir accuellie à l'Imperial College et ETH Zurich, ce qui m'a permis de gagner une riche experience.

Je tiens à remercier très chaleureusement l'équipe de la Fondation de Sciences Mathématiques, avec laquelle j'ai eu une très belle collaboration dès ma premiére année de master et également à Juliette Bavard et Camille Male pour leur soutien et implication dans la création de l'ALUMNI de la FSMP.

Mes meilleures pensées se dirigent aussi vers Adélaide Olivier, avec laquelle j'ai organisé le Séminaire des Jeunes Chercheurs du Ceremade et j'ai partagé de nombreux beaux moments. Evidemment, je ne peux pas oublier les autres doctorants qui one été proches de moi pendant ces trois années de thèses, pour leur chaleureuse amitié (je pense en particulier à Kaniav, Yiyi, Nicolas, Viviana, Géraldine, Ahmed, Fang, Jean-Luc ...).

Je n'aurais pas pu arriver à ce point sans la contribution essentielle de mes professeurs de la Faculté de Mathématiques de l'Université "Al. I. Cuza", Iasi (Roumanie) et je les remercie pour toute leur confiance. Ma plus sincère reconaissance se dirige vers l'Académicien Viorel Barbu, qui a dirigé mon travail de recherche pendant la licence et qui m'a toujours encouragée de continuer avec une carrière de haut niveau en mathématiques.

Je ne peux pas finir ces remerciements sans penser à ma famille, qui a toujours cru en moi et m'a toujours offert un amour inconditionnel. v Résumé Cette thèse se compose de deux parties indépendantes qui portent sur le contrôle stochastique avec des espérances non linéaires et les équations stochastiques rétrogrades (EDSR), ainsi que sur les méthodes numériques de résolution de ces équations. Dans la première partie on étudie une nouvelle classe d'équations stochastiques rétrogrades, dont la particularité est que la condition terminale n'est pas fixée mais vérifie une contrainte non linéaire exprimée en termes de "f -espérances". Ce nouvel objet mathématique est étroitement lié aux problèmes de couverture approchée des options européennes où le risque de perte est quantifié en termes de mesures de risque dynamiques, induites par la solution d'une EDSR non linéaire. Dans le chapitre suivant on s'intéresse aux problèmes d'arrêt optimal pour les mesures de risque dynamiques avec sauts. Plus précisément, on caractérise dans un cadre markovien la mesure de risque minimale associée à une position financière comme l'unique solution de viscosité d'un problème d'obstacle pour une équation intégro-différentielle. Dans le troisième chapitre, on établit un principe de programmation dynamique faible pour un problème mixte de contrôle stochastique et d'arrêt optimal avec des espérances non linéaires, qui est utilisé pour obtenir les EDP associées. La spécificité de ce travail réside dans le fait que la fonction de gain terminal ne satisfait aucune condition de régularité (elle est seulement considérée mesurable), ce qui n'a pas été le cas dans la littérature précédente. Dans le chapitre suivant, on introduit un nouveau problème de jeux stochastiques, qui peut être vu comme un jeu de Dynkin généralisé (avec des espérances non linéaires). On montre que ce jeu admet une fonction valeur et on obtient des conditions suffisantes pour l'existence d'un point selle. On prouve que la fonction valeur correspond à l'unique solution d'une équation stochastique rétrograde doublement réfléchie avec un générateur non linéaire général. Cette caractérisation permet d'obtenir de nouveaux résultats sur les EDSR doublement réfléchies avec sauts. Le problème de jeu de Dynkin généralisé est ensuite étudié dans un cadre markovien. Dans la deuxième partie, on s'intéresse aux méthodes numériques pour les équations stochastiques rétrogrades doublement réfléchies avec sauts et barrières irrégulières, admettant des sauts prévisibles et totalement inaccessibles. Dans un premier chapitre, on propose un schéma numérique qui repose sur la métode de pénalisation et l'approximation de la solution d'une EDSR par une suite d'EDSR discrètes dirigées par deux arbres binomiaux indépendants (un qui approxime le mouvement brownien et l'autre le processus de Poisson composé). Dans le deuxième chapitre, on construit un schéma en discrétisant directement l'équation stochastique rétrograde doublement réfléchie, schéma qui présente l'avantage de ne plus dépendre du paramètre de pénalisation. On prouve la convergence des deux schémas numériques et on illustre avec des exemples numériques les résultats théoriques.

Abstract

This thesis consists of two independent parts which deal with stochastic control with nonlinear expectations and backward stochastic differential equations (BSDE), as well as with the numerical methods for solving these equations. We begin the first part by introducing and studying a new class of backward stochastic differential equations, whose characteristic is that the terminal condition is not fixed, but only satisfies a vi nonlinear constraint expressed in terms of "f -expectations". This new mathematical object is closely related to the approximative hedging of an European option, when the shortfall risk is quantified in terms of dynamic risk measures, induced by the solution of a nonlinear BSDE. In the next chapter we study an optimal stopping problem for dynamic risk measures with jumps. More precisely, we characterize in a Markovian framework the minimal risk measure associated to a financial position as the unique viscosity solution of an obstacle problem for partial integrodifferential equations. In the third chapter, we establish a weak dynamic programming principle for a mixed stochastic control problem / optimal stopping with nonlinear expectations, which is used to derive the associated PDE. The specificity of this work consists in the fact that the terminal reward does not satisfy any regularity condition (it is considered only measurable), which was not the case in the previous literature. In the next chapter, we introduce a new game problem, which can be seen as a generalized Dynkin game ( with nonlinear expectations ). We show that this game admits a value function and establish sufficient conditions ensuring the existence of a saddle point . We prove that the value function corresponds to the unique solution of a doubly reflected backward stochastic equation (DRBSDE) with a nonlinear general driver. This characterisation allows us to obtain new results on DRBSDEs with jumps. The generalized Dynkin game is finally addressed in a Markovian framework. In the second part, we are interested in numerical methods for doubly reflected BSDEs with jumps and irregular barriers, admitting both predictable and totally inaccesibles jumps. In the first chapter we provide a numerical scheme based on the penalisation method and the approximation of the solution of a BSDE by a sequence of discrete BSDEs driven by two independent random walks (one approximates the Brownian motion and the other one the compensated Poisson process). In the second chapter, we construct an alternative scheme based on the direct discretisation of the DRBSDE, scheme which presents the advantage of not depending anymore on the penalisation parameter. We prove the convergence of the two schemes and illustrate the theoretical results with some numerical examples.

Chapter 1

General Introduction 1.1 BSDEs with nonlinear weak terminal condition

This chapter is based on a paper written under the coordination of B. Bouchard1 and submitted for publication: "BSDEs with nonlinear weak terminal condition" [START_REF] Dumitrescu | BSDEs with nonlinear weak terminal condition[END_REF].

Preliminaries and overview of the literature

We start by recalling that a Backward Stochastic Differential Equation (in short BSDE) is an equation which takes the following form

Y t = ξ + T t g(s, Y s , Z s )ds - T t Z s dW s , 0 ≤ t ≤ T, (1.1.1)
where {W t } 0≤t≤T is a Brownian motion defined on a probability space endowed with the natural complete filtration denoted by {F t } 0≤t≤T . The data of a such equation are given by the terminal condition ξ, which is a random variable F T -measurable, valued in R and a driver g, a random map defined on [0, T ] × Ω × R × R and valued in R, which is measurable with respect to the σ-algebras P ⊗ B(R) ⊗ B(R) and B(R), where P represents the predictable σ-algebra. To solve this equation means to find a couple of processes {(Y t , Z t )} 0≤t≤T satisfying equation (1.1.1) and adapted with respect to the filtration generated by the Brownian motion. We give below a more precise definition.

Definition 1.1.1. The solution of a BSDE is a couple of processes (Y, Z) valued in R×R such that Y is continuous and adapted, Z is predictable and P-a.s, t → Z t belongs to L 2 (0, T ), t → g(t, Y t , Z t ) belongs to L 1 (0, T ) and

Y t = ξ + T t g(s, Y s , Z s )ds - T t Z s dW s , 0 ≤ t ≤ T. (1.1.2)
The BSDEs have been first introduced in the case of a linear driver by J.-M.Bismut [START_REF] Bismut | Conjugate convex functions in optimal stochastic control[END_REF]. The starting point of the theory of nonlinear backward equations is the paper of E. Pardoux and S. Peng [START_REF] Pardoux | Adapted solution of a backward stochastic differential equation[END_REF], since the authors consider BSDEs with nonlinear generator in (y, z). Let us recall this result:

Theorem 1.1.2 (E.Pardoux -S. Peng). Suppose that the driver g is lipschitz in (y, z), uniformly with respect to (t, ω) and

E[|ξ| 2 + T 0 |g(s, 0, 0)| 2 ds] < +∞.
Then the BSDE (1.1.1) admits an unique solution (Y, Z) belonging to S 2 × H 2 .

Expressed in a "forward form", the resolution of such equations boils down in finding an initial condition Y 0 and a process Z such that the controlled process (Y Y 0 ,Z t ) 0≤t≤T satisfies the SDE:

Y Y 0 ,Z t = Y 0 - t 0 g(s, Y Y 0 ,Z s , Z s )ds + t 0 Z s dW s (1.1.3)
and the condition Y Y 0 ,Z T = ξ at the terminal time T . From a financial application point of view, the study of these equations is related to the pricing of European options in complete markets, since Y gives the price and Z provides the associated hedging strategy. However, since in incomplete markets it is not always possible to construct a replicating portfolio such that its terminal value coincides with the price of the claim ξ, a weaker formulation is to find an initial condition Y 0 and a control Z such that

Y Y 0 ,Z T ≥ ξ.
(1. 1.4) In this case, one is interested in finding the minimal initial condition Y 0 , which corresponds to the cost of the cheapest super-replication strategy for the contingent claim ξ and the associated control Z (see e.g. [START_REF] Karoui | Backward Stochastic Differential Equations in Finance[END_REF]). Since in most cases, the super-hedging price leads to an unbearble cost for the buyer, which is not reasonable in practice, it was suggested to relax the strong constraint (1.1.4) into a weaker one of the form

E[l(Y Y 0 ,Z T -ξ)] ≥ m, (1.1.5)
where m is a given threshold and l is a non-decreasing map. For l(x) = 1 {x≥0} , this corresponds to matching the criteria Y Y 0 ,Z T ≥ ξ at least with probability m and corresponds to the quantile hedging problem introduced by Föllmer and Leukert [START_REF] Follmer | Quantile hedging[END_REF]. Then, this problem has been studied by Bouchard, Elie and Touzi [START_REF] Bouchard | Stochastic target problems with controlled loss[END_REF] in a Markovian framework, using the stochastic target techniques developed by Soner and Touzi (see [START_REF] Soner | Stochastic target problems, dynamic programming, and viscosity solutions[END_REF]). This approach, based on the primal formulation of the value function and the geometric dynamic programming, allows one for a treatement of this problem in a more general framework, e.g. when the strategy of the agent may influence the value of the risky assets (large investor model). The original treatment of the problem by Föllmer and Leukert relies on the fact that this strategy is linear in the control.

More generally, l may represent a loss function, a classical example being l(x) := -(x -) q with q ≥ 1, see [START_REF] Follmer | Efficient hedging: cost versus shortfall risk[END_REF] for general non-Markovian but linear dynamics. Another example in financial mathematics could be represented by the case when l plays the role of an utility function. Very recently, Bouchard, Elie and Reveillac [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF] have addressed this problem in a nonlinear non-Markovian setting and to this purpose they introduce a new class of BSDEs whose terminal condition is not fixed as a random variable, but only satisfies the following weak constraint

E[Ψ(Y Y 0 ,Z T )] ≥ m.
(1. 1.6) The problem can be thus formulated as follows:

Find the minimal Y 0 such that (1.1.3) and (1.1.6) hold for some Z.

(1.1.7)

The key idea is to "transpose" problem (1.1.7) written in terms of BSDEs with weak terminal condition into an equivalent one, expressed as an optimization problem on the solutions of a family of BSDEs with strong terminal conditions, indexed by an additional control α, as we shall explain in the sequel. In order to do it, the authors appeal to the martingale representation theorem. More precisely, if Y 0 and Z are such that (1.1.6) holds, then the martingale Theorem implies that we can find an element α in the set A 0 of predictable square integrable processes, such that

Ψ(Y Y 0 ,Z T ) ≥ M m,α T := m + T 0 α s dW s . (1.1.8)
Since Ψ is non-decreasing, one can define its left-continuous inverse Φ and we get that the solution (Y α , Z α ) of the following BSDE

Y α t = Φ(M m,α T ) + T t g(s, Y α s , Z α s )ds - T t Z α s dW s , 0 ≤ t ≤ T, (1.1.9)
solves (1.1.3) and (1.1.6). It is finally proved that the solution of (1.1.7) is given by

Y 0 (m) := inf{Y α 0 , α ∈ A 0 }. (1.1.10)
We would like to point out that in a Markovian setting, it is used the same idea of introducing an additional process M and control α and the difficulty relies on the fact that α can take unbounded values, since it comes from a martingale representation Theorem. Now, in order to study (1.1.10), the authors make the problem dynamic and define Y α (τ ) := essinf{Y α ′ τ , α ′ ∈ A 0 s.t. α ′ = α on [[0, τ ]]}, 0 ≤ τ ≤ T.

(1. 1.11) It is shown that the family {Y α , α ∈ A 0 } satisfies a dynamic programming principle, which can be seen as a counterpart of the geometric dynamic programming principle. It is then provided a representation of the family {Y α , α ∈ A 0 } in terms of minimal supersolutions to a family of BSDEs with driver g and (strong) terminal conditions {Φ(M m,α T ), α ∈ A 0 }, as well as the existence of an optimal control in the case when g and Φ are convex. Some main properties of the value function given by (1.1.10), as continuity and convexity with respect to the threshold m are obtained. Finally, by using only probabilistic arguments, it is shown that Problem (1.1.10) admits a dual representation which takes the form of a stochastic control problem in Meyer form, extending the results obtained in the case when the driver g is linear (see [START_REF] Bouchard | Stochastic target problems with controlled loss[END_REF], [START_REF] Follmer | Quantile hedging[END_REF] and [START_REF] Follmer | Efficient hedging: cost versus shortfall risk[END_REF]).

Contributions

In this Chapter, we introduce a more general class of BSDEs than the one considered by Bouchard, Elie and Reveillac [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF], whose terminal condition satisfies the following nonlinear weak constraint:

E f 0,T [Ψ(Y Y 0 ,Z T )] ≥ m, (1.1.12)
where E f [ξ] is the nonlinear operator which gives the solution of the BSDE associated to the terminal condition ξ and the nonlinear driver f . We can easily remark that (1.1.6) represents a particular case of (1.1.12) for f = 0. The problem under study in this paper is the following: inf{Y 0 such that ∃Z : (1.1.3) and (1.1.12) hold}.

(1. 1.13) Following the key idea of [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF], we rewrite our problem (1.1.13) into an equivalent one expressed in terms of a family of BSDEs with strong terminal condition. The main difference with respect to [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF] is given by the fact that in our case we have to introduce a new controlled diffusion process, which is an f -martingale, contrary to [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF] where it is a classical martingale. Indeed, for a given Y 0 and Z such that (1.1.3) and (1.1.12) are satisfied, using the BSDE representation of Ψ(Y Y 0 ,Z T ), we can find α ∈ A 0 such that:

Ψ(Y Y 0 ,Z T ) ≥ M m,α T = m - T 0 f (s, M m,α s , α s )ds + T 0 α s dW s .
(1. 1.14) Thanks to this observation, we show that Problem (1.1.13) is equivalent to (1.1.10), where, in our more general framework, Y α t corresponds to the solution at time t of the BSDE with (strong) terminal condition Φ(M m,α T ). We study the dynamical counterpart of (1.1.10):

Y α (τ ) := essinf{Y α ′ τ , α ′ ∈ A 0 s.t. α ′ = α on [[0, τ ]]}. (1.1.15)
We carry out a similar analysis as in [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF] of the family {Y α , α ∈ A 0 }. We start by studying for each α ∈ A 0 the regularity of the family {Y α (τ ), τ ∈ T }. More precisely, we show that it can be aggregated into a right-continuous process, result which requires in our case some subtle arguments of stochastic analysis due to the nonlinearity of the driver f . We then provide that Y α corresponds to the unique minimal solution of a BSDE. We show that our value function is continuous and convex (in a probabilistic sense) with respect to the threshold m. In the case of a concave driver f , we obtain the existence of an optimal control, as well as a dual a representation. Indeed, we prove that Y 0 (m) (defined as in (1.1.10)) corresponds to the Fenchel transform of the value function of the following stochastic control problem, that is Y 0 (m) = sup l>0 (lm -X 0 (l)), where

X 0 (l) := inf (λ,γ)∈U ×V X l,λ,γ 0 , (1.1.16)
with

X l,λ,γ 0 := E T 0 L λ s g(s, λ s )ds - T 0 A l,γ s f (s, γ s )ds + L λ T Φ( A l,γ T L λ T ) ,
with f ( respectively g, Φ) the concave conjugate of f ( respectively the convex conjugates of g and Φ).

The additional nonlinearity f raises important and subtle technical difficulties, since most of the results in [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF] are provided using specific techniques to the case of linear constraints, which cannot be adapted to our nonlinear setting. Besides the mathematical interest of our study, this work is also motivated by some financial applications. Indeed, our problem is closely related to the approximative hedging under dynamic risk measures contraints of an European option, which can be expressed in the following form:

inf{Y 0 such that ∃ Z : ρ 0,T [-(Y T -ξ) -] ≤ m}, (1.1.17)
where ρ t,T [ξ] represents the risk measure at time t of ξ which is defined as -E f t,T [ξ]. Note that in the case of a nonlinear concave driver, the associated dynamic risk measure is convex. More details concerning the design of risk measures in a dynamic setting by means of backward stochastic differential equations are presented in the next chapter.

Optimal Stopping for Dynamic Risk measures with jumps and obstacle problems

This chapter is based on the paper "Optimal Stopping for Dynamic Risk measures with jumps and obstacle problems" [START_REF] Dumitrescu | Optimal Stopping for Dynamic Risk Measures with Jumps and Obstacle Problems[END_REF], joint work with M.C. Quenez and A. Sulem, J. Optim. Theory Applic.(2014) DOI 10.1007/s10957-014-0636-2.

Preliminaries and overview of the literature

In the first chapter, we have introduced the Backward Stochastic Differential Equations in the case of a Brownian filtration, which can be seen as a generalization of the conditional expectation of a random variable ξ, since when the driver g is the null function, we have Y t = E[ξ|F t ], and in that case, Z is the process appearing in (F t ) t≥0 -martingale representation property of (E[ξ|F t ]) t≥0 .

In the case of a filtered probability space generated by both a Brownian Motion W and a Poisson random measure N with compensator ν, the martingale representation of (E[ξ|F t ]) t≥0 becomes:

E[ξ|F t ] = ξ + T t Z s dW s + T t R *
U s (e)(Nν)(de, ds), Pa.s.,

where U is a predictable function. This leads to the following natural generalization of equation (1.1.1) to the case of jumps. We will say that (Y, Z, U ) is a solution of the BSDE with jumps (BSDEJ in short) with generator f and terminal condition ξ if for all t ∈ [0, T ] we have P-a.s.

Y t = ξ + T t g(s, Y s , Z s , U s )ds - T t Z s dW s - T t R *
U s (e)(Nν)(de, ds), 0 ≤ t ≤ T. (1.2.1)

In 1994, Tang and Li [START_REF] Tang | Necessary conditions for optimal control of stochastic systems with random jumps[END_REF] were the first to prove existence and uniqueness of a solution for (1.2.1) in the case when g is Lipschitz in (y, z, u).

The case of a discontinuous framework is more involved, especially concerning the comparison theorem which requires an additional assumption. In 1995, Barles, Buckdahn, Pardoux [START_REF] Barles | Backward stochastic differential equations and integralpartial differential equations[END_REF] provided a comparison theorem as well as some links between BSDEs and non-linear parabolic integral-partial differental equations, generalizing some result of [START_REF] Pardoux | Backward Stochastic Differential equations and Quasilinear Parabolic Partial Differential equations[END_REF] to the case of jumps. In 2006, Royer [START_REF] Royer | Backward stochastic differential equations with jumps and related non-linear expectations[END_REF] proved a comparison theorem under weaker assumptions, and introduced the nonlinear expectations in this framework.

Furthermore, in 2004-2005, various authors have introduced dynamic risk measures in a Brownian framework, defined as the solutions of BSDEs. More precisely, given a Lipschitz driver g(t, x, π) and a terminal time T , the risk measure ρ at time t of a position ξ is given by -X t , where X is the solution of the BSDE driven by a Brownian motion, associated with g and terminal condition ξ. By the comparison theorem, ρ satisfies the monotonicity property, which is usually required for a risk measure. Many studies have been recently done on such dynamic risk measures, especially concerning robust optimization problems and optimal stopping problems, in the case of a Brownian filtration and a concave driver (see, among others, Bayraktar and coauthors in [START_REF] Bayraktar | Optimal stopping for Dynamic Convex Risk Measures[END_REF]). In the case with jumps, the links between BSDEs and dynamic risk measures have been recently studied by Quenez-Sulem in [START_REF] Quenez | BSDEs with jumps, optimization and applications to dynamic risk measures[END_REF].

Reflected backward stochastic differential equations (RBSDEs in short) have been introduced in 1997 by the five authors El Karoui, Kapoudjian, Pardoux, Peng and Quenez [START_REF] El Karoui | Reflected solutions of backward SDE's, and related obstacle problems for PDE's[END_REF] in the case of a filtration generated by the Brownian motion. These equations are generalisations of the deterministic Skorokhod problem. Indeed, given an adapted process ξ := (ξ t ) t≤T which plays the role of the barrier, the solution of a RBSDE associated to data (η, g, ξ) is a triplet of square integrable processes {(Y t , Z t , A t ); 0 ≤ t ≤ T } which satisfy:

             Y t = η + T t g(s, ω, Y s , Z s )ds + A T -A t - T t Z s dW s , 0 ≤ t ≤ T, Y t ≥ ξ t , T 0 (Y t -ξ t )dA t = 0. (1.2.2)
where A is a continuous, increasing process whose role is to push the solution Y such it remains above the barrier ξ. The condition T 0 (Y tξ t )dA t = 0 ensures that the process A acts in a minimal way. More precisely, A increases only on the set {Y = ξ}.

The development of reflected BSDEs has been motivated in particular by two applications: the pricing and hedging of American options, especially in markets with constraints, and the probabilistic representation of solutions of obstacle problems for nonlinear PDEs .

Concerning the first application in financial mathematics, El Karoui, Pardoux and Quenez were the first to show that in a complete market, the price of an American option with underlying asset (ξ t ) t≤T and exercise price γ is given by Y 0 where (Y t , π t , A t ) t≤T is the solution of the following reflected BSDE:

     -dY t = b(t, Y t , π t )dt + dA t -π t dW t , Y T = (ξ T -γ) + , Y t ≥ (ξ t -γ) + and T 0 (Y t -(ξ t -γ) + )dA t = 0, (1.2.3)
for a particular choice of b. The process π gives us the replication strategy and A is the buyer's consumption process. In a standard financial market, the function b is given by b(t, ω, y, z) = r t y +zθ t , where θ t is the risk premium and r t represents the interest rate of investement or borrowing.

The generalization to the case of reflected BSDEs with jumps, which is a standard reflected BSDE driven by a Brownian motion and an independent Poisson random measure, has been established by Hamadène and Ouknine in [START_REF] Hamadène | Reflected backward stochastic differential equation with jumps and random obstacle[END_REF]. A solution for such equation, associated with a coefficient f , terminal value η and a barrier ξ, is a quadruple of process (Y, Z, U, A) of adapted solutions which satisfy the following equation:

                     Y t = η + T t g(s, ω, Y s , Z s , U s )ds + A T -A t - T t Z s dW s - T t R *
U s (e) Ñ (ds, de), 0 ≤ t ≤ T, Y t ≥ ξ t , T 0 (Y tξ t )dA t = 0.

(1.2.4)

Using two methods -the first one based on the penalization argument and the second one on the snell envelope theory -, the authors have shown the existence and uniqueness of solutions if η is square integrable, g is uniformly lipschitz with respect to (y, z, u) and the barrier ξ is right continuous left-limited (RCLL for short) whose jumping times are inaccessible stopping times. Note that this later condition played a crucial role in their proofs. In this case, the jumping times of the process Y come only from those of its Poisson process and then they are inaccessible.

The general case of RBSDEs with jumps and irregular obstacles has been considered e.g. in [START_REF] Essaky | Reflected backward stochastic differential equation with jumps and RCLL obstacle[END_REF] and more recently by Quenez-Sulem [START_REF] Quenez | Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps[END_REF]. The barrier ξ is just rcll and thus the jumping times of process Y come not only from those of its Poisson process (inaccessible jumps) but also from those of the process ξ (predictable jumps), which means that the process Y has two types of jumps: inaccessible and predictable ones. The difficulty here lies in the fact that since the barrier ξ is allowed to have predictable jumps then the reflecting process A is no longer continuous but just RCLL. In this case, the difference with respect to (1.2.4) only appears in the Skorokhod condition which becomes: T 0 (Y t -ξ t -)dA t = 0.

An important application of reflected BSDEs is its connection to optimal stopping problems and its associated variational inequalities in the Markovian case. More precisely, given an RCLL process (ξ t , 0 ≤ t ≤ T ) and a Lipschitz driver g satisfying the additional assumption such that the comparison theorem holds, the solution Y of the associated RBSDE satisfies: for each stopping time S ∈ T 0 , Y S = esssup τ ∈T S X S (ξ τ , τ ), a.s.

(1.2.5)

where for τ ∈ T S , X • (ξ τ , τ ) is the solution of the BSDE associated with terminal time τ , terminal condition ξ τ , and driver g (see [START_REF] Quenez | Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps[END_REF]). Note that T S represents the set of stopping times with values in [0, T ], a.s. greater than S.

Contributions

In this chapter, we study the optimal stopping problem for dynamic risk measures with jumps in a Markovian framework. Let us first formulate our problem. Let T > 0 be the terminal time and f be a Lipschitz driver. For each T ′ ∈ [0, T ] and η ∈ L 2 (F T ′ ), set:

ρ f t (η, T ′ ) = ρ t (η, T ′ ) := -X t (η, T ′ ), 0 ≤ t ≤ T ′ , (1.2.6)
where X t (η, T ′ ) denotes the solution of the BSDE with driver g and terminal conditions (T ′ , η). If T ′ represents a given maturity and η a financial position at time T ′ , then ρ(η, T ′ ) is interpreted as the risk of η at time t. The functional ρ : (η, T ′ ) → ρ • (η, T ′ ) thus represents a dynamic risk measure induced by the BSDE with driver f . Let (ξ t , 0 ≤ t ≤ T ) be an RCLL adapted process in S 2 ( which denotes the set of processes φ such that E[sup t≤T φ 2 t ] ≤ +∞ ), representing a dynamic financial position. Let S ∈ T 0 . The problem is to minimize the risk measure at time S. Let v(S) be the associated value function, equal to the F S -measurable random variable (unique for the equality in the almost sure sense) defined by v(S) := essinf τ ∈T S ρ S (ξ τ , τ ).

(1.2.7)

This random variable v(S) corresponds to the minimal risk measure at time S. Since by definition ρ S (ξ τ , τ ) = -X S (ξ τ , τ ), we have, for each stopping time S ∈ T 0 ,

v(S) = essinf τ ∈T S -X S (ξ τ , τ ) = -esssup τ ∈T S X S (ξ τ , τ ).
(1.2.8)

Now, using the link between reflected BSDEs and optimal stopping (3.2.4), one can relate the value function of the problem defined by (2.2.8) to the solution of the reflected BSDE. More precisely, we have:

v(S) = -Y S .
(1.2.9)

Since our aim is to characterize this value function in a Markovian framework, we consider the terminal condition, obstacle and driver of the following form:

       ξ t,x
s := h(s, X t,x s ), s < T, ξ t,x T := g(X t,x T ), g(s, ω, y, z, k) := g(s, X t,x s (ω), y, z, k), s ≤ T, (1.2.10) where (t, x) is a fixed initial condition and X t,x is a state process which has the following dynamic:

X t,x s = x + s t b(X t,x r )dr + s t σ(X t,x r )dW r + s t R *
β(X t,x r -, e) Ñ (dr, de).

(1.2.11)

The maps f, h, g, b, σ, β are deterministic functions satisfying usual Lipschitz assumptions (the reader is refered to the corresponding chapter). In the Markovian setting, for each (t, x), the minimal risk measure v(t, x) is defined as:

v(t, x) = -Y t,x t , (1.2.12)
where Y t,x is the reflected BSDE with data given by (1.2.10).

Our main constribution consists in establishing the link between the value function of our optimal stopping problem and parabolic partial integro-differential variational inequalities (PIDVIs). We prove that the minimal risk measure is a viscosity solution of a PIDVI. This provides an existence result for the obstacle problem under relatively weak assumptions. In the Brownian case, this result was obtained by using a penalization method via non-reflected BSDEs. This method could also be adapted to our case with jumps, but would involve heavy computations in order to prove the convergence of the solutions of the penalized BSDEs to the solution of the reflected BSDE. It would also require some convergence results of the viscosity solutions theory in the integro-differential case. We provide a direct and much shorter proof.

Under some additional asumptions, we provide a comparison theorem, relying on a non-local version of Jensen-Ishii Lemma, from which the uniqueness of the viscosity solution follows. We extend the results of [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions theory revisited[END_REF] to the case of nonlinear BSDEs, which leads to a more complex integrodifferential operator in the associated PDE. In the case of integro-differential equations, the difficulty arises from the treatement of nonlocal operators. The main idea is to split them in one operator corresponding to the small jumps and one corresponding to the big jumps and to use a less classical definition of viscosity solution introduced in [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions theory revisited[END_REF], adapted to integro-differential equations and equivalent to the two classical ones, which combines the approach with test-functions and sub-superjets (the solution is replaced by the test function only around the singularity of the measure in the nonlocal operator).

Generalized Dynkin Games and Doubly Reflected BS-DEs with jumps

This chapter is based on the paper "Generalized Dynkin games and Doubly Reflected BSDEs with jumps" [START_REF] Dumitrescu | Generalized Dynkin games and Doubly reflected BSDEs with Jumps[END_REF], joint with M.C. Quenez and A. Sulem and submitted for publication.

Preliminaries and overview of the literature

The Dynkin game is a zero-sum, optimal stopping game between two players. Each player can either stop the game or continue. The game is stopped as soon as either player stops, and the payoff depends on who stops first. This stochastic stopping game, nowadays known as the Dynkin game, was first introduced by Dynkin [START_REF] Dynkin | Game variant of a problem on optimal stopping[END_REF] as a generalization of optimal stopping problems. Since then, there has been a considerable amount of research on Dynkin games and related problems. Some examples include [START_REF] Dynkin | Theorems and Problems in Markov Processes[END_REF] [START_REF] Dynkin | Theorems and Problems in Markov Processes[END_REF], [START_REF] Bensoussan | Non-linear variational inequalities and differential games with stopping times[END_REF] [START_REF] Bensoussan | Non-linear variational inequalities and differential games with stopping times[END_REF], [START_REF] Neveu | Discrete Parameter Martingales[END_REF] [START_REF] Neveu | Discrete Parameter Martingales[END_REF], [START_REF] Bismut | Sur un problème de Dynkin[END_REF] [START_REF] Bismut | Sur un problème de Dynkin[END_REF], [START_REF] Stettner | Zero-sum Markov games with stopping and impulsive strategies[END_REF] [START_REF] Stettner | Zero-sum Markov games with stopping and impulsive strategies[END_REF], Alario, Lepeltier et Marechal (1982) [START_REF] Alario-Nazaret | Dynkin games[END_REF], [START_REF] Morimoto | Dynkin games and martingale methods[END_REF] [START_REF] Morimoto | Dynkin games and martingale methods[END_REF], [START_REF] Lepeltier | Le jeu de Dynkin en theorie generale sans lhypothse de Mokobodski[END_REF] [START_REF] Lepeltier | Le jeu de Dynkin en theorie generale sans lhypothse de Mokobodski[END_REF], Cvitanic and Karatzas (1996) [START_REF] Cvitanic | Backward stochastic differential equations with reflection and Dynkin games[END_REF], [START_REF] Karatzas | Connections between bounded variation control and Dynkin games, Optimal Control and Partial Differential Equations[END_REF] [START_REF] Karatzas | Connections between bounded variation control and Dynkin games, Optimal Control and Partial Differential Equations[END_REF], Ekstrom and Peskir [START_REF] Ekstrom | Optimal stopping games for Markov processes[END_REF], Laraki and Solan [START_REF] Lions | Differential Games, Optimal Control and Directional Derivatives of Viscosity Solutions of Bellman's and Isaacs' Equations[END_REF], Peskir [START_REF] Peskir | Optimal stopping games and Nash equilibrium[END_REF], Rosenberg and al. [START_REF] Rosenberg | Stopping games with randomized strategies[END_REF], [START_REF] Touzi | Continuous-time Dynkin games with mixed strategies[END_REF] [START_REF] Touzi | Continuous-time Dynkin games with mixed strategies[END_REF] etc. Most of the literature focuses on establishing the existence of optimal stopping times as well as value under various models and payoff assumptions. In discrete-time, it is easy to show the existence of optimal stopping times and value using backward induction arguments. In continuous-time, perhaps the most important result is due to Lepeltier and Maingueneau [START_REF] Lepeltier | Le jeu de Dynkin en theorie generale sans lhypothse de Mokobodski[END_REF], who proved the existence of ε-optimal stopping times as well as the value.

Let us recall the mathematical formulation of a classical Dynkin Game.

The setting of the problem is very simple. There are two players, labeled Player 1 and Player 2, who observe two payoff processes ξ and ζ defined on a probability space (Ω, F, P). Player 1 (resp., 2) chooses a stopping time τ (resp. σ) as a control for this optimal stopping problem. At (stopping) time σ ∧ τ the game is over, and Player 2 pays the amount ζ σ 1 τ >σ + ξ τ 1 τ ≤σ to Player 1. Therefore the objective of Player 1 is to maximize this payment, while Player 2 wishes to minimize it. It is then natural to introduce the lower and upper values of the game

V := sup σ inf τ E[ξ σ 1 τ >σ + ζ τ 1 τ ≤σ ]; V := inf τ sup σ E[ξ σ 1 τ >σ + ζ τ 1 τ ≤σ ].
(1.3.1)

If the two value functions defined above coincide, then the game is said to admit a value function.

An interesting financial application of the Dynkin game is in the study of game options, also known as Israeli options, as defined by Kifer [START_REF] Kifer | Game options[END_REF]. A game option is a contract between an issuer and a holder, in which the holder may exercise the option at any time for a payoff and the issuer may cancel the option at any time for a fee. It is one of the few financial contracts in which the issuer also makes meaningful decisions affecting the payoff. If we ignore the dependence on the underlying assets and focus on the relationship between decisions and payoffs, the game option is comparable to a Dynkin game. Moreover, the cancellation fee is typically assumed to be greater than or equal to the exercise payoff, echoing the standard payoff inequalities found in Dynkin games. In both discrete-time and continuous-time models, it was shown by Kifer [START_REF] Kifer | Game options[END_REF] that the game option has a unique arbitrage price. Further research on game options, as well as more sophisticated game-type financial contracts, includes papers by Bielecki and al. [START_REF] Bielecki | Defaultable game options in a hazard process model[END_REF], and Dolinsky Kifer [START_REF] Dolinsky | Hedging with risk for game options in discrete time, Stochastics[END_REF], Dolinsky and al. [START_REF] Dolinsky | Perfect and partial hedging for swing game options in discrete time[END_REF], Hamadene and Zhang [START_REF] Hamadène | The continuous time nonzero-sum Dynkin game problem and application in game options[END_REF], Kallsen and Kuhn [START_REF] Kallsen | Pricing derivatives of American and game type in incomplete markets[END_REF][START_REF] Kallsen | Convertible bonds: financial derivatives of game type[END_REF], and Kifer [START_REF] Kifer | Game options[END_REF].

We now focus on the relationship between Classical Dynkin Games and Doubly Reflected BSDEs (in short DRBSDEs), which have been introduced by Cvitanic and Karatzas [START_REF] Cvitanic | Backward stochastic differential equations with reflection and Dynkin games[END_REF] in the case of a Brownian filtration. The solution is forced to remain between two upper and lower barriers ξ and ζ and it is represented by a quadruple of square integrable processes {(Y t , Z t , A t , A ′ t ); 0 ≤ t ≤ T } satisfying:

             Y t = ξ + T t g(s, ω, Y s , Z s )ds + A T -A t -(A ′ T -A ′ t ) - T t Z s dW s , 0 ≤ t ≤ T, ξ t ≤ Y t ≤ ζ t , T 0 (Y t -ξ t )dA t = 0 et T 0 (ζ t -Y t )dA ′ t = 0. (1.3.2)
with A and A ′ two continuous processes, increasing, whose role is to keep the process Y between the two barriers. They proved existence and uniqueness of the solution in the case when the barriers are regular and satisfy the so-called Mokobodski condition which turns into the existence of the difference of two non-negative supermartingales between ξ and ζ.

In the case of a process driver g which only depends on (t, ω), Cvitanic-Karatzas have shown that the existence of a solution (Y, Z, A) to the above BSDE implies that Y corresponds to the value function of a Classical Dynkin Game. We give below their result:

Theorem 1.3.1 (Cvitanic-Karatzas). Let (Y, Z, A, A ′ ) be a solution of the BSDE with g(t, ω, y, z) = g(t, ω). For any 0 ≤ t ≤ T and any two stopping times (τ, σ) ∈ T × T , consider the payoff: of a corresponding Dynkin game. This game has value V (t), given by the state process Y of the solution to the BSDE, that is,

I t (
V (t) = V (t) = V (t) = Y t a.s. ∀0 ≤ t ≤ T,
as well as a saddlepoint (σ t , τt ) ∈ T t × T t given by

σt := inf{s ∈ [t, T ), Y s = ζ s } ∧ T ; τt := inf{s ∈ [t, T ), Y s = ξ s } ∧ T.
namely

E[I t (τ t , σt )] ≤ E[I t (τ t , σt )] = Y t ≤ E[I t (τ t , σ t )],
for every (σ, τ ) ∈ T t × T t .

Since the seminal paper of Cvitanic-Karatzas, many authors have explored the existence and the uniqueness of the solution as well as the links with classical Dynkin Games under different assumptions on the coefficient g and regularity of the barriers (see for e.g. Lepeltier-San Martin and [START_REF] Lepeltier | Reflected backward stochastic differential equations with two rcll barriers[END_REF]).These results have also been extended to the case of DRBSDEs driven by both a Brownian motion and a random Poisson measure (see for e.g. [START_REF] Hamadène | BSDEs with two reacting barriers driven by a Brownian motion and an independent Poisson noise and related Dynkin game[END_REF], [START_REF] Hamadène | Backward SDEs with two rcll reflecting barriers without Mokobodski's hypothesis[END_REF], [START_REF] Crépey | Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison[END_REF]).

The above link between classical Dynkin games and DRBSDEs can be extended to the case of general nonlinear DRBSDEs, since given the solution Y of the DRBSDE, it is shown to coincide with the value function of the classical Dynkin game with payoff:

I S (τ, σ) = σ∧τ S g(u, Y u , Z u , k u )du + ξ τ 1 τ ≤σ + ζ σ 1 σ<τ , (1.3.3)
where Z, k are the associated processes with Y . However, this characterization is not really tractable because the instantaneous payoff g(u, Y u , Z u , k u ) depends on the value function Y of the associated Dynkin game. We shall see in our contribution that we can define a well-posed game problem (in the sense that the criterium does not involve the value function itself), which is shown to admit a value coinciding with the solution of a DRBSDE with general nonlinear driver.

Contributions

In Chapter 3, we introduce a new game problem, which can be seen as a generalization of the classical Dynkin game. More precisely, the linear expectation in the performance is replaced by a nonlinear g-conditional expectation, induced by a backward stochastic differential equation (BSDE) with jumps. We describe below very briefly this new game problem.

Let ξ and ζ be two adapted processes only supposed to be RCLL with

ζ T = ξ T a.s., ξ ∈ S 2 , ζ ∈ S 2 , ξ t ≤ ζ t , 0 ≤ t ≤ T a.s.
For each τ, σ ∈ T 0 , the payoff at the stopping time τ ∧ σ is given by:

I(τ, σ) := ξ τ 1 τ ≤σ + ζ σ 1 σ<τ . (1.3.4)
Let S ∈ T 0 . For each τ ∈ T S and σ ∈ T S , the associated criterium is given by E S,τ ∧σ (I(τ, σ)), the g-conditional expectation of I(τ, σ). At the stopping time S, the first (resp. the second) player chooses a stopping time τ (resp. σ) after S, in order to maximize (resp. minimize) the criterium.

For each stopping time S ∈ T 0 , the upper and lower value functions at time S are defined as follows:

V (S) := essinf

σ∈T S esssup τ ∈T S E S,τ ∧σ (I(τ, σ)); (1.3.5) V (S) := ess sup τ ∈T S essinf σ∈T S E S,τ ∧σ (I(τ, σ)). (1.3.6)
The game admits a value function if V (S) = V (S). Under Mokobodski's condition, we show the existence of a value function for this game, which can be characterized by the unique solution of a nonlinear doubly reflected BSDE (DRBSDE). Up to now, no interpretation of general nonlinear doubly reflected BSDEs in terms of control or game problems (with nonlinear expectation) had been given in the literature.

Using this characterization, we obtain some properties of these DRBSDEs, such as a general comparison theorem and a strict comparison theorem. We also establish new a priori estimates with universal constant for DRBSDEs, and the proof is based on the characterization of the solution as the value function of this new game problem. When both obstacles are left upper semicontinuous along stopping times, we show the existence of a saddle point of the generalized Dynkin game. We point out that we do not assume the strict separability of the barriers, assumption which is crucial in the previous literature. We can get rid of it by imposing an additional constraint on the increasing processes A, A ′ which appear in Definition 1.3.2 (note that in our setting the increasing processes A and A ′ are no longer continuous). More precisely, we assume that the measures dA and dA ′ are mutually singular in the probabilistic sense, i.e. there exists D ∈ P such that

E[ T 0 1 D dA t ] = E[ T 0 1 D c dA ′ t ] = 0.
This constraint is also important in order to obtain the uniqueness of the increasing processes A and A ′ . Moreover, it allows us to identify the positive and negative jumps of the solution of the DRBSDE, this identification being used in the proof of the existence of saddle point without assuming the strict separability of the barriers.

We continue by studying a generalized mixed zero-sum game under the g-conditional expectation, in which two players compete by taking two actions: continuous control and stopping. We provide some sufficient conditions (such as the controlled drivers g u,v have a saddle point g u,v ), which ensure the existence of a value function of the generalized mixed game and characterize the common value function as the solution of a DRBSDE with driver g u,v . When both obstacles are left upper semicontinuous along stopping times, the associated generalized mixed game admits a saddle point.

We then address the generalized Dynkin game in the Markovian framework and study its links with parabolic partial integro-differential variational inequalities (PIDVI) with two obstacles. More precisely, we show that the value function of the generalized Dynkin game in the Markovian case is the unique viscosity solution of the corresponding PIDVI. From a PDE point of view, this result provides a new probabilistic interpretation of semi linear PDEs with two barriers in terms of game problems.

A Weak Dynamic Programming Principle for Combined Stochastic Control/Optimal Stopping with E f -Expectations

This chapter is based on the paper "A weak dynamic programming principle for Combined Stochastic Control / Optimal Stopping with E f -expectations" [START_REF] Dumitrescu | A Weak Dynammic Programming for Combined Stochastic Control/Optimal Stopping with E f -Expectations[END_REF], joint with M.C. Quenez and A. Sulem and submitted for publication.

Preliminaries and overview of the literature

The Dynamic Programming Principle (in short DPP) is the main tool in the theory of stochastic control. The basic idea of the method is to consider a family of stochastic control problems with different initial states and to establish relationships between the associated value functions. It was initiated in the fifties by Bellman ([28], [START_REF] Bellman | Introduction to the mathematical theory of control processes[END_REF]), who says that "an optimal policy has the property that, whatever the initial state and control are, the remaining decisions must constitue an optimal policy with regard to the state resulting from the first decision". Typically, a stochastic control problem in a finite horizon time T can be written as follows:

V (0, x) := sup α∈A E[ T 0 f (s, X α s , α s )ds + g(X α T )], (1.4.1)
where f is the instantaneous reward and g the terminal payoff.

A formal statement of the DPP is

V (0, x) = v(0, x) := sup α∈A E[ τ 0 f (s, X α s , α s )ds + V (τ, X α τ )], (1.4.2)
where τ is an arbitrary stopping time such that τ ∈ [0, T ) a.s.

In the case of controlled Markov jump-diffusions, the DPP is used in order to derive the corresponding dynamic programming equation in the sense of viscosity solutions. In the literature, this principle is classically established under assumptions which ensure that the value function satisfies some regularity/measurability properties, see e.g. Fleming-Rischel, Krylov, El Karoui, Bensoussan-Lions, Lions P.-L., Fleming-Soner, Touzi for the case of controlled diffusions and Oksendal and Sulem for the case of Markov jump-diffusions. The statement (1.4.2) of the DPP is very intuitive and can be easily proved in the deterministic framework, or in discrete-time with finite probability space. However, its proof is in general not trivial and requires on the first stage that V is measurable.

The case of a discontinuous value function has been studied in a deterministic framework in the eighties: a weak dynamic programming principle has been established for deterministic control by [START_REF] Barles | Discontinuous viscosity solutions of first-order Hamilton-Jacobi equations: a guided visit[END_REF] (see [START_REF] Barles | Discontinuous viscosity solutions of first-order Hamilton-Jacobi equations: a guided visit[END_REF]) (see also [START_REF] Barles | Discontinuous viscosity solutions of deterministic optimal control problems Decision and Control[END_REF] [START_REF] Barles | Discontinuous viscosity solutions of deterministic optimal control problems Decision and Control[END_REF]). More precisely, he proves that the upper semicontinuous envelope V * and the lower semicontinuous envelope V * of the value function V satisfy, respectively, the sub-and super-optimality principle of dynamic programming of [START_REF] Lions | Differential Games, Optimal Control and Directional Derivatives of Viscosity Solutions of Bellman's and Isaacs' Equations[END_REF] [START_REF] Lions | Differential Games, Optimal Control and Directional Derivatives of Viscosity Solutions of Bellman's and Isaacs' Equations[END_REF]. He then derived that the (discontinuous) value function is a weak viscosity solution of the associated Bellman equation in the sense that V * is a viscosity subsolution and V * is a supersolution of the Bellman equation.

More recently, Bouchard and Touzi (2011) (see [START_REF] Bouchard | Weak Dynamic Programming Principle for Viscosity Solutions[END_REF]) have proved a weak dynamic programming principle in a stochastic framework, when the value function is not necessarily continuous, not even measurable. They prove that the upper semicontinuous envelope V * satisfies the sub-optimality principle of dynamic programming, and under an additional regularity (lower semi continuity) assumption of the reward g, they obtain that the lower semicontinuous envelope V * satisfies the super-optimality principle.

A weak dynamic principle has been further established, under some specific regularity assumptions, for problems with state constraints by [START_REF] Bouchard | Weak Dynamic Programming for Generalized State Constraints[END_REF] in [START_REF] Bouchard | Weak Dynamic Programming for Generalized State Constraints[END_REF], and for zero-sum stochastic games by [START_REF] Bayraktar | A Weak Dynamic Programming Principle for Zero-Sum Stochastic Differential Games with Unbounded Controls[END_REF] in [START_REF] Bayraktar | A Weak Dynamic Programming Principle for Zero-Sum Stochastic Differential Games with Unbounded Controls[END_REF].

In the sequel, we present the classical statement of the problem for both stochastic control and optimal stopping problems (in a finite horizon time T ), in the case when the value function is not a priori continuous, not even measurable. We recall the weak dynamic programming principle obtained by Bouchard and Touzi ([35]), as well as the associated HJB equations.

(i) Stochastic control and weak dynamic programming in the case of classical expectations

We denote by A the set of all progressively measurable processes α = {α t , t < T } valued in A, a subset of R, belonging to H 2 (the set of processes φ such taht E[ T 0 φ 2 s ds] < +∞). The elements of A are called control processes. For each control process α ∈ A, we consider the following controlled stochastic differential equation:

dX t,x,α s = b(X t,x,α s , α s )ds + σ(X t,x,α s , α s )dW s , (1.4.3)
where the coefficients b and σ satisfy the usual Lipschitz and linear growth conditions so that the above SDE has a unique strong solution.

For a given initial data (t, x) and control α ∈ A, the process X t,x,α is called the controlled process, as its dynamic is driven by the action of the control process α.

We define the cost functional J on [0, T ] × R × A by:

J(t, x, α) = E T t f (s, X α,t,x s , α s )ds + g(X α,t,x T ) ,
where f is Lipschitz continuous and g Borelian, with quadratic growth.

The purpose is to study the following stochastic control problem:

v(t, x) = sup α∈At J(t, x, α), (1.4.4)
where A t represents the set of t-admissible controls, which are independent of F t .

In order to describe the local behavior of the value function V by means of the so-called dynamic programming equation or Hamilton-Jacobi-Bellman, the key point is the Dynamic Programming Principle. Since the DPP involves the value function itself, which may not be measurable under these assumptions, Bouchard and Touzi [START_REF] Bouchard | Weak Dynamic Programming Principle for Viscosity Solutions[END_REF] propose a Weak version of the Dynamic Programming Principle, which is shown to be sufficient for the derivation of the dynamic programming equation. This weak DPP involves the upper semicontinuous envelope of the value function V , respectively the lower semicontinuous one, which are defined as follows: for each t ∈ [0, T ], for each x ∈ R,

V * (t, x) := lim inf (t ′ ,x ′ )→(t,x) V (t ′ , x ′ ) and V * (t, x) := lim sup (t ′ ,x ′ )→(t,x) V (t ′ , x ′ ).
(

Let us now recall the Weak dynamic Programming Principle.

Theorem 1.4.1 (Weak Dynamic Programming Principle). 1. Let {θ α , α ∈ U t } be a family of finite stopping times independent of F t , with values in [t, T ]. Then:

V (t, x) ≤ sup α∈At E[ θ α t f (s, X α,t,x s , α s )ds + V * (θ α , X α,t,x θ α )], (1.4.6)
2. Assume further that g is lower-semicontinuous and

X α t,x 1 t,θ α is L ∞ -bounded for all ν ∈ A t . Then V (t, x) ≥ sup α∈At E[ θ α t f (s, X α,t,x s , α s )ds + V * (θ α , X α,t,x θ α )]. (1.4.7)
The above weak DPP is shown without using the abstract theorems of measurable selection. The authors use instead to Vitali's covering lemma. The inequality which is the most difficult to provide is the second one and it requires a lower semicontinuity assumption on the criterium J (which is satisfied in the case when the terminal reward g is lower semicontinuous).

We also point out that, in the case when V is continuous, then V = V * = V * , and the above weak dynamic programming principle reduces to the classical dynamic programming principle:

V (t, x) = sup α∈At E θ α t f (s, X α,t,x s , α s )ds + V (θ α , X α,t,x θ α ) .
As mentioned previously, the dynamic programming principle represents the main step for the derivation of the dynamic programming equation, corresponding to the infinitesimal counterpart of the DPP. It is widely called the Hamilton-Jacobi-Bellman equation.The associated HJB equation is provided in the following theorem: Theorem 1.4.2. Assume that the value function V ∈ C 1,2 ([0, T ), R), and let f (•, •, a) be continuous in (t, x) for all fixed a ∈ A. Then, for all (t, x) ∈ [0, T ) × R:

-∂ t V (t, x) -sup a∈A {b(t, x, a)∂ x V (t, x) + 1 2 T r[σσ(t, x, a)D 2 xx V (t, x)] + f (t, x, a)} = 0. (1.4.8)
Note that in the case when the value function V is not continuous, then it satisfies the above PDE in the viscosity sense.

We now present the main results concerning optimal stopping, which represent a particular case of stochastic control problems when the control takes the form of a stopping time.

(ii) Optimal stopping and weak dynamic programming in the case of classical expectations For 0 ≤ t ≤ T < +∞, we denote by T [t,T ] the collection of all F-stopping times with values in [t, T ]. The underlying state process X t,x , with initial condition (t, x), is defined by the stochastic differential equation: dX t,x s = b(s, X t,x s )ds + σ(s, X t,x s )dW s , where b and σ satisfy the usual Lipschitz and linear growth conditions so that the above SDE has a unique strong solution. Let g be a measurable function, with polynomial growth, and assume that:

E[ sup 0≤t≤T |g(X t )|] < ∞.
For an admissible stopping time, the criterium is defined as follows:

J(t, x, τ ) = E g(X t,x τ ) .
(1.4.9)

We now consider the subset of stopping rules:

T t t := {τ ∈ T [t,T ] : τ independant of F t }. (1.4.10)
The optimal stopping problem is defined by:

V (t, x) = sup τ ∈T t t J(t, x, τ ). (1.4.11)
Using the same arguments as for the stochastic control problem presented above, Bouchard and Touzi show the following Weak Dynamic Programming Principle:

Theorem 1.4.3. For (t, x) ∈ [0, T ] × R, let θ ∈ T t t be a stopping time such that X t,x θ is bounded. Then: V (t, x) ≤ sup τ ∈T t t E 1 {τ <σ} g(X t,x τ ) + 1 {τ ≥θ} V * (θ, X t,x θ ) , (1.4.12) V (t, x) ≥ sup τ ∈T t t E 1 {τ <σ} g(X t,x τ ) + 1 {τ ≥θ} V * (θ, X t,x θ ) . (1.4.13)
In the case when the value function V is a priori known to be smooth, the infinitesimal counterpart of the dynamic programming principle is the following: Theorem 1.4.4. Assume that V ∈ C 1,2 ([0, T ), R) and let g : R → R be continuous. Then V solves the obstacle problem:

min{-(∂ t + L)V, V -g} = 0, (1.4.14)
where LV represents the infinitesimal generator of the Markov diffusion process X.

The classical stochastic control problem (1.4.1) has been generalized by Peng to the case when the cost functional is defined through a nonlinear controlled backward stochastic differential equation (see [START_REF] Peng | A generalized dynamic programming principle and Hamilton-Jacobi-Bellman-Equation[END_REF] and [START_REF] Peng | BSDE and stochastic optimizations, Topic in Stochastic Analysis[END_REF]), under assumptions which ensure that the value function is continuous. He establishes a dynamic programming by using the backward semigroup method and derives the associated HJB equations. These results allow him to obtain a stochastic interpretation for a larger class of nonlinear HJB equations, since the coefficient f also depends on (y, z).

At the end of this section, we would like to mention some developments in the case when the uncertainty impacts only the volatility of the model. Soner,Touzi and Zhang ([145]) recently introduced the notion of second order BSDEs (2BSDEs), whose basic idea is to require that the solution verifies the equations P α a.s. for every probability measure in a non dominated class of mutually singular measures. This theory is closely related to the notion of G-expectation of Peng ( [START_REF] Peng | G-Brownian motion and related stochastic calculus of Ito type[END_REF]) and provides a different probabilitic representation of the solutions of fully nonlinear HJB equations.

Contributions

In this chapter, we are interested in generalizing the results obtained by Bouchard and Touzi ([35]) to the case when the linear expectation E is replaced by a nonlinear expectation induced by a Backward Stochastic Differential Equation with jumps. In a Markovian setting, the value function of our problem is the following:

V (t, x) := sup α∈A E α 0,T [g(X α,t,x T )], (1.4.15)
where E α is the nonlinear conditional expectation associated with a BSDE with jumps with controlled driver f (α t , X α t , y, z, k). We address this study in the case when the reward function g is only Borelian. Moreover, in this chapter, we consider the combined problem when there is an additional control in the form of a stopping time. We thus consider mixed generalized stochastic control/ optimal stopping problems of the form

V (t, x) := sup α sup τ E α 0,τ [h(X α,t,x τ )], (1.4.16)
where h(X α,t,x τ

) is an irregular payoff. In order to characterize the value function as the solution of a HJB variational inequality, we first establish a Dynamic Programming Principle, which is obtained using sofisticated techniques of stochastic analysis. We point out that, due to the weak assumptions on the coefficients, the value function of our problem is not necessarily continuous, not even measurable.

As mentioned in the introductory section, since for fixed t, the value function x → V (t, x) is not necessarily measurable, we cannot a priori establish a classical dynamic programming. We thus provide a weak dynamic programming involving the map V * and the map V * defined by

V * (t, x) := lim sup (t ′ ,x ′ )→(t,x) V (t ′ , x ′ ), ∀(t, x) ∈ [0, T ) × R and V * (T, x) = g(x), ∀x ∈ R; V * (t, x) := lim inf (t ′ ,x ′ )→(t,x) V (t ′ , x ′ ), ∀(t, x) ∈ [0, T ) × R and V * (T, x) = g(x), ∀x ∈ R.
Remark that in our case, the map V * (resp. V * ) is not necessarily upper (resp. lower) semicontinuous on [0, T ] × R, because the terminal reward g is only Borelian ( it is not supposed to satisfy any regularity assumption). This is not the case in the previous literature even in the linear case, where g is supposed to be lower-semicontinuous (see [START_REF] Bouchard | Weak Dynamic Programming Principle for Viscosity Solutions[END_REF]). We give below the sub-(resp. super-) optimality principle of dynamic programming satisfied by V * (resp. V * ), one of the main results of this chapter.

Theorem 1.4.5 (A weak dynamic programming principle). The function V * satisfies the suboptimality principle of dynamic programming, that is for each t ∈ [0, T ] and for each stopping time θ ∈ T t t , that is

V (t, x) ≤ sup α∈A t t sup τ ∈T t t E α,t,x t,θ∧τ h(τ, X α,t,x τ )1 τ <θ + V * (θ, X α,t,x θ )1 τ ≥θ , (1.4.17)
The function V * satisfies the super-optimality principle of dynamic programming, that is for each t ∈ [0, T ] and for each stopping time θ ∈ T t t , that is

V (t, x) ≥ sup α∈A t t sup τ ∈T t t E α,t,x t,θ∧τ h(τ, X α,t,x τ )1 τ <θ + V * (θ, X α,t,x θ )1 τ ≥θ .
(1.4.18)

In the above theorem, A t t represents the set of controls independent on F t and restricted to [t, T ]. Similarly, T t t denotes the set of stopping times independent on F t , with values in [t, T ]. The sub-optimality principle is the easiest to prove. It is based on the flow property for both backward and forward SDEs and a splitting property, which basically states that given an intermediary time t ≤ T and a fixed path up to time t (corresponding to the realization of the Brownian motion and Poisson random measure), the BSDE can be solved with respect to the t-translated Brownian motion and Poisson random measure. This result is needed in order to be able to use the definition of the value function, which is a deterministic map.

The second inequality is considerably more difficult and relies on the existence of weak εoptimal controls for our mixed control/optimal stopping problem (result requiring some subtle arguments, as an abstract measurable selection theorem), as well as on some new properties of BSDEs ( for e.g. a Fatou lemma for reflected BSDEs where the limit involves both terminal time and terminal condition).

Using this weak dynamic programming principle and a new comparison theorem between BS-DEs and reflected BSDEs, we derive that the value function is a weak viscosity solution of a nonlinear generalized HJB variational inequality. More precisely, the result is the following: Theorem 1.4.6. The function V , defined by (1.4.16), is a weak viscosity solution of the HJBVI

         min(V (t, x) -h(t, x), inf a∈A (- ∂V ∂t (t, x) -L a V (t, x) -f (a, t, x, V (t, x), (σ ∂V ∂x )(t, x), B a V (t, x))) = 0, (t, x) ∈ [0, T ) × R V (T, x) = g(x), x ∈ R (1.4.19)
with L a := A a + K a , and for φ ∈ C 2 (R),

• A a φ(x) := 1 2 σ 2 (x, α) ∂ 2 φ ∂x 2 (x) + b(x, α) ∂φ ∂x (x) • K a φ(x) := E φ(x + β(x, α, e)) -φ(x) - ∂φ ∂x (x)β(x, α, e) ν(de) • B a φ(x) := φ(x + β(x, α, •)) -φ(x),
in the sense that V * is a viscosity subsolution of (1.4.19) and V * is a viscosity supersolution of (1.4.19).

We conclude this chapter with some financial applications of the theoretical part.

Numerical methods for Doubly Reflected BSDEs with Jumps and irregular obstacles

This part of the thesis is dedicated to the study of numerical methods for DRBSDEs with jumps and irregular obstacles and is based on two papers written in collaboration with C. Labart: Numerical approximation for DRBSDEs with jumps and RCLL obstacles [START_REF] Dumitrescu | Numerical approximation for Doubly reflected BSDEs with Jumps and RCLL obstacles[END_REF] (accepted for publication in Journal of Mathematical Analysis and Applications) and Reflected scheme for DRBSDEs with jumps and RCLL obstacles [START_REF] Dumitrescu | Reflected Scheme for Doubly reflected BSDEs with Jumps and RCLL obstacles[END_REF] (accepted for publication in Journal of Computational and Applied Mathematics).

We start this section with a short presentation of the existing numerical methods for backward SDEs.

Preliminaries and overview of the literature

Since Backward Stochastic differential equations provide probabilistic representations of solutions of semilinear PDEs, there are many works on numerical schemes in the Markovian setting, in the case of a filtration generated by a Brownian motion. Among them, we recall the four step algorithm developed by J. Ma, P. Protter and J. Yong ( [START_REF] Ma | Solving forward-backward stochastic differential equations explicitly a four step scheme[END_REF], see also [START_REF] Douglas | Numerical methods for forward-backward stochastic differential equations[END_REF]), Bouchard-Touzi (see [START_REF] Bouchard | Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations[END_REF]), Zhang ([152]) etc. In the case of reflected and doubly reflected BSDEs, see e.g. [START_REF] Bouchard | Discrete-time approximation for continuously and discretely reflected BSDEs[END_REF] and [START_REF] Chassagneux | A discrete-time approximation for doubly reflected BSDEs[END_REF]. A relevant problem in the theory of BSDEs is to propose implementable numerical methods to approximate the solutions of such equations and the complexity is due to the computation of conditional expectations. Several efforts have been made in this direction. In [START_REF] Bouchard | Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations[END_REF], Bouchard and Touzi use the Malliavin calculus to rewrite the conditional expectations as the ratio of two unconditional expectations which can be estimated by standard Monte Carlo methods. In the reflected case, where the driver does not depend on Z, Bally and Pages (see [START_REF] Bally | Error analysis of the quantization algorithm for obstacle problems[END_REF], [START_REF] Bally | A quantization algorithm for solving multi-dimensional optimal stopping problems[END_REF]) use a quantization approach. This method is based on the approximation of the continuous time processes on a finite grid, and requires a further estimation of the transition probabilities on the grid. Gobet et al. ( [START_REF] Lemor | Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations[END_REF]) have suggested an adaptation of the so-called Longstaff-Schwartz algorithm based on non-parametric methods and very recently Ph. Briand and C. Labart ( [START_REF] Ph | Simulation of BSDEs by Wiener Chaos Expansion[END_REF]) have proposed the Wiener chaos expansion, which, in the spirit, is not so far from the regression techniques. We also recall the cubature methods, used by T. Lyons, D. Crisan and K. Manolarakis (see e.g. [START_REF] Crépey | Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison[END_REF]).

In the non-markovian setting, in the case of standard BSDEs ( [START_REF] Peng | Numerical algorithms for BSDEs with 1-d Brownian motion: convergence and simulation[END_REF]), as well as in the case of reflected BSDEs [START_REF] Xu | Numerical algorithms and Simulations for Reflected Backward Stochastic Differential Equations with Two Continuous Barriers[END_REF], the authors propose another technique which is based on the approximation of the Brownian motion by a random walk. This method allows them to simplify the computation of the conditional expectations involved at each time step and to obtain fully implementable schemes. The BSDE is thus replaced by an appropriate discrete backward stochastic differential equation, which is shown to convege by a result of Briand, Delyon and Memin [START_REF] Briand | Donsker-type theorem for BSDEs[END_REF] ( see also [START_REF] Briand | On the robustness of backward stochastic differential equations[END_REF]).

While many authors studied discrete schemes for the approximation of solutions of BSDEs in a purely Brownian setting, in a setting with jumps there is considerably less literature available, and only in the case of nonreflected BSDEs. In the Markovian setting, Bouchard and Elie ( [START_REF] Bouchard | Discrete-time approximation of decoupled Forward-Backward SDE with jumps[END_REF]) considered numerical schemes for BSDEs in a pure finite activity jump setting based on the dynamic programming equation. Recently, [START_REF] Lejay | Numerical approximation of Backward Stochastic Differential Equations with Jumps[END_REF] [109] have extended the results of Briand, Delyon and Memin to the case of jumps. Their method thus relies on the construction of a discrete BSDE with jumps driven by a complete system of three orthogonal discrete time-space martingales.

Contributions

In Chapter 5, we study in a non-markovian setting a discrete time approximation for the solution of Doubly Reflected BSDEs with Jumps, driven by a Brownian motion (denoted by W ) and an independant compensated Poisson process of intensity λ (denoted by Ñ ). Moreover, we assume that the barriers are right continuous left limited processes and admit both totally inaccesible and predictable jumps. The DRBSDE we solve numerically has the dynamics:

Y t = ξ T + T t g(s, Y s , Z s , U s )ds + (A T -A t ) -(K T -K t ) - T t Z s dW s - T t U s d Ñs , (1.5.1)
and satisfies the following constraints:

               (i) ∀t ∈ [0, T ], ξ t ≤ Y t ≤ ζ t a.s. (ii) T 0 (Y t --ξ t -)dA c t = 0 and T 0 (ζ t --Y t -)dK c t = 0 a.s. (iii) ∀τ predictable stopping time, ∆A d τ = ∆A d τ 1 Y τ -=ξ τ - and ∆K d τ = ∆K d τ 1 Y τ -=ζ τ -.
(1.5.2)

As we have mentioned in the previous chapters, since we consider the general setting when the jumps of the obstacles can be either predictable or totally inaccesible, the increasing processes A and K, whose role is to keep the solution Y between the barriers, are no longer continous. We can thus rewrite the Skorokhod condition separately, for the continuous part A c (resp. K c ) of A (resp. K) and the discontinuous one, denoted by A d (resp. K d ).

Our aim is to propose a fully implementable scheme to the above DRBSDE, based on two random binomial trees and the penalization method, which is then shown to converge to the solution of the DRBSDE. We present below the main ideas:

(i) We first introduce a sequence of penalized BSDEs in order to approximate the doubly reflected BSDE (1.5.1; 1.5.2), satisfying:

Y p t = ξ + T t g(s, Y p s , Z p s , U p s )ds + A p T -A p t -(K p T -K p t ) - T t Z p s dW s - T t U p s d Ñs , (1.5.3) with A p t := p t 0 (Y p s -ξ s ) -ds and K p t := p t 0 (ζ s -Y p s ) -ds.
We provide the convergence of the penalized equations in the case of a general Poisson random measure and, since we have to deal with a driver which depends on the solution, the penalization method used in previous literature (which treates only the case of a driver process, the general case being obtained by a fixed point argument) cannot be adapted to our general setting. We propose instead a proof which is based on a combination of penalization, Snell envelope theory, comparison theorem for BSDEs with jumps, a generalized monotonic theorem under Mokobodski's condition and stochastic games.

(ii) We approximate the Brownian motion and the Poisson process by two independent random walks, denoted by W n respectively Ñ n and defined as follows:

W n 0 = 0; W n t = √ δ [t/δ] i=1 e n i , Ñ n 0 = 0, Ñ n t = [t/δ] i=1 η n i ,
with e n i , i = 1, n independent identically distributed random variables taking the values {-1; 1}, both with probability 1 2 and η n i , i = 1, n defined similarly to (e n i ), but taking the values {κ n -1; κ n } with probability 1-κ n , resp. κ n , where κ n = e -λ n . In the above definition, δ n := T n represents the time step. The couple (W n , Ñ n ) converges to (W, Ñ ) in probability for the J 1 -Skorokhod topology. Using these approximations, we get the following discrete approximation of the penalized equation defined by (1.5.3):

           y p,n j = y p,n j+1 + g(t j , y p,n j , z p,n j , u p,n j )δ n + a p,n j -k p,n j -(z p,n j √ δ n e n j+1 + u p,n j η n j+1 + v p,n j µ n j+1 ) a p,n j = pδ n (y p,n j -ξ n j ) -; k p,n j = pδ n (ζ n j -y p,n j ) -, y p,n n := ξ n n , (1.5.4)
where the third martingale increments sequence {µ n j = e n j η n j , j = 0, ..., n} is needed in order to obtain the martingale representation (see [109]).

Then, using the above discrete implicit scheme, we can derive the expressions of the coefficients (z p,n j , u p,n j , v p,n j ) j=1,n , involving conditional expectations, which are easy to compute in our framework, thanks to the above approximations of W and Ñ . However, the value of (y p,n j ) j=1,n is not so easy to deduce, since we have to introduce an operator whose numerical inversion is quite difficult and time consuming. In order to overcome this issue, we introduce an explicit discrete backward equation, which is obtained by replacing in (1.5.4) y p,n j by E[y p,n j+1 |F n j ] in the generator g:

           y p,n j = y p,n j+1 + g(t j , E[y p,n j+1 |F n j ], z p,n j , u p,n j )δ n + a p,n j -k p,n j -(z p,n j √ δ n e n j+1 + u p,n j η n j+1 + v p,n j µ n j+1 ) a p,n j = pδ n (y p,n j -ξ n j ) -; k p,n j = pδ n (ζ n j -y p,n j ) -, y p,n n := ξ n n , (1.5.5)
where F n represents the discrete filtration generated by (e n j , η n j ) j=1,n . We then introduce the continuous time version (Y p,n t , Z p,n t , U p,n t , A p,n t , K p,n t ) 0≤t≤T of the solution of this explicite scheme and show its convergence in n to the solution of (1.5.3). Coupling this result with the convergence in p of the penalized equation (see (i)), we obtain the convergence of our scheme in (p, n) to the solution of the DRBSDE.

We finally study numerically our theoretical results, in the case of barriers admiting both predictable and totally inaccesible jumps. The difficulty in the choice of the examples is due to the Mokobodski's condition, that we have to assume and which is difficult to check in practice. We point out that the practical use of our scheme is restricted to low dimensional cases. Indeed, since we use a random walk to approximate the Brownian motion and the Poisson process, the complexity of the algorithm grows very fast in the number of time steps n (more precisely, in n d , d being the dimension) and, as we will see in the numerical part, the penalization method requires many time steps to be stable.

In Chapter 6, we propose an alternative scheme to (1.5.4) and respectively to (1.5.5) in order to solve the DRBSDE given by (1.5.1; 1.5.2). Compared to the discrete backward equations (1.5.4) and (1.5.5) , the schemes we present in chapter 6, called implicit reflected scheme and explicit reflected scheme are based on a direct discretization of (1.5.1; 1.5.2). More precisely, there is no penalization step. Then, this method only depends on one parameter of approximation (the number of time steps n), contrary to the schemes proposed in Chapter 5 (see (1.5.4) and (1.5.5)), which also depends on the penalization parameter. We provide the convergence of both schemes. The explicit reflected scheme is the following:

y n j = E[y n j+1 |F n j ] + g(t j , E[y n j+1 |F n j ], z n j , u n j )δ n + a n j -k n j a n j ≥ 0, k n j ≤ 0, a n j k n j = 0, ξ n j ≤ y n j ≤ ζ n j , (y n j -ξ n j )a n j = (y n j -ζ n j )k n j = 0.
(1.5.6)

We illustrate numerically the theoretical results and show they coincide with the ones obtained by using the penalized scheme (1.5.5), for large values of the penalization parameter p.

Conclusions and perspectives

In this Phd thesis we have investigated some new problems of interest in stochastic analysis, stochastic control, game theory and financial mathematics, from both theoretical and numerical point of views. The main results are the following:

• In Chapter 3 (based on paper [START_REF] Dumitrescu | BSDEs with nonlinear weak terminal condition[END_REF]), we introduce and study a new class of BSDEs with nonlinear weak terminal condition, related to the approximative hedging under dynamic risk measures constraints.

• In Chapter 4 (based on paper [START_REF] Dumitrescu | Optimal Stopping for Dynamic Risk Measures with Jumps and Obstacle Problems[END_REF]), we study an optimal stopping problem for dynamic risk measures induced by BSDEs with jumps and show that the value function corresponds to the unique viscosity solution of an obstacle problem for partial integro-differential equations.

• In Chapter 5 (based on paper [START_REF] Dumitrescu | Generalized Dynkin games and Doubly reflected BSDEs with Jumps[END_REF]), we introduce a new game problem, which can be seen as a generalization of the classical Dynkin game to the case of nonlinear expectations , allowing us to obtain a representation of the solution of general nonlinear doubly reflected BSDEs in terms of stochastic games.

• In Chapter 6 (based on paper [START_REF] Dumitrescu | A Weak Dynammic Programming for Combined Stochastic Control/Optimal Stopping with E f -Expectations[END_REF]), we study in a Markovian framework a mixed stochastic control/optimal stopping problem in the case when the classical expectation in the criterium is replaced by a nonlinear one induced by a solution of a BSDE with jumps and the terminal reward is only measurable. We establish a weak dynamic programming principle and derive the associated nonlinear HJB equations.

• In Chapter 7 (based on paper [START_REF] Dumitrescu | Numerical approximation for Doubly reflected BSDEs with Jumps and RCLL obstacles[END_REF]), we introduce a numerical approximation for the solution of doubly reflected BSDEs with jumps and irregular obstacles, which admit both totally inaccesible and predictable jumps. We propose a fully implementable scheme, based on penalization method and the approximation of the Brownian motion and the Poisson process by two independent random walks, which is shown to converge to the solution of the DRBSDE. We illustrate the theoretical results with some numerical examples.

• In Chapter 8 (based on paper [START_REF] Dumitrescu | Reflected Scheme for Doubly reflected BSDEs with Jumps and RCLL obstacles[END_REF]), we introduce an alternative fully implementable scheme to the one presented in Chapter 6, in order to approximate the solution of doubly reflected BSDEs with jumps and irregular obstacles. This scheme is obtained by a direct discretization of the DRBSDE and it thus depends only on the time step n (no more on the penalization parameter p). We provide the convergence of the scheme, as well as some numerical examples.

Regarding the perspectives, there are many directions of research to follow, from both theoretical and numerical point of views. In collaboration with R. Elie and D. Possamai we are finishing a work on BSDEs with weak reflection ( [START_REF] Dumitrescu | Weak reflected BSDEs and approximative hedging for american options[END_REF]), which are related to the approximative hedging for American options. Together with M.C. Quenez and A. Sulem, we address a new mixed stochastic control/ optimal stopping game problem in the Markovian framework [START_REF] Dumitrescu | Mixed Stochastic Control/Optimal Stopping Games with E f -Expectations[END_REF] and study the links between Generalized Dynkin Games and nonlinear pricing, in complete and incomplete markets ( [START_REF] Dumitrescu | Nonlinear pricing and hedging in a market with default[END_REF]). From a numerical point of view, it would be useful to propose some numerical schemes for the solution of DRBSDEs with RCLL barriers in the case of a general Poisson measure, as well as for BSDEs with weak terminal condition.

Chapter 2

Introduction générale 2.1 EDSR avec condition terminale faible non linéaire

Ce chapitre repose sur un article écrit sous la coordination de Prof B. Bouchard1 et soumis pour publication: ≪BSDEs with nonlinear weak terminal condition≫ [START_REF] Dumitrescu | BSDEs with nonlinear weak terminal condition[END_REF].

Préliminaires et vue d'ensemble de la littérature

Nous commençons par rappeler qu'une équation différentielle stochastique rétrograde (EDSR) est une équation de la forme

Y t = ξ + T t g(s, Y s , Z s )ds - T t Z s dW s , 0 ≤ t ≤ T, ( 2
-p.s, t → Z t appartient à L 2 (0, T ), t → g(t, Y t , Z t ) appartient à L 1 (0, T ) et Y t = ξ + T t g(s, Y s , Z s )ds - T t Z s dW s , 0 ≤ t ≤ T. (2.1.2)
Les EDSR ont été introduites pour la première fois avec un générateur linéaire par J.-M.Bismut [START_REF] Bismut | Conjugate convex functions in optimal stochastic control[END_REF]. Le point de départ de la théorie des EDSR non linaires est l'article de E. Pardoux et S. Peng [START_REF] Pardoux | Adapted solution of a backward stochastic differential equation[END_REF] où les auteurs prouvent l'existence et l'unicité pour des EDSR avec un générateur non linéaire en (y, z). Rappelons ce résultat: Theorem 2.1.2 (E.Pardoux -S.Peng). Supposons le générateur g Lipschitz en (y, z), uniformément par rapport (t, ω) et

E[|ξ| 2 + T 0 |g(s, 0, 0)| 2 ds] < +∞. Alors l'EDSR (2.1.1) admet une unique solution (Y, Z) appartenant S 2 × H 2 .
Exprimée dans le ≪sens forward≫, la résolution de ces équations revient à trouver une condition initiale Y 0 et un processus Z tels que le processus contrôlé (Y Y 0 ,Z t ) 0≤t≤T satisfait l'EDS:

Y Y 0 ,Z t = Y 0 - t 0 g(s, Y Y 0 ,Z s , Z s )ds + t 0 Z s dW s (2.1.3)
et la condition Y Y 0 ,Z T = ξ à la date terminale T . D'un point de vue des application en finance, l'étude de ces équations se rapporte à l'évaluation d'options européennes en marché complet car Y donne le prix et Z fournit la stratégie de couverture associée. Cependant, puisqu'en marché incomplet il n'est pas toujours possible de construire un portefeuille de réplication tel que la valeur terminale coïncide avec le prix de la créance ξ, une formulation plus faible est de chercher une condition initiale Y 0 et un contrôle Z tels que

Y Y 0 ,Z T ≥ ξ. (2.1.4)
Dans ce cas, nous sommes intéressés à trouver la condition initiale Y 0 minimale qui correspond au coût de la stratégie de sur-réplication la moins onéreuse pour la créance éventuelle ξ, et le contrôle associé Z (voir e.g. [START_REF] Karoui | Backward Stochastic Differential Equations in Finance[END_REF]). Comme dans la plupart des cas, le prix de sur-réplication conduit à coût trop élevé pour l'acheteur, il a été suggéré d'assouplir la forte contrainte (2.1.4) en une version plus faible de la forme

E[l(Y Y 0 ,Z T -ξ)] ≥ m, (2.1.5)
où m est un seuil fixé et l est une fonction croissante. En particulier, l(x) = 1 {x≥0} correspond au critère Y Y 0 ,Z T ≥ ξ avec probabilité m au moins ce qui correspond au problème de couverture de quantile introduit par Föllmer et Leukert [START_REF] Follmer | Quantile hedging[END_REF]. Ce problème a ensuite été étudié par Bouchard, Elie et Touzi [START_REF] Bouchard | Stochastic target problems with controlled loss[END_REF] dans un cadre markovien en utilisant les techniques de cible stochastique développées par Soner et Touzi (voir [START_REF] Soner | Stochastic target problems, dynamic programming, and viscosity solutions[END_REF]). Cette approche, reposant sur la formulation primale de la fonction valeur et sur la programmation dynamique géométrique, permet un traitement de ce problème dans un cadre plus général, par exemple quand la stratégie de l'agent peut influencer la valeur des actifs risqués (modèle avec un grand investisseur). La résolution initiale du problème par Föllmer et Leukert s'appuie sur le fait que la stratégie est linéaire dans le contrôle.

Plus généralement, l peut représenter une fonction de perte, un exemple classique est l(x) := -(x -) q avec q ≥ 1, voir [START_REF] Follmer | Efficient hedging: cost versus shortfall risk[END_REF] pour des dynamiques générales non markoviennes mais linéaires. Un autre exemple en mathématiques financières peut être représenté avec l jouant le rôle d'une fonction d'utilité. Très récemment, Bouchard, Elie et Reveillac [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF] ont traité le problème dans un cadre non linéaire et non markovien. Dans ce but, ils ont introduit une nouvelle classe d'EDSR pour lesquelles la condition terminale n'est pas une variable aléatoire fixée mais satisfait seulement la faible contrainte suivante ), α ∈ A 0 }, ainsi que l'existence d'un contrôle optimal lorsque g et Φ sont convexes. Quelques propriétés de la fonction valeur décrite par (2.1.10), comme la continuité et la convexité par rapport au seuil m, sont obtenues. Finalement, par des arguments probabilistiques, il est montré que le problème (2.1.10) admet une représentation duale qui prend la forme d'un problème de contrôle stochastique sous la forme de Meyer, étendant les résultats obtenus lorsque le générateur g est linéaire (voir [START_REF] Bouchard | Stochastic target problems with controlled loss[END_REF], [START_REF] Follmer | Quantile hedging[END_REF] et [START_REF] Follmer | Efficient hedging: cost versus shortfall risk[END_REF]).

E[Ψ(Y Y 0 ,Z T )] ≥ m. ( 2
Y α t = Φ(M m,α T ) + T t g(s, Y α s , Z α s )ds - T t Z α s dW s , 0 ≤ t ≤ T, (2.1.9
Y α (τ ) := essinf{Y α ′ τ , α ′ ∈ A 0 t.q. α ′ = α sur [[0, τ ]]}, 0 ≤ τ ≤ T. ( 2

Contributions

Dans ce chapitre, nous introduisons une classe d'EDSR plus générale que celle considérée par Bouchard, Elie et Reveillac [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF] dont la condition terminale satisfait la contrainte faible et non linéaire suivante:

E f 0,T [Ψ(Y Y 0 ,Z T )] ≥ m, ( 2
Ψ(Y Y 0 ,Z T ) ≥ M m,α T = m - T 0 f (s, M m,α s , α s )ds + T 0 α s dW s . ( 2
Y α (τ ) := essinf{Y α ′ τ , α ′ ∈ A 0 t.q. α ′ = α sur [[0, τ ]]}. ( 2
(m) = sup l>0 (lm -X 0 (l)), où X 0 (l) := inf (λ,γ)∈U ×V X l,λ,γ 0 , (2.1.16) avec X l,λ,γ 0 := E T 0 L λ s g(s, λ s )ds - T 0 A l,γ s f (s, γ s )ds + L λ T Φ( A l,γ T L λ T ) ,
et f (respectivement g, Φ) la conjuguée concave de f (respectivement la conjuguée convexe de g et Φ).

La non-linéarité additionnelle de f soulève à des difficultés techniques, puisque la plupart des résultats de [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF] sont obtenus en utilisant des techniques exploitant la linéarité de la contrainte et ne sont donc pas adaptées à notre cadre non linéaire. Outre l'intérêt mathématique de notre étude, ce travail est aussi motivé par des applications financières. En effet, notre problème est étroitement lié à la couverture approximative sous des contraintes données par des mesures de risque dynamiques d'une créance, qui peut être exprimée par la forme suivante: [START_REF] Royer | Backward stochastic differential equations with jumps and related non-linear expectations[END_REF] prouve un théorème de comparaison sous des hypothèses plus faibles et introduit l'espérance non linéaire dans ce cadre.

inf{Y 0 such that ∃ Z : ρ 0,T [-(Y T -ξ) -] ≤ m}, ( 2
En outre, en 2004-2005, différents auteurs ont introduit des mesures de risque dynamiques dans un cadre brownien, définies comme des solutions d'EDSR. Plus précisément, étant donné un générateur Lipschtiz g(t, x, π) et une condition terminale T , la mesure de risque ρ à la date t d'une position ξ est donnée par -X t où X est la solution de l'EDSR dirige par un mouvement brownien, associée au driver g et à la condition terminale ξ. Par le théorème de comparaison, ρ satisfait la propriété de monotonicité qui est habituellement exigée pour une mesure de risque. Beaucoup d'études ont été faites récemment sur ce type de mesure de risque dynamique, particulièrement concernant les problèmes d'optimisation robuste et les problèmes d'arrêt optimal, dans le cas d'une filtration brownienne et d'un générateur concave (voir, par exemple, Bayraktar et co-auteurs de [START_REF] Bayraktar | Optimal stopping for Dynamic Convex Risk Measures[END_REF]). Dans le cas avec sauts, les liens entre les EDSR et les mesures de risque dynamiques ont été étudiés recemment par Quenez-Sulem dans [START_REF] Quenez | BSDEs with jumps, optimization and applications to dynamic risk measures[END_REF].

Les Équations Différentielles Stochastiques Rétrogrades Réfléchies (EDSRR) ont été introduites en 1997 par El Karoui, Kapoudjian, Pardoux, Peng et Quenez [START_REF] El Karoui | Reflected solutions of backward SDE's, and related obstacle problems for PDE's[END_REF] dans le cas d'une filtration générée par un mouvement brownien. Ces équations sont des généralisations du problème de Skorokhod déterministe. En effet, étant donné un processus adapté ξ := (ξ t ) t≤T qui joue le rôle d'une barrière, la solution d'une EDSRR associée aux données (η, g, ξ) est un triplet de processus de carré intégrable {(Y t , Z t , A t ); 0 ≤ t ≤ T } qui satisfont: [START_REF] Hamadène | Reflected backward stochastic differential equation with jumps and random obstacle[END_REF]. Une solution pour ce type d'équation, associée à un générateur f , une valeur terminale η et une barrière ξ, est un quadruplet de processus (Y, Z, U, A) de solutions adaptées qui satisfont l'équation suivante: [START_REF] Quenez | Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps[END_REF]. La barrière ξ est seulement càdlàg et donc les dates des sauts du processus Y ne proviennent pas uniquement de ceux du processus de Poisson associé (sauts inaccessibles) mais aussi de ceux du processus ξ (sauts prévisibles) qui implique que le processus Y a deux types de sauts: inaccesibles et prévisibles. La difficulté vient du fait que, puisque la barrière ξ peut avoir des sauts prévisibles, le processus réfléchissant A n'est plus continu mais seulement càdlàg. Dans ce cas, la différence par rapport à (2.2.4) apparaît seulement dans la condition de Skorokhod qui devient:

             Y t = η + T t g(s, ω, Y s , Z s )ds + A T -A t - T t Z s dW s , 0 ≤ t ≤ T, Y t ≥ ξ t , T 0 (Y t -ξ t )dA t = 0. ( 2
     -dY t = b(t, Y t , π t )dt + dA t -π t dW t , Y T = (ξ T -γ) + , Y t ≥ (ξ t -γ) + et T 0 (Y t -(ξ t -γ) + )dA t = 0, ( 2
                     Y t = η + T t g(s, ω, Y s , Z s , U s )ds + A T -A t - T t Z s dW s - T t R * U s (e) Ñ (ds, de), 0 ≤ t ≤ T, Y t ≥ ξ t , T 0 (Y t -ξ t )dA t = 0. ( 2
T 0 (Y t --ξ t -)dA t = 0.
Une application importante des EDSR réfléchies est leur connexion aux problèmes d'arrêt optimal et aux inégalités variationnelles associées dans le cas markovien. Plus précisément, étant donné un processus càdlàg (ξ t , 0 ≤ t ≤ T ) et un générateur Lipschitz g satisfaisant une hypothèse supplémentaire afin que le théorème de comparaison soit vérifié, la solution Y de l'EDSRR associée satisfait: pour tout temps d'arrêt S ∈ T 0 ,

Y S = esssup τ ∈T S X S (ξ τ , τ ), p.s.
(2.2.5) où pour τ ∈ T S , X • (ξ τ , τ ) est la solution de l'EDSR associée à la date terminale τ , condition terminale ξ τ et générateur g (voir [START_REF] Quenez | Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps[END_REF]). On précise que T S représente l'ensemble des temps d'arrêt à valeurs en [0, T ], p.s. plus grands que S.

Contributions

Dans ce chapitre, nous étudions le problème d'arrêt optimal pour des mesures de risque dynamiques avec sauts dans un cadre markovien. Formulons notre problème. Soit T > 0 la date terminale et f un générateur Lipschitz. Pour chaque T ′ ∈ [0, T ] et η ∈ L 2 (F T ′ ), définissons: Cette variable aléatoire v(S) correspond à la mesure de risque minimale à la date S.

ρ f t (η, T ′ ) = ρ t (η, T ′ ) := -X t (η, T ′ ), 0 ≤ t ≤ T ′ , ( 2
Puisque par définition ρ S (ξ τ , τ ) = -X S (ξ τ , τ ), nous avons, pour chaque temps d'arrêt S ∈ T 0 ,

v(S) = essinf τ ∈T S -X S (ξ τ , τ ) = -esssup τ ∈T S X S (ξ τ , τ ). (2.2.8)
Maintenant, en utilisant le lien entre les EDSR réfléchies et l'arrêt optimal (2.2.5), on peut relier la fonction valeur du problème défini par (2.2.8) à la solution de l'EDSR réfléchie. Plus précisément, nous avons:

v(S) = -Y S .
(2.2.9)

Puisque notre objectif est de caractériser cette fonction valeur dans un cadre markovien, nous considérons la condition terminale, l'obstacle et le générateur de la forme suivante:

       ξ t,x
s := h(s, X t,x s ), s < T, ξ t,x T := g(X t,x T ), g(s, ω, y, z, k) := g(s, X Sous quelques hypothèses supplémentaires, nous obtenons un théorème de comparaison, s'appuyant sur une version non locale du lemme de Jensen-Ishii, duquel l'unicité de la solution de viscosité est une conséquence. Nous étendons les résultats de [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions theory revisited[END_REF] dans le cas des EDSR non linéaires qui conduisent à un opérateur intégro-différentiel plus complexe dans l'EDP associée. Dans le cas d'équations intégro-différentielles, une difficulté significative réside dans le traitement d'opérateurs non locaux. L'idée principale est de les décomposer en un opérateur qui correspond aux petits sauts et un autre qui correspond aux grands sauts et d'utiliser une définition moins classique des solutions de viscosité introduite dans [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions theory revisited[END_REF], adaptée aux équations intégro-différentielles et équivalentes aux deux classiques qui combinent l'approche avec les fonctions tests et les sub-superjets (la solution est remplacée par la fonction test seulement autour de la singularité de la mesure dans l'opérateur non local).

Jeux de Dynkin généralisés et EDSR doublement réfléchies avec sauts

V := sup σ inf τ E[ξ σ 1 τ >σ + ζ τ 1 τ ≤σ ]; V := inf τ sup σ E[ξ σ 1 τ >σ + ζ τ 1 τ ≤σ ].
(2.3.1)

Si les deux fonctions valeurs définies ci-dessus coïncident, on dit que le jeu admet une fonction valeur.

Une application financière intéressante du jeu de Dynkin est l'étude des options jeu, aussi connus sous le nom d'options israéliennes, tels que définies par Kifer [START_REF] Kifer | Game options[END_REF]. Une option jeu est un contrat entre un émetteur et un détenteur dans lequel le détenteur peut exercer l'option à tout moment pour un gain tandis que l'émetteur peut la résilier à tout moment contre un paiement.

Ceci est l'un des quelques contrats financiers dans lesquels l'émetteur prend également des décisions qui affectent le gain. Si nous ignorons la dépendance par rapport aux actifs sous-jacents et nous concentrons sur les liens entre décisions et gains, l'option jeu est comparable au jeu de Dynkin. De plus, le paiement associé à la résiliation est typiquement supposé supérieur ou égal au gain de l'exercice, faisant écho aux inégalités standard sur le gain existant dans les jeux de Dynkin. À la fois dans les modèles à temps discret et continu, Kifer [START_REF] Kifer | Game options[END_REF] a montré que l'option jeu a un unique prix d'arbitrage. D'autres recherches associées aux options jeu ainsi qu'à des contrats financiers de type jeu ont été initiées par Bielecki et al. [START_REF] Bielecki | Defaultable game options in a hazard process model[END_REF], et Dolinsky Kifer [START_REF] Dolinsky | Hedging with risk for game options in discrete time, Stochastics[END_REF], Dolinsky et al. [START_REF] Dolinsky | Perfect and partial hedging for swing game options in discrete time[END_REF], Hamadene et Zhang [START_REF] Hamadène | The continuous time nonzero-sum Dynkin game problem and application in game options[END_REF], Kallsen et Kuhn [START_REF] Kallsen | Pricing derivatives of American and game type in incomplete markets[END_REF][START_REF] Kallsen | Convertible bonds: financial derivatives of game type[END_REF], et Kifer [START_REF] Kifer | Game options[END_REF], etc.

Nous nous concentrons maintenant sur le lien entre les Jeux de Dynkin Classiques et les EDSR Doublement Réfléchies (EDSRDR) introduites par Cvitanic et Karatzas [START_REF] Cvitanic | Backward stochastic differential equations with reflection and Dynkin games[END_REF] dans le cas d'une filtration brownienne. La solution est contrainte de rester entre une barrière supérieure ζ et une barrière inférieure ξ et est représentée par un quadruplet de processus de carré intégrable

{(Y t , Z t , A t , A ′ t ); 0 ≤ t ≤ T } satisfaisant:                      Y t = ξ + T t g(s, ω, Y s , Z s )ds + A T -A t -(A ′ T -A ′ t ) - T t Z s dW s , 0 ≤ t ≤ T, ξ t ≤ Y t ≤ ζ t , T 0 (Y t -ξ t )dA t = 0 et T 0 (ζ t -Y t )dA ′ t = 0. ( 2
V (t) = V (t) = V (t) = Y t a.s. ∀0 ≤ t ≤ T, ainsi qu'un point-selle (σ t , τt ) ∈ T t × T t donné par σt := inf{s ∈ [t, T ), Y s = ζ s } ∧ T ; τt := inf{s ∈ [t, T ), Y s = ξ s } ∧ T.
à savoir

E[I t (τ t , σt )] ≤ E[I t (τ t , σt )] = Y t ≤ E[I t (τ t , σ t )],
pour tout (σ, τ ) ∈ T t × T t .

Depuis l'article séminal de Cvitanic-Karatzas, beaucoup d'auteurs ont exploré l'existence et l'unicité de la solution ainsi que les liens avec les Jeux de Dynkin Classiques avec différentes hypothèses sur le coefficient g et sur la régularité des barrières (voir par exemple Lepeltier-San Martin et [START_REF] Lepeltier | Reflected backward stochastic differential equations with two rcll barriers[END_REF]). Ces résultats ont également été étendus au cas des EDSRDR dirigées par un mouvement brownien et une mesure aléatoire de Poisson (voir par exemple [START_REF] Hamadène | BSDEs with two reacting barriers driven by a Brownian motion and an independent Poisson noise and related Dynkin game[END_REF], [START_REF] Hamadène | Backward SDEs with two rcll reflecting barriers without Mokobodski's hypothesis[END_REF], [START_REF] Crépey | Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison[END_REF]).

Le lien ci-dessus entre les Jeux de Dynkin Classiques et les EDSRDR peut être étendu dans le cas plus général des EDSRDR non linéaires, puisque étant donné la solution Y de la EDSRDR, il a été montré qu'elle coïncide avec la fonction valeur d'un jeu de Dynkin classique de gain: Le jeu admet une fonction valeur si V (S) = V (S).

I S (τ, σ) = σ∧τ S g(u, Y u , Z u , k u )du + ξ τ 1 τ ≤σ + ζ σ 1 σ<τ , (2.3
Sous la condition de Mokobodski, nous montrons l'existence d'une fonction valeur pour ce jeu qui peut être caractérisée comme étant l'unique solution d'une EDSR non linéaire doublement réfléchie (EDSRDR). Jusqu'à maintenant, aucune interprétation des EDSR non linéaires doublement réfléchies en termes de contrôle ou de problèmes de jeu (avec espérance non linéaire) n'a été donnée dans la littérature.

Grâce à cette caractérisation, nous obtenons des propriétés de ces EDSRDR comme un théorème de comparaison général et un théorème de comparaison strict. Nous établissons également de nouvelles estimations à priori avec constante universelle pour EDSRDR et la preuve s'appuie sur la caractérisation de la solution comme la fonction valeur de ce nouveau problème de jeu. Lorsque les deux obstacles sont semi-continus supérieurement à gauche le long des temps d'arrêts, nous montrons l'existence d'un point-selle du jeu de Dynkin généralisé. Nous soulignons que nous ne supposons pas la stricte séparabilité des barrières, hypothèse qui est cruciale dans la littérature antérieure. Nous pouvons nous en passer en imposant une contrainte supplémentaire sur les processus croissants A, A ′ qui apparaissent dans la définition (2.3.2) (notons que dans notre cadre les processus croissants A et A ′ ne sont plus continus). Plus précisément, nous supposons que les mesures dA et dA ′ sont mutuellement singulières dans le sens probabiliste, i.e. qu'il existe D ∈ P tel que

E[ T 0 1 D dA t ] = E[ T 0 1 D c dA ′ t ] = 0.
Cette contrainte est également importante afin d'obtenir l'unicité des processus croissants A et A ′ . De plus, elle permet d'identifier les sauts positifs et négatifs de la solution de l'EDSRDR.

Nous continuons par l'étude d'un jeu à somme nulle mixte généralisé sous la g-espérance conditionnelle, dans lequel deux joueurs s'affrontent en prenant deux actions: contrôle continu et arrêt. Nous obtenons des conditions suffisantes (par exemple les générateurs contrôlés g u,v ont un pointselle g u,v ) qui assurent l'existence d'une fonction valeur du jeu mixte généralisé et caractérisent la fonction valeur commune comme la solution d'une EDSRDR avec générateur g u,v . Quand les deux obstacles sont semi-continus supérieurement à gauche le long des temps d'arrêt, le jeu mixte généralisé correspondant admet un point-selle.

Nous étudions ensuite le jeu de Dynkin généralisé dans le cadre markovien et ses liens avec les inégalités variationnelles des équations intégro-différentielles partielles paraboliques avec deux obstacles. Plus précisément, nous montrons que la fonction valeur d'un jeu de Dynkin généralisé dans le cas markovien est l'unique solution de viscosité de l'équation intégro-différentielle correspondant. Du point de vue des EDP, ce résultat donne une nouvelle interprétation probabiliste de l'EDP semi-linéaire avec deux barrières en terme de problèmes de jeux. [START_REF] Bellman | Introduction to the mathematical theory of control processes[END_REF]) qui disait que ≪an optimal policy has the property that, whatever the initial state and control are, the remaining decisions must constitue an optimal policy with regard to the state resulting from the first decision≫ qui pourrait se traduire en français par ≪une règle optimale a la propriété que, quel que soit l'état initial et le contrôle, les décisions ultérieures doivent constituer une règle optimale par rapport à l'état résultant de la première décision≫. Typiquement, un problème de contrôle stochastique avec un horizon fini T peut être écrit comme suit:

V (0, x) := sup α∈A E[ T 0 f (s, X α s , α s )ds + g(X α T )], (2.4.1)
où f est le gain instantané et g est le gain terminal.

Une écriture formelle du PPD est Le cas d'une fonction valeur discontinue a été étudié dans un cadre déterministe dans les années 80: un principe de programmation dynamique faible a été établi pour le contrôle déterministe par [START_REF] Barles | Discontinuous viscosity solutions of first-order Hamilton-Jacobi equations: a guided visit[END_REF] (voir [START_REF] Barles | Discontinuous viscosity solutions of first-order Hamilton-Jacobi equations: a guided visit[END_REF], voir aussi [START_REF] Barles | Discontinuous viscosity solutions of deterministic optimal control problems Decision and Control[END_REF] [START_REF] Barles | Discontinuous viscosity solutions of deterministic optimal control problems Decision and Control[END_REF]).

V (0, x) = v(0, x) := sup α∈A E[ τ 0 f (s, X α s , α s )ds + V (τ, X α τ )], (2.4
Plus précisément, il prouve que l'enveloppe semi-continue supérieure V * et l'enveloppe semicontinue inférieure V * de la fonction valeur V satisfait, respectivement, le principe de sous-et sur-optimalité de la programmation dynamique de Lions et Souganidis (1985) [START_REF] Lions | Differential Games, Optimal Control and Directional Derivatives of Viscosity Solutions of Bellman's and Isaacs' Equations[END_REF]. Il montre ensuite que la fonction valeur (discontinue) est une solution de viscosité faible de l'équation de Bellman associée dans le sens où V * est une sous-solution de viscosité et V * est une sur-solution de viscosité de l'équation de Bellman.

Plus récemment, Bouchard et Touzi (2011) (voir [START_REF] Bouchard | Weak Dynamic Programming Principle for Viscosity Solutions[END_REF]) ont prouvé un principe de programmation dynamique faible dans un cadre stochastique lorsque la fonction valeur n'est pas nécessairement continue, ni même mesurable. Ils prouvent que l'enveloppe semi-continue supérieure V * satisfait le principe de sous-optimalité de la programmation dynamique et avec une hypothèse supplémentaire de régularité (semi-continuité inférieure) du gain g, ils obtiennent que l'enveloppe semi-continue inférieure V * satisfait le principe de sur-optimalité.

Un principe de programmation dynamique faible a ensuite été établi, sous des hypothèses de régularité spécifiques, pour des problèmes avec contraintes d'état par Bouchard et Nutz (2012) dans [START_REF] Bouchard | Weak Dynamic Programming for Generalized State Constraints[END_REF] et pour les jeux stochastiques à somme nulle par Bayraktar et Yao (2013) dans [START_REF] Bayraktar | A Weak Dynamic Programming Principle for Zero-Sum Stochastic Differential Games with Unbounded Controls[END_REF].

Dans la suite, nous présentons les résultats classiques du problème pour le contrôle stochastique et l'arrêt optimal (pour des problèmes à horizon fini) dans le cas où la fonction valeur n'est pas a priori continue, ni même mesurable. Nous rappelons le principe de programmation dynamique faible obtenu par Bouchard et Touzi ( [START_REF] Bouchard | Weak Dynamic Programming Principle for Viscosity Solutions[END_REF]) ainsi que les équations HJB associées.

(i) Contrôle stochastique et programmation dynamique faible pour les espérances classiques

Nous notons A l'ensemble de tous les processus progessivement mesurables α = {α t , t < T } à valeurs dans A, un sous ensemble de R, appartenant à H 2 (l'ensemble des processus φ tels que E[ T 0 φ 2 s ds] < +∞). Les éléments de A sont appelés les processus de contrôle.

Pour chaque processus de contrôle α ∈ A, nous considérons l'équation différentielle stochastique suivante:

dX t,x,α s = b(X t,x,α s , α s )ds + σ(X t,x,α s , α s )dW s , (2.4.3)
où les coefficients b et σ satisfont les conditions habituelles de Lipschitz et de croissance linéaire afin que l'EDS ci-dessus admette une unique solution forte.

Étant donné une condition initiale (t, x), le processus X t,x,α est appelé le processus controlé car sa dynamique est dirigée par l'action du processus de contrôle α.

Nous définissos la fonctionnelle de coût J sur [0, T ] × R × A par: Afin de décrire le comportement local de la fonction valeur V au moyen de l'équation de programmation dynamique ou Hamilton-Jacobi-Bellman, le point clé est le Principe de Programmation Dynamique. Puisque le PPD implique la fonction valeur elle-même qui n'est peut-être pas mesurable sous ces hypothèses, Bouchard et Touzi [START_REF] Bouchard | Weak Dynamic Programming Principle for Viscosity Solutions[END_REF] proposent une version Faible du Principe de Programmation Dynamique laquelle est montrée être suffisante pour obtenir l'équation de programmation dynamique. Ce PPD faible fait apparaître l'enveloppe semi-continue supérieure de la fonction valeur V , respectivement celle semi-continue inférieure, qui sont définies comme suit:

J(t, x, α) = E T t f (s, X α,t
pour tout t ∈ [0, T ], pour tout x ∈ R, V * (t, x) := lim inf (t ′ ,x ′ )→(t,x) V (t ′ , x ′ ) et V * (t, x) := lim sup (t ′ ,x ′ )→(t,x) V (t ′ , x ′ ). ( 2
V (t, x) ≤ sup α∈At E[ θ α t f (s, X α,t,x s , α s )ds + V * (θ α , X α,t,x θ α )], (2.4.6)
2. Supposons de plus que g est semi-continue inférieurement et X α t,x 1 t,θ α est L ∞ -borné pout tout ν ∈ A t . Alors:

V (t, x) ≥ sup α∈At E[ θ α t f (s, X α,t,x s , α s )ds + V * (θ α , X α,t,x θ α )].
(2.4.7)

Le PPD faible ci-dessus est montré sans utiliser les théorèmes abstraits de sélection mesurable. Les auteurs appellent à la place le covering lemma de Vitali. L'inégalité qui est la plus difficile à obtenir est la seconde et elle nécessiste une hypothèse de semi-continuité inférieure sur le critère (qui est satisfaite dans le cas où le gain g est semi-continue inférieurement).

Nous faisons remarquer que lorsque V est continue alors V = V * = V * et le principe de programmation dynamique faible ci-dessus se réduit au principe de programmation dynamique classique:

V (t, x) = sup α∈At E θ α t f (s, X α,t,x s , α s )ds + V (θ α , X α,t,x θ α ) .
Comme mentionné précédemment, le principe de programmation dynamique faible représente l'étape principale pour obtenir l'équation de programmation dynamique qui correspond à la contrepartie infinitésimale du PPD. Elle est généralement appelée l'équation Hamilton-Jacobi-Bellman. L'équation HJB associée est obtenue grâce au théorème suivant:

Theorem 2.4.2. Supposons que la fonction valeur V ∈ C 1,2 ([0, T ), R) et soit f (•, •, a) continue en (t, x) pour tout a ∈ A fixé. Alors, pour tout (t, x) ∈ [0, T ) × R: -∂ t V (t, x) -sup a∈A {b(t, x, a)∂ x V (t, x) + 1 2 T r[σσ(t, x, a)D 2 xx V (t, x)] + f (t, x, a)} = 0. (2.4.8)
Notons que lorsque la fonction valeur V n'est pas continue, alors elle vérifie au sens des solutions de viscoité l'EDP ci-dessus.

Nous présentons maintenant les résultats principaux concernant l'arrêt optimal qui représentent un cas particulier des problèmes de contrôle stochastique lorsque le contrôle prend la forme d'un temps d'arrêt.

(ii) Arrêt optimal et principe de programmation dynamique faible pour les espérances classiques Pour 0 ≤ t ≤ T < +∞, nous notons T [t,T ] la collection de tous les F-temps d'arrêt à valeurs dans [t, T ]. Le processus d'état sous-jacent X t,x de condition initiale (t, x) est défini par l'équation différentielle stochastique:

dX t,x s = b(s, X t,x s )ds + σ(s, X t,x s )dW s ,
où b et σ satisfont les conditions habituelles de Lipschitz et de croissance linéaire afin que l'EDS ci-dessus ait une unique solution forte. Soit g une fonction mesurable à croissance polynômiale et supposons que:

E[ sup 0≤t≤T |g(X t )|] < ∞.
Pour un temps d'arrêt admissible, le critère est défini comme suit:

J(t, x, τ ) = E g(X t,x τ ) (2.4.9)
Nous considérons maintenant le sous-ensemble des temps d'arrêt:

T t t := {τ ∈ T [t,T ] : τ independents de F t }.
(2.4.10)

Le problème d'arrêt optimal est défini par:

V (t, x) = sup τ ∈T t t J(t, x, τ ). (2.4.11)
En utilisant les mêmes arguments que pour le problème de contrôle stochastique présenté cidessus, Bouchard et Touzi ont montré le Principe de Programmation Dynamique Faible suivant:

Theorem 2.4.3. Pour (t, x) ∈ [0, T ] × R, soit θ ∈ T t t un temps d'arrêt tel que X t,x θ soit borné. Alors: V (t, x) ≤ sup τ ∈T t t E 1 {τ <σ} g(X t,x τ ) + 1 {τ ≥θ} V * (θ, X t,x θ ) , (2.4
.12)

V (t, x) ≥ sup τ ∈T t t E 1 {τ <σ} g(X t,x τ ) + 1 {τ ≥θ} V * (θ, X t,x θ ) . (2.4.13)
Lorsqu'on sait a priori que la fonction valeur V est régulière, la contrepartie infinitésimale du principe de programmation dynamique est la suivante:

Theorem 2.4.4. Supposons que V ∈ C 1,2 ([0, T ), R) et soit g : R → R continue. Alors V est solution du problème d'obstacle: min{-(∂ t + L)V, V -g} = 0, (2.4.14)
où LV est le générateur infinitésimal du processus de diffusion markovien X.

Le problème de contrôle stochastique classique (2.4.1) a été généralisé par Peng lorsque la fonctionnelle de coût est définie au moyen d'une équation différentielle stochastique rétrograde non linéaire (voir [START_REF] Peng | A generalized dynamic programming principle and Hamilton-Jacobi-Bellman-Equation[END_REF] et [START_REF] Peng | BSDE and stochastic optimizations, Topic in Stochastic Analysis[END_REF]) sous des hypothèses qui assurent que la fonction valeur est continue. Il établit une programmation dynamique en utilisant la méthode du semi-groupe rétrograde et obtient les équations HJB associées. Ces résultats permettent d'obtenir une interprétation stochastique pour une plus large classe d'équations HJB non linéaires puisque le coefficient f dépend également de (y, z).

À la fin de cette section, nous aimerions mentionner quelques développements dans le cas où l'incertitude affecte uniquement la volatilité du modèle.

Contributions

Dans ce chapitre, nous nous intéressons à la généralisation des résultats obtenus par Bouchard et Touzi ( [START_REF] Bouchard | Weak Dynamic Programming Principle for Viscosity Solutions[END_REF]) lorsque l'espérance linéaire E est remplacée par une espérance non linéaire définie par une Équation Différentielle Stochastique avec sauts. Dans le cadre markovien, la fonction valeur de notre problème est la suivante:

V (t, x) := sup α∈A E α 0,T [g(X α,t,x T )], (2.4.15)
où E α est l'espérance conditionnelle non linéaire associée à l'EDSR avec sauts et le générateur contrôlé f (α t , X α t , y, z, k). Nous regardons cette étude dans le cas où la fonction de gain g est uniquement borélienne. De plus, dans ce chapitre, nous considérons le problème combiné lorsqu'il y a un contrôle supplémentaire prenant la forme d'un temps d'arrêt. Nous considérons ensuite des problèmes de contrôle stochastique et d'arrêt optimal généralisés de la forme

V (t, x) := sup α sup τ E α 0,τ [h(X α,t,x τ )], (2.4.16) où h(X α,t,x τ
) est un gain irrégulié. Afin de caractériser la fonction valeur comme la solution d'une inégalité variationnelle HJB, nous établissons tout d'abord un Principe de Programmation Dynamique qui est obtenu en utilisant des techniques sophistiquées d'analyse stochastique. Nous mettons en avant que, en conséquence des faibles hypothèses sur les coefficients, la fonction valeur de notre problème n'est pas nécessairement continue, ni même mesurable.

Comme mentionné dans la section introductive, puisque pour t fixé, la fonction valeur x → V (t, x) n'est pas nécessairement mesurable, nous ne pouvons pas établir une programmation dynamique classique. Nous obtenons à la place un Principe Programmation Dynamique faible qui implique la fonction V * et la fonction V * définies par

V * (t, x) := lim sup (t ′ ,x ′ )→(t,x) V (t ′ , x ′ ), ∀(t, x) ∈ [0, T ) × R et V * (T, x) = g(x), ∀x ∈ R; V * (t, x) := lim inf (t ′ ,x ′ )→(t,x) V (t ′ , x ′ ), ∀(t, x) ∈ [0, T ) × R et V * (T, x) = g(x), ∀x ∈ R.
Remarquons que dans notre cas, la fonction V * (respectivement V * ) n'est pas nécessairement semi-continue supérieurement (respectivement inférieurement) sur [0, T ] × R car le gain terminal g est seulement borélien (il n'est pas supposé satisfaire une quelconque hypothèse de régularité). Ce n'est pas le cas dans la littérature précédente même dans le cas linéaire où g est supposée être semicontinue inférieurement (voir [START_REF] Bouchard | Weak Dynamic Programming Principle for Viscosity Solutions[END_REF]). Nous donnons ci-dessous le principe de sous-(respectivement sur-) optimalité de la programmation dynamique satisfait par V * (respectivement V * ), l'un de nos principaux résultats de ce chapitre.

Theorem 2.4.5. (Un principe de programmation dynamique faible) La fonction V * satisfait le principe de sous-optimalité de la programmation dynamique, c'est à dire pour tout t ∈ [0, T ] et pour tout temps d'arrêt θ ∈ T t t ,

V (t, x) ≤ sup α∈A t t sup τ ∈T t t E α,t,x t,θ∧τ h(τ, X α,t,x τ )1 τ <θ + V * (θ, X α,t,x θ )1 τ ≥θ , (2.4.17)
La fonction V * satisfait le principe de sur-optimalité de la programmation dynamique, c'est à dire

pour tout t ∈ [0, T ] et pour tout temps d'arrêt θ ∈ T t t , V (t, x) ≥ sup α∈A t t sup τ ∈T t t E α,t,x t,θ∧τ h(τ, X α,t,x τ )1 τ <θ + V * (θ, X α,t,x θ )1 τ ≥θ . (2.4.18)
Dans le théorème ci-dessus, A t t représente l'ensemble des contrôles indépendents de F t et restrictionnés à [t, T ]. De manière similaire, T t t denote l'ensemble des temps d'arrêt indépendents de F t , à valeurs dans [t, T ].

Le principe de sous-optimalité est le plus facile à obtenir. Il repose sur la propriété de flot pour les EDS rétrogrades et classiques et une propriété de séparation qui dit essentiellement que, étant donné une date intermédiaire t ≤ T et une trajectoire fixée jusqu'à la date t (correspondant à la réalisation du mouvement brownien et de la mesure aléatoire de Poisson), l'EDSR peut être résolue par rapport au mouvement brownien et à la mesure de Poisson aléatoire les deux translatés de t. Ce résultat est nécessaire afin de pouvoir utiliser la définition de la fonction valeur qui est une fonction déterministe.

La seconde inégalité est considérablement plus difficile à établir et repose sur l'existence de contrôles ε-optimaux faibles pour notre problème mixte de contrôle et temps d'arrêt optimal (résultat qui necessite des arguments fins comme un théorème de sélection mesurable abstrait) ainsi que sur de nouvelles propriétés des EDSR (e.g. un lemme de Fatou pour les EDSR réfléchies où la limite concerne à la fois la date terminale et la condition terminale).

En utilisant ce principe de programmation dynamique faible et un nouveau théorème de comparaison entre les EDSR et les EDSR réfléchies, nous déduisons que la fonction valeur est une solution faible de viscosité d'une inégalité variationnelle HJB non linéaire généralisée. Plus précisément, le résultat est le suivant: Theorem 2.4.6. La fonction V , définie dans (1.4.16), est une solution de viscosité faible de HJBVI

             min(V (t, x) -h(t, x), inf α∈A (- ∂V ∂t (t, x) -L α V (t, x) -f (α, t, x, V (t, x), (σ ∂V ∂x )(t, x), B α V (t, x))) = 0, (t, x) ∈ [0, T ) × R V (T, x) = g(x), x ∈ R (2.4.19) avec L α := A α + K α , et pour φ ∈ C 2 (R), • A α φ(x) := 1 2 σ 2 (x, α) ∂ 2 φ ∂x 2 (x) + b(x, α) ∂φ ∂x (x) • K α φ(x) := E φ(x + β(x, α, e)) -φ(x) - ∂φ ∂x (x)β(x, α, e) ν(de) • B α φ(x) := φ(x + β(x, α, •)) -φ(x),
dans le sens où V * est une sous-solution de viscosité de (2.4.19) et V * est une sur-solution de viscosité de (2.4.19).

Nous concluons ce chapitre avec des applications financières de la partie théorique.

Méthodes numériques pour les EDSR Doublement Réfléchies avec Sauts et obstacles irréguliers

Cette partie de la thèse est dédiée aux méthodes numériques pour les EDSRDR avec sauts et obstacles irréguliers et repose sur deux articles écrits en collaboration avec C. Labart: Numerical approximation for DRBSDEs with jumps and RCLL obstacles [START_REF] Dumitrescu | Numerical approximation for Doubly reflected BSDEs with Jumps and RCLL obstacles[END_REF] (accepté pour publication dans Journal of Mathematical Analysis and Applications) et Reflected scheme for DRBSDEs with jumps and RCLL obstacles [START_REF] Dumitrescu | Reflected Scheme for Doubly reflected BSDEs with Jumps and RCLL obstacles[END_REF] (accepté pour publication dans Journal of Computational and Applied Mathematics).

Nous commençons cette section avec une courte présentation des méthodes numériques existantes pour les EDS rétrogrades.

Préliminaires et vue d'ensemble de la littérature

Les Équations Différentielles Stochastiques Rétrogrades offrant une représentation probabiliste de la solution des EDP semi-linéaires, de nombreux travaux concernent les schémas numériques dans le cadre markovien lorsque la filtration est générée par un mouvement brownien. Parmi ceux-ci, rappelons l'algorithme à quatre étapes developpé par J. Ma, P. Protter et J. Yong ( [START_REF] Ma | Solving forward-backward stochastic differential equations explicitly a four step scheme[END_REF], voir aussi [START_REF] Douglas | Numerical methods for forward-backward stochastic differential equations[END_REF]), ), Zhang ([152]) etc. Pour le cas des EDSR réfléchies et doublement réfléchies, voir [START_REF] Bouchard | Discrete-time approximation for continuously and discretely reflected BSDEs[END_REF] et [START_REF] Chassagneux | A discrete-time approximation for doubly reflected BSDEs[END_REF].

Une approche pertinente dans la théorie des EDSR est de proposer des méthodes numériques implémentables afin d'approximer les solutions de ces équations, la complexité étant due au calcul des espérances conditionnelles. Beaucoup d'efforts ont été faits dans cette direction. Dans [START_REF] Bouchard | Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations[END_REF], Bouchard et Touzi utilisent le calcul de Malliavin pour réécrire les espérances conditionnelles comme le ratio de deux espérances non conditionnelles qui peuvent être estimées par des méthodes standard de Monte Carlo. Dans le cas réfléchi où le générateur ne dépend pas de Z, Bally et Pagès (voir [START_REF] Bally | Error analysis of the quantization algorithm for obstacle problems[END_REF], [START_REF] Bally | A quantization algorithm for solving multi-dimensional optimal stopping problems[END_REF]) utilisent une approche de quantification. Cette méthode repose sur l'approximation des processus temporels continus sur un maillage fini et nécessite une autre estimation des probabilités de transition sur le maillage. Gobet et al. ( [START_REF] Lemor | Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations[END_REF]) ont suggéré une adaptation de l'algorithme de Longstaff-Schwartz qui repose sur des méthodes non paramétriques et très récemment Ph. Briand et C. Labart ( [START_REF] Ph | Simulation of BSDEs by Wiener Chaos Expansion[END_REF]) ont proposé un développement en chaos de Wiener qui, dans l'esprit, n'est pas très éloigné des techniques de régression. Nous rappelons également les méthodes de cubature, utilisées par T. Lyons, D. Crisan et K. Manolarakis. (voir [START_REF] Crépey | Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison[END_REF]).

Dans le cadre non markovien, dans le cas des EDSR standard ( [START_REF] Peng | Numerical algorithms for BSDEs with 1-d Brownian motion: convergence and simulation[END_REF]) et des EDSR réfléchies [START_REF] Xu | Numerical algorithms and Simulations for Reflected Backward Stochastic Differential Equations with Two Continuous Barriers[END_REF], les auteurs proposent une autre technique qui repose sur l'approximation du mouvement brownien par une marche aléatoire. Cette méthode permet de simplifier le calcul des espérances conditionnelles qui apparaissent à chaque étape et d'obtenir des schémas complètement implémentables. Les EDSRs ont alors été remplacées par une une équation stochastique rétrograde discrète appropriée dont la convergence est obtenue grâce à un résultat de Briand, Delyon et Memin [START_REF] Briand | Donsker-type theorem for BSDEs[END_REF] (voir aussi [START_REF] Briand | On the robustness of backward stochastic differential equations[END_REF]).

Alors que les schémas discrets pour l'approximation des solutions des EDSR dans un cadre purement brownien sont étudiés dans plusieurs travaux, le cadre avec saut a une littéraire moins abondante et réduite au cas des EDSR non réfléchies. Dans le cadre markovien, Bouchard et Elie ( [START_REF] Bouchard | Discrete-time approximation of decoupled Forward-Backward SDE with jumps[END_REF]) ont considéré des schémas numériques pour les EDSR avec une activité de saut purement finie reposant sur l'équation de la programmation dynamique. Récemment, [START_REF] Lejay | Numerical approximation of Backward Stochastic Differential Equations with Jumps[END_REF] [109] ont étendu les résultats de Briand, Delyon et Memin au cas avec sauts. Leur méthode s'appuie sur la construction d'une EDSR avec sauts discrète et dirigée par un système complet de trois martingales orthogonales discrètes en temps et en espace, la première étant une marche aléatoire qui converge vers un mouvement brownien, la deuxième une autre marche aléatoire indépendante de la première, et la troisième converge vers un processus de Poisson.

Contributions

Dans le chapitre 5, nous étudions dans un cadre non markovien un schéma d'approximation en temps discret de la solution des EDSR Doublement Réfléchies avec Sauts dirigées par un mouvement brownien (noté W ) et un processus de Poisson compensé indépendant d'intensité λ (noté N ). De plus, nous supposons que les barrières sont continues à droite, limitées à gauche et admettent à la fois des sauts inaccessibles et prévisibles. L'EDSRDR que nous résolvons numériquement a la dynamique:

Y t = ξ T + T t g(s, Y s , Z s , U s )ds + (A T -A t ) -(K T -K t ) - T t Z s dW s - T t U s d Ñs , (2.5.1)
et satisfait les contraintes suivantes:

         (i) ∀t ∈ [0, T ], ξ t ≤ Y t ≤ ζ t p.s. (ii) T 0 (Y t --ξ t -)dA c t = 0 et T 0 (ζ t --Y t -)dK c t = 0 p.s. (iii) ∀τ temps d'arrêt prévisible, ∆A d τ = ∆A d τ 1 Y τ -=ξ τ -et ∆K d τ = ∆K d τ 1 Y τ -=ζ τ -.
(2.5.2)

Comme nous avons mentionné dans le chapitre précédent, puisque nous considérons le cadre général dans lequel les sauts des obstacles peuvent être soit prévisibles soit totalement inacessibles, les processus croissants A et K, dont le rôle est de garder la solution Y entre les barrières, ne sont plus continus. Nous pouvons alors réécrire la condition de séparabilité de Skorokhod, pour la partie continue A c (respectivement

K c ) de A (respectivement K) et la discontinue, notée A d (respective- ment K d ).
Notre objectif est de proposer une schéma complètement implémentable de l'EDSRDR cidessus, reposant sur deux arbres binomiaux aléatoires et la métode de pénalisation, qui est alors convergente vers la solution de l'EDSRDR. Nous présentons ci-dessous l'idée principale:

(i) Nous introduisons dans un premier temps une suite d'EDSR prénalisées afin d'approximer l'EDSR doublement réfléchie (1.5.1; 1.5.2) satisfaisant:

Y p t = ξ + T t g(s, Y p s , Z p s , U p s )ds + A p T -A p t -(K p T -K p t ) - T t Z p s dW s - T t U p s d Ñs , (2.5.3) avec A p t := p t 0 (Y p s -ξ s ) -ds et K p t := p t 0 (ζ s -Y p s ) -ds.
Nous obtenons la convergence des équations pénalisées dans le cas d'une mesure aléatoire de Poisson générale et, puisque nous devons traiter avec un générateur qui dépend de la solution, la méthode de pénalisation utilisée dans la littérature précédente (qui traite seulement le cas d'un générateur sous la forme d'un processus, le cas général étant obtenu par un argument de point fixe) ne peut être utilisée dans notre cadre général. Nous proposons à la place une preuve qui repose sur une combinaison de pénalisations, la théorie de l'enveloppe de Snell, le théorème de comparaison pour les EDSR avec sauts, un théorème de monotonicité généralisé sous la condition de Mokobodski et des jeux stochastiques.

(ii) Nous approximons le mouvement brownien et le processus de Poisson par deux marches aléatoires indépendantes, notées W n , respectivement Ñ n , et définies comme suit:

W n 0 = 0; W n t = √ δ [t/δ] i=1 e n i , Ñ n 0 = 0, Ñ n t = [t/δ] i=1 η n i , avec e n i , i = 1, n des varables aléatoires indépendantes et identiquement distribuées qui pren- nent les valeurs {-1; 1}, chacune avec probabilité 1 2 et η n i , i = 1, n définies similairement à (e n i ) mais prenant les valeurs {κ n -1; κ n } avec probabilités 1 -κ n , respectivement κ n , où κ n = e -λ n
. Dans la définition ci-dessus, δ n := T n représente le pas de temps. Le couple (W n , Ñ n ) converge vers (W, Ñ ) en probabilité pour la topologie J 1 de Skorokhod. En utilisant ces approximations, nous obtenons le schéma d'approximation discret suivant de l'équation pénalisée définie par (2.5.3):

           y p,n j = y p,n j+1 + g(t j , y p,n j , z p,n j , u p,n j )δ n + a p,n j -k p,n j -(z p,n j √ δ n e n j+1 + u p,n j η n j+1 + v p,n j µ n j+1 ) a p,n j = pδ n (y p,n j -ξ n j ) -; k p,n j = pδ n (ζ n j -y p,n j ) -, y p,n n := ξ n n , (2.5.4)
où la troisième suite d'incréments de martingale {µ n j = e n j η n j , j = 0, ..., n} est nécessaire afin d'obtenir la représentation des martingales (voir [109]).

Ensuite, en utilisant le schéma discret implicite ci-dessus, nous pouvons déduire les expressions des coefficients (z p,n j , u p,n j , v p,n j ) j=1,n faisant apparaître les espérances conditionnelles qui sont faciles à calculer dans notre cadre grâce aux approximations ci-dessus de W et de Ñ . Cependant, la valeur de (y p,n j ) j=1,n n'est pas facile à déduire puisque nous devons introduire un opérateur dont l'inversion numérique est difficile et consommatrice de temps. Afin de surmonter ce problème, nous introduisons une équation rétrograde discrète explicite qui est obtenue en remplaçant dans (2.5.4) y p,n j par E[y p,n j+1 |F n j ] dans le générateur g:

           y p,n j = y p,n j+1 + g(t j , E[y p,n j+1 |F n j ], z p,n j , u p,n j )δ n + a p,n j -k p,n j -(z p,n j √ δ n e n j+1 + u p,n j η n j+1 + v p,n j µ n j+1 ) a p,n j = pδ n (y p,n j -ξ n j ) -; k p,n j = pδ n (ζ n j -y p,n j ) -, y p,n n := ξ n n , (2.5.5)
où F n représente la filtration discrète générée par (e n j , η n j ) j=1,n . Nous introduisons ensuite les versions à temps continu (Y p,n t , Z p,n t , U p,n t , A p,n t , K p,n t ) 0≤t≤T de la solution de ce schéma explicite et nous montrons sa convergence en n vers la solution de (2.5.3). Reliant ce résultat avec la convergence en p de l'équation pénalisée (voir (i)), nous obtenons la convergence de notre schéma en (p, n) de la solution de l'EDSRDR.

Nous terminons par étudier numériquement nos résultats théoriques dans le cas où les barrières admettent à la fois des sauts prévisibles et totalement inaccessibles. La difficulté dans le choix des exemples provient de la condition de Mokobodski que nous devons supposer et qui est difficile à vérifier en pratique. Nous faisons remarquer que l'utilité pratique de nos schémas est restreinte aux petites dimensions. En effet, puisque nous utilisons une marche aléatoire pour approcher le mouvement brownien et le processus de Poisson, la complexité de l'algorithme croît très vite avec le nombre de pas n (plus précisément, en n d , d étant la dimension) et, comme nous le verrons dans la partie numérique, la méthode de pénalisation nécessite un pas de temps faible afin d'être stable.

Dans le chapitre 6, nous proposons un schéma alternatif à (2.5.4) et respectivement à (2.5.5) afin de résoudre l'EDSRDR donnée par (2.5.1; 2.5.2). Comparés aux équations rétrogrades discrètes (1.5.4) et (1.5.5), les schémas que nous présentons dans le chapitre 6, appelés schéma réfléchi implicite et schéma réfléchi explicite reposent sur une discrétisation directe de (2.5.1; 2.5.2). Il n'y a pas d'étape de pénalisation. Cette méthode ne dépend que d'un paramètre d'approximation (le nombre de pas de temps n) contrairement aux schémas proposés dans le chapitre 5 (voir (2.5.4) et (2.5.5)) qui dépendent également du paramètre de pénalisation. Nous obtenons la convergence des deux schémas. Le schéma réfléchi explicite est le suivant:

y n j = E[y n j+1 |F n j ] + g(t j , E[y n j+1 |F n j ], z n j , u n j )δ n + a n j -k n j a n j ≥ 0, k n j ≤ 0, a n j k n j = 0, ξ n j ≤ y n j ≤ ζ n j , (y n j -ξ n j )a n j = (y n j -ζ n j )k n j = 0.
(2.5.6)

Nous illustrons numériquement les résultats théoriques et nous montrons qu'ils coïncident avec ceux obtenus en utilisant le schéma pénalisé (2.5.5) pour de grandes valeurs du paramètre de pénalisation p.

Conclusions et perspectives

Dans cette thèse, nous avons exploré de nouveaux problèmes en analyse stochastique, contrôle stochastique, théorie de jeux et mathématiques financières, à la fois d'un point de vue théorique et numérique. Les résultats principaux sont les suivants:

• Dans le chapitre 3 (article [START_REF] Dumitrescu | BSDEs with nonlinear weak terminal condition[END_REF]), nous introduisons une nouvelle classe d'EDSR avec condition terminale faible non linéaire, associées à la couverture approximative sous contraintes de mesures de risque dynamiques.

• Dans le chapitre 4 (article [START_REF] Dumitrescu | Optimal Stopping for Dynamic Risk Measures with Jumps and Obstacle Problems[END_REF]), nous étudions un problème d'arrêt optimal pour des mesures de risque dynamiques induites par des EDSR acec sauts et nous montrons que la fonction valeur correspond à l'unique solution de viscosité d'un problème d'obstacle pour les équation partielles intégro-différentielles.

• Dans le chapitre 5 (article [START_REF] Dumitrescu | Generalized Dynkin games and Doubly reflected BSDEs with Jumps[END_REF]), nous introduisons un nouveau problème de jeu, qui généralise le jeu classique de Dynkin au cas d'espérance non linéaire, permettant d'obtenir une représentation de la solution des EDSR doublement réfléchies non linéaires en termes de jeux stochastiques.

• Dans le chapitre 6 (article [START_REF] Dumitrescu | A Weak Dynammic Programming for Combined Stochastic Control/Optimal Stopping with E f -Expectations[END_REF]), nous étudions dans un cadre markovien un problème de contrôle stochastique et d'arrêt optimal mixte dans le cas où l'espérance classique dans le critère est remplacée par une espérance non linéaire induite par la solution d'une EDSR avec sauts et la fonction de profit terminal est seulement mesurable. Nous établissons un principe de programmation dynamique faible et en déduisons les équations HJB non linéaires associées.

• Dans le chapitre 7 (article [START_REF] Dumitrescu | A Weak Dynammic Programming for Combined Stochastic Control/Optimal Stopping with E f -Expectations[END_REF]), nous introduisons une approximation numérique pour la solution d'une EDSRDR avec sauts et obstacles irréguliers qui admet à la fois des sauts totalement inacessibles et prévisibles. Nous proposons un schéma complètement implémentable, reposant sur une méthode de pénalisation et l'approximation du mouvement brownien et du processus de Poisson par deux marches aléatoires indépendantes dont on montre la convergence vers la solution de l'EDSRDR. Nous illustrons les résultats théoriques avec des exemples numériques.

• Dans le chapitre 8 (article [START_REF] Dumitrescu | Reflected Scheme for Doubly reflected BSDEs with Jumps and RCLL obstacles[END_REF]), nous introduisons un schéma complètement implémentable alternatif à celui présenté dans le chapitre 6 afin d'approximer la solution de l'EDSR doublement réfléchie avec sauts et obstacles irréguliers. Ce schéma est obtenu par une discrétisation directe des EDSRDR et ne dépend alors que du nombre de pas de temps n (et plus du paramètre de pénalisation p). Nous obtenons la convergence du schéma et donnons des exemples numériques.

Concernant les perspectives, ils y a plusieurs orientations de recherche à venir, à la fois d'un point de vue théorique et numérique. En collaboration avec R. Elie et D. Possamai nous sommes en train de finir un travail sur les EDSR avec réflexion faible ( [START_REF] Dumitrescu | Weak reflected BSDEs and approximative hedging for american options[END_REF]) qui sont reliées à la couverture approximative des options américaines. Avec M.C. Quenez et A. Sulem, nous travaillons sur un nouveau problème de jeux mixte avec controle stochastique et temps d'arrêt dans le cadre markovien [START_REF] Dumitrescu | Mixed Stochastic Control/Optimal Stopping Games with E f -Expectations[END_REF] et on étudie les liens entre les jeux de Dynkin Généralisés et le pricing non linéaire, dans des marchés complets et incomplets [START_REF] Dumitrescu | Nonlinear pricing and hedging in a market with default[END_REF].

D'un pont de vue numérique, il serait utile de proposer des schémas numériques pour la solution des EDSRDR avec barrières càdlàg dans le cas d'une mesure de Poisson générale ainsi que des EDSR avec condition terminale faible.

E f 0,T [Ψ(Y T )] ≥ m
, where E f represents the f -conditional expectation associated to a nonlinear driver f . These BSDEs are called BSDEs with nonlinear weak terminal solution. We carry out a similar analysis as in [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF] of the value function corresponding to the minimal solutions Y of the BSDE with nonlinear weak terminal condition: we study the regularity, establish the main properties, in particular continuity and convexity with respect to the parameter m, and finally provide a dual representation in the case of concave constraints. From a financial point of view, our study is closely related to the approximative hedging of an European option under dynamic risk measures constraints. The nonlinearity f raises subtle difficulties, highlighted through out the paper, which cannot be handled by the arguments used in the case of classical expectations constraints studied in [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF]. Key words : Backward stochastic differential equations, g-expectation, dynamic risk measures, optimal control, stochastic targets.

Introduction

Linear backward stochastic differential equations (BSDEs) were introduced by Bismut as the adjoint equations associated with Pontryagin maximum principles in stochastic control theory. The general case of non-linear BSDEs was then studied by Pardoux and Peng [START_REF] Peng | Monotonic limit theorem of BSDE and nonlinear decomposition theorem of doobmeyer's type[END_REF]. They provided Feynman-Kac representations of solutions of non-linear parabolic partial differential equations. The solution of a BSDE consists in a pair of predictable processes (Y, Z) satisfying

-dY t = g(t, Y t , Z t )dt -Z t dW t ; Y T = ξ. (3.1.1)
These equations appear as an useful mathematical tool in various problems in finance, for example in the theory of derivatives pricing. In a complete market -when it is possible to construct a portfolio which attains as final wealth the payoff-the value of the replicating portfolio is given by Y and the hedging strategy by Z. Since in incomplete markets is not always possible to construct a portfolio which attains exactly as final wealth the amount ξ, it was suggested to replace the terminal condition into a weaker one of the form Y T ≥ ξ. In this case, the minimal initial value Y 0 defines the smallest initial investment which allows one to superhedge the contingent claim ξ.

Recently, Bouchard, Elie and Reveillac [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF] introduced a new class of BSDEs, the so called BSDEs with weak terminal condition, in which the T -terminal value Y T only satisfies a weak constraint. More precisely, a couple of predictable processes (Y, Z) is said to be a solution of such a BSDE if it satisfies:

-dY t = g(t, Y t , Z t )dt -Z t dW t ; (3.1.2) E[Ψ(Y T )] ≥ m, (3.1.3)
where m is a given threshold and Ψ a non-decreasing map. The main question in [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF] is the following:

Find the minimal Y 0 such that (3.1.2) and (3.1.3) hold for some Z.

(3.1.4)
From a financial point of view, this study is related to the hedging in quantile or more generally to the hedging with expected loss constraints. This problem was addressed in the literature for the first time by Follmer and Leukert [START_REF] Follmer | Efficient hedging: cost versus shortfall risk[END_REF] and then further studied in a Markovian framework in [START_REF] Bouchard | Stochastic target problems with controlled loss[END_REF] and [START_REF] Moreau | Stochastic target problems with controlled expected loss in jump diffusion models[END_REF], using stochastic target techniques .

In [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF], the key point of the analysis is the reformulation of the problem written in terms of BSDE with weak terminal condition into an optimization problem on a family of BSDEs with strong terminal condition, by using the martingale representation theorem. The main observation is that if Y 0 and Z are such that (3.1.3) holds, then the martingale representation Theorem implies that it exists an element α ∈ A 0 , the set of predictable square integrable processes, such that:

Ψ(Y T ) ≥ M m,α T = m + T 0 α s dW s , (3.1.5)
It is then shown that the initial problem (3.1.4) is equivalent to:

inf{Y α 0 , α ∈ A 0 }, (3.1.6)
where Y α t corresponds to the solution at time t of the BSDE with (strong) terminal condition Φ(M α T ), Φ representing the left-continuous inverse of Ψ.

The aim of this paper is to introduce a new class of BSDEs with weak nonlinear terminal condition . We extend the results of [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF] to a more general class of constraints which take the form:

E f 0,T [Ψ(Y T )] ≥ m, (3.1.7)
where f is a nonlinear driver and E f •,T [ξ] the solution of the BSDE with generator f and terminal condition ξ.

We can easily remark that the constraint (3.1.3) is a particular case of (3.1.7) for f = 0. The problem under study in this paper is the following: inf{Y 0 such that ∃Z : (3.1.2) and (3.1.7) hold}.

(3.1.8)

Following the key idea of [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF], we rewrite our problem (3.1.8) into an equivalent one expressed in terms of BSDEs with strong terminal condition. The main difference with respect to [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF] is given by the fact that in our case we have to introduce a new controlled diffusion process, which is an f -martingale, contrary to [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF] where it is a classical martingale. Indeed, for a given Y 0 and Z such that (3.1.2) and (3.1.7) are satisfied, appealing to the BSDE representation of Ψ(Y T ), we can find α ∈ A 0 such that:

Ψ(Y T ) ≥ M m,α T = m - T 0 f (s, M m,α s , α s )ds + T 0 α s dW s . (3.1.9)
Thanks to this observation, we show that Problem (3.1.8) is equivalent to (3.1.6), where, in our more general framework, Y α t corresponds to the solution at time t of the BSDE with (strong) terminal condition Φ(M α T ). We study the dynamical counterpart of (3.1.6):

Y α (τ ) := essinf{Y α ′ τ , α ′ ∈ A 0 s.t.α ′ = α on [[0, τ ]]}. (3.1.10)
We carry out a similar analysis as in [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF] of the family {Y α , α ∈ A 0 }. We start by studying the regularity of the family Y α and show that it can be aggregated into a RCLL process, proof which becomes considerably more technical in our context with respect to [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF], because we have to deal with the nonlinearity f . We then provide a BSDE representation of Y α and show that, under a concavity assumption on the driver f , there exists an optimal control. We also study the main properties of the value function, as continuity and convexity with respect to the threshold m, and propose proofs specific to the nonlinear case. We finally get, in the case of concave constraints, a dual representation of the value function, related to a stochastic control problem in Meyer's form.

Besides the mathematical interest of our study, this work is also motivated by some financial applications, as it provides the approximative hedging under dynamic risk measures contraints of an European option, when the shortfall risk is quantified in terms of dynamic risk measures induced by BSDEs.

The paper is organized as follows. In Section 2 we introduce notation, assumptions and the BSDEs with nonlinear weak terminal condition. In Section 3, we study the regularity and the BSDE representation of the value function Y α . In Section 4, we establish the main properties of the value function and finally we provide a dual representation in Section 5.

Problem formulation

Notation

Let (Ω, F, P) supporting a d-dimensional Brownian motion W and F := (F t ) t≤T the completed associated filtration. Fix T > 0.

In the sequel, we adopt the following notation:

-P denotes the predictable σ-algebra on [0, T ] × Ω;

-L 2 (F T ) is the set of random variables ξ which are F T -measurable and square-integrable;

-H 2 denotes the set of R d -valued predictable processes φ such that φ 2 H 2 := E[( T 0 φ 2 t dt)] < ∞; -S 2 is the set of real-valued RCLL adapted processes φ such that φ 2 S 2 := E[sup 0≤t≤T |φ t | 2 ] < ∞; -I 2 is the set of non-decreasing adapted processes φ such that φ 2 S 2 < ∞; -For any σ-algebra G ⊂ F T , L 0 (G)
denotes the set of random variables measurable with respect to G; -T denotes the set of stopping times τ such that τ ∈ [0, T ] a.s.

BSDEs with nonlinear weak terminal condition Definition and Assumptions

In this section, we introduce the main object of this paper, the BSDEs with nonlinear weak terminal condition. It is well known that, in the classical case of nonlinear BSDEs introduced by Pardoux-Peng, the data of the BSDE is represented by a driver g and a terminal condition ξ.

In the recent paper [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF], the authors define a new class of BSDEs called BSDEs with weak terminal condition. The particularity consists in the fact that the terminal condition is not fixed as a F T -measurable random variable, but only satisfies a weak constraint expressed in terms of classical expectations. The data of this new class of BSDEs is given by four elements: a driver g and a triplet (Ψ, µ, τ ) describing the constraint on the terminal condition.

The aim of this work is to introduce a more general class of BSDEs, named BSDEs with nonlinear weak terminal condition, whose terminal value verifies a weak constraint defined via a BSDE with a nonlinear driver f , satisfying the following hypothesis:

Assumption 3.2.1. Let f : (ω, t, y, z) ∈ Ω × [0, T ] × R × R d → f t (ω, y, z) ∈ R be a driver such that (f t (•, y, z)) t≤T is P-measurable for every (y, z) ∈ R × R d and |f t (ω, y, z) -f t (ω, y ′ , z ′ )| ≤ C f (|y -y ′ | + z -z ′ R d ) , ∀(y, z), (y ′ , z ′ ) ∈ R × R d , for dt ⊗ dP -a.e. (t, ω) ∈ [0, T ] × Ω, for some constant number C f > 0.
We also assume that (f t (0, 0, 0)) satisfies the following condition

E T 0 |f t (0, 0)| 2 dt < ∞.
Note that the data of this new BSDE are (f, Ψ, µ, τ, g) and the particular case when f = 0 corresponds to the class of BSDEs studied in [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF]. In the sequel, we shall denote this BSDE with nonlinear weak terminal condition by BSDE(f, Ψ, µ, τ, g).

Before defining this new mathematical object, we introduce the nonlinear conditional expectation E f associated with f , defined for each stopping time τ ∈ T and for each η ∈ L 2 (F τ ) as:

E f t,τ [η] := Y r , 0 ≤ t ≤ τ, (3.2.1)
where (Y t ) t≤τ is the unique solution in S 2 of the BSDE associated with driver f , terminal time τ and terminal condition η, that is satisfying:

-dY t = f (t, Y t , Z t )dt -Z t dW t ; Y T = η, (3.2.2)
with Z the associated process belonging to

H 2 . Moreover, set E f σ,τ [η] := -∞, for any η ∈ L 0 (F τ ) such that E[η -] = +∞, for any σ ∈ T with σ ≤ τ a.s.
We are now in position to define the so-called BSDEs with nonlinear weak terminal condition.

Definition 3.2.2 (BSDEs with nonlinear weak terminal condition). Given a measurable map

Ψ : R × Ω → U , with U ⊂ A ∪ {-∞}, A a bounded subset of R, τ ∈ T , µ ∈ L 0 (R, F τ ), a driver f satisfying Assumption 3.2.1 and a measurable function g, we say that (Y, Z) ∈ S 2 × H 2 is a solution of the BSDE (f, Ψ, µ, τ, g) if Y t = Y T + T t g(s, Y s , Z s )ds - T t Z s dW s , 0 ≤ t ≤ T ; (3.2.3) E f τ,T [Ψ(Y T )] ≥ µ. (3.2.4)
Throughout the paper, we shall assume that the driver g satisfies Assumption 3.2.1, with C g instead of C f . To the coefficient g, we associate the nonlinear operator E g defined as E f , with g instead of f . Let us now precise the hypothesis on the map Ψ and the threshold µ. We then discuss the wellposedness of the BSDE(f, Ψ, τ, µ, g) under these assumptions. Assumption 3.2.3. For a.e. ω ∈ Ω, the map y ∈ R → Ψ(ω, y) is non-decreasing and valued in

[0; 1] ∪ {-∞} and its right-inverse Φ(ω, •) is such that Φ : Ω × [0, 1] → [0, 1] and it is measurable. This means that Ψ(ω, •) ∈ [0, 1] on [0, ∞) and Ψ(ω, •) = -∞ on (-∞, 0).
In view of the definition of the operator E f , this implies that Y T ≥ 0 a.s. Note that for notational simplicity we have considered the compact [0, 1], as in [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF], which can be obviously replaced by an arbitrary compact set belonging to R. Moreover, our analysis is the same if for a.e. ω the map Ψ(ω,

•) is valued in [G 1 (ω), G 2 (ω)], with G 1 , G 2 ∈ L ∞ (F T ).
The threshold µ is assumed to belong to D τ , where D τ corresponds to the set of random vari-

ables {η ∈ L 2 (F τ ) such that η ∈ [E f τ,T [0], E f τ,T [ 1
]] a.s.}. throughout the paper, we shall denote by (Y i , Z i ), for i = 1, 2, the solution of the BSDE associated to driver f and terminal condition ξ i , where ξ 1 = 0 and ξ 2 = 1. Concerning the existence of a solution, remark that any random variable Φ(ξ), with ξ ∈ [0, 1] a.s. and E f τ,T [ξ] ≥ µ could serve as terminal condition. However, the constraint is too weak to expect uniqueness.

We now introduce the value function V : D → L 2 ; (τ, µ) → V(τ, µ), where D := {(τ, µ); τ ∈ T and µ ∈ D τ } as follows:

V(τ, µ) := essinf{Y τ : (Y, Z) ∈ S 2 × H 2 is a solution of BSDE(f, Ψ, µ, τ, g)}.
(3.2.5)

The rest of the paper is dedicated to the study of the above map. In order to do it, we shall first establish the link with a control problem for BSDEs with strong terminal condition.

Link with a control problem for BSDEs with strong terminal condition

In the spirit of [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF] or [START_REF] Bouchard | Stochastic target problems with controlled loss[END_REF], we introduce an additional process M which allows to transform the weak constraint

E f 0,T [Ψ(Y T )] ≥ µ into a strong one of the form Y T ≥ Φ(M µ T )
. Since our constraint is expressed in terms of nonlinear BSDEs, the process M is an f -martingale, contrary to [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF] and [START_REF] Bouchard | Stochastic target problems with controlled loss[END_REF] where M is a classical martingale. For each α ∈ H 2 , stopping time τ ∈ T and µ ∈ D τ , let M τ,µ,α be the R-valued solution of the SDE:

M τ,µ,α t∨τ = µ - t∨τ τ f (s, M τ,µ,α s , α s )ds + t∨τ τ α ⊤ s dW s , 0 ≤ t ≤ T.
We introduce the set of admissible controls A τ,µ , which is defined as follows:

A τ,µ := {α ∈ H 2 such that M τ,µ,α t ∈ E f t,T [0], E f t,T [1] dP ⊗ dt a.s. on [[τ, T ]]}.
Notice that for all α ∈ A τ,µ , Φ(M τ,µ,α T ) could serve as terminal condition, since satisfies (3.2.4). We thus introduce for all α ∈ A τ,µ the BSDE with strong condtion Φ(M τ,µ,α T ) and driver g and define the value function Y(τ, µ) as follows:

Y(τ, µ) := essinf α∈Aτ,µ E g τ,T [Φ(M τ,µ,α T )]. (3.2.6)
Our aim now is to link Y(τ, µ) to V(τ, µ), i.e. to prove that for all τ ∈ T and µ ∈ D τ :

V(τ, µ) = Y(τ, µ) a.s. (3.2.7)
In order to explain the above equality between V and Y, we state the following proposition:

Proposition 3.2.4. Fix τ ∈ T , µ ∈ D τ . Then (Y, Z) ∈ S 2 ×H 2 is a solution of BSDE(f, Ψ, µ, τ, g) if and only if (Y, Z) satisfies (3.2.3) and there exists α ∈ A τ,µ such that Y t ≥ E g t,T [Φ(M τ,µ,α T )] for t ∈ [0, T ], P-a.s.
A sketch of proof is given in Appendix. We come back to the explanation of equality (3.2.7).

(i) Let (Y, Z) ∈ S 2 ×H 2 be a solution of the BSDE(f, Ψ, µ, τ, g). Then the above Proposition implies that it exists

α ∈ A τ,µ such that Y τ ≥ E τ,T [Φ(M τ,µ,α T )] ≥ Y(τ, µ)
, where the last inequality follows from definition (3.2.6). By arbitrariness of (Y, Z), we get V(τ, µ) ≥ Y(τ, µ) a.s.

(ii) Fix α ∈ A τ,µ . Let Z α be the associated process to the BSDE representation of Φ(M τ,µ,α T ). Since Φ(M τ,µ,α T ) is admissible as a terminal condition, we obtain, by the Proposition 3.2.4 that (E •,T [Φ(M τ,µ,α )], Z α ) is a solution, and thus E τ,T [Φ(M τ,µ,α )] ≥ V(τ, µ). By arbitrariness of α, we deduce V(τ, µ) ≤ Y(τ, µ) a.s.

From now on, we fix an initial condition µ 0 ∈ D 0 at time 0. For each α ∈ A 0,µ 0 (denoted for simplicity A 0 ), we introduce the process (M α t ) t≤T , representing a dynamic threshold controlled by the action of α, which is defined as follows:

M α t := M 0,µ 0 ,α t .
We introduce for each τ ∈ T the set of admissible controls coinciding with α up to the stopping time τ :

A α τ := {α ′ ∈ A τ,M α τ : α ′ = α dt ⊗ dP on [[0, τ ]]}.
The associated value is defined by:

Y α (τ ):= essinf α ′ ∈A α τ E g τ,T [Φ(M τ,M α τ ,α ′ T )].
In the following section, we shall investigate the time regularity of the above function and provide a BSDE representation. Before doing this, note that |Y α (τ )| ≤ η τ a.s. for all τ ∈ T , where η belongs to S 2 .

(3.2.8)

Note that η is given by η

t := |E g t,T [Φ(1)]| + |E g t,T [Φ(0)]|, t ≤ T .

Time regularity of the value function Y α and BSDE representation

In this section, we study the regularity of the family {Y α (τ ), τ ∈ T }. More precisely, we show that it can be aggregated into a right continuous left limited process. The proof of this result becomes considerably more technical in our nonlinear case. Some comments regarding the main difficulties with respect to the case of linear constraints are provided in Remark 3.3.4. We first state the following dynamic programming principle.

Lemma 3.3.1. For any α ∈ A 0 , Y α satisfies the following dynamic programming principle: for all τ 1 ∈ T , τ 2 ∈ T with τ 1 ≤ τ 2 a.s. it holds:

Y α (τ 1 ) = essinf α∈A α τ 1 E g τ 1 ,τ 2 [Y α (τ 2 )].
Since the proof of the dynamic programming principle is based on classical arguments, we refer the reader to [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF]. We now make the following hypothesis on the map Φ, under which we provide the time-regularity of our value function Y α . Proof. By Lemma 3.3.1, we easily obtain that the family {-Y α (τ ), τ ∈ T } is a -g(-) supermartingale system. Since moreover 3.2.8 holds, we can apply Lemma A.2 in [START_REF] Bouchard | A general Doob-Meyer-Mertens decomposition for gsupermartingale systems[END_REF] and obtain the existence of an optional ladlag process, denoted by (Y α t ) t≤T which aggregates the family, that is

Y α (τ ) = Y α τ ,
Y α s .
are well-defined and finite. Now, we define:

Y α t := lim s∈(t,T ]↓t Y α s , t ∈ [0, T [, Y α T := Y α T . (3.3.1)
which is by definition a real-valued RCLL process.

In order to prove the desired regularity property, we have to show that for every stopping time τ ∈ T , it holds that:

Y α τ = Y α τ a.s.

The above relation implies that the processes Y

α and Y α are indistinguishable. The proof is divided in two steps.

Step 1. Fix τ ∈ T . We first prove that

Y α τ ≤ Y α τ a.s. a. Let α ′ ∈ A α τ . Fix k ∈ N * . Define Mk,α ′ T := 1 k + M α ′ T (1 -1 k ). Note that Mk,α ′ T ≥ M α ′ T and Mk,α ′ T → M α ′ T when k → ∞.
In the sequel, we denote by (E f

•,T [ Mk,α ′ T ], Zk ) the solution of the BSDE associated to ( Mk,α ′ T , f ). Recall that M α ′ T belongs for a.e. ω to [0, 1]. Hence, by construction, we have:

0 ≤ M α ′ T ≤ Mk,α ′ T ≤ 1 a.s.
By applying the comparison theorem for BSDEs and since α ′ ∈ A α τ , we obtain:

E f τ,T [0] ≤ M α τ ≤ E f τ,T [ Mk,α ′ T ] a.s. (3.3.2)
We claim that it exists a sequence of stopping times (τ n,k ) n and an admissible control αk ∈ A α

Y α τ = lim n→∞ Y α τ n,k . (3.3.3)
Using the definition of Y α , we get:

Y α τ n,k ≤ E g τ n,k ,T [Φ(M αk T )]. (3.3.4) As M αk T ≤ Mk,α ′ T a.
s. and Φ is nondecreasing, by applying the comparison theorem for BSDEs, we get for all n:

E g τ n,k ,T [Φ(M αk T )] ≤ E g τ n,k ,T [Φ( Mk,α ′ T )] a.s.
The above inequality together with (3.3.3), (3.3.4) and the continuity of the process

E f •,T [Φ( Mk,α ′ T )] lead to: Y α τ ≤ E g τ,T [Φ( Mk,α ′ T )]. Since Mk,α ′ T → M α ′ T a.
s. and Φ is a.s. continuous, by letting k tend to ∞, we obtain:

Y α τ ≤ E g τ,T [Φ(M α ′ T )] a.s.
By arbitrariness of α ′ ∈ A α τ , we conclude:

Y α τ ≤ Y α τ a.s. b. i) We first construct, for each k ∈ N * , the sequence of stopping times (τ n,k ) n such that τ n,k → τ when n → ∞ and τ n,k > τ a.s. on {τ > T } for all n ∈ N.
To do this, we start by defining the following stopping time:

σ k := inf{τ ≤ t ≤ T ; M α t = E f t,T [ Mk,α ′ T ]}. (3.3.5)
We use the convention inf ∅ = +∞. We introduce (τ n ) n a sequence of stopping times which take values in [0, T ] a.s. such that τ n > τ on {τ < T } for all n and τ n → τ a.s. when n tends to +∞. For each n, we define τ n,k as follows:

τ n,k := τ n 1 A k + (τ n ∧ σ k ) 1 A c k , (3.3.6)
with

A k := {E f τ,T [ Mk,α ′ T ] -M α τ = 0} ∈ F τ ; A c k := {E f τ,T [ Mk,α ′ T ] -M α τ > 0} ∈ F τ . Remark that by (3.3.2), P (A k ∪ A c k ) = 1 and thus τ n,k ↓ τ a.s. when n → ∞.
We precise that we have to introduce the sets

A k and A c k because σ k = τ on A k . It thus remains to prove that τ < σ k on A c
k . The definition of σ k together with the continuity of the processes M α and

E f •,T [ Mk,α ′ ], imply that almost surely, σ k = +∞ or E f σ k ,T [ Mk,α ′ T ] ≤ M α τk . Moreover, since on A c k we have E f τ,T [ Mk,α ′ T ] > M α τ and τ ≤ σ k a.s., one can thus conclude that τ < σ k a.s. on A c k .
ii) We provide the existence of an admissible control αk ∈ A α τ n,k such that M αk T ≤ Mk,α ′ T . The control αk is defined as follows:

αk := α s 1 {s≤σ k } + Zk s 1 {s>σ k } ,
where σk = σ k ∧ T. Recall that Zk is the process associated to the BSDE representation of Mk,α ′ T . Note that the above construction ensures that 0 ≤ M αk T ≤ Mk,α ′ T a.s. It remains to show that αk ∈ A α τ n,k . It is clear that we have:

M α τn∧σ k = M αk τn∧σ k a.s. (3.3.7)
and hence

M α τn∧σ k = M αk τn∧σ k a.s. on A c k . (3.3.8) Since σ k = τ on A k , it remains to show that M α τn = M αk τn a.s. on A k . Recall that α ′ ∈ A α τ . Hence, by definition of the set A k , we obtain M α ′ τ = E f τ,T [ Mk,α ′ T ] a.s. on A k . A strict comparison theorem for BSDEs and the definition of Mk,α ′ T lead to Mk,α ′ T = M α ′ T = 1, a.s. on A k . (3.3.9)
The comparison theorem for BSDEs implies:

E f τn,T [ Mk,α ′ T ] = M α ′ τn = E f τn,T [1], a.s. on A k . (3.3.10)
Moreover, by (3.3.9) and the comparison theorem for BSDEs, we have

M α ′ τ = E f τ,T [1] a.s. on A k and since α ′ ∈ A α τ , we get M α τ = E f τ,T [1] a.s. on A k .
The strict comparison theorem for BSDEs allows us to conclude that: . This inequality and the above convergence lead to the desired result. Remark 3.3.4. In [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF], it is provided the existence of a control α n ∈ A α τn , with τ n → τ and τ n > τ for all n, such that M αn T remains "sufficiently close" to M α ′ T . The control α n is obtained by scaling α in an appropriate way. This approach cannot be applied in the case of nonlinear constraints, as being clearly specific to the linear setting.

M α τn = E f τn,T [1] a.s. on A k . ( 3
||Y α τ -E g τ,τn [Y α τn ]|| L 2 ≤ C ||Y α τ -Y α τn || L 2 + E[ τ n τ |g(s, Y α τ , 0)| 2 ds] . ( 3
Using similar arguments as in Theorem 2.1 in [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF] (points (iii), (iv)) one can show the following BSDE representation for Y α : Theorem 3.3.5. Assume that Assumption 3.3.2 holds. Then there exists a family (Z α , K α ) α∈A 0 satisfying sup

α∈A 0 Y α , Z α , K α S 2 ×H 2 ×I 2 < +∞. (3.3.14)
and such that for all α ∈ A 0 , we have

Y α t = Φ(M α T ) + T t g(s, Y α s , Z α s )ds - T t Z α s dW s + K α t -K α T . (3.3.15) K α τ 1 = essinf ᾱ∈A α τ 1 E[K ᾱ τ 2 |F τ 1 ], ∀τ 1 ∈ T , τ 2 ∈ T τ 1 , (3.3.16
)

and (Y α , Z α , K α )1 [0,τ ] = (Y α , Z α , K α )1 [0,τ ] , ∀τ ∈ T , ᾱ ∈ A α τ . ( 3

Existence of optimal controls in the case of concave constraints

We show that in the case of concave constraints and under convexity assumptions on Φ and g, we get the existence of an optimal control α, that is

Y α t = E g t,T [Φ(M α T )]. For all (λ, m 1 , m 2 , t, y 1 , y 2 , z 1 , z 2 ) ∈ [0, 1] × [0, 1] 2 × [0, T ] × R 2 × [R d ] 2
, we assume a.s. the following:

(H conc ) λf (t, y 1 , z 1 ) + (1 -λ)f (t, y 2 , z 2 ) ≤ f (t, λy 1 + (1 -λ)y 2 , λz 1 + (1 -λ)z 2 ). (H conv ) Φ(λm 1 + (1 -λ)m 2 ) ≤ λΦ(m 1 ) + (1 -λ)Φ(m 2 ) g(t, λy 1 + (1 -λ)y 2 , λz 1 + (1 -λ)z 2 ) ≤ λg(t, y 1 , z 1 ) + (1 -λ)g(t, y 2 , z 2 ). Proposition 3.4.1. Under Hypothesis (H conv ) and (H conc ), for any (τ, α) ∈ T × H 2 , there exists ατ,α ∈ A α τ such that Y α τ = E g τ Φ(M ατ,α T ) = E g τ,τ ′ Y ατ,α τ ′ , ∀τ ′ ∈ T τ .
Proof. By Lemma 3.7.1 in the Appendix, there exists (α n ) n ∈ A α τ such that:

Y α τ = lim n→∞ E g t,T [Φ(M α n T )]. (3.4.1) Recall that (M α n T ) n is valued in [0,1]
. By Komlos Theorem, Mn T := 1 n i≤n M α i T converges a.s. to a random variable MT which belongs a.s. to [0, 1]. From the concavity assumption on the driver f and the comparison theorem for BSDEs we get:

E f τ,T [ Mn T ] ≥ 1 n i≤n E f τ,T [M α i T ] = M α τ , (3.4.2)
since α n ∈ A α τ for all n. The a priori estimates for BSDEs lead to:

|E f τ,T [ Mn T ] -E f τ,T [ MT ]| 2 ≤ E t [| Mn T -MT | 2 ]. (3.4.3)
The a.s. convergence Mn T → MT and the boundness of the sequence ( Mn T ) n allow us to apply the Lebesgue's theorem and to derive that the right hand side of the above inequality tends to 0 when n goes to +∞. We thus derive that:

E f τ,T [ Mn T ] → E f τ,T [ MT ] a.s. (3.4.4) Hence, inequality (3.4.2) combined with (3.4.4) lead to E f τ,T [ MT ] ≥ M α τ .
Let us denote by α the control associated to the BSDE with terminal condition MT and driver f . We define the following stopping time:

θ α := inf{τ ≤ s ≤ T : M τ,M α τ , α s = E f s,T [0]} ∧ T,
with the convention inf ∅ = +∞. We recall that (Y 0 , Z 0 ) represents the solution of the BSDE associated to driver f and terminal condition 0 and we define the control α as follows:

αs := α s 1 s≤τ + αs 1 {τ <s≤θ α} + Z 0 s 1 {s>θ α} . (3.4.5)
Note that α belongs to A α τ . Moreover, by construction, we have:

M α T ≤ MT a.s. (3.4.6)
Now, by using hypothesis (H conv ) and the comparison theorem, we obtain:

Ỹn τ := 1 n i≤n E g τ,T Φ(M α i T ) ≥ E g τ,T Φ( Mn T ) . ( 3
Y α τ ≥ E g τ,T Φ(M α T ) . (3.4.9)
The equality follows by definition of Y α τ and α is hence the optimal control. In order to show the second equality

Y α τ = E g τ,τ ′ Y ατ,α τ ′ , ∀τ ′ ∈ T τ , we first observe that Y α τ = E g τ,τ ′ E g τ ′ ,T [Φ(M α T )] ≥ E g τ,τ ′ Y α τ ′ ,
) ∈ A α τ ′ such that E g τ ′ ,T [Φ(M αn T )] → Y α τ ′ a.s. By (3.2.8
), the convergence also holds in L 2 . The a priori estimates on BSDEs give:

Y α τ ≤ E g τ,τ ′ E g τ ′ ,T [Φ(M αn T )] → E g τ,τ ′ [Y α τ ′ ]. Remark 3.4.2.
Note that in [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF], the optimal control is obtained directly by using the martingale representation of MT , due to the linearity of the expectation. In our nonlinear case, that is no longer possible and we need a more complicated construction.

Properties of the value function

In this section, we study the continuity and the convexity (defined in a probabilistic sense) of the map Y t (µ) := Y(t, µ) with respect to the threshold µ, for any t < T .

Continuity

Fix t ∈ [0, T ]. We give below an estimate on the map µ → Y t (µ), ensuring its continuity under some weak assumptions on the map Φ ( e.g. Φ is Lipschitz continuous with respect to x, uniformly in ω or deterministic continuous). We obtain a nicer and more natural bound for |Y t (µ 1 ) -Y t (µ 2 )| than the one provided in the case of classical expectations constraints ( [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF]), which is expressed through the spread

|µ 1 -µ 2 | 1 2 ( in [31] it depends on (1-µ 1 µ 2 )1 µ 1 <µ 2 + µ 1 -µ 2 1-µ 2 1 µ 1 >µ 2 ; (1-µ 2 µ 1 )1 µ 2 <µ 1 + µ 2 -µ 1 1-µ 1 1 µ 1 <µ 2
and on other two terms related to the case when the thresholds take the boundary values 0 and 1). Moreover, our proof is based on BSDEs techniques, allowing to treat the nonlinear case, contrary to [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF], where the arguments hold only in the case of linear constraints. Theorem 3.5.1. Let t < T , and

µ 1 , µ 2 ∈ D t . Then |Y t (µ 1 ) -Y t (µ 2 )| ≤ Err t (∆(µ 1 , µ 2 )), where ∆(µ 1 , µ 2 ) = C|µ 1 -µ 2 | 1 2 , with C a constant depending only on (C f , T ) and Err t (ξ) := ess sup{R t (M, M ′ ) : M, M ′ ∈ L 0 ([0, 1]), E t [|M -M ′ | 2 ] ≤ ξ}, (3.5.1)
where ξ ∈ L 2 (R, F t ) and R t (M, M ′ ) := |E g t,T [Φ(M )] -E g t,T [Φ(M ′ )]|. Proof.
We define μ1 := µ 1 ∨ µ 2 and μ2 := µ 1 ∧ µ 2 . By the monotonocity property of the map µ → Y t (µ) (3.7.2), we have Y t ( μ1 ) ≥ Y t ( μ2 ) a.s.

By Lemma 3.7.1, it exists α n ∈ A t,μ 2 such that lim n→∞ E g t,T [Φ(M μ2 ,α n )] = Y t (μ 2 ) a.s. Fix n ∈ N. We now construct an admissible control αn ∈ A t,μ 1 such that M μ1 , αn s ∈ M μ2 ,α n s , E f s,T [1] , t ≤ s ≤ T, a.s. It is defined as follows: αn s := α n s 1 {s≤τ } + Z 1 s 1 {s>τ } , where τ := inf{s ∈ [t, T ] : M μ1 ,α n s = E f s [1]} ∧ T , with the convention inf ∅ = +∞.
Recall that Z 1 corresponds to the control associated to the BSDE of terminal condition 1 and driver f . By definition of the value function Y t , we get:

Y t ( μ1 ) ≤ E g t,T [Φ(M μ1 , αn T )] = E g t,T [Φ(M μ1 , αn T )] -E g t,T [Φ(M μ2 ,α n T )] + E g t,T [Φ(M μ2 ,α n T )]. (3.5.2) Let us now estimate E t [|M μ1 , αn T -M μ2 ,α n T | 2 ]. Since M μ1 , αn
E t [|M μ1 , αn T -M μ2 ,α n T | 2 ] ≤ E t [M μ1 , αn T -M μ2 ,α n T ].
(3.5.3)

A similar linearization technique as in the proof of the Comparison Theorem for BSDEs (see for e.g. [START_REF] Peng | Nonlinear expectations, nonlinear evaluations and risk measures[END_REF]) yields:

μ1 -μ2 ≥ E t H n t,T (M μ1 , αn T -M μ2 ,α n T )|F t a.s., (3.5.4)
where (H n t,s ) s∈[t,T ] is the square integrable process satisfying

dH n t,s = H n t,s [δ n s ds + β n s dW s ] ; H n t,t = 1, with        δ n t := f (t, M μ1 , αn t , αn t ) -f (t, M μ2 ,α n t , αn t ) M μ1 , αn t -M μ2 ,α n t 1 {M μ1 , αn t =M μ2 ,α n t } ; β n t := f (t, M μ2 ,α n t , αn t ) -f (t, M μ2 ,α n t , α n t ) |α n t -α n t | 2 (α n t -α n t )1 αn t =α n t .
Now, from (1.4.7) and the Holder inequality, we obtain:

E t [M μ1 , αn T -M μ2 ,α n T ] = E t [(H n t,T ) -1 2 (H n t,T ) 1 2 (M μ1 , αn T -M μ2 ,α n T )] ≤ E t (H n t,T ) -1 1 2 E t H n t,T (M μ1 , αn T -M μ2 ,α n T ) 2 1 2 . (3.5.5)
Note that (δ n ) n , (β n ) n are predictable process bounded by C f , the Lipschitz constant of f . We thus have for all n ∈ N, E t (H n t,T ) -1 ≤ C, for some C > 0 depending on C f and T (by the properties of exponential martingales). The above relation together with (3.5.4) and the fact that

M μ1 , αn T -M μ2 ,α n T takes values in [0, 1] a.s., imply: E t [|M μ1 , αn T -M μ2 ,α n T | 2 ] ≤ C(μ 1 -μ2 ) 1 2 , (3.5.6)
where C is a constant depending on the Lipschitz constant of the driver f . By letting n tend to infinity in inequality (3.5.2) and using (3.5.6), we get:

|Y t (μ 1 ) -Y t (μ 2 )| ≤ Err t (∆(μ 1 , μ2 )) . (3.5.7)
Same arguments as in Step 2 of the previous theorem lead to:

|Y t (µ 1 ) -Y t (µ 2 )| ≤ Err t (∆(µ 1 , µ 2 )) . (3.5.8)

Convexity

In this section, we provide a convexity result adapted to the non-markovian setting which is established for the map µ → Y t (µ), for any t < T . We extend the results of [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF] to the case of nonlinear constraints, which lead to nontrivial additional technicalities. More precisely, in [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF] it is used the fact that a threshold µ "admissible" at time t (as it belongs to [0, 1] a.s.), it is "admissible" at any time between 0 and T . In our case, due to nonlinearity, the admissibility set is not the same for all t, as it is given by the two processes

E f [0] and E f [1].
We first recall the notion of F t -convexity introduced in [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF].

Definition 3.5.2 (F t -convexity). (i) We say that a subset D ⊂ L 2 (R, F t ) is F t -convex if for all µ 1 , µ 2 ∈ D and λ ∈ L 0 ([0, 1], F t ), λµ 1 + (1 -λ)µ 2 ∈ D. (ii) Let D be an F t -convex subset of L 2 (R, F t ). A map J : D → L 2 (R, F t ) is said to be F t -convex if Epi(J ) := {(µ, Y ) ∈ D × L 2 (R, F t ) : Y ≥ J (µ)} is F t -convex.
(iii) Let Epi c (J ) be the set of elements of the form

n≤N λ n (µ n , Y n ) with (µ n , Y n , λ n ) n≤N ⊂ Epi(J ) × L 0 ([0, 1], F t ) such that n≤N λ n = 1, for some N ≥ 1.
We then denote by Epi c (J ) its closure in L 2 . The F t -convex envelope of J t is defined as

J c t (µ) := ess inf{Y ∈ L 2 (R, F t ) : (µ, Y ) ∈ Epi c (J t )}.
(3.5.9)

Assumption 3.5.3. We assume that the map Φ is Lipschitz continuous in x, uniformly with respect to ω.

Proposition 3.5.4. Under Assumption 3.5.3, the map µ ∈ D t → Y t (µ) is F t -convex, for all t < T .
The proof is divided in several steps. We follow the arguments used in the proof of Proposition 3.2 in [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF] up to non trivial modifications due to the nonlinearity of the driver f . The technical arguments specific to the nonlinear case are mostly needed in Step 5 of the proof. For convienence of the reader, we also present the main ideas of Steps 1-4.

Proof. 1. (µ, Y c t (µ)) ∈ Epi c (Y t ), for all µ ∈ D t .
For every fixed element µ ∈ D t , the family

F := {Y ∈ L 2 (R, F t ) : (µ, Y ) ∈ Epi c (Y t )} is direct downward since Y 1 1 {Y 1 ≤Y 2 } + Y 2 1 {Y 1 >Y 2 } ∈ F for all Y 1 , Y 2 ,
Y c t (µ)) ∈ Epi c (Y t ) is obtaind as L 2 -limit of elements of the form n≤N λ n (µ n , Y n ) with (µ n , Y n , λ n ) ⊂ Epi(Y t ) × L 0 ([0, 1], F t ), such that n≤N λ n = 1.
, Y ≥ Y c t . 3. The map µ ∈ D t → Y c t (µ) is F t -convex. We have the show that Epi(Y c t ) is F t -convex. Let us fix µ 1 , µ 2 ∈ D t and λ ∈ L 0 ([0, 1], F t ). Since Epi c (Y t ) is F t -convex and (µ i , Y c t (µ i )) ∈ Epi c (Y t ), for i = 1, 2, it follows that (λµ 1 + (1 - λ)µ 2 , λY c t (µ 1 ) + (1 -λ)Y c t (µ 2 )) ∈ Epi c (Y t ), and thus λY c t (µ 1 ) + (1 -λ)Y c t (µ 2 ) ≥ Y c t (λµ 1 + (1 -λ)µ 2 , by definition of Y c t (µ). We obtain that λY 1 + (1 -λ)Y 2 ≥ Y c t (λµ 1 + (1 -λ)µ 2 ), for any Y 1 , Y 2 such that (µ i , Y i ) ∈ Epi(Y c t ), i = 1, 2. The result follows. 4. Y t (µ) ≥ Y c t (µ), for all µ ∈ D t . Let (µ n ) n ∈ D t be such that µ n → µ a.s. when n → ∞. Recall that under Assumption 3.5.3, the map µ → Y t (µ) is a.s. continuous and hence Y t (µ n ) → Y t (µ) a.s. when n → ∞. Moreover, by 3.2.8 we have Y t (µ n ) → Y t (µ) in L 2 . Note that Epi(Y t ) ⊂ Epi c (Y t ) and thus (µ, Y t (µ)) ∈ Epi c (Y t ).
The result follows by using the definition of Y c t .

5. Y c t (µ) ≥ Y t (µ), for all µ ∈ D t .

(i) It follows from Point 1, that there exists a sequence

(µ n , Y n , λ N n ) n≥1,N ≥1 ⊂ Epi(Y t ) × L 0 ([0, 1], F t )
such that n≤N λ N n = 1, for all N , and

(μ N , ŶN ) := n≤N λ N n (µ n , Y n ) → (µ, Y c t ) ∈ L 2 . (3.5.10) Fix N ≥ 1 and M ≥ 1. We claim that Y t (μ N ) ≤ ŶN .
The proof is postponed to Step 5, point (ii). We deduce:

lim inf N →∞ Y t (μ N ) ≤ Y c t (µ). (3.5.11)
We now define:

Z M (µ) := ess inf{Y t (µ ′ ) : |µ ′ -µ| ≤ 1 M }. (3.5.12)
and set

D M µ := {µ ′ ∈ D t : |µ ′ -µ| ≤ 1 M }. By Lemma 3.7.1, it exists a sequence (µ M n ) n with µ M n ∈ D M µ for all n such that Y t (µ M n ) → Z M (µ) a.s. when n → ∞. (3.5.13)
One can easily remark that under Assumption 3.5.3, the estimate given in Theorem 3.5.1 becomes:

|Y t (µ M n ) -Y t (µ)| ≤ Err t (∆|µ M n -µ|) ≤ K|µ M n -µ| 1 4 ≤ K 1 M 1 4 , (3.5.14)
where K is a constnat depending on C f , T and the Lipschitz constant of Φ.

Coupling the inequality (3.5.14) with (3.5.13) and letting first n and then M to ∞, we get

Z M (µ) → Y t (µ) a.s. when M → +∞. (3.5.15)
Now, the convergence μN → µ a.s. and Lemma 3.7.3 imply that: (3.5.16) where: μN := μN

Z M (µ) ≤ lim inf N →∞ Y t (μ N ) = lim inf N →∞ Y t (μ N )1 |μ N -µ|≤ 1 M + Y t (µ)1 |μ N -µ|> 1 M ≤ Y c t (µ),
1 |μ N -µ|≤ 1 M + µ1 |μ N -µ|> 1 M ∈ D M µ .
Also, since by (3.5.15), Z M (µ) ↑ Y t (µ) as M goes to +∞, the result follows.

(ii) It remains to prove:

Y t (μ N ) ≤ ŶN . (3.5.17) Fix ε > 0. Let us consider a random variable, F t+ε measurable ζ ε N such that P [ζ ε N = M µn,αn t+ε |F t ] = λ N n , where α n ∈ A t,µn . Clearly, by construction, ζ ε N belongs to E f t+ε,T [0], E f t+ε,T [1]
a.s. We set:

µ ε N := E f t,t+ε [ζ ε N ] . (3.5.18)
We rewrite Y t (μ N ) as follows:

Y t (μ N ) = Y t (μ N ) -Y t (µ ε N ) + Y t (µ ε N ) (3.5.19)
and by appealing to Theorem 3.5.1, we obtain:

Y t (μ N ) ≤ Err t (∆(μ N -µ ε N )) + Y t (µ ε N ). (3.5.20) We now show that lim sup ε→0 [Err t (∆(μ N -µ ε N )) + Y t (µ ε N )] ≤ ŶN .
To this purpose, we split the proof in several steps:

Step a. We prove that lim ε→0 Err t (∆(μ Nµ ε N )) = 0 a.s.

We start by showing that lim ε→0 µ ε N = μN a.s. Since (µ n ) n≤N are F t -measurable and

P [ζ ε N = M µn,αn t+ε |F t ] = λ N n , we have μN = E t [ n≤N 1 ζ ε N =M µn,αn t+ε µ n ]
a.s. We split the difference between µ ε N and µ N in two terms as follows:

|µ ε N -μN | = |E f t,t+ε [ n≤N 1 ζ ε N =M µn,αn t+ε M µn,αn t+ε ] -E t [ n≤N 1 ζ ε N =M µn,αn t+ε µ n ]| 2 ≤ 2|E f t,t+ε [ n≤N 1 ζ ε N =M µn,αn t+ε M µn,αn t+ε ] -E f t,t+ε [ n≤N 1 ζ ε N =M µn,αn t+ε µ n ]| 2 (3.5.21) + 2|E f t,t+ε [ n≤N 1 A ε n µ n ] -E t [ n≤N 1 ζ ε N =M µn,αn t+ε µ n ]| 2 .
From the a priori estimations on BSDEs, we obtain:

|E f t,t+ε [ n≤N 1 ζ ε N =M µn,αn t+ε M µn,αn t+ε ] -E f t,t+ε [ n≤N 1 ζ ε N =M µn,αn t+ε µ n ]| 2 ≤ E t [ n≤N 1 ζ ε N =M µn,αn t+ε (M µn,αn t+ε -µ n ) 2 ] ≤ n≤N E t [(M µn,αn t+ε -µ n ) 2 ]. (3.5.22)
Since for all n ≤ N the processes M µn,αn

• are continuous and belong to S 2 , we can apply Lebesgue's theorem and obtain that the right member of (3.5.22) tends to 0 when ε → 0. Moreover, by applying Proposition 3.7.4 with ξ ε = n≥1 1 A ε n µ n , we derive that it exists η ε , with η ε → 0 a.s. when ε → 0 such that: Step b. We prove that for each Step c. Recall that by (3.5.18) we have

|E f t,t+ε [ n≤N 1 ζ ε N =M µn,αn t+ε µ n ] -E t [ n≤N 1 ζ ε N =M µn,αn t+ε µ n ]| 2 ≤ η ε . ( 3
n ≤ N , lim ε→0 E t [|Y t+ε (M µn,αn t+ε ) -Y t (µ n )|] = 0 a.
µ ε N = E f t,t+ε [ζ ε N ] . Lemma 3.3.1 leads to: Y t (µ ε N ) ≤ E g t,t+ε [Y t+ε (ζ ε N )] = E g t,t+ε Y t+ε ( n≤N 1 ζ ε N =M µn,αn t+ε M µn,αn t+ε ) .
By Lemma 3.7.3, we obtain:

Y t (µ ε N ) ≤ E g t,t+ε n≤N 1 ζ ε N =M µn,αn t+ε
Y t+ε (M µn,αn t+ε ) .

(3.5.24)

We now apply Proposition 3.7.4 with

ξ ε := n≥1 1 ζ ε N =M µn,αn t+ε
Y t+ε (M µn,αn t+ε ) and derive that it exists η ′ ε , with η ′ ε → 0 a.s. when ε → 0 such that:

Y t (µ ε N ) ≤ E g t,t+ε n≤N 1 ζ ε N =M µn,αn t+ε Y t+ε (M µn,αn t+ε ) ≤ E t n≤N 1 ζ ε N =M µn,αn t+ε Y t+ε (M µn,αn t+ε ) + η ′ ε .
We finally get:

Y t (µ ε N ) ≤ E t n≤N 1 ζ ε N =M µn,αn t+ε Y t (µ n ) + n≤N E t [|Y t+ε (M µn,αn t+ε ) -Y t (µ n )|] + η ′ ε .
Letting ε tend to 0 in the above inequality, we obtain, by Step b:

lim sup ε→0 Y t (µ ε N ) ≤ n≤N λ N n Y t (µ n ) ≤ n≤N λ N n Y n = Ŷn , ( 3

Dual representation in the case of concave constraints

We now provide a dual representation of the value function defined by (3.2.6), which takes the form of a stochastic control problem in Meyer form. The results of this section extend the ones given in [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF], but involve technical additional proofs, due to the nonlinearity of the coefficient f . For each (ω, t), let f (ω, t, •, •, •) be be the concave conjugate of f with respect to (x, π), defined for each (p, q) in R × R d as follows:

f : (ω, t, p, q) ∈ Ω × [0, T ] × R × R d → inf (x,π)∈R×R d xp + π ⊤ q -f (ω, t, x, π) .
For each (ω, t), we denote by g(ω, t, •, •, •) the convexe conjugate of g with respect to (y, z), defined for each (u, v) in R × R d as follows:

g : (ω, t, u, v) ∈ Ω × [0, T ] × R × R d → sup (y,z)∈R×R d yu + z ⊤ v -g(ω, t, y, z) .
We also introduce for each ω, the polar function of Φ with respect to m:

Φ : (ω, l) ∈ Ω × R → sup m∈[0,1] (ml -Φ(ω, m)) .
In the sequel, we denote by U the set of predictable processes valued in D 1 , respectively by V the set of predictable processes valued in D 2 t , where for each (t, ω) ∈ [0, T ] × Ω, D 1 t (ω) and D 2 t (ω) are defined:

D 1 t (ω) := {(p, q) : f (t, ω, p, q) > -∞}; D 2 t (ω) := {(u, v) : g(t, ω, u, v) < +∞}. (3.6.1) Remark 3.6.1. For each (t, ω), D 1 t (ω) ⊂ U , where U is the closed subset of R × R d of elements α = (α 1 , α 2 ) such that |α 1 | ≤ C g and |α i 2 | ≤ C g , ∀i = 1, d.
The same remark holds for the elements belonging to D 2 t (ω), with C f instead of C g .

To each l > 0, γ = (κ, ϑ) ∈ V (resp. λ = (µ, ν) ∈ U), we associate the processes A l,γ (resp. L λ ) defined by

A l,γ t = l + t 0 A l,γ s κ s ds + t 0 A l,γ s ϑ s dW s , t ∈ [0, T ]; L λ t = 1 + t 0 L λ s µ s ds + t 0 L λ s ν s dW s , t ∈ [0, T ].
The dual formulation of Y 0 is expressed in terms of

X 0 (l) := inf (λ,γ)∈U ×V X l,λ,γ 0 where X l,λ,γ 0 := E T 0 L λ s g(s, λ s )ds - T 0 A l,γ s f (s, γ s )ds + L λ T Φ( A l,γ T L λ T ) . Proposition 3.6.2. Y 0 (m) ≥ sup l>0 (lm -X 0 (l)), for all m ∈ E f 0,T [0], E f 0,T [1] . Proof. Fix α ∈ A 0,m , λ = (ν, µ) ∈ U, l > 0 and γ = (κ, ϑ) ∈ V.
The definition of Φ, together with Ito formula imply:

E[Y m,α T L λ T ] ≤ Y m,α 0 + E[ T 0 L λ s g(s, λ s )ds] (3.6.2)
and

E[Y m,α T L λ T ] = E[Φ(M m,α T )L λ T ] ≥ E[A l,γ T M m,α T -L λ T Φ( A l,γ T L λ T )] ≥ E[lm + T 0 A l,γ s f (s, γ s )ds -L λ T Φ( A l,γ T L λ T )]. (3.6.3) Note that since Y m,α , L λ , M m,α , A l,γ ∈ S 2 , Z m,α
, α ∈ H 2 and Remark 3.6.1 holds, by applying Burkholder-Davis-Gundy inequality, we obtain that the local martingales

• 0 Y m,α s L λ s ν ⊤ s dW s , • 0 L λ s Z m,α,⊤ s dW s , • 0 M m,α s A l,γ s ϑ ⊤ s dW s , • 0 A l,γ s α ⊤
s dW s are in fact martingales. Hence we can cancel their expectations. From the two above inequalities, we derive that:

Y m,α 0 ≥ lm -E T 0 L λ s g(s, λ s )ds - T 0 A l,γ s f (s, γ s )ds + L λ T Φ( A l,γ T L λ T ) .
By arbitrariness of (λ, γ) ∈ U × V, we get:

Y m,α 0 ≥ lm -X 0 (l).
We then take the essential infimum on α ∈ A m 0 and the supremum on l > 0. The result follows.

We now show that equality holds under some additional assumptions.

Assumption 3.6.3. We make the following assumptions:

(a) For each (t, ω) ∈ Ω × [0, T ], the maps Φ(ω, •), f (t, ω, •) and g(t, ω, •) are of class C 1 b . Also D 1 t (ω)and D 2 t (ω) are closed. (b) |∇ Φ(ω, •)| + ||∇ f (ω, t, •)|| R×R d + ||∇g(ω, t, •)|| R×R d ≤ C Φ, f ,g , for some C Φ, f ,g ∈ L 2 (R); (c) Φ(ω, m) = sup l>0 lm -Φ(ω, l) , for all m ∈ [0, 1]; (d) f (ω, t, x, π) = min (u,v)∈D 2 t (ω) px + π ⊤ q -f (ω, t, p, q) , for all (x, π) ∈ R × R d ; (e) g(ω, t, y, z) = max (u,v)∈D 1 t (ω) yu + z ⊤ v -g(ω, t, u, v)
, for all (y, z) ∈ R × R d . Proposition 3.6.4. Assume that there exists l > 0, λ ∈ U and γ ∈ V such that sup(lm -X 0 (l)) = lm -X 0 ( l) = lm -X l, λ,γ 0 .

(3.6.4)

Then there exists α ∈ H 2 such that Y 0 (m) = Y m, α 0 = lm -X 0 ( l).
(3.6.5)

Also it satisfies      f (•, M m, α, α) = κM m, α + θ⊤ α -f (•, γ); M m, α T = ∇ Φ A l,γ T L λ T ; g(•, Y m, α, Z m, α) = μY m, α + ν⊤ Z m, α -g(•, λ); Φ(M m, α T ) = M m, α T A l,γ T L λ T -Φ( A l,γ T L λ T ). (3.6.6)
Proof. The proof is divided in two steps.

Step 1. We denote by

E f •,T ∇ Φ( A l,γ T L λ T
) , α the solution of the BSDE associated to the terminal

condition ∇ Φ A l,γ T L λ T
and driver f . We first need to show that

E f 0,T ∇ Φ A l,γ T L λ T = m.
By the optimality of l, we get:

lm -E L λ T Φ A l,γ T L λ T - T 0 A l,γ s f (s, γs )ds ≥ m( l + ε) -E L λ T Φ( A l+ε,γ T L λ T ) - T 0 A l+ε,γ s f (s, γs )ds ,
for all ε > -l. Note that A l,γ = lA 1,γ for all l ∈ R. Since by construction Φ is a.s. convex, we deduce that:

mε ≤ E - T 0 f (s, γs )A 1,γ s + A 1,γ T ∇ Φ A l+ε,γ T L λ T ε.
We take in the above inequality ε = 1 n and ε = -1 n . By letting n tend to ∞ and using (3.6.3) (a) and Lebesgue's Theorem, we finally get:

m = E - T 0 f (s, γs )A 1,γ s + A 1,γ T ∇ Φ A l,γ T L λ T . (3.6.7)
We now introduce the processes ( M , N ) ∈ S 2 × H 2 , solution of the BSDE associated to the

terminal condition ∇ Φ A l,γ T L λ T and driver h(s, ω, y, z) := -f (s, κs (ω), θs (ω)) + yκ s (ω) + z ⊤ θs (ω). (3.6.8)
Note that h is Lipschitz continuous with respect to (y, z), uniformly in (s, ω) (see Remark 3.6.1).

Existence and uniqueness of the solution of the above BSDE is thus guaranteed. We apply Itô formula to A 1,γ M and obtain:

A 1,γ t Mt = A 1,γ T ∇ Φ A l,γ T L λ T - T t f (s, γs )A 1,γ s ds - T t A 1,γ s Ñs dW s , (3.6.9)
where Ñ is defined by Ñ := N + M θ. Clearly, Ñ belongs to

H 2 since N ∈ H 2 , M ∈ S 2 and || θ|| R d ≤ C, by Remark 3.6.1 .
Let us now fix γ = (κ, ϑ) ∈ V. Since V is convex, we get that for all ε ∈ [0, 1],

γ ε := (1 -ε)(κ, θ) + ε(κ, ϑ) ∈ V. Using now the optimality condition X l,γ ε , λ 0 ≥ X l,γ, λ 0
, the fact that l > 0, the Lagrange's and Lebesgue's Theorems, one can easily show that ∇ f (•, γ) satisfies:

0 ≥ E T 0 -A 1,γ s Ks f (s, γs ) + ∇ p f (s, γs )δκ s + ∇ q f (s, γs ) ⊤ δϑ s ds + KT A 1,γ T ∇ Φ A l,γ T L λ T , (3.6.10)
where (δκ, δϑ) := (κκ, ϑ -θ) and K :=

• 0 δκ s -δϑ s θs ds + • 0 δϑ s dW s . By (3.6.9) we have A 1,γ T ∇ Φ( A l,γ T L λ T ) = A 1,γ
T MT . Hence inequality (3.6.10) can be re-written as follows:

0 ≥ E T 0 -A 1,γ s Ks f (s, γs ) + ∇ p f (s, γs )δκ s +∇ q f (s, γs ) ⊤ δϑ s ds + KT A 1,γ T MT . (3.6.11)
The definition of K together with (3.6.11) and Itô formula implies:

0 ≤ E T 0 A 1,γ s (∇ p f (s, γs ) -Ms )δκ s + (∇ q f (s, γs ) -Ns ) ⊤ δϑ s ds . (3.6.12)
We introduce the map Θ :

[0, T ] × Ω × R × R d → R defined as follows: Θ : (ω, t, u, v) → (∇ p f (ω, t, γt (ω)) -Mt (ω))(u -κt (ω)) + (∇ q f (ω, t, γt (ω)) -Nt (ω)) ⊤ (v -θt (ω)).
By Remark 3.6.1, Assumption 3.6.3 (a) and Theorem 18.19, p.605 in [START_REF] Aliprantis | Infinite dimensional analysis: a hitchhiker's guide[END_REF], there exists a predictable γ belonging to V such that γ =argmin{Θ(•, u, v), (u, v) ∈ D 2 }. For each (t, ω) ∈ [0, T ] × Ω, define the map F as follows:

(p, q) ∈ D 2 t (ω) → F (ω, t, p, q) := f (ω, t, p, q)p Mt (ω)q ⊤ Nt (ω).

(3.6.13)

Note that we have:

Θ(t, ω, u, v) = ∇ p F (t, ω, γt (ω))(u -κt (ω)) + ∇ q F (t, ω, γt (ω)) ⊤ (v -θt (ω)).
Since (3.6.12) holds for all γ ∈ V, we can take γ1 Θ(•,γ)>0 + γ1 Θ(•,γ)≤0 . Hence we derive that, for dt ⊗ dP -a.e. (ω, t) ∈ Ω × [0, T ], we have:

Θ(t, ω, u, v) ≤ 0, ∀ (u, v) ∈ D 2 t (ω).
By a result of convex analysis, this implies that γt (ω) maximizes F (ω, t, •) for dt ⊗ dP -a.e. (ω, t) ∈ Ω × [0, T ] and thus by Assumption 3.6.3 (d) we get:

f (•, γ) = κ M + θ⊤ N -f (•, M , N ).
(3.6.14)

The above relation together with the definition of h (see (3.6.8)) leads to:

h(•, M , N ) = f (•, M , N ).
Recall that ( M , N ) represents the solution of the BSDE of terminal condition ∇ Φ(

A l,γ T L λ T
) and driver h. Hence by applying the comparison theorem for BSDEs, we get

( M , N ) = (E f •,T [∇ Φ( A l,γ T L λ T )], α). (3.6.15)
Now, we take the conditional expectation in (3.6.9) and we get:

Mt := (A 1,γ t ) -1 E[- T t f (s, γs )A 1,γ s + A 1,γ T ∇ Φ( A l,γ T L γ T )|F t ]. (3.6.16)
We have cancelled the expectation of

E f 0,T ∇ Φ( A l,γ T L λ T ) = m and moreover,
that the first statement of (3.6.6) holds .

Since Φ is a.s.incresing, we derive that ∇ Φ(

A l,γ T L λ T
) ≥ 0 a.s. Also, by construction, Φ is a.s.

1-Lipschitz, which implies that ∇ Φ(

A l,γ T L λ T ) ∈ [-1, 1] a.s. We thus conclude that ∇ Φ( A l,γ T L λ T ) ∈ [0, 1] a.s. and E f 0,T [∇ Φ( A l,γ T L λ T )] = m.
Step 2. First, recall that (Y m, α, Z m, α) represents the solution of the BSDE with terminal condition Φ(M m, α T ) and driver g, where by

Step 1, M m,

α T = ∇ Φ( A l,γ T L λ T ). Now, Assumption 3.6.3 (c) yields Φ(M m, α T ) = M m, α T A l,γ T L λ T -Φ( A l,γ T L λ T
).

(3.6.17)

Using the optimality of λ, i.e. for all ε > 0, X l,γ,λ ε 0 ≥ X l,γ, λ 0 and similar arguments as in Step 1, we get:

(Y m, α, Z m, α) = ( Ŷ , Ẑ), (3.6.18)
where ( Ŷ , Ẑ) corresponds to the solution of the BSDE associated to the terminal condition Φ(M m, α T ) and driver -g(s, μs (ω), νs (ω)) + yµ s (ω) + z ⊤ νs (ω). Also by the same arguments given at Step 1, Ŷ satisfies:

Ŷ = (L λ) -1 E • [- T • g(s, λs )L λ s + L λ T Φ(M m, α T )]. ( 3
Y m, α 0 = E L λ T Φ(M m, α T ) - T 0 L λ s g(s, λ)ds = E M m, α T A l,γ T -E L λ T Φ( A l,γ T L λ T ) + T 0 L λ s g(s, λ)ds . (3.6.20)
Now, we appeal to (3.6.7) and since by Step 1, M m,

α T = ∇ Φ( A l,γ T L λ T ), we get E M m, α T A l,γ T = l m + E T 0 Â1,γ s f (s, γ)ds =m l+E T 0
Âl ,γ s f (s, γ)ds . From the two above equalities, we finally obtain

Y m, α 0 = lm -E L λ T Φ(M m, α, β T ) - T 0 Âl ,γ s f (s, γ)ds + T 0 L λ s g(s, λ)ds .
The above equality together with Proposition 3.6.2 give the desired result.

We now show that the existence of an optimal control in the primal problem implies the existence of an optimal control in the dual problem, under the following assumptions: Assumption 3.6.5.

(a) For each (t, ω), the maps Φ(ω), f (ω, t, •) and g(ω, t,

•) are C 1 b on [0, 1] and R × R d respectively; (b) |∇Φ(ω, •)| ≤ C Φ (ω), for some C Φ ∈ L 2 (R).
Proposition 3.6.6. Let l > 0 be fixed and assume that there exists m ∈ [E f 0,T [0], E f 0,T [START_REF] Alario-Nazaret | Dynkin games[END_REF]] and α ∈ A 0, m such that

sup m∈[E f 0,T [0],E f 0,T [1]] (ml -Y 0 (m)) = ml -Y m, α 0 .
(3.6.21)

Then, there exists ( λ, γ)

∈ U × V such that Y 0 ( m) = ml -X 0 (l) = ml -X l,γ, λ 0 . (3.6.22)
Proof. The proof is divided in three steps.

Step

1. Let M • be an arbitrary f -martingale valued in [E f • [0], E f • [1]] and ε ∈ [0, 1]. We denote by M ε the process defined as M ε • := E f •,T MT + ε(M T -MT )
, where M := M m, α. We set m ε := M ε 0 and (δM, δα) := (M -M, αα). We now consider the BSDE associated to δM T and generator:

h 1 (t, ω, u, v) := ∇ x f (t, ω, Mt (ω), αt (ω))u + ∇ π f (t, ω, Mt (ω), αt (ω)) ⊤ v.
Since δM T belongs to L 2 (F T ) and since by Assumption 3.6.5 on the coefficient f , h is uniformly Lipschitz in (u, v) with respect to (t, ω) , we conclude that the above BSDE admits an unique solution. This unique solution will be denoted by (∇M, ∇α).

Our aim is to show that ε

-1 (δM ε , δα ε ) converges in S 2 × H 2 as ε → 0 to (∇M, ∇α).
First, observe that ε -1 (δM ε s , δα ε s ) solves the following equation: We now introduce the processes Ξ ε := ε -1 δM ε -∇M and Π ε := ε -1 δα ε -∇α. We can remark that (Ξ ε , Π ε ) solves the BSDE associated to terminal condition 0 and driver:

δM ε t ε = δM T + T t B M,ε s δM ε s ε + B α,ε,⊤ s δα ε s ε ds - T t δα ε s ε ⊤ dW s , (3.6
h 2 (t, ω, u, v) := B M,ε t (ω)u + B α,ε t (ω) ⊤ v + D ε t (ω),
where

D ε t := ∇M t B M,ε t -∇ x f (t, Mt , αt ) + ∇α ⊤ t B α,ε t -∇ π f (t, Mt , αt ) .
We apply the stability result with BSDE(ξ, h 2 ) and BSDE(ξ, 0), where ξ = 0. We thus get:

||Ξ ε || S 2 + ||Π ε || H 2 ≤ C||D ε || H 2 .
(3.6.24)

In order to show the convergence of

||D ε || H 2 to 0 when ε → 0, we prove that (M ε , α ε ) converges to (M, α) in S 2 × H 2 .
In order to do this, we apply again the stability result for BSDEs and obtain:

||M ε -M|| 2 S 2 + ||α ε -α|| 2 H 2 ≤ C(||M ε T -M T || 2 L 2 ) → ε→0 0. (3.6.25)
By (3.6.25), Assumption 3.6.5 and the Lebesgue's Theorem, we get that ||D ε || H 2 → 0 when ε → 0. Finally, by (3.6.24), we derive that ε -1 (δM ε , δα ε ) converges in S 2 × H 2 to (∇M, ∇α) as ε → 0.

Step 2. We denote by (Y ε , Z ε ) the solution of the BSDE(g, Φ(M ε T )) and we set ( Ŷ , Ẑ) := (Y m, α, Z m, α). Using the same arguments as in Step 2, one can show that (

δY ε ε , δZ ε ε ) := ( Y ε - Ŷ ε , Z ε - Ẑ ε ) converges in S 2 × H 2 to
the unique solution (∇Y, ∇Z) of the following BSDE:

∇Y t = ∇Φ( MT )δM T + T t ∇ y g(s, Ŷs , Ẑs )∇Y s ds + T t ∇ z g(s, Ŷs , Ẑs ) ⊤ ∇Z s ds - T t ∇Z ⊤ s dW s .
(3.6.26)

Step 3. Since ( m, α) is optimal, we have Y ε 0m ε -Ŷ0 + ml ≥ 0, for any ε > 0. Dividing now by ε > 0 and sending ε → 0, we get

0 ≤ ∇Φ( MT )δM T + T 0 ∇g(s, Ŷs , Ẑs ) ⊤ (∇Y s , ∇Z s )ds - T 0 ∇Z ⊤ s dW s (3.6.27) -l δM T + T 0 ∇f (s, Ms , αs ) ⊤ (∇M s , ∇α s )ds - T 0 ∇α ⊤ s dW s = ∇Y 0 -l∇M 0 .
We set γt := ∇f (s, Mt , αt ) and λt := ∇g(s, Ŷt , Ẑt ), which belong to V and, respectively, U . Since γt ( resp. λt ) belongs to the subdifferential of f at (M t , αt ) (resp. the subdifferential of g at ( Ŷt , Ẑt )) we have (see [START_REF] Barbu | Convexity and Optimization in Banach Spaces[END_REF]):

f (•, M, α) = κ M + θ⊤ α -f (•, γ).
(3.6.28) and

g(•, Ŷ , Ẑ) = μ Ŷ + ν⊤ Ẑ -g(•, λ). (3.6.29)
Now, by applying Ito's formula, we obtain that A l,γ ∇M and L λ∇Y are martingales. As L λ 0 = 1 and (3.6.27) holds, we thus obtain:

L0 ∇Y 0 -l∇M 0 = E L λ T ∇Y T -A l,γ T ∇M T = E L λ T δM T ∇Φ( MT ) - A l,γ T L λ T ≥ 0. (3.6.30)
Since M T can be arbitrary choses with values in [0, 1], we obtain that MT (ω) minimizes the map

m ∈ [0, 1] → Φ(ω, m) -m A l,γ T L λ T (ω). Thus, we obtain: MT A l,γ T -L λ T Φ( MT ) = L λ T Φ( A l,γ T L λ T
). This inequality together with (6.5.2), (3.6.29) and Ito's formula allow to conclude that l m -Ŷ0 = X l, λ,γ 0 . The conclusion follows by Proposition 3.6.2.

Appendix

Proof of Proposition 3.2.4. The proof is standard. We provide it for completeness. Let (Y, Z) be a supersolution of BSDE(g, f, Ψ, µ, τ ). Now, the BSDE representation of Ψ(Y T ) implies that it exists ᾱ

∈ A τ,ρ such that Ψ(Y T ) = M τ,ρ, ᾱ T , where ρ := E f τ,T [Ψ(Y T )]. Since condition (3.2.4
) is satisfied, we have ρ ≥ µ a.s. We define the following stopping time

σ ᾱ := inf{τ ≤ s ≤ T : M τ,µ, ᾱ s = E f s,T [0]} ∧ T,
with the convention inf ∅ = -∞. Recall that (Y 0 , Z 0 ) represents the solution of the BSDE associated to driver f and terminal condition 0. We define the control α as follows:

αs := ᾱs 1 {s≤σ ᾱ} + Z 0 s 1 {s>σ ᾱ} . (3.7.1)
Note that α belongs to A τ,µ . The control is constructed in such a way that M τ,µ,

α • belongs to [E f •,T [0], E f •,T [1]].
We have not considered the hitting time of the process

E f •,T [1], since clearly M τ,µ, ᾱ • ≤ M τ,ρ, ᾱ • . We can easily remark that M τ,ρ, α T ≥ M τ,µ,α T a.s. The monotonocity of Φ and the identity Ψ(Y T ) = M τ,ρ, α T imply that Y T ≥ (Φ • Ψ)(Y T ) ≥ Φ(M τ,µ,α T ). (3.7.2)
Hence, by the comparison theorem for BSDEs, we obtain that

Y t ≥ E g t,T [Φ(M τ,µ,α T )] for t ∈ [0, T ]. Conversely, let α ∈ A τ,µ be such that Y t ≥ E g t,T [Φ(M τ,µ,α T
)] for t ∈ [0, T ] and suppose that (Y, Z) satifies (3.2.3). We thus get

Ψ(Y T ) ≥ (Ψ • Φ)(M τ,µ,α T ) ≥ M τ,µ,α T .
Taking the f -conditional expectation on both sides, the result follows.

Lemma 3.7.1. Fix θ, ν ∈ T , with θ ≥ τ, µ ∈ D τ and α ∈ A τ,µ . Then there exists a sequence

(α ′ n ) ⊂ A θ,α τ,µ := {α ′ ∈ A τ,µ , α ′ 1 [0,θ) = α1 [0,θ) } such that lim n→∞ ↓ E g θ,T [Φ(M τ,µ,α ′ n T )] = Y α θ (M τ,µ,α θ ) a.s.
Proof. In order to obtain the desired result, we only have to prove that

{J(α ′ ) := E g θ,T [Φ(M τ,µ,α ′ T )], α ′ ∈ A θ,α τ,µ } is directed downward. Set A := {J(α ′ 1 ) ≤ J(α ′ 2 )} ∈ F θ and fix α ′ 1 , α ′ 2 ∈ A θ,α τ,µ . We denote α′ := α1 [0,θ) + 1 [θ,T ] (α ′ 1 1 A + α ′ 2 1 A c ). Note that α′ ∈ A θ,α τ,µ . We get: J(α ′ ) = E θ,T [Φ(M τ,µ,α ′ 1 θ )1 A + Φ(M τ,µ,α ′ 2 θ )1 A c ] = min{J(α ′ 1 ), J(α ′ 2 )}. Theorem 3.7.2. Fix t ∈ [0, T ]. The map Y t : µ → Y t (µ); D t → L 2 ; is non-decreasing, i.e. for all µ 1 , µ 2 ∈ D t , we have Y t (µ 1 ) ≤ Y t (µ 2 ) on {µ 1 ≤ µ 2 } and Y t (µ 1 ) ≥ Y t (µ 2 ) on {µ 1 ≥ µ 2 }.
Proof. The proof is divided in two steps.

Step 1. We set μ1 := µ 1 ∧ µ 2 and μ2 := µ 1 ∨ µ 2 . Remark that μ1 and μ2 belong to D t . By Lemma 3.7.1, we know that it exists

α n ∈ A t,μ 2 s.t. E g t,T [Φ(M μ2 ,α n T )] → Y t (μ 2 ) a.s. Fix n ∈ N.
We define αn ∈ A t,μ 1 as follows:

αn s := α n s 1 s≤τ + Z 0 s 1 s>τ , where τ := inf{t ≤ s ≤ T : M μ1 ,α n s = E f s,T [0]} ∧ T , with the convention inf ∅ = +∞.
Recall that Z 0 is the associated control to the the BSDE with terminal condition 0 and driver f . By construction of αn , we have M μ1 ,α n T ∈ [0, 1] a.s. Now, by using the fact that Φ in nondecreasing and the comparison theorem for BSDEs, we obtain:

E g t,T [Φ(M μ1 , αn T )] ≤ E g t,T [Φ(M μ2 ,α n T )] a.s. which implies Y t (μ 1 ) ≤ E g t,T [Φ(M μ2 ,α n T )] a.s. (3.7.3)
By letting n → ∞ in the above relation, we obtain Y t (μ 1 ) ≤ Y t (μ 2 ) a.s.

Step 2. We define

A := {µ 1 ≤ µ 2 } ∈ F t . Let us show that Y t (μ 1 ) = Y t (µ 1 )1 A + Y t (µ 2 )1 A c . For all α i ∈ A t,µ i , i = 1, 2, we set α := 1 [t,T ] (α 1 1 A + α 2 1 A c ) ∈ A t,μ 1 .
Bt the zero-one law for f -conditional expectations, we get

E g t,T [Φ(M μ1 , α T )] = E g t,T [Φ(M µ 1 ,α 1 T )]1 A + E g t,T [Φ(M µ 2 ,α 2 T )]1 A c and by arbitrariness of α i , i = 1, 2, we derive that Y t (μ 1 ) ≤ Y t (µ 1 )1 A + Y t (µ 2 )1 A c . In order to show that Y t (μ 1 ) ≥ Y t (µ 1 )1 A + Y t (µ 2 )1 A c , we use the previous equality with α 1 := α1 A + α1 1 A c and α 2 := α2 1 A + α1 A c , for all α ∈ A t,μ 1 , α1 ∈ A t,µ 1 and α2 ∈ A t,µ 2 . Similarly, one can prove that Y t (μ 2 ) = Y t (µ 2 )1 A + Y t (µ 1 )1 A c .
From Step 1 and Step 2, the result follows.

Using the same arguments as in Step 2 of the above proof, one can easily show:

Lemma 3.7.3. Fix t ∈ [0, T ]. We have Y t (µ 1 1 A + µ 2 1 A c ) = Y t (µ 1 )1 A + Y t (µ 2 )1 A c , for all A ∈ F t , µ 1 , µ 2 ∈ D t .
We now recall the following result, which can be found in [START_REF] Bouchard | BSDEs with weak terminal condition[END_REF].

Proposition 3.7.4. Let the Assumption 3.2.1 (with g instead f ) holds. Then:

(i) There exist χ g ∈ L 2 and C > 0 which only depends on C g and T such that:

ess sup ξ∈L 0 ([0,1]) |E g t,T [ξ] ≤ C(1 + E t [|χ g | 2 ])| 1
2 ), 0 ≤ t ≤ T.

(ii) For some ξ ∈ L 2 and t ∈ [0, T ], consider a family (ξ ε ) ε≥0 ⊂ L 0 (R d ) satisfying |ξ ε | ≤ ξ and ξ ε ∈ L 0 (F (t+ε)∧T ), for any ε > 0. Then, there exists a family (η ε ) ε>0 ⊂ L 0 (R) which converges to 0 P -a.s. as ε → 0 such that:

E g t,t+ε [ξ ε ] -E t [ξ ε ] ≤ η ε , ∀ε ∈ [0, T -t].
(iii) Let (ξ ε ) ε>0 and t ∈ [0, T ] be as in (ii). Then, there exists a family (η ε ) ε>0 ⊂ L 0 (R) which converges to 0 a.s. as ε → 0 such that

E g t-ε,t [ξ ε ] -E t [ξ ε ] ≤ η ε , ∀ε ∈ [0, t].
Chapter 4

Optimal stopping for dynamic risk measures with jumps and obstacle problems

Abstract. We study the optimal stopping problem for a monotonous dynamic risk measure induced by a BSDE with jumps in the Markovian case. We show that the value function is a viscosity solution of an obstacle problem for a partial integro-differential variational inequality, and we provide an uniqueness result for this obstacle problem.

Introduction

In the last years, there has been several studies on dynamic risk measures and their links with nonlinear backward stochastic differential equations (BSDEs). We recall that nonlinear BSDEs have been introduced in [START_REF] Pardoux | Backward Stochastic Differential equations and Quasilinear Parabolic Partial Differential equations[END_REF] in a Brownian framework, in order to provide a probabilistic representation of semilinear parabolic partial-differential equations. BSDEs with jumps and their links with partial integro-differential equations are studied in [START_REF] Barles | Backward stochastic differential equations and integralpartial differential equations[END_REF]. A comparison theorem is established in [START_REF] Royer | Backward stochastic differential equations with jumps and related non-linear expectations[END_REF] and generalized in [START_REF] Quenez | BSDEs with jumps, optimization and applications to dynamic risk measures[END_REF], where properties of dynamic risk measures induced by BSDEs with jumps are also provided. An optimal stopping problem for such risk measures is addressed in [START_REF] Quenez | Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps[END_REF], and the value function is characterized as the solution of a reflected BSDE with jumps and RCLL obstacle process.

In the present chapter, we focus on the optimal stopping problem for dynamic risk measures induced by BSDEs with jumps in a Markovian framework. In this case the driver of the BSDE depends on a given state process X, which can represent, for example, an index or a stock price. This process will be assumed to be driven by a Brownian motion and a Poisson random measure.

Our main contribution consists in establishing the link between the value function of our optimal stopping problem and parabolic partial integro-differential variational inequalities (PIDVIs). We prove that the minimal risk measure, which corresponds to the solution of a reflected BSDE with jumps, is a viscosity solution of a PIDVI. This provides an existence result for the obstacle problem under relatively weak assumptions. In the Brownian case, this result was obtained in [START_REF] El Karoui | Reflected solutions of backward SDE's, and related obstacle problems for PDE's[END_REF] by using a penalization method via non-reflected BSDEs. Note that this method could also be adapted to our case with jumps, but would involve heavy computations in order to prove the convergence of the solutions of the penalized BSDEs to the solution of the reflected BSDE. It would also require some convergence results of the viscosity solutions theory in the integro-differential case. We provide here instead a direct and shorter proof.

Furthermore, under some additional assumptions, we prove a comparison theorem in the class of bounded continuous functions, relying on a non-local version of Jensen-Ishii Lemma (see [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions theory revisited[END_REF]), from which the uniqueness of the viscosity solution follows. We point out that our problem is not covered by the study in [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions theory revisited[END_REF], since we are dealing with nonlinear BSDEs, and this leads to a more complex integro-differential operator in the associated PDE.

The chapter is organized as follows: In Section 4.2 we give the formulation of our optimal stopping problem. In Section 4.3, we prove that the value function is a solution of an obstacle problem for a PIDVI in the viscosity sense. In Section 4.4, we establish an uniqueness result. In the Appendix, we prove some estimates, from which we derive that the value function is continuous and has polynomial growth and provide some complementary results.

Optimal stopping problem for dynamic risk measures with jumps in the Markovian case

Let (Ω, F , P ) be a probability space. Let W be a one-dimensional Brownian motion and N (dt, du) be a Poisson random measure with compensator ν(du)dt such that ν is a σ-finite measure on IR * equipped with its Borel field B(I R * ), and satisfies R * (1 ∧ e 2 )ν(de) < ∞. Let Ñ (dt, du) be its compensated process. Let IF = {F t , t ≥ 0} be the natural filtration associated with W and N . We consider a state process X which may be interpreted as an index, an interest rate process, an economic factor, an indicator of the market or the value of a portfolio, which has an influence on the risk measure and the position. For each initial time t ∈ [0, T ] and each condition x ∈ R, let X t,x be the solution of the following stochastic differential equation (SDE):

X t,x s = x + s t b(X t,x r )dr + s t σ(X t,x r )dW r + s t R * β(X t,x r -, e) Ñ (dr, de), (4.2.1)
where b, σ : R → R are Lipschitz continuous, and β : R × R * → R is a measurable function such that for some non negative real C, and for all e ∈ R

|β(x, e)| ≤ C(1 ∧ |e|), x ∈ R |β(x, e) -β(x ′ , e)| ≤ C|x -x ′ |(1 ∧ |e|), x, x ′ ∈ R.
We introduce a dynamic risk measure ρ induced by a BSDE with jumps. For this, we consider two functions γ and f satisfying the following assumption:

Assumption H 1 • γ : R × R * → R is B(R) ⊗ B(R * )-measurable, |γ(x, e) -γ(x ′ , e)| < C|x -x ′ |(1 ∧ |e|), x, x ′ ∈ R, e ∈ R * -1 ≤ γ(x, e) ≤ C(1 ∧ |e|), e ∈ R * • f : [0, T ] × R 3 × L 2 ν → R
is continuous in t uniformly with respect to x, y, z, k, and continuous in x uniformly with respect to t, y, z, k.

(i) |f (t, x, 0, 0, 0)| ≤ C(1 + x p ), ∀x ∈ R (ii) |f (t, x, y, z, k) -f (t, x ′ , y ′ , z ′ , k ′ )| ≤ C(|y -y ′ | + |z -z ′ | + k -k ′ L 2 ν ), ∀ t ∈ [0, T ], y, y ′ , z, z ′ ∈ R, k, k ′ ∈ L 2 ν (iii) f (t, x, y, z, k 1 ) -f (t, x, y, z, k 2 ) ≥< γ(x, •), k 1 -k 2 > ν , ∀t, x, y, z, k 1 , k 2 .
Here, L 2 ν denotes the set of Borelian functions ℓ :

IR * → IR such that ℓ 2 ν := IR * |ℓ(u)| 2 ν(du) < +∞. It
ρ t,x s (ζ, S) := -E t,x s,S (ζ), t ≤ s ≤ S, (4.2.2)
where E t,x •,S (ζ) denotes the f -conditional expectation, starting at (t, x), defined as the solution in S 2 of the BSDE with Lipschitz driver f (s, X t,x s , y, z, k), terminal condition ζ and terminal time S, that is the solution (E t,x s ) of

-dE s = f (s, X t,x s , E s , π s , l s (•))ds -π s dW s - R * l s (u) Ñ (dt, du) ; E S = ζ, (4.2.3)
where (π s ), (l s ) are the associated processes, which belong to H 2 and H 2 ν respectively. The functional ρ : (ζ, S) → ρ • (ζ, S) defines then a dynamic risk measure induced by the BSDE with driver f (see [START_REF] Quenez | BSDEs with jumps, optimization and applications to dynamic risk measures[END_REF]). Assumption H 1 implies that the driver f (s, X t,x s , y, z, k) satisfies Assumption 3.1 in [START_REF] Quenez | Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps[END_REF], which ensures the monotonocity property of ρ with respect to ζ. More precisely, for each maturity S and for each positions

ζ 1 , ζ 2 ∈ L 2 (F S ), with ζ 1 ≤ ζ 2 a.s., we have ρ t,x s (ζ 1 , S) ≥ ρ t,x s (ζ 2 , S) a.s.
We now formulate our optimal stopping problem for dynamic risk measures. For each (t, x) ∈ [0, T ] × R, we consider a dynamic financial position given by the process (ξ t,x s , t ≤ s ≤ T ), defined via the state process (X t,x s ) and two functions g and h such that

• g ∈ C(R) with at most polynomial growth at infinity,

• h : [0, T ] × R → R is continuous in t,
|h(t, x)| ≤ C(1 + |x| p ), ∀t ∈ [0, T ], x ∈ R, (4.2.4)
• h(T, x) ≤ g(x), ∀x ∈ R.

For each initial condition (t, x) ∈ [0, T ] × R, the dynamic position is then defined by: ξ t,x s := h(s, X t,x s ), s < T ξ t,x T := g(X t,x T ).

Let t ∈ [0, T ] be the initial time and let x ∈ R be the initial condition. The minimal risk measure at time t is given by: ess inf The continuity of u implies that Y t,x s = u(s, X t,x s ), t ≤ s ≤ T a.s. Moreover, the stopping time τ * ,t,x (also denoted by τ * ), defined by

τ ∈Tt ρ t,x t (ξ t,x τ , τ ) = -ess sup τ ∈Tt E t,
                               Y t,x s = g(X t,x T ) + T s f (r, X t,x r , Y t,x r , Z t,x r , K t,x r (•))dr + A t,x T -A t,x s - T s Z t,x r dW r - T s R * K t,x (r, e) Ñ (dr, de) Y t,x s ≥ ξ t,x s , 0 ≤ s ≤ T a.s. A t,
τ * := inf{s ≥ t, Y t,x s = ξ t,x s } = inf{s ≥ t, u(s, X t,x s ) = h(s, X t,x s )}
is an optimal stopping time for (4.2.5) (see Th. 3.6 in [START_REF] Quenez | Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps[END_REF]). Here, the function h is defined by h(t, x) := h(t, x)1 t<T + g(x)1 t=T , so that ξ t,x s = h(s, X t,x s ), 0 ≤ t ≤ T a.s. In the next section, we prove that the value function is a viscosity solution of an obstacle problem.

The value function, viscosity solution of an obstacle problem

We consider the following related obstacle problem for a parabolic PIDE:

         min(u(t, x) -h(t, x), - ∂u ∂t (t, x) -Lu(t, x) -f (t, x, u(t, x), (σ ∂u ∂x )(t, x), Bu(t, x)) = 0, (t, x) ∈ [0, T [×R u(T, x) = g(x), x ∈ R (4.3.1
) where

L := A + K, Aφ(t, x) := 1 2 σ 2 (x) ∂ 2 φ ∂x 2 (t, x) + b(x) ∂φ ∂x (t, x), Kφ(t, x) := R * φ(t, x + β(x, e)) -φ(t, x) - ∂φ ∂x (t, x)β(x, e) ν(de), (4.3.2)
Bφ(t, x)(•) := φ(t, x + β(x, •)) -φ(t, x) ∈ L 2 ν .
The operator B and K are well defined for φ ∈ C We prove below that the value function u defined by (4.2.8) is a viscosity solution of the above obstacle problem. Definition 4.3.1. • A continuous function u is said to be a viscosity subsolution of (4.3.1) iff u(T, x) ≤ g(x), x ∈ R, and iff for any point (t 0 , x 0 ) ∈ (0, T ) × R and for any φ ∈ C 1,2 ([0, T ] × R) such that φ(t 0 , x 0 ) = u(t 0 , x 0 ) and φu attains its minimum at (t 0 , x 0 ), we have min(u(t 0 , x 0 )h(t 0 , x 0 ),

- ∂φ ∂t (t 0 , x 0 ) -Lφ(t 0 , x 0 ) -f (t 0 , x 0 , u(t 0 , x 0 ), (σ ∂φ ∂x )(t 0 , x 0 ), Bφ(t 0 , x 0 )) ≤ 0.
In other words, if u(t 0 , x 0 ) > h(t 0 , x 0 ), then

- ∂φ ∂t (t 0 , x 0 ) -Lφ(t 0 , x 0 ) -f (t 0 , x 0 , u(t 0 , x 0 ), (σ ∂φ ∂x )(t 0 , x 0 ), Bφ(t 0 , x 0 )) ≤ 0.
• A continuous function u is said to be a viscosity supersolution of (4.3.1) iff u(T, x) ≥ g(x), x ∈ R, and iff for any point (t 0 , x 0 ) ∈ (0, T ) × R and for any φ ∈ C 1,2 ([0, T ] × R) such that φ(t 0 , x 0 ) = u(t 0 , x 0 ) and φu attains its maximum at (t 0 , x 0 ), we have min(u(t 0 , x 0 )h(t 0 , x 0 ),

- ∂φ ∂t (t 0 , x 0 ) -Lφ(t 0 , x 0 ) -f (t 0 , x 0 , u(t 0 , x 0 ), (σ ∂φ ∂x )(t 0 , x 0 ), Bφ(t 0 , x 0 )) ≥ 0.
In other words, we have both u(t 0 , x 0 ) ≥ h(t 0 , x 0 ), and Proof. • We first prove that u is a subsolution of (4.3.1).

- ∂φ ∂t (t 0 , x 0 ) -Lφ(t 0 , x 0 ) -f (t 0 , x 0 , u(t 0 , x 0 ), (σ ∂φ ∂x )(t 0 , x 0 ), Bφ(t 0 , x 0 )) ≥ 0.
Let (t 0 , x 0 ) ∈ (0, T ) × R and φ ∈ C 1,2 ([0, T ] × R) be such that φ(t 0 , x 0 ) = u(t 0 , x 0 ) and φ(t, x) ≥ u(t, x), ∀(t, x) ∈ [0, T ] × R. Suppose by contradiction that u(t 0 , x 0 ) > h(t 0 , x 0 ) and that - ∂φ ∂t (t 0 , x 0 ) -Lφ(t 0 , x 0 ) -f (t 0 , x 0 , φ(t 0 , x 0 ), (σ ∂φ ∂x )(t 0 , x 0 ), Bφ(t 0 , x 0 )) > 0.
By continuity of Kφ (which can be shown using Lebesgue's theorem) and that of Bφ : [0, T ] × R → L 2 ν , we can suppose that there exists ε > 0 and η ε > 0 such that:

∀(t, x) such that t 0 ≤ t ≤ t 0 + η ε < T and |x -x 0 | ≤ η ε , we have: u(t, x) ≥ h(t, x) + ε and - ∂φ ∂t (t, x) -Lφ(t, x) -f (t, x, φ(t, x), (σ ∂φ ∂x )(t, x), Bφ(t, x)) ≥ ε. (4.3.3) Note that Y t 0 ,x 0 s = Y s,X t 0 ,x 0 s s = u(s, X t 0 ,x 0 s ) a.s. because X t 0 ,
x 0 is a Markov process and u is continuous. We define the stopping time θ as:

θ := (t 0 + η ε ) ∧ inf{s ≥ t 0 , |X t 0 ,x 0 s -x 0 | > η ε }. (4.3.4)
By definition of the stopping time θ,

u(s, X t 0 ,x 0 s ) ≥ h(s, X t 0 ,x 0 s ) + ε > h(s, X t 0 ,x 0 s ), t 0 ≤ s < θ a.s.
This means that for a.e. ω the process (Y t 0 ,x 0 s (ω), s ∈ [t 0 , θ(ω)[) stays strictly above the barrier. It follows that for a.e. ω, the function

s → A c s (ω) is constant on [t 0 , θ(ω)]. In other words, Y t 0 ,x 0 s = E t 0 ,x 0 s,θ (Y θ ), t 0 ≤ s ≤ θ a.s, that is (Y t 0 ,x 0 s , s ∈ [t 0 , θ]
) is the solution of the classical BSDE associated with driver f , terminal time θ and terminal value Y t 0 ,x 0 θ . Applying Itô's lemma to φ(t, X t 0 ,x 0 t ), we get:

φ(t, X t 0 ,x 0 t ) = φ(θ, X t 0 ,x 0 θ ) - θ t ψ(s, X t 0 ,x 0 s )ds - θ t (σ ∂φ ∂x )(s, X t 0 ,x 0 s )dW s - θ t R * Bφ(s, X t 0 ,x 0 s -) Ñ (ds, de) (4.3.5)
where ψ(s, x) := ∂φ ∂s (s, x) + Lφ(s, x).

Note that (φ(s, X t 0 ,x 0 s ), (σ ∂φ ∂x )(s, X t 0 ,x 0 s ), Bφ(s, X t 0 ,x 0 s -); s ∈ [t 0 , θ]) is the solution of the BSDE associated to terminal time θ, terminal value φ(θ, X t 0 ,x 0 θ ) and driver process -ψ(s, X t 0 ,x 0 s ).

By (4.3.3) and the definition of the stopping time θ, we have a.s. that for each s ∈ [t 0 , θ]:

- ∂φ ∂t (s, X t 0 ,x 0 s ) -Lφ(s, X t 0 ,x 0 s ) -f s, X t 0 ,x 0 s , φ(s, X t 0 ,x 0 s ), (σ ∂φ ∂x )(s, X t 0 ,x 0 s ), Bφ(s, X t 0 ,x 0 s ) ≥ ε. (4.3.6)
Using the definition of the function ψ, (4.3.6) can be rewritten: for all s ∈ [t 0 , θ],

-ψ(s, X t 0 ,x 0 s ) -f s, X t 0 ,x 0 s , φ(s, X t 0 ,x 0 s ), (σ ∂φ ∂x )(s, X t 0 ,x 0 s ), Bφ(s, X t 0 ,x 0 s ) ≥ ε.
This gives a relation between the drivers -ψ(s, X t 0 ,x 0 s

) and f (s,

X t 0 ,x 0 s , •) of the two BSDEs. Also, φ(θ, X t 0 ,x 0 θ ) ≥ u(θ, X t 0 ,x 0 θ ) = Y t 0 ,x 0 θ a.s.
Consequently, the extended comparison result for BSDEs with jumps given in the Appendix (see Proposition 4.6.2) implies that:

φ(t 0 , x 0 ) = φ(t 0 , X t 0 ,x 0 t 0 ) > Y t 0 ,x 0 t 0 = u(t 0 , x 0 ),
which leads to a contradiction.

• We now prove that u is a viscosity supersolution of (4.3.1).

Let (t 0 , x 0 ) ∈ (0, T ) × R and φ ∈ C 1,2 ([0, T ] × R) be such that φ(t 0 , x 0 ) = u(t 0 , x 0 ) and φ(t, x) ≤ u(t, x), ∀(t, x) ∈ [0, T ] × R. Since the solution (Y t 0 ,x 0 s
) stays above the obstacle, we have: u(t 0 , x 0 ) ≥ h(t 0 , x 0 ).

We must prove that:

- ∂φ ∂t (t 0 , x 0 ) -Lφ(t 0 , x 0 ) -f t 0 , x 0 , φ(t 0 , x 0 ), (σ ∂φ ∂x )(t 0 , x 0 ), Bφ(t 0 , x 0 ) ≥ 0.
Suppose by contradiction that:

- ∂φ ∂t (t 0 , x 0 ) -Lφ(t 0 , x 0 ) -f t 0 , x 0 , φ(t 0 , x 0 ), (σ ∂φ ∂x )(t 0 , x 0 ), Bφ(t 0 , x 0 ) < 0.
By continuity, we can suppose that there exists ε > 0 and η ε > 0 such that for each (t, x) such that t 0 ≤ t ≤ t 0 + η ε < T and |xx 0 | ≤ η ε , we have:

- ∂φ ∂t (t, x) -Lφ(t, x) -f t, x, φ(t, x), (σ ∂φ ∂x )(t, x), Bφ(t, x) ≤ -ε. (4.3.7)
We define the stopping time θ as:

θ := (t 0 + η ε ) ∧ inf{s ≥ t 0 /|X t 0 ,x 0 s -x 0 | > η ε }.
Applying as above Itô's lemma to φ(s, X t 0 ,x 0 s ), we get that (φ(s, X t 0 ,x 0 s ), (σ ∂φ ∂x )(s, X t 0 ,x 0 s ), Bφ(s, X t 0 ,x 0 s -); s ∈ [t 0 , θ]) is the solution of the BSDE associated with terminal value φ(θ, X t 0 ,x 0 θ ) and driver -ψ(s, X t 0 ,x 0 s ).

The process (Y t 0 ,x 0 , s ∈ [t 0 , θ]) is the solution of the classical BSDE associated with terminal condition Y t 0 ,x 0 θ = u(θ, X t 0 ,x 0 θ ) and generalized driver f (s, X t 0 ,x 0 s , y, z, q)ds + dA t 0 ,x 0 s . By (5.6.3) and the definition of the stopping time θ, we have :

(- ∂φ ∂t (s, X t 0 ,x 0 s ) -Lφ(s, X t 0 ,x 0 s ) -f (s, X t 0 ,x 0 s , φ(s, X t 0 ,x 0 s ), (σ ∂φ ∂x )(s, X t 0 ,x 0 s ), Bφ(s, X t 0 ,x 0 s )))ds -dA t 0 ,x 0 s ≤ -ε ds, t 0 ≤ s ≤ θ a.s.
or, equivalently, ψ(s, X t 0 ,x 0 s )ds

≤ (f (s, X t 0 ,x 0 s , φ(s, X t 0 ,x 0 s ), (σ ∂φ ∂x )(s, X t 0 ,x 0 s ), Bφ(s, X t 0 ,x 0 s )))ds + dA t 0 ,x 0 s -ε ds, t 0 ≤ s ≤ θ a.s.
This gives a relation between the drivers of the two BSDEs. Also, φ(θ, X t 0 ,x 0 θ ) ≤ u(θ, X t 0 ,x 0 θ ) = Y t 0 ,x 0 θ a.s. Consequently, Proposition 6.4.1 in the Appendix implies that:

φ(t 0 , x 0 ) = φ(t 0 , X t 0 ,x 0 t 0 ) < Y t 0 ,x 0 t 0 = u(t 0 , x 0 ),
which leads to a contradiction.

Uniqueness result for the obstacle problem

We provide a uniqueness result for (4.3.1) in the particular case when for each φ

∈ C 1,2 ([0, T ] × R), Bφ is a map valued in R instead of L 2 ν . More precisely, Bφ(t, x) := R * (φ(t, x + β(x, e)) -φ(t, x))γ(x, e)ν(de), (4.4.1)
which is well defined since |φ(t,

x + β(x, e)) -φ(t, x)| ≤ C|β(x, e)|.
We suppose that Assumption H 1 holds and we make the additional assumptions: Assumption H 2 :

1. f (s, X t,x s (ω), y, z, k) := f s, X t,x s (ω), y, z, R * k(e)γ(X t,x s (ω), e)ν(de) 1 s≥t ,
where f : [0, T ] × R 4 → R is continuous in t uniformly with respect to x, y, z, k, continuous in x uniformly with respect to y, z, k, and satisfies:

(i) |f (t, x, 0, 0, 0)| ≤ C, for all t ∈ [0, T ], x ∈ R. (ii) |f (t, x, y, z, k) -f (t, x ′ , y ′ , z ′ , k ′ )| ≤ C(|y -y ′ | + |z -z ′ | + |k -k ′ |), for all t ∈ [0, T ], y, y ′ , z, z ′ , k, k ′ ∈ R. (iii) k → f (t, x, y, z, k) is non-decreasing, for all t ∈ [0, T ],
x, y, z ∈ R.

2.

For each R > 0, there exists a continuous function m R : R

+ → R + such that m R (0) = 0 and |f (t, x, v, p, q) -f (t, y, v, p, q)| ≤ m R (|x -y|(1 + |p|)), for all t ∈ [0, T ], |x|, |y| ≤ R, |v| ≤ R, p, q ∈ R. 3. |γ(x, e) -γ(y, e)| ≤ C|x -y|(1 ∧ e 2
) and 0 ≤ γ(x, e) ≤ C(1 ∧ |e|), for all x, y ∈ R, e ∈ R * . 4. There exists r > 0 such that for all t ∈ [0, T ], x, u, v, p, l ∈ R:

f (t, x, v, p, l) -f (t, x, u, p, l) ≥ r(u -v) when u ≥ v. 5. |h(t, x)| + |g(x)| ≤ C, for all t ∈ [0, T ], x ∈ R.
To simplify notation, f is denoted by f in the sequel.

We state below a comparison theorem, which uses results of three lemmas. The proofs of these lemmas are given in Subsection 4.4.1.

Theorem 4.4.1 (Comparison principle). Under the above hypotheses, if U is a viscosity subsolution and V is a viscosity supersolution of the obstacle problem (4.3.1) in the class of continuous bounded functions, then

U (t, x) ≤ V (t, x), for each (t, x) ∈ [0, T ] × R. Proof. Set M := sup [0,T ]×R (U -V ).
It is sufficient to prove that M ≤ 0. For each ε, η > 0, we introduce the function:

ψ ε,η (t, s, x, y) := U (t, x) -V (s, y) - (x -y) 2 ε 2 - (t -s) 2 ε 2 -η 2 (x 2 + y 2 ), for t, s, x, y in [0, T ] 2 × R 2 . Let M ε,η := max [0,T ] 2 ×R 2 ψ ε,η
. This supremum is reached at some point (t ε,η , s ε,η , x ε,η , y ε,η ).

Using that ψ ε,η (t ε,η , s ε,η , x ε,η , y ε,η ) ≥ ψ ε,η (0, 0, 0, 0), we obtain:

U (t ε,η , x ε,η ) -V (s ε,η , y ε,η ) - (t ε,η -s ε,η ) 2 ε 2 - (x ε,η -y ε,η ) 2 ε 2 -η 2 ((x ε,η ) 2 + (y ε,η ) 2 ) ≥ U (0, 0) -V (0, 0), (4.4.2)
or, equivalently,

(t ε,η -s ε,η ) 2 ε 2 + (x ε,η -y ε,η ) 2 ε 2 + η 2 ((x ε,η ) 2 + (y ε,η ) 2 ) ≤ U ∞ + V ∞ -U (0, 0) -V (0, 0). (4.4.3)
Consequently, we can find a constant C such that:

|x ε,η -y ε,η | + |t ε,η -s ε,η | ≤ Cε (4.4.4) |x ε,η | ≤ C η , |y ε,η | ≤ C η . (4.4.5)
Extracting a subsequence if necessary, we may suppose that for each η the sequences (t ε,η ) ε and (s ε,η ) ε converge to a common limit t η when ε tends to 0, and from (4.4.4) and (4.4.5) we may also suppose, extracting again, that for each η, the sequences (x ε,η ) ε and (y ε,η ) ε converge to a common limit x η .

Lemma 4.4.2. We have:

lim ε→0 (x ε,η -y ε,η ) 2 ε 2 = 0 ; lim ε→0 (t ε,η -s ε,η ) 2 ε 2 = 0 lim η→0 lim ε→0 M ε,η = M.
We now introduce the functions:

Ψ 1 (t, x) := V (s ε,η , y ε,η ) + (x -y ε,η ) 2 ε 2 + (t -s ε,η ) 2 ε 2 + η 2 (x 2 + (y ε,η ) 2 ) Ψ 2 (s, y) := U (t ε,η , x ε,η ) - (x ε,η -y) 2 ε 2 - (t ε,η -s) 2 ε 2 -η 2 ((x ε,η ) 2 + y 2 ).
As (t, x) → (U -Ψ 1 )(t, x) reaches its maximum at (t ε,η , x ε,η ) and U is a subsolution we have two cases:

• t ε,η = T and then U (t ε,η , x ε,η ) ≤ g(x ε,η ),

• t ε,η = T and then min U (t ε,η , x ε,η ) -h(t ε,η , x ε,η ), ∂Ψ 1 ∂t (t ε,η , x ε,η ) -LΨ 1 (t ε,η , x ε,η )- -f t ε,η , x ε,η , U (t ε,η , x ε,η ), (σ ∂Ψ 1 ∂x )(t ε,η , x ε,η ), BΨ 1 (t ε,η , x ε,η ) ≤ 0. (4.4.6)
As (s, y) → (Ψ 2 -V )(s, y) reaches its maximum at (s ε,η , y ε,η ) and V is a supersolution we have the two following cases:

• s ε,η = T and then V (s ε,η , y ε,η ) ≥ g(y ε,η ),

• s ε,η = T and then

min(V (s ε,η , y ε,η ) -h(s ε,η , y ε,η ), ∂Ψ 2 ∂t (s ε,η , y ε,η ) -LΨ 2 (s ε,η , y ε,η ) -f (s ε,η , y ε,η , V (s ε,η , y ε,η ), (4.4.7) (σ ∂Ψ 2 ∂x )(s ε,η , y ε,η ), BΨ 2 (s ε,η , y ε,η )) ≥ 0.
We now prove that M ≤ 0. Three cases are possible. 1st case: There exists a subsequence of (t η ) such that t η = T for all η (of this subsequence). As U is continuous, for all η and for ε small enough

U (t ε,η , x ε,η ) ≤ U (t η , x η ) + η ≤ g(x η ) + η,
and as V is continuous, for all η and for ε small enough

V (s ε,η , y ε,η ) ≥ V (t η , x η ) -η ≥ g(x η ) -η. Hence U (t ε,η , x ε,η ) -V (s ε,η , y ε,η ) ≤ 2η and M ε,η = U (t ε,η , x ε,η ) -V (s ε,η , y ε,η ) - (x ε,η -y ε,η ) 2 ε 2 - (t ε,η -s ε,η ) 2 ε 2 -η 2 ((x ε,η ) 2 + (y ε,η ) 2 ) ≤ U (t ε,η , x ε,η ) -V (s ε,η , y ε,η ) ≤ 2η.
Letting ε → 0 and then η → 0 one gets, using Lemma 4.4.2, that M ≤ 0.

2nd case: There exists a subsequence such that t η = T , and for all η belonging to this subsequence, there exists a subsequence of (x ε,η ) η such that

U (t ε,η , x ε,η ) -h(t ε,η , x ε,η ) ≤ 0.
As from (4.4.7) one has

V (s ε,η , y ε,η ) -h(s ε,η , y ε,η ) ≥ 0, it comes that M ε,η ≤ U (t ε,η , x ε,η ) -V (s ε,η , y ε,η ) ≤ h(t ε,η , x ε,η ) -h(s ε,η , y ε,η ).
Letting ε → 0 and then η → 0, using the equality lim η→0 lim ε→0 M ε,η = M (see Lemma 4.4.2), we derive that M ≤ 0.

Last case: We are left with the case when, for a subsequence of η, we have t η = T and for all η belonging to this subsequence there exists a subsequence of (x ε,η ) ε such that:

U (t ε,η , x ε,η ) -h(t ε,η , x ε,η ) > 0. Set ϕ(t, s, x, y) := (x -y) 2 ε 2 + (t -s) 2 ε 2 + η 2 (x 2 + y 2 ). (4.4.8)
The maximum of the function ψ ε,η (t, s, x, y) := U (t, x) -V (s, y)ϕ(t, s, x, y) is reached at the point (t ε,η , s ε,η , x ε,η , y ε,η ). Let us fix δ > 0 and consider the ball B δ = B(0, δ). We introduce the operators K δ , Kδ , B δ , Bδ corresponding to the operators K and B defined in (4.3.2) and (4.4.1), but integrating on B δ or R\B δ (also denoted by B c δ ) only. They are defined respectively for all φ ∈ C 1,2 , Φ ∈ C by Here C denotes the set of bounded continuous functions.

K δ [t, x, φ] := B δ φ(t, x + β(x, e)) -φ(t, x) - ∂φ ∂x (t, x)β(x,
We apply the non-local version of Jensen Ishii's lemma [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions theory revisited[END_REF] ( see also Corollary 1 in [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions theory revisited[END_REF]) and we obtain that there exists ᾱ, such that for any α such that 0 < α ≤ ᾱ there exist: (a, p, X) ∈ P 2,+ U (t ε,η , x ε,η ), (b, q, Y ) ∈ P 2,-V (s ε,η , y ε,η ) such that

           F (t ε,η , x ε,η , U (t ε,η , x ε,η ), a, p, X, K δ [t ε,η , x ε,η , ϕ α (•, s ε,η , y ε,η )] + Kδ [t ε,η , x ε,η , p, U ], B δ [t ε,η , x ε,η , ϕ α (•, s ε,η , y ε,η )] + Bδ [t ε,η , x ε,η , U ]) ≤ 0 F (s ε,η , y ε,η , V (s ε,η , y ε,η ), a, q, Y, K δ [s ε,η , y ε,η , -ϕ α (t ε,η , x ε,η , •)] + Kδ [s ε,η , y ε,η , q, V ], B δ [s ε,η , y ε,η , -ϕ α (t ε,η , x ε,η , •)] + Bδ [s ε,η , y ε,η , V ]) ≥ 0 (4.4.13)
where

F (t, x, u, a, p, X, l 1 , l 2 ) := -a - 1 2 σ 2 (x)X -b(x)p -l 1 -f (t, x, u, pσ(x), l 2 ). (4.4.14)
and such that

           p = p + 2η 2 x ε,η ; q = p -2η 2 y ε,η ; p = 2(x ε,η -y ε,η ) ε 2 a = b = 2(t ε,η -s ε,η ) ε 2 X 0 0 -Y ≤ 2 ε 2 1 -1 -1 1 + (2η 2 + O(α)) 1 0 0 1 .
Here, P 2,+ (resp. P 2,-) is the set of superjets (resp. subjets) defined in [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions theory revisited[END_REF] (see Definition 3). Note that the operators K δ , Kδ , B δ and Bδ satisfy the hypotheses (NLT) of [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions theory revisited[END_REF] (see Section 2.2 in [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions theory revisited[END_REF]). Hence we can use the alternative definition for sub-superviscosity solutions expressed in terms of sub-supersolutions and super-subjets given by Definition 4 in [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions theory revisited[END_REF]. By Lemma 1 in [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions theory revisited[END_REF], we have

ϕ α := R α [ϕ], with R α [ϕ][(x, p)] := sup Z∈B(x,κ) ϕ(Z) -r(Z -x) - |Z -x| 2 2α , (x, p) := ((t ε,η , s ε,η , x ε,η , y ε,η ), (a, b, p, q))
and κ is assumed to be sufficiently small. Proposition 3 in [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions theory revisited[END_REF] together with the Lipschitz continuity of F with respect to l 1 , l 2 lead to:

           F (t ε,η , x ε,η , U (t ε,η , x ε,η ), a, p, X, K δ [t ε,η , x ε,η , ϕ x ] + Kδ [t ε,η , x ε,η , p, U ], B δ [t ε,η , x ε,η , ϕ x ] + Bδ [t ε,η , x ε,η , U ]) ≤ O(α) F (s ε,η , y ε,η , V (s ε,η , y ε,η ), a, q, Y, K δ [s ε,η , y ε,η , -ϕ y ] + Kδ [s ε,η , y ε,η , q, V ], B δ [s ε,η , y ε,η , -ϕ y ] + Bδ [s ε,η , y ε,η , V ]) ≥ O(α), (4.4.15)
where we denote by ϕ x the function (t, x) → ϕ(t, x, s ε,η , y ε,η ) and by ϕ y the function (s, y) → ϕ(t ε,η , x ε,η , s, y).

Since (t ε,η , s ε,η , x ε,η , y ε,η ) is a global maximum of ψ ε,η ,we have:

ψ ε,η (t ε,η , s ε,η , x ε,η + β(x ε,η , e), y ε,η + β(y ε,η , e)) ≤ ψ ε,η (t ε,η , s ε,η , x ε,η , y ε,η ) ⇔ U (t ε,η , x ε,η + β(x ε,η , e)) -V (s ε,η , y ε,η + β(y ε,η , e)) - (x ε,η + β(x ε,η , e) -y ε,η -β(y ε,η , e)) 2 ε 2 - (t ε,η -s ε,η ) 2 ε 2 -η 2 ((x ε,η + β(x ε,η , e)) 2 + (y ε,η + β(y ε,η , e)) 2 ) ≤ U (t ε,η , x ε,η ) -V (s ε,η , y ε,η ) - (x ε,η -y ε,η ) 2 ε 2 - (t ε,η -s ε,η ) 2 ε 2 -η 2 ((x ε,η ) 2 + (y ε,η ) 2 ).
Consequently, we get:

U (t ε,η , x ε,η + β(x ε,η , e)) -U (t ε,η , x ε,η ) ≤ V (s ε,η , y ε,η + β(y ε,η , e)) -V (s ε,η , y ε,η ) + (β(x ε,η , e) -β(y ε,η , e)) 2 ε 2 + p(β(x ε,η , e) -β(y ε,η , e)) + η 2 (β 2 (x ε,η , e) + 2x ε,η β(x ε,η , e) + 2y ε,η β(y ε,η , e) + β 2 (y ε,η , e)). (4.4.16)
The two following lemmas hold.

Lemma 4.4.3. Let

l K := K δ [t ε,η , x ε,η , ϕ x ] + Kδ [t ε,η , x ε,η , p, U ] l ′ K := K δ [s ε,η , y ε,η , -ϕ y ] + Kδ [s ε,η , y ε,η , q, V ]. ( 4
.4.17)

We have

l K ≤ l ′ K + O( (x ε,η -y ε,η ) 2 ε 2 ) + O(η 2 ) + ( 1 ε 2 + η 2 )O(δ). ( 4
l B := B δ [t ε,η , x ε,η , ϕ x ] + Bδ [t ε,η , x ε,η , U ] l ′ B := B δ [s ε,η , y ε,η , -ϕ y ] + Bδ [s ε,η , y ε,η , V ]. (4.4.19)
We have

l B ≤ l ′ B + (η 2 + 1 ε 2 )O(δ) + O( (x ε,η -y ε,η ) 2 ε 2 ) + O(|x ε,η -y ε,η |) + O(η 2 ). ( 4

.4.20)

We argue now by contradiction by assuming that

M > 0. (4.4.21)
Using Assumption (H 2 ).4, we get

0 < r 2 M ≤ rM ε,η ≤ r(U (t ε,η , x ε,η ) -V (s ε,η , y ε,η )) ≤ F (s ε,η , y ε,η , U (t ε,η , x ε,η ), a, q, Y, l ′ K , l ′ B ) -F (s ε,η , y ε,η , V (s ε,η , y ε,η ), a, q, Y, l ′ K , l ′ B ) = F (s ε,η , y ε,η , U (t ε,η , x ε,η ), a, q, Y, l ′ K , l ′ B ) -F (s ε,η , y ε,η , U (s ε,η , y ε,η ), a, q, Y, l ′ K , l ′ B ) + F (s ε,η , y ε,η , U (s ε,η , y ε,η ), a, q, Y, l ′ K , l ′ B ) -F (s ε,η , y ε,η , U (s ε,η , y ε,η ), a, q, Y, l K , l B ) + F (s ε,η , y ε,η , U (s ε,η , y ε,η ), a, q, Y, l K , l B ) -F (t ε,η , x ε,η , U (t ε,η , x ε,η ), a, p, X, l K , l B ) + F (t ε,η , x ε,η , U (t ε,η , x ε,η ), a, p, X, l K , l B ) -F (s ε,η , y ε,η , V (s ε,η , y ε,η ), a, q, Y, l ′ K , l ′ B ) ≤ K|U (t ε,η , x ε,η ) -U (s ε,η , y ε,η )| + F (s ε,η , y ε,η , U (s ε,η , y ε,η ), a, q, Y, l K , l B ) -F (t ε,η , x ε,η , U (t ε,η , X ε,η ), a, p, X, l K , l B ) + (η 2 + 1 ε 2 )O(δ) + O( (x ε,η -y ε,η ) 2 ε 2 ) + O(|x ε,η -y ε,η |) + O(η 2 ) + O(α). (4.4.22)
We have used here the (nonlocal) ellipticity of F , the Lipschitz property of F , (4.4.15) and the estimates proven in Lemma 4.4.3 and Lemma 4.4.4. From the hypothesis on b and σ, we have:

σ 2 (x ε,η )X -σ 2 (y ε,η )Y ≤ C(x ε,η -y ε,η ) 2 ε 2 + O(η 2 ) + O(α), b(x ε,η )p -b(y ε,η )q ≤ C|x ε,η -y ε,η | ε 2 + O(η 2 ).
We thus obtain the inequality:

F (s ε,η , y ε,η , U (s ε,η , y ε,η ), a, q, Y, l K , l B ) -F (t ε,η , x ε,η , U (t ε,η , x ε,η ), a, p, X, l K , l B ) ≤ C(x ε,η -y ε,η ) 2 ε 2 + O(η 2 ) + f (t ε,η , x ε,η , U (t ε,η , x ε,η ), (p + 2η 2 )σ(x ε,η ), l B ) -f (s ε,η , y ε,η , U (s ε,η , y ε,η ), (p -2η 2 )σ(y ε,η ), l B ) ≤ f (t ε,η , x ε,η , U (t ε,η , x ε,η ), (p + 2η 2 )σ(x ε,η ), l B ) -f (s ε,η , x ε,η , U (t ε,η , x ε,η ), (p + 2η 2 )σ(x ε,η ), l B ) + m R (|x ε,η -y ε,η |(1 + (p + 2η 2 )σ(x ε,η ))) + K|U (t ε,η , x ε,η ) -U (s ε,η , y ε,η )| + O( (x ε,η -y ε,η ) 2 ε 2 ) + O(η 2 ) + O(α). (4.4.23)
The last equality is obtained by some computations similar to those in (4.

< r 2 M ≤ rM ε,η ≤ f (t ε,η , x ε,η , U (t ε,η , x ε,η ), (p + 2η 2 )σ(x ε,η ), l B ) -f (s ε,η , x ε,η , U (t ε,η , x ε,η ), (p + 2η 2 )σ(x ε,η ), l B ) + m R (|x ε,η -y ε,η |(1 + (p + 2η 2 )σ(x ε,η )) + K|U (t ε,η , x ε,η ) -U (s ε,η , y ε,η )|+ + O( (x ε,η -y ε,η ) 2 ε 2 ) + O(|x ε,η -y ε,η |) + (η 2 + 1 ε 2 )O(δ) + O(η 2 ) + O(α). ( 4

Proofs of the lemmas

Proof of Lemma 4.4.2. For η > 0, we introduce the functions:

Ũ η (t, x) = U (t, x) -η 2 x 2 and Ṽ η (t, x) = V (t, x) + η 2 x 2 . Set M η := sup [0,T ]×R ( Ũ η -Ṽ η ).
The maximum M η is reached at some point ( tη , xη ). From the form of ψ ε,η , we have that for fixed η, there exists a subsequence (t ε,η , s ε,η , x εη , y ε,η ) ε which converges to some point (t η , s η , x η , y η ) when ε tends to 0.

Since M ε,η is reached at (t ε,η , s ε,η , x ε,η , y ε,η ), we have:

( Ũ η -Ṽ η )( tη , xη ) = (U -V )( tη , xη ) -η 2 ((x η ) 2 + (ŷ η ) 2 ) ≤ M ε,η = U (t ε,η , x ε,η ) -V (s ε,η , y ε,η ) - (t ε,η -s ε,η ) 2 ε 2 - (x ε,η -y ε,η ) 2 ε 2 -η 2 ((x ε,η ) 2 + (y ε,η ) 2 ).
Setting

l η := lim sup ε→0 (x ε,η -y ε,η ) 2 ε 2 , l η := lim inf ε→0 (x ε,η -y ε,η ) 2 ε 2 we get 0 ≤ l η ≤ l η ≤ ( Ũ η -Ṽ η )(t η , x η ) -( Ũ η -Ṽ η )( tη , xη ) ≤ 0. (4.4.25)
We derive that, up to a subsequence, lim ε→0

(x ε,η -y ε,η ) 2 ε 2
= 0 and lim ε→0 M ε,η = M η . Similarly, we get lim ε→0

(t ε,η -s ε,η ) 2 ε 2 = 0.
Let us prove that lim η→0 M η = M . First, note that M η ≤ M, for all η. By definition of M , for all δ > 0 there exists (t

δ , x δ ) ∈ [0, T ] × R such that M -δ ≤ (U -V )(t δ , x δ ). Consequently, we get M -2η 2 x 2 δ -δ ≤ (U -V )(t δ , x δ ) -2η 2 x 2 δ = ( Ũ η -Ṽ η )(t δ , x δ ) ≤ M η ≤ M.
By letting η and then δ tend to 0, the result follows.

Proof of Lemma 4.4.3. We have:

K δ [t ε,η , x ε,η , ϕ x ] = B δ ( 1 ε 2 + η 2 )β 2 (x ε,η , e)ν(de) (4.4.26) K δ [s ε,η , y ε,η , -ϕ y ] = B δ (- 1 ε 2 -η 2 )β 2 (y ε,
K δ [t ε,η , x ε,η , ϕ x ]≤K δ [s ε,η , y ε,η , -ϕ y ]+( 1 ε 2 +η 2 ) B δ β 2 (y ε,η , e)ν(de) + ( 1 ε 2 +η 2 ) B δ β 2 (x ε,η , e)ν(de)≤K δ [s ε,η , y ε,η , -ϕ y ]+( 1 ε 2 +η 2 )O(δ). (4.4.28)
Using inequality (4.4.16) and integrating on B c δ , we obtain:

Kδ [t ε,η , x ε,η , p, U ] = B c δ U (t ε,η , x ε,η + β(x ε,η , e)) -U (t ε,η , x ε,η ) -(p + 2η 2 x ε,η )β(x ε,η , e) ν(de) ≤ B c δ V (s ε,η , y ε,η + β(y ε,η , e)) -V (s ε,η , y ε,η ) -(p -2η 2 y ε,η )β(y ε,η , e) ν(de) + B c δ (β(x ε,η , e) -β(y ε,η , e)) 2 ε 2 ν(de) + η 2 B c δ (β 2 (x ε,η , e) + β 2 (y ε,η , e))ν(de) ≤ Kδ [s ε,η , y ε,η , q, V ] + O( (x ε,η -y ε,η ) 2 ε 2 ) + O(η 2 ).
Using (4.4.17) and (4.4.28), we derive (4.4.18), which ends the proof of Lemma 4.4.3.

Proof of Lemma 4.4.4. From (4.4.11), we derive that:

B δ [t ε,η , x ε,η , ϕ x ] = B δ (η 2 + 1 ε 2 )β 2 (x ε,η , e) + 2β(x ε,η , e) ε 2 (x ε,η -y ε,η ) + 2η 2 x ε,η β(x ε,η , e) γ(x ε,η , e)ν(de) (4.4.29) B δ [s ε,η , y ε,η , -ϕ y ] = B δ (-η 2 - 1 ε 2 )β 2 (y ε,η , e) + 2β(y ε,η , e) ε 2 (x ε,η -y ε,η )
-2η 2 y ε,η β(y ε,η , e) γ(y ε,η , e)ν(de). (4.4.30)

After some computations, we obtain:

(η 2 + 1 ε 2 )β 2 (x ε,η , e) + 2β(x ε,η , e) ε 2 (x ε,η -y ε,η ) + 2η 2 x ε,η β(x ε,η , e) γ(x ε,η , e) = (-η 2 - 1 ε 2 )β 2 (y ε,η , e)γ(y ε,η , e) + 2β(y ε,η , e) ε 2 (x ε,η -y ε,η )γ(y ε,η , e)
-2η 2 y ε,η β(y ε,η , e)γ(y ε,η , e)

+ (η 2 + 1 ε 2 ) β 2 (y ε,η , e)γ(y ε,η , e) + β 2 (x ε,η , e)γ(x ε,η , e) + 2 ε 2 (x ε,η -y ε,η ) β(x ε,η , e)γ(x ε,η , e) -β(y ε,η , e)γ(y ε,η , e)
+ 2η 2 x ε,η β(x ε,η , e)γ(x ε,η , e) + y ε,η β(y ε,η , e)γ(y ε,η , e) .

B δ [t ε,η , x ε,η , ϕ x ] ≤ B δ [s ε,η , y ε,η , -ϕ y ] + (η 2 + 1 ε 2 )O(δ) + O( (x ε,η -y ε,η ) 2 ε 2 ) + O(η 2 ). ( 4

.4.32)

We now estimate the operator Bδ . Inequality (4.4.16) implies:

U (t ε,η , x ε,η + β(x ε,η , e)) -U (t ε,η , x ε,η ) γ(x ε,η , e) ≤ V (s ε,η , y ε,η + β(y ε,η , e)) -V (s ε,η , y ε,η ) + |β(x ε,η , e) -β(y ε,η , e)| 2 ε 2 + p(β(x ε,η , e) -β(y ε,η , e))
+ η 2 (β 2 (x ε,η , e) + 2x ε,η β(x ε,η , e) + 2y ε,η β(y ε,η , e) + β 2 (y ε,η , e) γ(x ε,η , e) = V (s ε,η , y ε,η + β(y ε,η , e)) -V (s ε,η , y ε,η ) γ(y ε,η , e)

+ V (s ε,η , y ε,η + β(y ε,η , e)) -V (s ε,η , y ε,η ) γ(x ε,η , e) -γ(y ε,η , e) + |β(x ε,η , e) -β(y ε,η , e)| 2 ε 2 γ(x ε,η , e) + p β(x ε,η , e) -β(y ε,η , e) γ(x ε,η , e)
+ η 2 β 2 (x ε,η , e) + 2x ε,η β(x ε,η , e) + 2y ε,η β(y ε,η , e) + β 2 (y ε,η , e) γ(x ε,η , e).

Bδ [t ε,η , x ε,η , U ] ≤ Bδ [s ε,η , y ε,η , V ] + O(|x ε,η -y ε,η |) + O( (x ε,η -y ε,η ) 2 ε 2 ) + O(η 2 ). ( 4

Conclusions

In this chapter, we have studied the optimal stopping problem for a monotonous dynamic risk measure defined by a Markovian BSDE with jumps. We have proven that, under relatively weak hypotheses, the value function is a viscosity solution of an obstacle problem for a partial integrodifferential variational inequality. To obtain the uniqueness of the solution under appropriate conditions, we have proven a comparison theorem, based on the nonlocal version of the Jensen Ishii Lemma, which extends some results established in [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions theory revisited[END_REF] (Section 5.1, Th.3) to the case of a nonlinear BSDE.

The links given in this paper between optimal stopping problems for BSDEs and obstacle problems for PDEs can be extended to a larger class of problems. Among them, we can mention generalized Dynkin games with nonlinear expectation (see [START_REF] Dumitrescu | Generalized Dynkin games and Doubly reflected BSDEs with Jumps[END_REF]), and mixed optimal stopping/stochastic control problems (see [START_REF] Dumitrescu | A Weak Dynammic Programming for Combined Stochastic Control/Optimal Stopping with E f -Expectations[END_REF]). However, the latter case requires to establish a weak dynamic programming principle, which does not follow from the flow property of reflected BSDEs only, and needs rather sophisticated techniques.

Appendix

Some useful estimates

Let T > 0 be a fixed terminal time.

A map f :

[0, T ] × Ω × R 2 × L 2 ν → R; (t, ω, y, z, k) → f (t, ω, y, z, k
) is said to be a Lipschitz driver if it is predictable, uniformly Lipchitz with respect to y, z, k and such that f (t, 0, 0, 0) ∈ H 2 .

Let ξ 1 t , ξ 2 t ∈ S 2 . Let f 1 , f 2 be two admissible Lipschitz drivers with Lipchitz constant C. For i = 1, 2, let E i be the f i -conditional expectation associated with driver f i , and let (Y i t ) be the adapted process defined for each t ∈ [0, T ],

Y i t := ess sup τ ∈Tt E i t,τ (ξ i τ ). (4.6.1) Proposition 4.6.1. For s ∈ [0, T ], denote Y s = Y 1 s -Y 2 s , ξ s = ξ 1 s -ξ 2 s and f s = sup y,z,k |f 1 (s, y, z, k) -f 2 (s, y, z, k)|. Let η, β > 0 be such that β ≥ 3 η + 2C and η ≤ 1 C 2 .
Then for each t, we have:

e βt Y 2 t ≤ e βT (E[sup s≥t ξ s 2 |F t ] + ηE[ T t f 2 s ds|F t ]) a.s. (4.6.2)
Proof. For i = 1, 2 and for each τ ∈ T 0 , let (X i,τ , π i,τ s , l i,τ s ) be the solution of the BSDE associated with driver f i , terminal time τ and terminal condition ξ i τ . Set X τ s = X 1,τ s -X 2,τ s . By a priori estimate on BSDEs (see Proposition A.4 in [START_REF] Quenez | Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps[END_REF]), we have: Proof. Let X τ t be the solution of the BSDE associated with driver f , terminal time τ and terminal condition ξ τ . By applying inequality (4.6.3) with f 1 = f , ξ 1 = ξ, f 2 = 0 and ξ 2 = 0, we get:

e βt (X τ t ) 2 ≤ e βT E[ξ 2 τ |F t ] + ηE[ T t e βs (f 1 (s, X 2,τ s , π 2,τ s , l 2,τ s ) -f 2 (s, X 2,τ s , π 2,τ s , l
e βt (X τ t ) 2 ≤ e βT E[ξ 2 τ |F t ] + ηE[ T t e βs (f (s, 0, 0, 0)) 2 |F t ]. (4.6.7)
The result follows.

Remark 4.6.3. If the drivers satisfy Assumption 3.1 in [START_REF] Quenez | Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps[END_REF], then Y (resp. Y i ) is the solution of the RBSDE associated with driver f (resp.f i ) and obstacle ξ (resp. ξ i ). Hence the above estimates provide some new estimates on RBSDEs. Note that η and β are universal constants, i.e. they do not depend on T , ξ, ξ 1 , ξ 2 , f, f 1 , f 2 . This was not the case for the estimates given in the previous literature (see e.g. [START_REF] El Karoui | Reflected solutions of backward SDE's, and related obstacle problems for PDE's[END_REF]).

Some properties of the value function u

We prove below the continuity and polynomial growth of the function u defined by (4.2.8).

Lemma 4.6.4. The function u is continuous in (t, x).

Proof. It is sufficient to show that, when (t n , x n ) → (t, x), |u(t n , x n )u(t, x)| → 0. Let h be the map defined by h(t, x) = h(t, x) for t < T and h(T, x) = g(x), so that, for each (t, x), we have ξ t,x s = h(s, X t,x s ), 0 ≤ s ≤ T a.s. By applying Proposition 4.6.1 with X 1 s = X tn,xn s , X 2 s = X t,x s , f 1 (s, ω, y, z, q) := 1 [t,T ] (s)f (s, X t,x s (ω), y, z, q) and f 2 (s, ω, y, z, q) := 1 [tn,T ] (s)f (s, X tn,xn s (ω), y, z, q), we obtain:

|u(t n , x n ) -u(t, x)| 2 ≤ K C,T E[ sup 0≤s≤T |h(s, X tn,xn s ) -h(s, X t,x s )| 2 + T 0 (f n s ) 2 ],
where

   K C,T := e (3C 2 +2C)T max(1, 1 C 2 ) f n s (ω) := sup y,z,q |1 [t,T ] f (s, X t,x s (ω), y, z, q) -1 [tn,T ] f (s, X tn,xn s (ω), y, z, q)|.
The continuity of u is then a consequence of the following convergences as n → ∞:

E( sup 0≤s≤T |h(s, X t,x s ) -h(s, X tn s (x n ))| 2 ) → 0 E[ T 0 (f n s ) 2 ds] → 0,
which follow from the Lebesgue's theorem, using the continuity assumptions and polynomial growth of f and h.

Lemma 4.6.5. The function u has at most polynomial growth at infinity.

Proof. By applying Prop. 4.6.2 , we obtain the following estimate:

u(t, x) 2 ≤ K C,T (E( T 0 f (s, X t,x s , 0, 0, 0) 2 ds + sup 0≤s≤T h(s, X t,x s ) 2 ). (4.6.8)
Using now the hypothesis of polynomial growth on f, h, g and the standard estimate

E[ sup 0≤s≤T |X t,x s | 2 ] ≤ C ′ (1 + x 2 ),
we derive that there exist C ∈ R and p ∈ N such that |u(t, x)| ≤ C(1+x p ), ∀t ∈ [0, T ], ∀x ∈ R.

Remark 4.6.6. By (4.6.8), if (t, x) → f (t, x, 0, 0), h and g are bounded, then u is bounded.

An extension of the comparison result for BSDEs with jumps

We provide here an extension of the comparison theorem for BSDEs given in [START_REF] Quenez | BSDEs with jumps, optimization and applications to dynamic risk measures[END_REF] which formally states that if two drivers f 1 , f 2 satisfy f 1 ≥ f 2 + ε, then the associated solutions X 1 and X 2 satisfy X 1 0 > X 2 0 .

Proposition 4.6.7. Let t 0 ∈ [0, T ] and let θ be a stopping time such that θ > t 0 a.s. Let ξ 1 and ξ 2 ∈ L 2 (F θ ). Let f 1 be a driver. Let f 2 be a Lipschitz driver. For i = 1, 2, let (X i t , π i t , l i t ) be a solution in

S 2 × IH 2 × IH 2 ν of the BSDE -dX i t = f i (t, X i t , π i t , l i t )dt -π i t dW t - R * l i t (u) Ñ (dt, du); X i θ = ξ i . ( 4
.6.9)

Assume that there exists a bounded predictable process (γ t ) such that dt ⊗ dP ⊗ ν(de)-a.s. γ t (e) ≥ -1 and |γ t (e)| ≤ C(1 ∧ |e|), and such that

f 2 (t, X 2 t , π 2 t , l 1 t ) -f 2 (t, X 2 t , π 2 t , l 2 t ) ≥ γ t , l 1 t -l 2 t ν , t 0 ≤ t ≤ θ, dt ⊗ dP a.s. (4.6.10)
Suppose also that

ξ 1 ≥ ξ 2 a.s. f 1 (t, X 1 t , π 1 t , l 1 t ) ≥ f 2 (t, X 1 t , π 1 t , l 1 t ) + ε, t 0 ≤ t ≤ θ, dt ⊗ dP a.s.
where ε is a real constant. Then, X 1 t 0 -X 2 t 0 ≥ εα a.s. where α is a non negative F t 0 -measurable r.v. which does not depend on ε, with P (α > 0) > 0.

Proof. From inequality (4.22) in the proof of the Comparison Theorem in [START_REF] Quenez | BSDEs with jumps, optimization and applications to dynamic risk measures[END_REF], we derive that

X 1 t 0 -X 2 t 0 ≥ e -CT E θ t 0 H t 0 ,s ε ds|F t 0 a.s. ,
where C is the Lipschitz constant of f 2 , and (H t 0 ,s ) s∈[t 0 ,T ] is the square integrable non negative martingale satisfying

dH t 0 ,s = H t 0 ,s -β s dW s + R * γ s (u) Ñ (ds, du) ; H t 0 ,t 0 = 1,
(β s ) being a predictable process bounded by C. We get

X 1 t 0 -X 2 t 0 ≥ e -CT ε E [H t 0 ,θ (θ -t 0 )|F t 0 ] a.s.
Since θ > t 0 a.s. , we have H t 0 ,θ (θt 0 ) ≥ 0 a.s. and P (H t 0 ,θ (θt 0 ) > 0) > 0. Setting α := e -CT E [H t 0 ,θ (θt 0 )|F t 0 ], the result follows.

Chapter 5

Generalized Dynkin Games and DRBSDEs with Jumps

Abstract. We introduce a new game problem which can be seen as a generalization of the classical Dynkin game problem to the case of a nonlinear E g -conditional expectation , induced by a Backward Stochastic Differential Equation (BSDE) with jumps. Let ξ, ζ be two RCLL adapted processes with ξ ≤ ζ. The criterium is given by

J τ,σ = E g 0,τ ∧σ ξ τ 1 {τ ≤σ} + ζ σ 1 {σ<τ }
where τ and σ are stopping times valued in [0, T ]. Under Mokobodski's condition, we establish the existence of a value function for this game, i.e. inf σ sup τ J τ,σ = sup τ inf σ J τ,σ . This value can be characterized via a doubly reflected BSDE. Using this characterization, we provide some new results on these equations, such as comparison theorems and a priori estimates. When ξ and ζ are left upper semicontinuous along stopping times, we prove the existence of a saddle point. We also study a generalized mixed game problem when the players have two actions: continuous control and stopping. We then study the generalized Dynkin game in a Markovian framework and its links with parabolic partial integro-differential variational inequalities with two obstacles.

Introduction

The classical Dynkin game has been widely studied: see e.g. Bismut [START_REF] Bismut | Sur un problème de Dynkin[END_REF], Alario-Nazaret et al.

J τ,σ = E ξ τ 1 {τ ≤σ} + ζ σ 1 {σ<τ } .
Under Mokobodski's condition, which states that there exist two supermartingales such that their difference is between ξ and ζ, there exists a value function for the Dynkin game, i.e. inf σ sup τ J τ,σ = sup τ inf σ J τ,σ . When the barriers ξ, ζ are left upper semicontinuous, and ξ t < ζ t , t < T , there exists a saddle point.

Using a change of variable, these results can be generalized to the case of a criterium with an instantaneous reward process (g t ), of the form

E τ ∧σ 0 g s ds + ξ τ 1 {τ ≤σ} + ζ σ 1 {σ<τ } .
(5.1.1)

In the Brownian case and when (ξ t ) and (ζ t ) are continuous processes, Cvitanić and Karatzas have established links between these Dynkin games and doubly reflected Backward stochastic differential equations with driver process (g t ) and barriers (ξ t ) and (ζ t ) (see [START_REF] Cvitanic | Backward stochastic differential equations with reflection and Dynkin games[END_REF]).

In this chapter, we introduce a new game problem, which generalizes the classical Dynkin game to the case of E g -conditional expectations. Nonlinear expectations induced by BSDEs have been introduced by S. Peng [START_REF] Peng | Backward SDE and related g-expectation, Backward stochastic differential equations[END_REF] in the Brownian framework . Given a Lipschitz driver g(t, y, z), a stopping time τ ≤ T and a square integrable F τ -measurable random variable η, the associated conditional E g -expectation process denoted by (E g t,τ (η), 0 ≤ t ≤ τ ) is defined as the solution of the BSDE with driver g, terminal time τ and terminal condition η. The extension to the case with jumps is studied in e.g. [START_REF] Quenez | BSDEs with jumps, optimization and applications to dynamic risk measures[END_REF]. We consider here a generalized Dynkin game, where the criterium is given, for each pair (τ, σ) of stopping times valued in [0, T ], by

J τ,σ = E g 0,τ ∧σ ξ τ 1 {τ ≤σ} + ζ σ 1 {σ<τ } with ξ, ζ two RCLL adapted processes satisfying ξ ≤ ζ.
When the driver g does not depend on the solution, that is, when it is given by a process (g t ), the criterium J τ,σ coincides with (5.1.1). It is well-known that in this case, under Mokobodski's condition, the value function for the Dynkin game problem can be characterized as the solution of the Doubly Reflected BSDE (DRBSDE) associated with driver process (g t ) and barriers (ξ t ) and (ζ t ) (see e.g. [START_REF] Cvitanic | Backward stochastic differential equations with reflection and Dynkin games[END_REF][START_REF] Hamadène | Reflected BSDEs and mixed game problem[END_REF][START_REF] Lepeltier | Reflected backward stochastic differential equations with two rcll barriers[END_REF]). We generalize this result to the case of a nonlinear driver g(t, y, z, k) depending on the solution. More precisely, under Mokobodski's condition, we prove that

inf σ sup τ J τ,σ = sup τ inf σ J τ,σ
and we characterize this common value function as the solution of the DRBSDE associated with driver g and barriers ξ and ζ. Moreover, when ξ and ζ are left-upper semicontinuous along stopping times, we show that there exist saddle points. Note that, contrary to the previous existence results given in the case of classical Dynkin games, we do not assume the strict separability of the barriers. We point out that the approach used in the classical case cannot be adapted to our case because of the nonlinearity of the driver.

Using the characterization of the solution of a DRBSDE as the value function of a generalized Dynkin game, we prove some results on DRBSDEs, such as a comparison and a strict comparison theorem, and a priori estimates, which complete those given in the previous literature.

Moreover, we introduce a new mixed game problem expressed in terms of E g -conditional expectations, when the players have two possible actions: continuous control and stopping. The first (resp. second) player chooses a pair (u, τ ) (resp. (v, σ)) of control and stopping time, and aims at maximizing (resp. minimizing) the criterium. This problem has been studied by [START_REF] Ph | Simulation of BSDEs by Wiener Chaos Expansion[END_REF] and [START_REF] Hamadène | Reflected BSDEs and mixed game problem[END_REF] in the classical case, that is when the criterium is given, for each quadruple (u, τ, v, σ) of controls and stopping times, by

E Q u,v τ ∧σ 0 c(t, u t , v t )dt + ξ τ 1 {τ ≤σ} + ζ σ 1 {σ<τ } , (5.1.2)
where Q u,v are a priori probability measures and c(t, u t , v t ) represents the instantenous reward associated with controls u, v. In [START_REF] Hamadène | Reflected BSDEs and mixed game problem[END_REF], Hamadène and Lepeltier have established some links between this mixed game problem and DRBSDEs, when ξ and ζ are regular. Here, we consider a generalized mixed game problem, where, for a given family of Lipschitz drivers g u,v , the criterium is defined by

E u,v 0,τ ∧σ ξ τ 1 {τ ≤σ} + ζ σ 1 {σ<τ } , (5.1.3)
where E u,v denotes the g u,v -conditional expectation. Note that the criterium (5.1.3) corresponds to a criterium of the form (5.1.2) when the drivers g u,v are linear. We generalize the results of [START_REF] Hamadène | Reflected BSDEs and mixed game problem[END_REF] to the case of nonlinear expectations and irregular payoffs ξ and ζ. We provide some sufficient conditions which ensure the existence of a value function of our generalized mixed game problem, and show that the common value function can be characterized as the solution of a DRBSDE. Under additional regularity assumptions on ξ and ζ, we prove the existence of saddle points. The chapter is organized as follows. In Section 5.2 we introduce notation and definitions and provide some preliminary results. In Section 5.3, we consider a classical Dynkin game problem and study its links with a DRBSDE associated with a driver which does not depend on the solution. We provide an existence result for this game problem under relatively weak assumptions on ξ and ζ. Note that Section 5.3, although it contains new results, mainly situates our work and introduces the tools used in the sequel. In Section 5.4, we introduce the generalized Dynkin game with E g -conditional expectation. We prove the existence of a value function for this game problem. We show that the common value function can be characterized as the solution of a nonlinear DRBSDE with jumps and RCLL barriers ξ and ζ. We then study a generalized mixed game problem when the players have two actions: continuous control and stopping. In Section 5.5, using the characterization of the solution of a DRBSDE as the value function of a generalized Dynkin game, we prove comparison theorems and a priori estimates for DRBSDEs. Finally, we address the generalized Dynkin game in the Markovian case and its links with parabolic partial integro-differential variational inequalities (PIDVI) with two obstacles in Section 5.6. The value function of the generalized Dynkin game is a viscosity solution of a PIDVI. A uniqueness result is obtained under additional assumptions.

Notation and definitions

Let (Ω, F , P ) be a probability space. Let W be a one-dimensional Brownian motion. Let E := IR * and B(E) be its Borelian filtration. Suppose that it is equipped with a σ-finite positive measure ν and let N (dt, de) be a Poisson random measure with compensator ν(de)dt. Let Ñ (dt, de) be its compensated process. Let IF = {F t , t ≥ 0} be the completed natural filtration associated with W and N .

Notation. Let P be the predictable σ-algebra on [0, T ] × Ω.

For each T > 0, we use the following notation: L 2 (F T ) is the set of random variables ξ which are F T -measurable and square integrable; IH 2 is the set of real-valued predictable processes φ such that φ 2 IH 2 := E T 0 φ 2 t dt < ∞; S 2 denotes the set of real-valued RCLL adapted processes φ such that φ 2 S 2 := E(sup 0≤t≤T |φ t | 2 ) < ∞; A 2 (resp. A 1 ) is the set of real-valued non decreasing RCLL predictable processes A with A 0 = 0 and E(A 2 T ) < ∞ (resp. E(A T ) < ∞). We also introduce the following spaces:

• L 2
ν is the set of Borelian functions ℓ : E → IR such that E |ℓ(e)| 2 ν(de) < +∞. The set L 2 ν is a Hilbert space equipped with the scalar product ℓ ′ , ℓ ν := E ℓ(e)ℓ ′ (e)ν(de) for all ℓ, ℓ ′ ∈ L 2 ν × L 2 ν , and the norm ℓ 2 ν := E |ℓ(e)| 2 ν(de).

• IH 2 ν is the set of all mappings l : [0, T ]×Ω×E → IR that are P ⊗B(E)/B(IR) measurable and satisfy l 2

IH 2 ν := E T 0 l t 2 ν dt < ∞,
where l t (ω, e) = l(t, ω, e) for all (t, ω, e) ∈ [0, T ]×Ω×E.

Moreover, T 0 is the set of stopping times τ such that τ ∈ [0, T ] a.s. and for each S in T 0 , we denote by T S the set of stopping times τ such that S ≤ τ ≤ T a.s. Definition 5.2.1 (Driver, Lipschitz driver). A function g is said to be a driver if

• g : [0, T ] × Ω × IR 2 × L 2 ν → IR (ω, t, y, z, κ(•)) → g(ω, t, y, z, k(•)) is P ⊗ B(IR 2 ) ⊗ B(L 2 ν )-measurable, • g(., 0, 0, 0) ∈ IH 2 .
A driver g is called a Lipschitz driver if moreover there exists a constant C ≥ 0 such that dP ⊗ dta.s. , for each (y 1 , z 1 , k 1 ), (y 2 , z 2 , k 2 ),

|g(ω, t, y 1 , z 1 , k 1 ) -g(ω, t, y 2 , z 2 , k 2 )| ≤ C(|y 1 -y 2 | + |z 1 -z 2 | + k 1 -k 2 ν ).
Recall that for each Lipschitz driver g, and each terminal condition ξ ∈ L 2 (F T ), there exists a unique solution (X, π, l) ∈ S 2 × IH 2 × IH 2 ν satisfying

-dX t = g(t, X t -, π t , l t (•))dt -π t dW t - E l t (e) Ñ (dt, de); X T = ξ. (5.2.1)
The solution is denoted by (X(ξ, T ), π(ξ, T ), l(ξ, T )). This result can be extended when the terminal time T is replaced by a stopping time τ ∈ T 0 and when ξ is replaced by a random variable η ∈ L 2 (F τ ). The solution X • (η, τ ) corresponds to the so-called E g -conditional expectation of η, denoted by E g

•,τ (η). Definition 5.2.2. Let A = (A t ) 0≤t≤T and A ′ = (A ′ t ) 0≤t≤T belonging to A 1 . We say that the random measures dA t and dA ′ t are mutually singular, and we write dA t ⊥ dA ′ t , if there exists D ∈ P such that:

E[ T 0 1 D c dA t ] = E[ T 0 1 D dA ′ t ] = 0,
which can also be written as

T 0 1 D c t dA t = T 0 1 Dt dA ′ t = 0 a.s.
, where for each t ∈ [0, T ], D t is the section at time t of D, that is, D t := {ω ∈ Ω , (t, ω) ∈ D}.

We define now DRBSDEs with jumps, for which the solution is constrained to stay between two given RCLL processes called barriers ξ ≤ ζ. Two nondecreasing processes A and A ′ are introduced in order to push the solution Y above ξ and below ζ in a minimal way. This minimality property of A and A ′ is ensured by the Skorohod conditions (condition (iii) below) together with the additional constraint dA t ⊥ dA ′ t (condition (ii)). Definition 5.2.3 (Doubly Reflected BSDEs with Jumps). Let T > 0 be a fixed terminal time and g be a Lipschitz driver. Let ξ and ζ be two adapted RCLL processes with

ζ T = ξ T a.s., ξ ∈ S 2 , ζ ∈ S 2 , ξ t ≤ ζ t , 0 ≤ t ≤ T a.s. A process (Y, Z, k(.), A, A ′ ) in S 2 × IH 2 × IH 2 ν × A 2 × A 2 is
said to be a solution of the doubly reflected BSDE (DRBSDE) associated with driver g and barriers ξ, ζ if

-dY t = g(t, Y t , Z t , k t (•))dt + dA t -dA ′ t -Z t dW t - E k t (e) Ñ (dt, de); Y T = ξ T , (5.2.2) with (i) ξ t ≤ Y t ≤ ζ t , 0 ≤ t ≤ T a.s., (ii) dA t ⊥ dA ′ t (iii) T 0 (Y t -ξ t )dA c t = 0 a.s. and T 0 (ζ t -Y t )dA ′ c t = 0 a.s. ∆A d τ = ∆A d τ 1 {Y τ -=ξ τ -} and ∆A ′ d τ = ∆A ′ d τ 1 {Y τ -=ζ τ -} a.s. ∀τ ∈ T 0 predictable.
Here A c (resp A ′ c ) denotes the continuous part of A (resp A ′ ) and A d (resp A ′ d ) its discontinuous part.

Remark 5.2.4. Note that when A and A ′ are not required to be mutually singular, they can simultaneously increase on {ξ t -= ζ t -}. The constraint dA t ⊥ dA ′ t will allow us to obtain the uniqueness of the non decreasing RCLL processes A and A ′ , without the usual strict separability condition ξ < ζ (see Theorem 5.3.5).

We introduce the following definition. Definition 5.2.5. A progressively measurable process (φ t ) (resp. integrable) is said to be left-upper semicontinuous (l.u.s.c.) along stopping times (resp. along stopping times in expectation ) if for all τ ∈ T 0 and for each non decreasing sequence of stopping times (τ n ) such that τ n ↑ τ a.s. ,

φ τ ≥ lim sup n→∞ φ τn a.s. (resp. E[φ τ ] ≥ lim sup n→∞ E[φ τn ]).
(5.2.3) Remark 5.2.6. Note that when (φ t ) is left-limited, then (φ t ) is left-upper semicontinuous (l.u.s.c.) along stopping times if and only if for all predictable stopping time τ ∈ T 0 , φ τ ≥ φ τ -a.s.

Classical Dynkin games and links with doubly reflected BSDEs with a driver process

In this section, we are given a predictable process g = (g t ) in H 2 . Let ξ and ζ be two adapted processes only supposed to be RCLL with

ζ T = ξ T a.s., ξ ∈ S 2 , ζ ∈ S 2 , ξ t ≤ ζ t , 0 ≤ t ≤ T a.s.
We prove below that the doubly reflected BSDE associated with the driver process (g t ) and the barriers ξ and ζ admits a unique solution (Y, Z, k(•), A, A ′ ), which is related to a classical Dynkin game problem. Our results complete previous works on classical Dynkin games and DRBSDEs (see e.g. [START_REF] Cvitanic | Backward stochastic differential equations with reflection and Dynkin games[END_REF], [START_REF] Hamadène | BSDEs with two reacting barriers driven by a Brownian motion and an independent Poisson noise and related Dynkin game[END_REF]). In particular, we provide an existence result of saddle points under weaker assumptions than those made in the previous literature.

For any S ∈ T 0 and any stopping times τ, σ ∈ T S , consider the gain (or payoff): We clearly have the inequality V (S) ≤ V (S) a.s. By definition, we say that there exists a value function at time S for the Dynkin game if V (S) = V (S) a.s.

I S (τ, σ) = σ∧τ S g(u)du + ξ τ 1 {τ ≤σ} + ζ σ 1 {σ<τ } . ( 5
Definition 5.3.1 (S-saddle point). Let S ∈ T 0 . A pair (τ * , σ * ) ∈ T 2 S is called an S-saddle point if for each (τ, σ) ∈ T 2 S , we have E[I S (τ, σ * )|F S ] ≤ E[I S (τ * , σ * )|F S ] ≤ E[I S (τ * , σ)|F S ] a.s.
We introduce the following RCLL adapted processes which depend on the process g:

ξg t := ξ t -E[ξ T + T t g(s)ds|F t ], ζg t := ζ t -E[ζ T + T t g(s)ds|F t ], 0 ≤ t ≤ T. (5.3.4)
They satisfy the property ξg T = ζg T = 0 a.s. Moreover, this change of variables allows us to get rid of the term g(t)dt, and thus to simplify the notation.

Definition 5.3.2. A nonnegative process φ . = (φ t ) valued in [0, +∞] is said to be a strong super- martingale if for any θ, θ ′ ∈ T 0 such that θ ≥ θ ′ a.s., E[φ θ | F θ ′ ] ≤ φ θ ′ a.s.
Lemma 5.3.3. There exist two strong supermartingales (J g t ) and (J

′ g t ) valued in [0, +∞] such that for all θ ∈ T 0 , J g θ = ess sup τ ∈T θ E J ′g τ + ξg τ |F θ a.s. and J ′ g θ = ess sup σ∈T θ E J g σ -ζg σ |F θ a.s. , (5.3

.5)

and satisfying the following minimality property: J g • and J

′ g

• are the smallest strong supermartingales valued in [0, +∞] such that

J g • ≥ J ′ g • + ξg • and J ′ g • ≥ J g • -ζg • .
(5.3.6)

If J g 0 < +∞ and J ′ g 0 < +∞, J g • and J ′g • are indistinguishable from RCLL supermartingales.
The proof is given in the Appendix. Using this lemma, we derive the following result.

Theorem 5.3.4. Let ξ and ζ be two adapted RCLL processes in S 2 with ζ T = ξ T a.s. and ξ t ≤ ζ t , 0 ≤ t ≤ T a.s. Suppose that J g , J ′ g ∈ S 2 . Let Y be the RCLL adapted process defined by

Y t := J g t -J ′ g t + E[ξ T + T t g(s)ds|F t ]; 0 ≤ t ≤ T.
(5.3.7)

There exist (Z, k, A, A ′ ) ∈ IH 2 × IH 2 ν × A 2 × A 2 such that (Y , Z, k, A, A ′
) is a solution of DRBSDE (5.2.2) associated with the driver process g(t).

Proof. By assumption, J g and J ′ g are square integrable supermartingales. The process Y is thus well defined. By Lemma 5.3.3, we have J g T = J ′g T a.s. Hence, Y T = ξ T a.s. By the Doob-Meyer decomposition, there exist two square integrable martingales M and M ′ and two processes B and B ′ ∈ A 2 such that:

dJ g t = dM t -dB t ; dJ ′ g t = dM ′ t -dB ′ t . (5.3.8) Set M t := M t -M ′ t + E[ξ T + T 0 g(s)ds|F t ].
By (5.3.8), (5.3.7), we derive dY t = dM tdα tg(t)dt, with α := B -B ′ . Now, by the martingale representation theorem, there exist Z ∈ H 2 and k ∈ H 2 ν such that dM t = Z t dW t + E k t (e) Ñ (de, dt). Hence,

-dY t = g(t)dt + dα t -Z t dW t - E k t (e) Ñ (dt, de).
By the optimal stopping theory (see e.g. Proposition B.7 or B.11 in [START_REF] Kobylanski | Optimal stopping in a general framework[END_REF]), the process B c increases only when the value function J g is equal to the corresponding reward J ′ g + ξg . Now,

{J g t = J ′ g t + ξg } = {Y t = ξ t }. Hence, T 0 (Y t -ξ t )dB c t = 0 a.s. Similarly the process B ′ c satisfies T 0 (Y t -ζ t )dB ′ c
t = 0 a.s. and for each predictable stopping time τ ∈ T 0 we have

∆B d τ = 1 J g τ -=J ′ g τ -+ ξg τ - ∆B d τ = 1 Y τ -=ξ τ -∆B d τ a.s. and ∆B ′ d τ = 1 Y τ -=ζ τ -∆B
′ d τ a.s. By the canonical decomposition of an RCLL process with integrable variation (see Proposition 5.7.9), there exist A, A ′ ∈ A 2 such that α = A -A ′ with dA t ⊥ dA ′ t . Also, dA t < < dB t . Hence, since T 0 1 Y t ->ξ t -dB t = 0 a.s. , we get T 0 1 Y t ->ξ t -dA t = 0 a.s. Similarly, we obtain T 0 1 Y t -<ζ t -dA ′ t = 0 a.s. The processes A and A ′ thus satisfy conditions (5.2.2)(iii).

From this theorem, we derive the following uniqueness and existence result for the DRBSDE associated with the driver process (g t ), as well as the characterization of the solution as the value function of the above Dynkin game problem.

Theorem 5.3.5. Let ξ and ζ be two adapted RCLL processes in S 2 with ζ T = ξ T a.s. and ξ t ≤ ζ t , 0 ≤ t ≤ T a.s. Suppose that J g t , J ′g t ∈ S 2 . The doubly reflected BSDE (5.2.2) associated with driver process g(t) admits a unique solution

(Y, Z, k, A, A ′ ) in S 2 × IH 2 × IH 2 ν × (A 2 ) 2 .
For each S ∈ T 0 , Y S is the common value function of the Dynkin game, that is

Y S = V (S) = V (S) a.s.
(5.3.9)

Moreover, if the processes A, A ′ are continuous, then, for each S ∈ T 0 , the pair of stopping times (τ * s , σ * s ) defined by

σ * S := inf{t ≥ S, Y t = ζ t }; τ * S := inf{t ≥ S, Y t = ξ t } (5.3.10)
is an S-saddle point for the Dynkin game problem associated with the gain I S .

A short proof is given in the Appendix.

Remark 5.3.6. The condition dA t ⊥ dA ′ t ensures that for each predictable stopping time τ ∈ T 0 , we have ∆A d τ = (∆Y τ ) -and ∆A

′ d τ = (∆Y τ ) + a.s.
We now provide a sufficient condition on ξ and ζ for the existence of saddle points. By the last assertion of Theorem 5.3.5, it is sufficient to give a condition which ensures the continuity of A and A ′ . Theorem 5.3.7 (Existence of S-saddle points). Suppose that the assumptions of Theorem 5.3.5 are satisfied and that ξ and -ζ are l.u.s.c. along stopping times. Let (Y, Z, k(.), A, A ′ ) be the solution of DRBSDE (5.2.2). The processes A and A ′ are then continuous. Also, for each S ∈ T 0 , the pair of stopping times (τ * S , σ * S ) defined by (5.3.10) is an S-saddle point.

Remark 5.3.8. The assumptions made on ξ and ζ are weaker than the ones made in the literature where it is supposed ξ t < ζ t , t < T a.s. (see e.g. [START_REF] Alario-Nazaret | Dynkin games[END_REF], [START_REF] Cvitanic | Backward stochastic differential equations with reflection and Dynkin games[END_REF], [START_REF] Kobylanski | Dynkin games in a general framework[END_REF]).

Proof. By the second assertion of Theorem 5.3.5, it is sufficient to prove that A and A ′ are continuous. Let τ ∈ T 0 be a predictable stopping time. Let us show ∆A τ = 0 a.s. By Remark 5.3.6, we have ∆A τ = (∆Y τ ) -a.s. Since dA t ⊥ dA ′ t , there exists D ∈ P such that:

T 0 1 D c t dA t =
T 0 1 Dt dA ′ t = 0 a.s. We introduce the set D τ := {ω, (τ (ω), ω) ∈ D}. Since A satisfies the Skorohod condition, we thus have

∆A τ = 1 Dτ ∩{Y τ -=ξ τ -} (Y τ --Y τ ) + = 1 Dτ ∩{Y τ -=ξ τ -} (ξ τ --Y τ ) + ≤ 1 Dτ ∩{Y τ -=ξ τ -} (ξ τ -Y τ ) +
a.s. , where the last inequality follows from the inequality ξ τ -≤ ξ τ a.s. (see Remark 5.2.6). Since ξ ≤ Y , we derive that ∆A τ ≤ 0 a.s. Hence, ∆A τ = 0 a.s. , and this holds for each predictable stopping time τ . Consequently, A is continuous. Similarly, A ′ is continuous. The saddle point property of (τ * S , σ * S ) follows from the second assertion of Theorem 5.3.5.

Definition 5.3.9 (Mokobodski's condition). Let ξ and ζ be adapted RCLL processes in S 2 with ζ T = ξ T a.s. and ξ t ≤ ζ t , 0 ≤ t ≤ T a.s. Mokobodski's condition is said to be satisfied when there exist two nonnegative RCLL supermartingales H and H ′ in S 2 such that:

ξ t ≤ H t -H ′ t ≤ ζ t 0 ≤ t ≤ T a.s. ( 5
(i) J g ∈ S 2
(ii) J 0 ∈ S 2 (iii) Mokobodski's condition holds.

(iv) DRBSDE (5.2.2) with driver process (g t ) has a solution.

A short proof is given in the Appendix.

Generalized Dynkin games and links with nonlinear doubly reflected BSDEs

In this section, we are given a Lipschitz driver g.

I S (τ, σ) = σ∧τ S g(u, Y u , Z u , k u )du + ξ τ 1 {τ ≤σ} + ζ σ 1 {σ<τ } .
(5.4.1)

where Z, k are the associated processes with Y . However, this characterization is not really usable and exploitable because the instantaneous reward g(u, Y u , Z u , k u ) depends on the value function Y of the associated Dynkin game.

We now introduce a new game problem, which can be seen as a generalized Dynkin game expressed in terms of E g -conditional expectations. In order to ensure that the E g -conditional expectation is non decreasing, we make the following assumption.

Assumption 5.4.3. Assume that dP ⊗ dt-a.s for each (y, z, k 1 , k 2 ) ∈ R 2 × (L 2 ν ) 2 , g(t, y, z, k 1 ) -g(t, y, z, k 2 ) ≥ γ y,z,k 1 ,k 2 t , k 1 -k 2 ν , with γ : [0, T ] × Ω × R 2 × (L 2 ν ) 2 → L 2 ν ; (ω, t, y, z, k 1 , k 2 ) → γ y,z,k 1 ,k 2 t
(ω, .)

P ⊗B(IR 2 )⊗B((L 2 ν ) 2 )-measurable, bounded, and satisfying dP ⊗dt⊗dν(e)-a.s. , for each (y, z, k

1 , k 2 ) ∈ R 2 × (L 2 ν ) 2 , γ y,z,k 1 ,k 2 t (e) ≥ -1 and |γ y,z,k 1 ,k 2 t (e)| ≤ ψ(e), (5.4.2)
where ψ ∈ L 2 ν .

For example, this assumption is satisfied if g is C 1 with respect to k with ∇ k g ≥ -1 and |∇ k g| ≤ ψ, where ψ ∈ L 2 ν . Also if g is of the form g(ω, t, y, z, k) := g(ω, t, y, z, E k(e)ψ(e)ν(de)) where ψ is a nonnegative function in L 2 ν and g : Ω × [0, T ] × R 3 → R is Borelian and non-decreasing with respect to k, then g satisfies Assumption 6.3.9. (see Proposition 5.7.2 in the Appendix for details). Assumption 6.3.9 ensures the non decreasing property of E g by the comparison theorem for BSDEs with jumps (see Theorem 4.2 in [START_REF] Quenez | BSDEs with jumps, optimization and applications to dynamic risk measures[END_REF]). When in (6.3.12), γ t > -1, the strict comparison theorem (see Theorem 4.4 in [START_REF] Quenez | BSDEs with jumps, optimization and applications to dynamic risk measures[END_REF]) implies that E g is strictly monotonous.

For each τ, σ ∈ T 0 , the reward at time τ ∧ σ is given by the random variable

I(τ, σ) := ξ τ 1 τ ≤σ + ζ σ 1 σ<τ . (5.4.3) Note that I(τ, σ) is F τ ∧σ -measurable.
Let S ∈ T 0 . For each τ ∈ T S and σ ∈ T S , the associated criterium is given by E g S,τ ∧σ (I(τ, σ)), the E g -conditional expectation of the reward I(τ, σ).

Recall that E g •,τ ∧σ (I(τ, σ)) = X τ,σ • , where (X τ,σ • , π τ,σ • , l τ,σ
• ) is the solution of the BSDE associated with driver g, terminal time τ ∧ σ and terminal condition I(τ, σ), that is

-dX τ,σ s = g(s, X τ,σ s , π τ,σ s , l τ,σ s )ds -π τ,σ s dW s - E l τ,σ
s (e) Ñ (ds, de); X τ,σ τ ∧σ = I(τ, σ).

There are two players with antagonistic objectives. At time S, the first player chooses a stopping time τ greater than S, and aims at maximizing the criterium. The other player chooses a stopping time σ greater than S, and aims at the opposite, that is, minimizing the criterium. For each stopping time S ∈ T 0 , the upper and lower value functions at time S are defined respectively by We first provide a sufficient condition for the existence of an S-saddle point and for the characterization of the common value function as the solution of the DRBSDE. Lemma 5.4.5. Suppose that the driver g satisfies Assumption (6.3.9). Let ξ and ζ be RCLL adapted processes in S 2 such that ξ T = ζ T a.s. and ξ t ≤ ζ t , 0 ≤ t ≤ T a.s. Suppose that Mokobodski's condition is satisfied.

Let (Y, Z, k(•), A, A ′ ) be the solution of the DRBSDE (5.2.2). Let S ∈ T 0 . Let (τ , σ) ∈ T S . Suppose that (Y t , S ≤ t ≤ τ ) is a strong E g -submartingale and that (Y t , S ≤ t ≤ σ) is a strong E g -supermartingale with Y τ = ξ τ and Y σ = ζ σ a.s. The pair (τ , σ) is then an S-saddle point for the generalized Dynkin game (5.4.4)- (5.4.5) and

Y S = V (S) = V (S) a.s.
Proof. Since the process (Y t , S ≤ t ≤ τ ∧ σ) is a strong E g -martingale (see Definition 5.7.7) and since Y τ = ξ τ and Y σ = ζ σ a.s. , we have

Y S = E g S,τ ∧σ (Y τ ∧σ ) = E g S,τ ∧σ (ξ τ 1 τ ≤σ + ζ σ1 σ<τ ) = E g S,
Y τ ∧σ = Y τ 1 τ ≤σ + Y σ1 σ<τ ≥ ξ τ 1 τ ≤σ + ζ σ1 σ<τ = I(τ, σ) a.s.
By inequality (5.4.7) and the monotonicity property of E g , we derive inequality (5.4.6). Similarly, one can show that for each σ ∈ T S , we have:

Y S ≤ E g S,τ ∧σ (I(τ , σ)) a.s.
The pair (τ , σ) is thus an S-saddle point and Y S = V (S) = V (S) a.s.

We now provide an existence result under an additional assumption.

Theorem 5.4.6 (Existence of S-saddle points). Suppose that g satisfies Assumption 5.4.3. Let ξ and ζ be RCLL adapted processes in S 2 such that ξ T = ζ T a.s. and ξ t ≤ ζ t , 0 ≤ t ≤ T a.s. Suppose that Mokobodski's condition is satisfied. Let (Y, Z, k, A, A ′ ) be the solution of the DRBSDE (5.2.2). Suppose that A, A ′ are continuous (which is the case if ξ and -ζ are l.u.s.c. along stopping times). For each S ∈ T 0 , let

τ * S := inf{t ≥ S, Y t = ξ t }; σ * S := inf{t ≥ S, Y t = ζ t }. τ S := inf{t ≥ S, A t > A S }; σ S := inf{t ≥ S, A ′ t > A ′ S }.
Then, for each S ∈ T 0 , the pairs of stopping times (τ * S , σ * S ) and (τ S , σ S ) are S-saddle points for the generalized Dynkin game and ) is an S-saddle point and Y S = V (S) = V (S) a.s. It remains to show that (τ S , σ S ) is an S-saddle point. By definition of τ S , σ S , we have A τ S = A S a.s. and A ′ σ S = A ′ S a.s. because A and A ′ are continuous and τ S , σ S are predictable stopping times. Moreover, since the continuous process A increases only on {Y t = ξ t }, we have Y τ S = ξ τ S a.s. Similarly, Y σ S = ζ σ S a.s. The result then follows from Lemma 5.4.5.

Y S = V (S) = V (S) a.s. Moreover, Y σ * S = ζ σ * S , Y τ * S = ξ τ * S , A τ * S = A S and A ′ σ * S = A ′ S a
In the case of irregular payoffs ξ and ζ, there does not generally exist a saddle point. However, we will now see that it is not necessary to have the existence of an S-saddle point to ensure the existence of a common value function and its characterization as the solution of a DRBSDE. Theorem 5.4.8 (Existence of the value function). Suppose that g satisfies Assumption (6.3.9). Let ξ and ζ be RCLL adapted processes in S 2 such that ξ T = ζ T a.s. and ξ t ≤ ζ t , 0 ≤ t ≤ T a.s. Suppose that Mokobodski's condition is satisfied. Let (Y,Z,k,A,A ′ ) be the solution of the DRBSDE (5.2.2). Then, there exists a value function for the generalized Dynkin game, and for each stopping time S ∈ T 0 , we have

Y S = V (S) = V (S) a.s.
(5.4.8)

Proof. For each S ∈ T 0 and for each ε > 0, let τ ε S and σ ε S be the stopping times defined by

τ ε S := inf{t ≥ S, Y t ≤ ξ t + ε}.
(5.4.9)

σ ε S := inf{t ≥ S, Y t ≥ ζ t -ε}.
(5.4.10)

We first prove two lemmas.

Lemma 5.4.9.

• We have

Y τ ε S ≤ ξ τ ε S + ε a.s. (5.4.11) Y σ ε S ≥ ζ σ ε S -ε a.s.
(5.4.12)

• We have A τ ε S = A S a.s. and A ′ σ ε S = A ′ S a
Y τ ∧σ ε S ≥ ξ τ 1 τ ≤σ ε S + (ζ σ ε S -ε)1 σ ε S <τ ≥ I(τ, σ ε S ) -ε a.s.
where the last inequality follows from the definition of I(τ, σ). Hence, using (5.4.14) and the monotonicity property of E g , we get

Y S ≥ E g S,τ ∧σ ε S (I(τ, σ ε S ) -ε) a.s. (5.4.15)
Now, by a priori estimates on BSDEs (see Proposition A.4, [START_REF] Quenez | BSDEs with jumps, optimization and applications to dynamic risk measures[END_REF]), we have End of proof of Theorem 5.4.8. Using Lemma 5.4.11, we derive that for each ε > 0,

|E g S,τ ∧σ ε S (I(τ, σ ε S ) -ε) -E g S,τ ∧σ ε S (I(τ, σ ε S ))| ≤ Kε a.s. It follows that Y S ≥ E g S,τ ∧σ ε S (I(τ, σ ε S )) -Kε a.
ess sup τ ∈Ts E g S,τ ∧σ ε S (I(τ, σ ε S )) -Kε ≤ Y S ≤ ess inf σ∈T S E g S,τ ε S ∧σ (I(τ, σ ε S )) + Kε a.s. , which implies V (S) -Kε ≤ Y S ≤ V (S) + Kε a.s.
Since V (S) ≤ V (S) a.s. , we get V (S) = Y S = V (S) a.s. The proof of Theorem 5.4.8 is thus complete.

Remark 5.4.12. Inequality (5.4.13) shows that (τ ε S , σ ε S ) defined by (5.4.9) and (5.

Generalized mixed game problems

We now introduce a new game problem, which can be seen as a generalization of a mixed game problem studied in [START_REF] Ph | Simulation of BSDEs by Wiener Chaos Expansion[END_REF] and [START_REF] Hamadène | Reflected BSDEs and mixed game problem[END_REF] to the case of nonlinear E g -conditional expectations. The players have two actions: continuous control and stopping. Let (g u,v ; (u, v) ∈ U × V) be a family of Lipschitz drivers satisfying Assumption 5.4.3 . Let S ∈ T 0 . For each quadruple (u, τ, v, σ) ∈ U × T S × V × T S , the criterium at time S is given by E u,v S,τ ∧σ (I(τ, σ)), where E u,v corresponds to the g u,v -conditional expectation. The first (resp. second) player chooses a pair (u, τ ) (resp. (v, σ)) of control and stopping time, and aims at maximizing (resp. minimizing) the criterium. For each stopping time S ∈ T 0 , the upper and lower value functions at time S are defined respectively by V (S) := essinf (5.4.17)

We say that there exists a value function at time S for the game problem if V (S) = V (S) a.s. We now introduce the definition of an S-saddle point for this game problem.

Definition 5.4.14.

Let S ∈ T 0 . A quadruple (u, τ , v, σ) ∈ U × T S × V × T S is called an S-saddle point for the generalized mixed game problem if for each (u, τ, v, σ) ∈ U × T S × V × T S we have E u,v S,τ ∧σ (I(τ, σ)) ≤ E u,v S,τ ∧σ (I(τ ∧ σ)) ≤ E u,v S,τ ∧σ (I(τ , σ)) a.s.
We prove below that when the obstacles are l.u.s.c. along stopping times, there exist saddle points for the above generalized mixed game problem.

Theorem 5.4.15. Let (g u,v ; (u, v) ∈ U ×V) be a family of Lipschitz drivers satisfying Assumptions 5.4.3. Let ξ and ζ be RCLL adapted processes in S 2 and l.u.s.c. along stopping times, such that ξ T = ζ T a.s. and ξ t ≤ ζ t , 0 ≤ t ≤ T a.s. Suppose that Mokobodski's condition is satisfied and that there exist controls u ∈ U and v ∈ V such that for each (u, v) ∈ U × V, 5.4.18) where (Y, Z, k, A, A ′ ) is the solution of the DRBSDE (5.2.2) associated with driver g u,v . Consider the stopping times

g u,v (t, Y t , Z t , k t ) ≤ g u,v (t, Y t , Z t , k t ) ≤ g u,v (t, Y t , Z t , k t ) dt ⊗ dP a.s. , (
τ * S := inf{t ≥ S : Y t = ξ t } ; σ * S := inf{t ≥ S : Y t = ζ t }.
The quadruple (u, τ * S , v, σ * S ) is then an S-saddle point for the generalized mixed game problem (5.4.16)-(5.4.17), and we have Y S = V (S) = V (S) a.s.

Proof. By the last assertion of Theoreom 5.4.6, the process

(Y t , S ≤ t ≤ τ * S ∧ σ * S ) is a strong E u,v -martingale and Y τ * S = ξ τ * S , Y σ * S = ζ σ * S a.s. , which implies Y S = E u,v S,τ * S ∧σ * S (Y τ * S ∧σ * S ) = E u,v S,τ * S ∧σ * S (ξ τ * S 1 τ * S ≤σ * S + ζ σ * S 1 σ * S <τ * S ) = E u,v S,
Y S ≥ E u,v S,τ ∧σ * S (I(τ, σ * S )) a.s.
Similarly, one can prove that for each v ∈ V, σ ∈ T S , we have:

Y S ≤ E u,v S,τ * S ∧σ (I(τ * S , σ)) a.s.
The quadruple (u, τ * S , v, σ * S ) is thus an S-saddle point and Y S = V (S) = V (S) a.s.

Under less restricted assumptions on the obstacles, we prove below that there exists a value function for the above game problem which can be characterized as the solution of a DRBSDE. Theorem 5.4.16 (Existence of the value function). Let (g u,v ; (u, v) ∈ U ×V) be a family of drivers satisfying Assumptions 5.4.3 and uniformly Lipschitz with common Lipchitz constant C. Let ξ and ζ be RCLL adapted processes in S 2 such that ξ T = ζ T a.s. and ξ t ≤ ζ t , 0 ≤ t ≤ T a.s. Suppose that Mokobodski's condition is satisfied and that there exist controls u ∈ U and v ∈ V such that for each u ∈ U, v ∈ V:

g u,v (t, Y t , Z t , k t ) ≤ g u,v (t, Y t , Z t , k t ) ≤ g u,v (t, Y t , Z t , k t ), dt ⊗ dP a.s.
(5.4. [START_REF] Bellman | Introduction to the mathematical theory of control processes[END_REF] where (Y, Z, k, A, A ′ ) is the solution of the DRBSDE (5.2.2) associated with driver g u,v . Then, there exists a value function for the generalized mixed game problem (5.4.16)-(5.4.17), and for each stopping time S ∈ T 0 , we have

Y S = V (S) = V (S) a.s.
Proof. For each S ∈ T 0 and for each ε > 0, let τ ε S and σ ε S be the stopping times defined by

τ ε S := inf{t ≥ S, Y t ≤ ξ t + ε}; σ ε S := inf{t ≥ S, Y t ≥ ζ t -ε}. Let τ ∈ T S . Since Y ≥ ξ and Y σ ε S ≥ ζ σ ε S -ε a.
s. ( see Lemma 5.4.9), we have:

Y τ ∧σ ε S ≥ ξ τ 1 τ ≤σ ε S + (ζ σ ε S -ε)1 σ ε S <τ ≥ I(τ, σ ε S ) -ε a.s.
By Lemma 5.4.9, A ′ σ ε S = A ′ S a.s. which implies that:

-dY t = g u,v (t, Y t , Z t , k t )dt + dA t -Z t dW t - E k t (e) Ñ (dt, de), S ≤ t ≤ σ ε S , dt ⊗ dP a.s.
Hence, (Y t ) S≤t≤τ ∧σ ε is the solution of the BSDE associated with generalized driver f (•)dt + dA t and terminal condition Y τ ∧σ ε . By using Assumption (5.4.19), the inequality Y τ ∧σ ε ≥ I(τ, σ ε )ε and the comparison theorem for BSDEs with jumps, we obtain

Y S ≥ E u,v S (I(τ, σ ε ) -ε) ≥ E u,v S (I(τ, σ ε )) -Kε a.s. ,
where the second inequality follows from a priori estimates for BSDEs with jumps. Here, the constant K only depends on T and C, the common Lipschitz constant. Consequently, we get

Y S ≥ essinf v∈V,σ∈T S ess sup u∈U ,τ ∈T S E u,v S,τ ∧σ (I(τ, σ)) -Kε a.s.
Similarly, one can prove that for each ε > 0,

Y S ≤ ess sup u∈U ,τ ∈T S essinf v∈V,σ∈T S E u,v S,τ ∧σ (I(τ, σ)) + Kε a.s.
Hence, V (S) ≤ V (S) a.s. Since V (S) ≤ V (S) a.s., the equality follows.

Remark 5.4.17. Note that Theorem 5.4.16 still holds if g u,v is replaced by any Lipschitz driver g which satisfies (5.4.19).

Application:

Let U, V be compact Polish spaces. We are given a map

F : [0, T ] × Ω × U × V × IR 2 × L 2 ν → IR, (t, ω, u, v, y, z, k) → F (t, ω, u, v, y, z, k), supposed to be measurable with respect to P ⊗ B(U ) ⊗ B(V ) ⊗ B(IR 2 ) ⊗ B(L 2
ν ), continuous, concave (resp. convex) with respect to u (resp. v), and uniformly Lipchitz with respect to (y, z, k). Suppose that F is C 1 with respect to k with ∇ k F ≥ -1, and that F (t, ω, u, v, 0, 0, 0) is uniformly bounded. Let U (resp. V) be the set of predictable processes valued in U (resp. V ). For each (u, v) ∈ U × V, let g u,v be the driver defined by g u,v (t, ω, y, z, k) := F (t, ω, u t (ω), v t (ω), y, z, k).

( Since U and V are Polish spaces, there exist some dense countable subsets U (resp. V ) of U (resp. V ). Since F is continuous with respect to u, v, the sup and the inf can be taken over U (resp. V ). Hence, g is a Lipschitz driver.

Let (Y, Z, k, A, A ′ ) ∈ S 2 × H 2 × H 2 ν × (A 2
) 2 be the solution of the DRBSDE associated with driver g and obstacles ξ and ζ. By classical convex analysis, for each (t, ω) there exist (u * , v * ) ∈ (U, V ) such that

F (t, ω, u, v * , Y t -(ω), Z t (ω), k t (ω)) ≤ F (t, ω, u * , v * , Y t -(ω), Z t (ω), k t (ω)) (5.4.22) ≤ F (t, ω, u * , v, Y t -(ω), Z t (ω), k t (ω)), ∀(u, v) ∈ U × V ; g(t, ω, Y t -(ω), Z t (ω), k t (ω))) = F (t, ω, u * , v * , Y t -(ω), Z t (ω), k t (ω)) Let (u, v) ∈ U × V .
Since the processes Y t -, Z t and k t are predictable, the map (t, ω, u

* , v * ) → (t, ω, u, v * , Y t -(ω), Z t (ω), k t (ω)
) is measurable with respect to the σ-algebras P ⊗ B(U ) ⊗ B(V ) and P ⊗ B(U ) ⊗ B(V ) ⊗ B(IR 2 ) ⊗ B(L 2 ν ). By using the measurability property of F , it follows by composition that the map (t, ω, u

* , v * ) → F (t, ω, u, v * , Y t -(ω), Z t (ω), k t (ω)) is P ⊗ B(U ) ⊗ B(V )- measurable.
Similarly, the other maps which appear in (5.4.22) are P ⊗ B(U ) ⊗ B(V )-measurable, which implies that the set of all (t, ω, u * , v * ) ∈ [0, T ] × Ω × U × V satisfying conditions (5.4.22) belongs to P ⊗ B(U ) ⊗ B(V ). By applying a section theorem (see Section 81 in the Appendix of Ch. III in [START_REF] Dellacherie | Probabilités et Potentiel[END_REF]), we get that there exists a pair of predictable process (u * , v * ) ∈ U × V such that dt ⊗ dP a.s., for all (u, v) ∈ U × V we have dt ⊗ dP a.s.:

F (t, u t , v * t , Y t , Z t , k t ) ≤ F (t, u * t , v * t , Y t , Z t , k t ) ≤ F (t, u * t , v t , Y t , Z t , k t ) and g(t, Y t , Z t , k t ) = F (t, u * t , v * t , Y t , Z t , k t ).
Hence, Assumption (5.4.18) is satisfied. By applying Theorems 5.4.16 and 5.4.15, we derive the following result: Proposition 5.4.18. There exists a value function for the generalized mixed game problem associated with the controlled drivers g u,v given by (5.4.20). Let Y be the solution of the DRBSDE associated with obstacles ξ, ζ and the driver g defined by (5.4.21). For each stopping time S ∈ T 0 , we have Y S = V (S) = V (S) a.s. Suppose that ξ and ζ are l.u.s.c. along stopping times, and consider the stopping times

τ * S := inf{t ≥ S : Y t = ξ t } ; σ * S := inf{t ≥ S : Y t = ζ t }.
The quadruple (u * , τ * S , v * , σ * S ) is then an S-saddle point for this generalized mixed game problem.

We give now an example of application of the above proposition.

Example: Consider the particular case when F takes the following form:

F (t, ω, u, v, y, z, k) = β(t, ω, u, v)z+ < γ(t, ω, u, v, •), k > ν +c(t, ω, u, v)
, with β, γ, c bounded. By classical results on linear BSDEs (see [START_REF] Quenez | BSDEs with jumps, optimization and applications to dynamic risk measures[END_REF]), the criterium can be written

E u,v S,τ ∧σ (I(τ, σ)) = E Q u,v τ ∧σ S c(t, u t , v t )dt + I(τ, σ)|F S ,
with Q u,v the probability measure which admits Z u,v T as density with respect to P , where (Z u,v t ) is the solution of the following SDE:

dZ u,v t = Z u,v t [β(t, u t , v t )dW t + E γ(t, u t , v t , e) Ñ (dt, de)]; Z u,v 0 = 1.
The process c(t, u t , v t ) can be interpreted as an instantaneous reward associated with controls u, v. This linear model takes into account some ambiguity on the model via the probability measures Q u,v as well as some ambiguity on the instantaneous reward. This case corresponds to the classical mixed game problems studied in [START_REF] Ph | Simulation of BSDEs by Wiener Chaos Expansion[END_REF] and [START_REF] Quenez | BSDEs with jumps, optimization and applications to dynamic risk measures[END_REF], for which the above analysis provides some alternative short proofs.

• ξ 2 t ≤ ξ 1 t and ζ 2 t ≤ ζ 1 t , 0 ≤ t ≤ T a.s. • g 2 (t, y, z, k) ≤ g 1 (t, y, z, k), for all (y, z, k) ∈ IR 2 × L 2 ν ; dP ⊗ dt -a.s. Let (Y i , Z i , k i , A i , A ′ i ) be the solution of the DRBSDE associated with (ξ i , ζ i , g i ) , i = 1, 2.
Then,

Y 2 t ≤ Y 1 t , 0 ≤ t ≤ T a.
s. Remark 5.5.2. Note that a comparison theorem has been provided in [START_REF] Crépey | Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison[END_REF] in the case of jumps under stronger assumptions.Their proof is different and based on Itô's calculus.

Proof. We give a short proof based on the characterization of solutions of DRBSDEs (Theorem 5.4.8) via generalized Dynkin games. Let t ∈ [0, T ]. For each τ, σ ∈ T t , let us denote by E i .,τ ∧σ (I i (τ, σ)) the unique solution of the BSDE associated with driver g i , terminal time τ ∧ σ and terminal condition

I i (τ, σ) := ξ i τ 1 τ ≤σ + ζ i σ 1 σ<τ for i = 1, 2.
Since g 2 ≤ g 1 , and I 2 (τ, σ) ≤ I 1 (τ, σ), by the comparison theorem for BSDEs, the following inequality

E 2
t,τ ∧σ (I 2 (τ, σ)) ≤ E 1 t,τ ∧σ (I 1 (τ, σ)) a.s. holds for each τ , σ in T t . Hence, by taking the essential supremum over τ in T t and the essential infimum over σ in T t , and by using Theorem 5.4.8, we get

Y 2 t = ess inf σ∈Tt ess sup τ ∈Tt E 2 t,τ ∧σ (I 2 (τ, σ)) ≤ ess inf σ∈Tt ess sup τ ∈Tt E 1 t,τ ∧σ (I 1 (τ, σ)) = Y 1 t a.s.
We now provide a strict comparison theorem. Note that no strict comparison theorem exists in the literature even in the Brownian case. The first assertion addresses the particular case when the non decreasing processes are continuous and the second one deals with the general case.

Theorem 5.5.3 (Strict comparison.). Suppose that the assumptions of Theorem 5.5.1 hold and that the driver g 1 satisfies Assumption 5.4.3 with γ t > -1 in (5.4.2). Let S in T 0 and suppose that

Y 1 S = Y 2 S a.s. 1. Suppose that A i , A ′ i , i = 1, 2 are continuous. For i = 1, 2, let τ i = τ i,S := inf{s ≥ S; A i s > A i S } and σ i = σ i,S := inf{s ≥ S; A ′ i s > A ′ i S }. Then Y 1 t = Y 2 t , S ≤ t ≤ τ 1 ∧ τ 2 ∧ σ 1 ∧ σ 2 a.s. and g 2 (t, Y 2 t , Z 2 t , k 2 t ) = g 1 (t, Y 2 t , Z 2 t , k 2 t ) S ≤ t ≤ τ 1 ∧ τ 2 ∧ σ 1 ∧ σ 2 , dP ⊗ dt -a.s. (5.5.1)
2. Consider the case when A i , A ′ i , i = 1, 2 are not necessarily continuous. For i = 1, 2, define for each ε > 0,

τ ε i := inf{t ≥ S, Y i t ≤ ξ i t + ε} ; σ ε i := inf{t ≥ S, Y i t ≥ ζ i t -ε}.
Setting τi := lim ε↓0 ↑ τ ε i and σi := lim ε↓0 ↑ σ ε i , we have Proof. We adopt the same notation as in the proof of the comparison theorem.

Y 1 t = Y 2 t , S ≤ t < τ1 ∧ τ2 ∧ σ1 ∧ σ2 . a.s. ( 5
1. Suppose first that A i , A ′ i , i = 1, 2 are continuous. By Theorem 5.4.6, for i = 1, 2, (τ i , σ i ) is a saddle point for the game problem associated with g = g i , ξ = ξ i and ζ = ζ i . By Remark 5.4.7,

(Y i t , S ≤ t ≤ τ i ∧ σ i ) is an E i martingale.
Hence we have

Y i t = E i t,τ i ∧σ i (I(τ i , σ i )), S ≤ t ≤ τ i ∧ σ i a.s. Setting θ = τ 1 ∧ τ 2 ∧ σ 1 ∧ σ 2 ,
we thus have

Y i t = E i t,θ (Y i θ ), S ≤ t ≤ θ a.s. for i = 1, 2.
By hypothesis, Y 1 S = Y 2 S a.s. Now, we apply the strict comparison theorem for non reflected BSDEs with jumps (see [START_REF] Quenez | BSDEs with jumps, optimization and applications to dynamic risk measures[END_REF], Th 4.4) for terminal time θ. Hence, we get Y 1 t = Y 2 t , S ≤ t ≤ θ a.s. , as well as equality (5.5.1), which provides the desired result.

2. Consider now the general case. Let ε > 0. By Remark 5.4.10, (Y i t , S ≤ t ≤ τ ε i ∧ σ ε i ) is an E i martingale. Hence we have

Y i t = E i t,τ ε i ∧σ ε i (I(τ ε i , σ ε i )), S ≤ t ≤ τ ε i ∧ σ ε i a.s.
By the same arguments as above with

τ * 1 ,τ * 2 and σ * 1 ,σ * 2 replaced by τ ε 1 ,τ ε 2 and σ ε 1 ,σ ε 2 respectively, we derive Y 1 t = Y 2 t , S ≤ t ≤ τ ε 1 ∧ τ ε 2 ∧ σ ε 1 ∧ σ ε 2
a.s. , and equality (5.5.1) holds on [S,

τ ε 1 ∧ τ ε 2 ∧ σ ε 1 ∧ σ ε 2 ]
, dt ⊗ dP -a.s. By letting ε tend to 0, we obtain the desired result.

We now give an application of the above comparison theorem to a control game problem for DRBSDEs.

Proposition 5.5.4 (Control game problem for DRBSDEs). Suppose that the assumptions of Th. 5.4.16 hold. For each (u, v) ∈ U × V, let Y u,v be the solution of the DRBSDE (5.2.2) associated with driver g u,v . Then, for each

S ∈ T 0 , Y u,v S ≤ Y u,v S ≤ Y u,v S a.s.
Proof. By using Assumption (5.4.18) and by applying the comparison theorem for DRBSDEs (Th.

5.5.1), we get that for each

u ∈ U , Y u,v S ≤ Y u,v S a.s. Similarly, for all v ∈ V, we have Y u,v S ≤ Y u,v S a.s.
Remark 5.5.5. We point out that the above control game problem for DRBSDEs is different from the generalized mixed game problem studied in Section ??. However, from the above proposition, it follows that, under Assumption (5.4.18), the value functions of these two game problems coincide.

A priori estimates with universal constants

Using the characterization of the solution of the nonlinear DRBSDE as the value function of a generalized Dynkin games and DRBSDEs (see Theorem 5.4.8), we prove the following estimates on the spread of the solutions of two DRBSDEs. Proposition 5.5.6. Let ξ 1 , ξ 2 , ζ 1 , ζ 2 ∈ S 2 such that ξ i T = ζ i T a.s. and ξ i t ≤ ζ i t , 0 ≤ t ≤ T a.s. Suppose that for i = 1, 2, ξ i and ζ i satisfy Mokobodski's condition. Let g 1 , g 2 be Lipschitz drivers satisfying Assumption 5.4.3 with common Lipschitz constant C > 0. For i = 1, 2, let Y i be the solution of the DRBSDE associated with driver g i , terminal time T and barriers

ξ i , ζ i . For s ∈ [0, T ], let Y := Y 1 -Y 2 , ξ := ξ 1 -ξ 2 , ζ = ζ 1 -ζ 2 and g s := sup y,z,k |g 1 (s, y, z, k) -g 2 (s, y, z, k)|. Let η, β > 0 be such that β ≥ 3 η + 2C and η ≤ 1 C 2 .
Then for each t, we have:

Y 2 t ≤ e β(T -t) E[sup s≥t ξ s 2 + sup s≥t ζ s 2 |F t ] + ηE[ T t
e β(s-t) g 2 s ds|F t ] a.s.

(5.5.3) Remark 5.5.7. Note that here the constants η and β are universal, i.e. they only depend on the terminal time T and the common Lipschitz constant C. This is not the case for the a priori estimates on DRBSDEs given in the literature (for details see Proposition ?? and Remark 5.7.6 in the Appendix).

Proof. For i = 1, 2 and for each τ, σ ∈ τ 0 , let (X i,τ,σ , π i,τ,σ , l i,τ,σ ) be the solution of the BSDE associated with driver g i , terminal time τ ∧ σ and terminal condition I i (τ, σ), where

I i (τ, σ) = ξ i τ 1 τ ≤σ + ζ i σ 1 σ<τ . Set X τ,σ := X 1,τ,σ -X 2,τ,σ and I τ,σ := I 1 (τ, σ) -I 2 (τ, σ) = ξ τ 1 τ ≤σ + ζ σ 1 σ<τ .
By a priori estimate on BSDEs (see Proposition A.4 in [START_REF] Quenez | Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps[END_REF]), we have a.s.:

(X τ,σ t ) 2 ≤ e β(T -t) E[I(τ, σ) 2 | F t ] + ηE[ T t e β(s-t) [(g 1 -g 2 )(s, X 2,τ,σ s , π 2,τ,σ s , l 2,τ,σ s )] 2 ds | F t ] (5.5.4)
from which we derive that

(X τ,σ t ) 2 ≤ e β(T -t) E[sup s≥t ξ 2 s + sup s≥t ζ 2 s |F t ] + ηE[ T t
e β(s-t) g 2 s ds|F t ] a.s.

(5.5.5) Now, by using inequality (5.4.13), we obtain that for each ε > 0 and for all stopping times τ, σ,

Y 1 t -Y 2 t ≤ X 1,τ ǫ 1 ,σ t -X 2,τ,σ ǫ 2 t + 2Kǫ. Applying this inequality to τ = τ ǫ 1 , σ = σ ǫ 2 we get Y 1 t -Y 2 t ≤ X 1,τ ǫ 1 ,σ ǫ 2 t -X 2,τ ǫ 1 ,σ ǫ 2 t + 2Kǫ ≤ |X 1,τ ǫ 1 ,σ ǫ 2 t -X 2,τ ǫ 1 ,σ ǫ 2 t | + 2Kǫ.
(5.5.6) By (5.5.5) and (5.5.6), we have:

Y 1 t -Y 2 t ≤ e β(T -t) E[sup s≥t ξ s 2 + sup s≥t ζ s 2 |F t ] + ηE[ T t e β(s-t) g 2 s ds|F t ] + 2Kǫ.
By symmetry, the last inequality is also verified by Y 2 t -Y 1 t . The result follows.

Remark 5.5.8. Note that the arguments of the above proof are different from those used in the literature. Based on Theorem 5.4.8, they allow us to obtain universal constants.

We also state the following estimate on the common value function Y of our generalized Dynkin game problem (5.4.4)-(5.4.5) (or equivalently the solution of the DRBSDE associated with driver g). Proposition 5.5.9. For each t, we have:

Y 2 t ≤ e β(T -t) E[sup s≥t ξ s 2 + sup s≥t ζ s 2 |F t ] + ηE[ T t
e β(s-t) g(s, 0, 0, 0) 2 ds|F t ] a.s.

(5.5.7)

Proof. Let X τ,σ t be the solution of the BSDE associated with driver g, terminal time τ ∧ σ and terminal condition I(τ, σ). By applying inequality (5.5.4) with g 1 = g, ξ 1 = ξ, ζ 1 = ζ, g 2 = 0 , ξ 2 = 0 and ζ 2 = 0, we get:

(X τ,σ t ) 2 ≤ e β(T -t) E[I(τ, σ) 2 |F t ] + ηE[ T t e β(s-t) (g(s, 0, 0, 0)) 2 |F t ].
(5.5.8)

By using the same procedure as in the proof of Proposition 5.5.6, the result follows.

Relation with partial integro-differential variational inequalities (PIDVI)

We where ϕ ∈ L 2 ν . For each (t, x) ∈ [0, T ] × IR, let (X t,x s , t ≤ s ≤ T ) be the unique IR-valued solution of the SDE with jumps:

X t,x s = x + s t b(X t,x r )dr + s t σ(X t,x r )dW r + s t E β(X t,x r -, e) Ñ (dr, de),
and set X t,x s = x for s ≤ t. We consider the DRBSDE associated with obstacles ξ t,x , ζ t,x of the following form: ξ t,x s := h 1 (s, X t,x s ), ζ t,x s := h 2 (s, X t,x s ), s < T , ξ t,x T = ζ t,x T := g(X t,x T ). We suppose that g ∈ C(IR), h 1 , h 2 : [0, T ] × IR → IR are continuous with respect to t and Lipschitz continuous with respect to x, uniformly in t and that g, h 1 , h 2 have at most polynomial growth with respect to x. Moreover, the obstacles ξ t,x s and ζ t,x s are supposed to satisfy Mokobodski's condition, which holds if for example h 1 and h 2 are C 1,2 . We consider two functions γ and f satisfying Assumption 2.1 in [START_REF] Barles | Backward stochastic differential equations and integralpartial differential equations[END_REF]. More precisely, we are given a map γ :

IR × E → IR which is B(IR) ⊗ K-measurable, such that |γ(x, e) -γ(x ′ , e)| < C|x -x ′ |ϕ(e) and -1 ≤ γ(x, e) ≤ Cϕ(e) for each x, x ′ ∈ IR, e ∈ E. Let f : [0, T ] × IR 3 × L 2
ν → IR be a map supposed to be continuous in t uniformly with respect to x, y, z, k, uniformly Lipschitz with respect to x, y, z, k uniformly in t, such that f (t, x, 0, 0, 0) at most polynomial growth with respect to x, and such that for each t, x, y, z, k

1 , k 2 , f (t, x, y, z, k 1 ) - f (t, x, y, z, k 2 ) ≥ < γ(x, •), k 1 -k 2 > ν .
The driver is defined by f (s, X t,x s (ω), y, z, k). By Th. 5.4.1, for each (t, x) ∈ [0, T ] × IR, there exists an unique solution (Y t,x , Z t,x , K t,x , A t,x , A ′ t,x ) of the associated DRBSDE. We define:

u(t, x) := Y t,x t , t ∈ [0, T ], x ∈ IR. (5.6.1)
which is a deterministic quantity. In the following, the map u is called the value function of the generalized Dynkin game. By the a priori estimates (see Propositions 5.5.6 and 5.5.9) and the same arguments as those used in the proofs of Lemma 3.1 and Lemma 3.2 in [START_REF] Barles | Backward stochastic differential equations and integralpartial differential equations[END_REF], we derive that the value function u is continuous in (t, x) and has at most polynomial growth at infinity. It follows that the process Y t,x s = u(s, X t,x s ) admits only totally inaccessible jumps. Hence, the processes A t,x , A ′ t,x are continuous. A solution of the obstacle problem is a function u : [0, T ] × IR → IR which satisfies the equality u(T, x) = g(x) and

       h 1 (t, x) ≤ u(t, x) ≤ h 2 (t, x) if u(t, x) < h 2 (t, x) then Hu ≥ 0 if h 1 (t, x) < u(t, x) then Hu ≤ 0 (5.6.2)
where L := A + K and

• Aφ(x) := 1 2 σ 2 (x) ∂ 2 φ ∂x 2 (x) + b(x) ∂φ ∂x (x), Bφ(t, x)(•) := φ(t, x + β(x, •)) -φ(t, x),
• Kφ(x) := E φ(x + β(x, e))φ(x) -∂φ ∂x (x)β(x, e) ν(de),

• Hφ(t, x) := -∂φ ∂t (t, x) -Lφ(t, x)f (t, x, φ(t, x), (σ ∂φ ∂x )(t, x), Bφ(t, x)).

Definition 5.6.1. • A continuous function u is said to be a viscosity subsolution of (5.6.2) if u(T, x) ≤ g(x), x ∈ IR, and if for any point (t 0 , x 0 ) ∈ [0, T ) × IR, we have h 1 (t 0 , x 0 ) ≤ u(t 0 , x 0 ) ≤ h 2 (t 0 , x 0 ) and, for any φ ∈ C 1,2 ([0, T ] × IR) such that φ(t 0 , x 0 ) = u(t 0 , x 0 ) and φu attains its minimum at (t 0 , x 0 ), if u(t 0 , x 0 ) > h 1 (t 0 , x 0 ), then (Hφ)(t 0 , x 0 ) ≤ 0.

• A continuous function u is said to be a viscosity supersolution of (5.6.2) if u(T, x) ≥ g(x), x ∈ IR, and if for any point (t 0 , x 0 ) ∈ [0, T ) × IR, we have h 1 (t 0 , x 0 ) ≤ u(t 0 , x 0 ) ≤ h 2 (t 0 , x 0 ) and, for any φ ∈ C 1,2 ([0, T ] × IR) such that φ(t 0 , x 0 ) = u(t 0 , x 0 ) and φu attains its maximum at (t 0 , x 0 ), if u(t 0 , x 0 ) < h 2 (t 0 , x 0 ) then (Hφ)(t 0 , x 0 ) ≥ 0.

Theorem 5.6.2. The value function u defined by (5.6.1) is a viscosity solution (i.e. both a viscosity sub-and supersolution) of the obstacle problem (5.6.2).

Proof. The proof is given for the convenience of the reader. We prove that u is a viscosity supersolution of (5.6.2), the proof in the case of subsolution being similar.

Let (t 0 , x 0 ) ∈ (0, T ) × R and φ ∈ C 1,2 ([0, T ] × R) be such that φ(t 0 , x 0 ) = u(t 0 , x 0 ) and φ(t, x) ≤ u(t, x), ∀(t, x) ∈ [0, T ] × R. Suppose that u(t 0 , x 0 ) < h 2 (t 0 , x 0 ) and that -∂ ∂t φ(t, x) -Lφ(t, x)g t, x, φ(t, x), (σ ∂φ ∂x )(t, x), Bφ(t, x) < 0.

By continuity, we can suppose that there exists ǫ > 0 and η ǫ > 0 such that: ∀(t, x) such that t 0 ≤ t ≤ t + η ǫ < T and |xx 0 | ≤ η ǫ , we have: u(t, x) ≤ h 2 (t, x)ǫ and -∂ ∂t φ(t, x) -Lφ(t, x)g t, x, φ(t, x), (σ ∂φ ∂x )(t, x), Bφ(t, x) ≤ -ǫ.

(5.6.3) Let θ be the stopping time defined as follows:

θ := (t 0 + η ǫ ) ∧ inf{s ≥ t 0 /|X t 0 ,x 0 ) ≤ u(θ, X t 0 ,x 0 θ ) = Y t 0 ,x 0 θ
. By applying the extended comparison theorem for BSDEs with jumps given in [START_REF] Barles | Backward stochastic differential equations and integralpartial differential equations[END_REF] (Proposition A.3) we get:

φ(t 0 , x 0 ) = φ(t 0 , X t 0 ,x 0 t 0 ) < Y t 0 ,x 0 t 0 = u(t 0 , x 0 ),
which provides a contradiction.

In the sequel, we suppose that the function ϕ is defined by ϕ(e) := 1 ∧ |e| and belongs to L 2 ν . We also suppose that g, h 1 and h 2 are bounded, and that Assumption 4.1 in [START_REF] Barles | Backward stochastic differential equations and integralpartial differential equations[END_REF] holds. More precisely, we assume:

(i) f (s, X t,x s (ω), y, z, k) := f s, X t,x s (ω), y, z, IR * k(e)γ(X t,x s (ω), e)ν(de) 1 s≥t ,
where f : [0, T ] × IR 4 → IR is a map which is continuous with respect to t uniformly in x, y, z, k, and continuous with respect x uniformly in y, z, k. It is also uniformly Lipschitz with respect to y, z, k and the map f (t, x, 0, 0, 0) is uniformly bounded. The map k → f (t, x, y, z, k) is also non-decreasing, for all t ∈ [0, T ], x, y, z ∈ IR.

(ii) For each R > 0, there exists a continuous function m R :

IR + → IR + with m R (0) = 0 and |f (t, x, y, z, k)-f (t, x ′ , y, z, k)| ≤ m R (|x-x ′ |(1+|z|)), for all t ∈ [0, T ], |x|, |x ′ | ≤ R, |y| ≤ R, z, k ∈ IR. (iii) |γ(x, e) -γ(y, e)| ≤ C|x -y|(1 ∧ e 2 ); 0 ≤ γ(x, e) ≤ C(1 ∧ |e|), x, y ∈ IR, e ∈ IR * . (iv) f (t, x, y, z, l) -f (t, x, y, z, l) ≥ r(u -v), u ≥ v, t ∈ [0, T ],
x, u, v, p, l ∈ IR, where r > 0.

To simplify notation, in the sequel, f is denoted by f . The operator B has now the following form: Bφ(x) := IR * (φ(x + β(x, e))φ(x))γ(x, e)ν(de).

Theorem 5.6.3 (Comparison principle). Suppose that Assumptions (i) to (iv). If U is a bounded viscosity subsolution and V is a bounded viscosity supersolution of the obstacle problem (5.6.2), then

U (t, x) ≤ V (t, x), for each (t, x) ∈ [0, T ] × IR.
Proof. The proof is similar to the proof given in [START_REF] Barles | Backward stochastic differential equations and integralpartial differential equations[END_REF] (in the case of one barrier). For the convenience of the reader, we give a sketch of proof, where we draw attention to some points which differ from the proof in [START_REF] Barles | Backward stochastic differential equations and integralpartial differential equations[END_REF]. Set

ψ ǫ,η (t, s, x, y) := U (t, x) -V (s, y) - |x -y| 2 ǫ 2 - |t -s| 2 ǫ 2 -η 2 (|x| 2 + |y| 2 ),
where ǫ, η are small parameters devoted to tend to 0. Let M ǫ,η be a maximum of ψ ǫ,η (t, s, x, y). This maximum is reached at some point (t ǫ,η , s ǫ,η , x ǫ,η , y ǫ,η ). We define:

Ψ 1 (t, x) := V (s ǫ,η , y ǫ,η ) + |x -y ǫ,η | 2 ǫ 2 + |t -s ǫ,η | 2 ǫ 2 + η 2 (|x| 2 + |y ǫ,η | 2 ); Ψ 2 (s, y) := U (t ǫ,η , x ǫ,η ) - |x ǫ,η -y| 2 ǫ 2 - |t ǫ,η -s| 2 ǫ 2 -η 2 (|x ǫ,η | 2 + |y| 2 ).
As (t, x) → (U -Ψ 1 )(t, x) reaches its maximum at (t ǫ,η , x ǫ,η ) and U is a subsolution, we have the two following cases:

• t ǫ,η = T and then

U (t ǫ,η , x ǫ,η ) ≤ g(x ǫ,η ), • t ǫ,η = T , h 1 (t ǫ,η , x ǫ,η ) ≤ U (t ǫ,η , x ǫ,η ) ≤ h 2 (t ǫ,η , x ǫ,η ) and, if U (t ǫ,η , x ǫ,η ) > h 1 (t ǫ,η , x ǫ,η
), we then have:

- ∂Ψ 1 ∂t (t ǫ,η , x ǫ,η ) -LΨ 1 (t ǫ,η , x ǫ,η ) -f t ǫ,η , x ǫ,η , U (t ǫ,η , x ǫ,η ), (σ ∂Ψ 1 ∂x )(t ǫ,η , x ǫ,η ), BΨ 1 (t ǫ,η , x ǫ,η ) ≤ 0.
(5.6.4)

As (s, y) → (Ψ 2 -V )(s, y) reaches its maximum at (s ǫ,η , y ǫ,η ) and V is a supersolution, we have the two following cases:

• s ǫ,η = T and V (s ǫ,η , y ǫ,η ) ≥ g(y ǫ,η ),

• s ǫ,η = T , h 1 (s ǫ,η , y ǫ,η ) ≤ V (s ǫ,η , y ǫ,η ) ≤ h 2 (s ǫ,η , y ǫ,η ) and, if V (s ǫ,η , y ǫ,η ) < h 2 (s ǫ,η , y ǫ,η ) then

- ∂Ψ 2 ∂t (s ǫ,η , y ǫ,η ) -LΨ 2 (s ǫ,η , y ǫ,η ) -f (s ǫ,η , y ǫ,η , V (s ǫ,η , y ǫ,η ), (σ ∂Ψ 2 ∂x )(s ǫ,η , y ǫ,η )), BΨ 2 (s ǫ,η , y ǫ,η ) ≥ 0.
As in [START_REF] Barles | Backward stochastic differential equations and integralpartial differential equations[END_REF], we have:

|x ǫ,η -y ǫ,η | + |t ǫ,η -s ǫ,η | ≤ Cǫ, |x ǫ,η | ≤ C η and |y ǫ,η | ≤ C η .
Extracting a subsequence if necessary, we may suppose that for each η the sequences (t ǫ,η ) ǫ and (s ǫ,η ) ǫ converge to a common limit t η , and the sequences (x ǫ,η ) ǫ and (y ǫ,η ) ǫ converge to a common limit x η . Here, we have to consider four cases. 1st case: there exists a subsequence of (t η ) such that t η = T for all η ( of this subsequence) 2nd case: there exists a subsequence of (t η ) such that t η = T and for all η belonging to this subsequence, there exist a subsequence of (x ǫ,η ) ǫ and a subsequence of (t ǫ,η ) ǫ , such that U (t ǫ,η , x ǫ,η )h 1 (t ǫ,η , x ǫ,η ) = 0. 3rd case: there exists a subsequence such that t η = T , and for all η belonging to this subsequence, there exist a subsequence of (y ǫ,η ) ǫ and a subsequence of (s ǫ,η ) ǫ , such that V (s ǫ,η , y ǫ,η )h 2 (s ǫ,η , y ǫ,η ) = 0. Last case: we are left with the case when, for a subsequence of η we have t η = T , and for all η belonging to this subsequence, there exists a subsequence of (x ǫ,η ) ǫ , (y ǫ,η ) ǫ , (t ǫ,η ) ǫ and (s ǫ,η ) ǫ such that

U (t ǫ,η , x ǫ,η ) -h 1 (t ǫ,η , x ǫ,η ) > 0; h 2 (s ǫ,η , y ǫ,η ) -V (s ǫ,η , y ǫ,η ) > 0.
We are thus in the case when the solution if strictly between the barriers, that is when there is no reflection. We can then use the same arguments as in the case of one barrier when there is no reflection. For convenience of the reader, we recall below the main arguments. We argue by contradiction by assuming that M > 0. We set

ϕ(t, s, x, y) := |x -y| 2 ǫ 2 + |t -s| 2 ǫ 2 + η 2 (|x| 2 + |y| 2 ).
(5.6.5)

We know that he maximum of the function ψ ǫ,η := U (t, x) -V (s, y)ϕ(t, s, x, y) is reached at the point (t ǫ,η , s ǫ,η , x ǫ,η , y ǫ,η ). We can thus apply the non-local version of Jensen Ishii's lemma in [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions theory revisited[END_REF], which leads to the desired result, by using exactly the same arguments as in [START_REF] Barles | Backward stochastic differential equations and integralpartial differential equations[END_REF] (see Theorem 4.1, last case). Note that the first, second and fourth case are identical to the three cases considered for reflected BSDEs (see [START_REF] Barles | Backward stochastic differential equations and integralpartial differential equations[END_REF]). The third one, which didn't appear in the case of reflected BSDEs, can be treated similarly to the second one.

We derive that under Assumptions (i) to (iv), there exists an unique solution of the obstacle problem (5.6.2) in the class of bounded continuous functions.

Appendix

Remark 5.7.1. Note that L 2 ν is a separable Hilbert space. Indeed, by a result of Measure Theory (see e.g. Proposition 3.4.5 of Cohn's book on Measure Theory [START_REF] Cohn | Measure Theory[END_REF]), given a measurable space where γ(α, k 1 , k 2 ) :=

1 0 ∇ k f (α, k 1 + t(k 2 -k 1 ))dt. Here, for each continuous map F : [0, 1] → L 2 ν ; t → F (t), the integral 1 0 F (t)dt is defined as 1 0 F (t)dt := i∈N ( 1 0 F i (t)dt)e i .
The desired result follows.

2. Suppose f is convex. By Proposition 5.4 in [START_REF] Ekeland | Convex Analysis and Variational Problems[END_REF], since f is convex and Gâteaux-differentiable, f is sub-differentiable. By Proposition 5.3 in [START_REF] Ekeland | Convex Analysis and Variational Problems[END_REF], the Gâteaux-gradiant ∇ g k f (α, k) coincides with the sub-differential at k. Hence, for each k, h in L 2 ν , we have:

f (α, k+h) ≥ f (α, k) + < ∇ g k f (α, k), h > ν .
By definition of the Gâteaux-gradiant (see Definition 5.2. in [START_REF] Ekeland | Convex Analysis and Variational Problems[END_REF]), we have that for each

i ∈ N, < ∇ g k f (α, k), e i > ν = lim t→0 f (α, k + te i ) -f (α, k) t .
Setting γ(α, k 1 , k 2 ) := ∇ g k f (α, k 1 ), the result follows. Suppose f is concave. By applying the previous property to the convex map -f and with (k

2 , k 1 ) instead of (k 1 , k 2 ), we get -f (α, k 1 ) + f (α, k 2 ) ≥ < -∇ g k f (α, k 2 ) , k 1 -k 2 > ν , for each (α, k 1 , k 2 ) ∈ X × (L 2 ν ) 2 . Setting γ(α, k 1 , k 2 ) := ∇ g k f (α, k 2 ), the result follows. 3. Setting γ(α, k 1 , k 2 ) := Cψ(e)1 { E (k 2 (e)-k 1 (e))
ψ(e)ν(de) ≤ 0} , the result follows.

Proof of Lemma 5.3.3: The results of this lemma can be derived from the results of [START_REF] Kobylanski | Dynkin games in a general framework[END_REF] obtained in the general framework of admissible families of random variables indexed by stopping times. We give here a sketch of the proof. Set J (0) • = 0 and J ′(0) • = 0, and define recursively for each n ∈ N, the RCLL supermartingale processes:

J (n+1) • := R(J ′(n) • + ξg • ); J ′(n+1) • := R(J (n) • -ζg • ) (5.7.2)
which belong to S 2 , where R is the classical Snell envelop operator. For sake of simplicity, in the above definition we have omitted the exponent g in the definition of J (n) . Since ξg T = ζg T = 0 a.s. , it follows that, for each n, J

(n+1) • , J ′ (n+1) • are well defined since (J ′ (n) • + ξg • ) -and (J (n) • -ζg • ) -belong to S 2 . Also, J (n+1) t ≥ E[J ′ (n) T + ξg T |F t ] ≥ 0 a.s. since ξg T = 0 a.s. Similarly, because ζg T = 0 a.s., J ′ (n+1) t ≥ 0 a.s. By classical results, J (n) • and J ′ (n) • are RCLL supermartingales. Let us prove that J (n) • and J ′ (n) •
are non decreasing sequences of processes. We have

J (1) • ≥ 0 = J (0) • and J ′ (1) • ≥ 0 = J ′ (0) • . Suppose that J (n) • ≥ J (n-1) • and J ′ (n) • ≥ J ′ (n-1) • . We then have: R(J ′(n) • + ξg • ) ≥ R(J ′ (n-1) • + ξg • ); R(J (n) • -ζg • ) ≥ R(J (n-1) • -ζg • ), (5.7.3)
which leads to

J (n+1) • ≥ J (n) • and J ′ (n+1) • ≥ J ′ (n) •
. Let us introduce the following optional processes valued in [0, +∞] defined by J g

• := lim ↑ J (n) • and J ′g • := lim ↑ J ′(n) • . Since for each n, J (n) T = J ′(n) T = 0 a.s. we have J g T = J ′ g
T = 0 a.s. By the monotone convergence theorem, one can show that J g

• and J ′g • are strong supermartingales. We now show equalities (5.3.5). In the following, we use the Snell envelope operator R which acts on admissible families of random variables (r.v.). The reader is referred to Section 1.1 in [START_REF] Kobylanski | Dynkin games in a general framework[END_REF] for the definition of an admissible family of r.v. indexed by stopping times, as well as the definition of a supermartingale family. Recall that for each admissible family φ = (φ(θ)) θ∈T 0 valued in R ∪ {+∞} with E [ess sup θ∈T φ(θ) -] < +∞, R(φ) is defined as the smallest supermartingale family greater than φ. Note that by some results of optimal stopping (see Section 1.1 in [START_REF] Kobylanski | Dynkin games in a general framework[END_REF]), we have

R(φ)(θ) = ess sup τ ∈T θ E[φ(τ ) | F θ ] a.s. (5.7.4)
for each stopping time θ. In the following, for each optional process φ • = (φ t ) 0≤t≤T valued in R ∪ {+∞}, we denote by φ its associated family of r.v. defined by

φ := (φ θ ) θ∈T 0 . If φ • ∈ S 2 , we then have R(φ)(θ) = ess sup τ ∈T θ E[φ τ | F θ ] = R(φ • ) θ a.s. (5.7.5)
for each stopping time θ. This property and equalities (5.7.3) lead to the following equalities written in terms of families and the operator R:

J (n+1) := R(J ′(n) + ξg ); J ′(n+1) := R(J (n) -ζg )
As the operator R is nondecreasing, for each n ∈ IN , we have

J (n+1) = R(J ′ (n) + ξg ) ≤ R(J ′ g + ξg )
. By letting n tend to +∞, we get that

J g ≤ R(J ′ g + ξg ). (5.7.6)
Now, for each n ∈ N, J (n+1) ≥ J ′ (n) + ξg . By letting n tend to +∞, we derive that J g ≥ J ′ g + ξg . By the supermartingale property of the family of r.v. J g = (J g θ ) θ∈T 0 and the characterization of R(J ′ g + ξg ) as the smallest supermartingale family greater than J ′ g + ξg , it follows that J g ≥ R(J ′ g + ξg ). This with (5.7.6) yields that J g = R J ′ g + ξ . Similarly, J ′ g = R(J gζg ), which, by the property (5.7.4), leads to the desired equalities (5.3.5). Note that the supermartingale property of the families J g and J ′ g corresponds to the strong supermartingale property of the optional processes J g

• and J ′ g

• . We have

J g • ≥ J ′ g • + ξg • and J ′ g • ≥ J g • -ζg • .
The proof of the minimality of J g • and J

′ g

• follows from Proposition 5.1 in [START_REF] Kobylanski | Dynkin games in a general framework[END_REF].

Moreover, if J g 0 < +∞ and J ′ g 0 < +∞, by Th.18 ch. VI in [START_REF] Dellacherie | Probabilités et Potentiel, Théorie des Martingales[END_REF], J g • and J ′g • are indistinguishable from nonnegative RCLL supermartingales, as the non decreasing limits of nonnegative RCLL supermartingales.

Remark 5.7.3. The property ξg T = ζg T = 0 a.s. ensures that for each n, J

T = J ′(n) T (n)
= 0 a.s. Note that if we had not made the change of variable (5.3.4), then ξg , ζg would be replaced by ξ, ζ in the definitions of J (n) and J ′ (n) . In that case, ξ T = ζ T a.s. but would not necessarily be equal to 0, and we would have

J (n) T = -J ′(n) T = 0 a.s. if n is even,

ensures that

B = A. Indeed, set H t := E[A T -A t |F t ] (resp. H ′ t := E[A ′ T -A ′ t |F t ]). Since dA t << dB t (resp. dA ′ t << dB ′ t ), we have H t ≤ J t = E[B T -B t |F t ] (resp. H ′ t ≤ J ′ t = E[B ′ T -B ′ t |F t ]
). Moreover, H -H ′ = J -J ′ . Hence, we have H ≥ H ′ + ξg and H ′ ≥ H -ζg . By the minimality property of J, J ′ (5.3.6), we derive that J = H (resp. J ′ = H ′ ).

Proof of Theorem 5.3.5: Theorem 5.3.4 gives the existence. Let (Y, Z, k, A, A ′ ) be a solution of the DRBSDE associated with driver process g(t) and obstacles (ξ, ζ). Let us prove that it is unique. We first show the uniqueness of Y . For each S ∈ T 0 and for each ε > 0, let

τ ε S := inf{t ≥ S, Y t ≤ ξ t + ε} σ ε S := inf{t ≥ S, Y t ≥ ζ t -ε}. (5.7.7)
Note that σ ε S and τ ε S ∈ T S . Fix ε > 0. By the same arguments as in the proof of Lemma 4.8, the function t → A t is constant a.s. on [S, τ ε S ] and

Y τ ε S ≤ ξ τ ε S + ε a.s. Similarly, A ′ is constant on [S, σ ε S ] and Y σ ε S ≥ ζ σ ε S -ε a.s. Let τ ∈ T S . Since A ′ is constant on [S, σ ε S ], the process (Y t + t 0 g(s)ds, S ≤ t ≤ τ ∧ σ ε S ) is a supermartingale. Hence Y S ≥ E[Y τ ∧σ ε S + τ ∧σ ε S S g(s)ds | F S ] a.s.
We also have that

Y τ ∧σ ε S = Y τ 1 τ ≤σ ε S + Y σ ε S 1 σ ε S <τ ≥ ξ τ 1 τ ≤σ ε S + (ζ σ ε S -ε)1 σ ε S <τ a.s. We get Y S ≥ E[I S (τ, σ ε S ) | F S ] -ε a.s. Similarly, one can show that for each σ ∈ T S , Y S ≤ E[I S (τ ε S , σ) | F S ] + ε a.s. It follows that for each ε > 0, ess sup τ ∈Ts E[I S (τ, σ ε S ) | F S ] -ε ≤ Y S ≤ ess inf σ∈T S E[I S (τ ε S , σ) | F S ] + ε a.s., that is V (S) -ε ≤ Y S ≤ V (S) + ε a.s. Since V (S) ≤ V (S) a.s. we get V (S) = Y S = V (
S) a.s. This equality holds of each stopping time S ∈ T 0 , which implies the uniqueness of Y . It remains to show the uniqueness of (Z, k, A, A ′ ). By the uniqueness of the decomposition of the semimartingale Y t + t 0 g(s)ds, there exists an unique square integrable martingale M and an unique square integrable finite variation RCLL adapted process α with α 0 = 0 such that dY t + g(t)dt = dM tdα t . The martingale representation theorem applied to M ensures the uniqueness of the pair (Z, k) ∈ IH 2 × IH 2 ν . The uniqueness of the processes A, A ′ follows from the uniqueness of the canonical decomposition of an RCLL process with integrable variation (see Proposition 5.7.9).

Suppose Proof of Proposition 5.3.10. Since J g ≥ J ′ g + ξg and

J ′ g ≥ J g -ζg , J g ∈ S 2 ⇔ J ′ g ∈ S 2
. Using the minimality property of J and J ′ given in Lemma 5.3.3, one can show that J g ∈ S 2 if and only if there exist two non-negative supermartingales

H g , H ′ g ∈ S 2 such that ξg t ≤ H g t -H ′ g t ≤ ζg t 0 ≤ t ≤ T a.s. ( 5.7.8)
Since this equivalence holds for all g ∈ IH 2 , in particular when g = 0, we get (ii) ⇔ (iii).

To prove (i) ⇔ (ii), it is sufficient to show that (5.3.11) is equivalent to (5.7.8). Suppose that (5.3.11) is satisfied. By setting

H g t := H t -E[ξ + T ds|F t ] -E[ T t g + (s)ds|F t ], 0 ≤ t ≤ T H ′ g t := H ′ t -E[ξ - T ds|F t ] -E[ T t g -(s)ds|F t ], 0 ≤ t ≤ T, (5.7.8)
holds. Similarly, (5.7.8) implies (5.3.11). We have that (i) implies (iv). It remains to prove that (iv) implies (i). Let (Y, Z, k, A, A ′ ) be the solution of the DRBSDE (5.2.2) associated with driver process (g t ). Let

H g t := E[A T -A t |F t ] and H ′ g t := E[A ′ T -A ′ t |F t ]. We have H g t -H ′ g t = Y t -E[ T t g(s)ds|F t ]. Since ξ ≤ Y ≤ ζ,
IH 2 × IH 2 × IH 2 ν equipped with the norm Y, Z, k(•) 2 β := Y 2 β + Z 2 β + k 2 ν,β .
We define a mapping Φ from IH 2 β into itself as follows. Given (U, V, l) ∈ IH 2 β , by Theorem 5.3.5 there exists a unique process (Y, Z, k) = Φ(U, V, l) solution of the DRBSDE associated with driver process g(s) = g(s, U s , V s , l s ). Note that (Y, Z, k) ∈ IH 2 β . Let A, A ′ be the associated non decreasing processes. Let us show that Φ is a contraction and hence admits a unique fixed point (Y, Z, k) in IH 2 β , which corresponds to the unique solution of DRBSDE (5.2.2). The associated finite variation process is then uniquely determined in terms of (Y, Z, k) and the pair (A, A ′ ) corresponds to the unique canonical decomposition of this finite variation process. Let (U 2 , V 2 , l 2 ) be another element of

IH 2 β and define (Y 2 , Z 2 , k 2 ) = Φ(U 2 , V 2 , l 2 ). Let A 2 , A ′2 be the associated non decreasing processes. Set U = U -U 2 , V = V -V 2 , l = l -l 2 and, Y = Y -Y 2 , Z = Z -Z 2 , k = k -k 2 .
By Itô's formula, for any β > 0, we have

Y 2 0 + E T 0 e βs [βY 2 s + Z 2 s + k 2 s ]ds + E 0<s≤T e βs (∆A s -∆A 2 s -∆A ′ s + ∆A ′ 2 s ) 2 = 2E T 0 e βs Y s [g(s, U s , V s , l s ) -g(s, U 2 s , V 2 s , l 2 s )] ds + 2E[ T 0 e βs Y s -dA s - T 0 e βs Y s -dA 2 s ] -2E[ T 0 e βs Y s -dA ′ s - T 0 e βs Y s -dA ′2 s ].
(5.7.9)

Now, we have a.s.

Y s dA c s = (Y s -ξ s )dA c s -(Y 2 s -ξ s )dA c s = -(Y 2 s -ξ s )dA c s ≤ 0
and by symmetry, Y s dA 2c s ≥ 0 a.s. Also, we have a.s.

Y s -∆A d s = (Y s --ξ s -)∆A d s -(Y 2 s --ξ s -)∆A d s = -(Y 2 s --ξ s -)∆A d s ≤ 0 and Y s -∆A 2 d
s ≥ 0 a.s. Similarly, we have a.s.

Y s dA ′ c s = (Y s -ζ s )dA ′ c s -(Y 2 s -ζ s )dA ′ c s = -(Y 2 s -ζ s )dA ′ c s ≥ 0
and by symmetry, Y s dA ′ 2c s ≤ 0 a.s. Also, we have a.s.

Y s -∆A ′ d s = (Y s --ζ s -)∆A ′ d s -(Y 2 s --ζ s -)∆A ′ d s = -(Y 2 s --ζ s -)∆A ′ d s ≥ 0 and Y s -∆A ′ 2 d
s ≤ 0 a.s. Consequently, the second and the third term of (5.7.9) are non positive. By using the Lipschitz property of g and the inequality 2Cyu ≤ 2C 2 y 2 + 1 2 u 2 , we get

β Y 2 β + Z 2 β + k 2 ν,β ≤ 6C 2 Y 2 β + 1 2 ( U 2 β + V 2 β + l 2 ν,β ). Choosing β = 6C 2 + 1, we deduce (Y , Z, k) 2 β ≤ 1 2 (U , V , l) 2 β .
The last assertion of the theorem follows from Theorem 5.3.7.

By similar arguments as above, one can show the following estimate, which is expressed in terms of the associated increasing processes.

Proposition 5.7.5 (A classical estimate). Let ξ 1 , ξ 2 , ζ 1 , ζ 2 ∈ S 2 such that for i = 1, 2, ξ i T = ζ i T a.s. and ξ i t ≤ ζ i t , 0 ≤ t ≤ T a.s.
Suppose that for i = 1, 2, ξ i and ζ i satisfy Mokobodski's condition. Let g 1 , g 2 be Lipschitz drivers satisfying Assumption 6.3.9 with Lipschitz constant C > 0.

For i = 1, 2, let (Y i , Z i , k i , A i , A ′ i
) be the solution of the DRBSDE associated with driver g i , terminal time T and barriers

ξ i , ζ i . For s ∈ [0, T ], let Y s := Y 1 s -Y 2 s , ξ s := ξ 1 s -ξ 2 s , ζ s = ζ 1 s -ζ 2 s , g(s) := g 1 (s, Y 2 s , Z 2 s , k 2 s ) - g(s, Y 2 s , Z 2 s , k 2 s ).
We have:

Y 2 S 2 ≤ K E[ξ 2 T ] + E[ T 0 g 2 (s)ds] + A 1 T + A 2 T L 2 sup 0≤s<T |ξ s | L 2 + A ′ 1 T + A ′ 2 T L 2 sup 0≤s<T |ζ s | L 2 , (5.7.10)
where the real constant K > 0 is universal, that is, depends only on C and T .

For the proof, see proof of Th. 3.2 in Appendix of [START_REF] Crépey | Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison[END_REF].

Remark 5.7.6. In [START_REF] Crépey | Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison[END_REF], in the particular case when for each i = 1, 2, the lower barrier ξ i is of the form ξ i = M i + B i , where M i is a square integrable martingale and B i is a square integrable RCLL predictable non decreasing process with B i 0 = 0, the authors derive from (5.7.10) the following estimate:

Y 2 S 2 ≤ K E[ξ 2 T ] + E[ T 0 g 2 (s)ds] + φ( sup 0≤s<T |ξ s | L 2 + sup 0≤s<T |ζ s | L 2 ) , (5.7.11)
where the constant φ > 0 is not necessarily universal, depending in particular on ξ i S 2 , ζ i S 2 , g i (s, 0, 0, 0) IH 2 and B i , for i = 1, 2. For details, the reader is refered to estimate (14) of Th. 3.2 in [START_REF] Crépey | Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison[END_REF] proven in the Appendix.

We now easily show an E g -Doob-Meyer decomposition of E g -supermartingales, which generalizes the results given in [START_REF] Peng | Backward SDE and related g-expectation, Backward stochastic differential equations[END_REF] and [?] under stronger assumptions. Moreover, our proof gives an alternative short proof of these previous results.

Definition 5.7.7. Let Y ∈ S 2 . The process (Y t ) is said to be a strong E g -supermartingale (resp E g -submartingale), if E g σ,τ (Y τ ) ≤ Y σ (resp. E g σ,τ (Y τ ) ≥ Y σ ) a.
s. on σ ≤ τ , for all σ, τ ∈ T 0 . Proposition 5.7.8. Suppose that g satisfies Assumption (5.4.3).

• Let A be a non decreasing (resp non increasing) RCLL predictable process in S 2 with A 0 = 0.

Let (Y, Z, k) ∈ S 2 × H 2 × H 2 ν following the dynamics:

-dY s = g(s, Y s , Z s , k s )ds + dA s -Z s dW s -E k s (e) Ñ (ds, de).

(5.7.12)

Then the process (Y t ) a strong E g -supermartingale (resp E g -submartingale).

• (E g -Doob-Meyer decomposition) Let (Y t ) be a strong E g -supermartingale (resp. E g -submartingale).

Then, there exists a non decreasing (resp non increasing) RCLL predictable process A in S 2 with A 0 = 0 and (Z, k) ∈ IH 2 × IH 2 ν such that (5.7.12) holds.

Proof. Suppose A is non decreasing. Let (X τ , π τ , l τ ) be the solution of the BSDE associated with driver g, terminal time τ , and terminal condition Y τ , Since g satisfies Assumption 5.4.3 and since g(s, y, z, k)ds+dA s ≥ g(s, y, z, k)ds, the comparison theorem for BSDEs (see Theorem 4.2 in [START_REF] Quenez | BSDEs with jumps, optimization and applications to dynamic risk measures[END_REF]) gives that Y σ ≥ X τ σ = E g σ,τ (Y τ ) a.s. on {σ ≤ τ }. The case when A is non-increasing can be shown similarly. Let us show the second assertion. Fix S ∈ T 0 . Since (Y t ) is a strong E g -supermartingale, we derive that for each τ ∈ T S , we have Y S ≥ E g S,τ (Y τ ) a.s. We thus get Y S ≥ ess sup τ ∈T S E g S,τ (Y τ ) a.s. Now, by definition of the essential supremum, Y S ≤ ess sup τ ∈T S E g S,τ (Y τ ) a.s. because S ∈ T S . Hence,

Y S = ess sup τ ∈T S E g S,τ (Y τ ) a.s.
By Theorem 3.3 in [START_REF] Quenez | Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps[END_REF], the process (Y t ) coincides with the solution of the reflected BSDE associated with the RCLL obstacle (Y t ). The result follows.

We now show the following result on RCLL adapted processes with integrable total variation, which can be seen as a probabilistic version of a well-known analysis result.

Proposition 5.7.9. Let (Ω, F, P ) be a probability space equipped with a completed right-continuous filtration (F t ) 0≤t≤T . Let α = (α t ) 0≤t≤T be a RCLL adapted process with integrable total variation, that is, E(

T 0 |dα t |) < ∞.
There exists an unique pair (A, A ′ ) ∈ (A 1 ) 2 such that α = A -A ′ with dA t ⊥ dA ′ t . This decomposition is called the canonical decomposition of the process α. Moreover, if (B, B ′ ) ∈ (A 1 ) 2 satisfies α = B -B ′ , then dA t << dB t in the (probabilistic) sense, that is, for each K ∈ P with T 0 1 K dB t = 0 a.s. , then

T 0 1 K dA t = 0 a.s.
Proof. By classical results, the process α can be written as

α = B -B ′ with B, B ′ ∈ A 1 . Let C t := B t +B ′
t . This process belongs to A 1 . For almost every ω, the measures dB • (ω) and dB ′ • (ω) on [0, T ] are absolutely continuous with respect to dC • (ω). By using the Radon-Nikodym Theorem for predictable RCLL non decreasing processes (see Th. 67, Chap. VI in [START_REF] Dellacherie | Probabilités et Potentiel, Théorie des Martingales[END_REF]), there exist nonnegative predictable processes H and H ′ such that for each t ∈ [0, T ],

B t = t 0 H s dC s and B ′ t = t 0 H ′ s dC s a.s
Let A and A ′ be the processes defined by

A t := t 0 (H s -H ′ s ) + dC s and A ′ t := t 0 (H s -H ′ s ) -dC s .
They belong to A 1 . Now, the set D defined by

D := {(t, ω) , H t (ω) -H ′ t (ω) ≥ 0}
belongs to P. We have

T 0 1 D c t dA t = T 0 1 {Ht-H ′ t <0} (H t -H ′ t ) + dC t = 0 a.s. Similarly T 0 1
Dt dA ′ t = 0 a.s. , which implies that dA t ⊥ dA ′ t . It remains to show the uniqueness of this decomposition. Since dA t ⊥ dA ′ t , it follows that, for almost every ω, the deterministic measures dA t (ω) and dA ′ t (ω) are mutually singular in the classical analysis sense. Hence, for almost every ω, the non decreasing maps A • (ω) and A ′ • (ω) correspond to the unique canonical decomposition of the RCLL bounded variational map α.(ω) by a well-known analysis result. This implies the uniqueness of A, A ′ . Moreover, since (H t -H ′ t ) + ≤ H t , the last assertion holds.

Remark 5.7.10. Note that it is obvious that, if the random measures dA t and dA ′ t are mutually singular in the probabilistic sense (see Definition 5.2.2), then for almost every ω, the deterministic measures on [0, T ] dA t (ω) and dA ′ t (ω) are mutually singular in the classical analysis sense. The converse is not so immediate. However, it holds by the above property. Abstract. We study combined optimal control/stopping problems with E f -expectations in the Markovian framework on a finite horizon of time T . We establish a weak dynamic programming principle, which extends to the nonlinear case the one obtained in [START_REF] Bouchard | Weak Dynamic Programming Principle for Viscosity Solutions[END_REF] in the case of linear expectations . To this purpose, we prove some measurability properties and a "splitting" result stating that, given an intermediary time t ≤ T , the problem can be decomposed into two independent parts, one corresponding to the past (before t) and one to the future (after t). Using this weak dynamic programming principle and properties of reflected backward stochastic differential equations, we prove that the value function of our combined control problem, which is not necessarily continuous, not even measurable, is a weak viscosity solution of a nonlinear Hamilton-Jacobi-Bellman variational inequality.

Some illustrating examples in mathematical finance are provided.

Introduction

Markovian stochastic control problems on a given horizon of time T can typically be written as

u(0, x) = sup α∈A E[ T 0 f (α s , X α s )ds + g(X α T )], (6.1.1)
where A is a set of admissible control processes α s , and (X α s ) is a controlled process of the form

X α s = x + s 0 b(X α u , α u )du + s 0 σ(X α u , α u )dW u + s 0 R n β(X α u , α u , e) Ñ (du, de).
The random variable g(X α T ) may represent a terminal reward and f (α s , X α s ) an instantaneous reward process. Formally, for all initial time t in [0, T ] and initial state y, the associated value function is defined by

u(t, y) = sup α∈A E[ T t f (α s , X α s )ds + g(X α T ) | X α t = y]. (6.1.2)
The dynamic programming principle can formally be stated as

u(0, x) = sup α∈A E[ t 0 f (α s , X α s )ds + u(t, X α t )], for t in [0, T ]. (6.1.3)
In the literature, this principle is classically established under assumptions which ensure that the value function u satisfies some regularity/ measurability properties. We refer e.g. to the books by Fleming-Rischel (1975) [START_REF] Fleming | Deterministic and Stochastic Optimal Control[END_REF], [START_REF] Krylov | Controlled Diffusion Processes[END_REF] [106], El [START_REF] El Karoui | Les aspects probabilistes du contrôle stochastique, École d'été de Probabilités de Saint-Flour IX[END_REF] [START_REF] El Karoui | Les aspects probabilistes du contrôle stochastique, École d'été de Probabilités de Saint-Flour IX[END_REF], Bensoussan-Lions J. (1988), Lions P.-L. (1983) [START_REF] Lions | Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, I,II,III[END_REF], [START_REF] Soner | Controlled Markov Processes and Viscosity solutions[END_REF] [START_REF] Soner | Controlled Markov Processes and Viscosity solutions[END_REF], [START_REF] Oksendal | Applied Stochastic Control of Jump Diffusions[END_REF] [START_REF] Oksendal | Applied Stochastic Control of Jump Diffusions[END_REF], [START_REF] Pham | Continuous-time Stochastic Control and Optimization with Financial Applications[END_REF] [START_REF] Peng | Monotonic limit theorem of BSDE and nonlinear decomposition theorem of doobmeyer's type[END_REF].

The case of a discontinuous value function has been studied in a deterministic framework in the eighties: a weak dynamic programming principle has been established for deterministic control by [START_REF] Barles | Discontinuous viscosity solutions of first-order Hamilton-Jacobi equations: a guided visit[END_REF] (see [START_REF] Barles | Discontinuous viscosity solutions of first-order Hamilton-Jacobi equations: a guided visit[END_REF]) (see also [START_REF] Barles | Discontinuous viscosity solutions of deterministic optimal control problems Decision and Control[END_REF] [START_REF] Barles | Discontinuous viscosity solutions of deterministic optimal control problems Decision and Control[END_REF]). More precisely, he proves that the upper semicontinuous envelope u * and the lower semicontinuous envelope u * of the value function u satisfy, respectively, the sub-and super-optimality principle of dynamic programming of [START_REF] Lions | Differential Games, Optimal Control and Directional Derivatives of Viscosity Solutions of Bellman's and Isaacs' Equations[END_REF] [START_REF] Lions | Differential Games, Optimal Control and Directional Derivatives of Viscosity Solutions of Bellman's and Isaacs' Equations[END_REF]. He then derived that the (discontinuous) value function is a weak viscosity solution of the associated Bellman equation in the sense that u * is a viscosity subsolution and u * is a supersolution of the Bellman equation.

More recently, Bouchard and Touzi (2011) (see [START_REF] Bouchard | Weak Dynamic Programming Principle for Viscosity Solutions[END_REF]) have proved a weak dynamic programming principle in a stochastic framework, when the value function is not necessarily continuous, not even measurable. They prove that the upper semicontinuous envelope u * satisfies the suboptimality principle of dynamic programming, and under an additional regularity (lower semi continuity) assumption of the reward g, they obtain that the lower semicontinuous envelope u * satisfies the super-optimality principle.

A weak dynamic principle has been further established, under some specific regularity assumptions, for problems with state constraints by [START_REF] Bouchard | Weak Dynamic Programming for Generalized State Constraints[END_REF] in [START_REF] Bouchard | Weak Dynamic Programming for Generalized State Constraints[END_REF], and for zero-sum stochastic games by [START_REF] Bayraktar | A Weak Dynamic Programming Principle for Zero-Sum Stochastic Differential Games with Unbounded Controls[END_REF] in [START_REF] Bayraktar | A Weak Dynamic Programming Principle for Zero-Sum Stochastic Differential Games with Unbounded Controls[END_REF].

In this chapter we are interested in generalizing these results to the case when the linear expectation E is replaced by a nonlinear expectation induced by a Backward Stochastic Differential Equation (BSDE). Typically, such problems in the Markovian case can be formulated as

sup α∈A E α 0,T [g(X α T )], (6.1.4)
where E α is the nonlinear conditional expectation associated with a BSDE with jumps with controlled driver f (α t , X α t , y, z, k). Note that Problem (6.1.1) is a particular case of (6.1.4) when the driver f does not depend on the solution of the BSDE, that is when f (α t , X α t , y, z, k) ≡ f (α t , X α t ). We study here the case when the reward function g is only Borelian. We provide a weak dynamic programming principle. To this purpose, we prove some measurability properties and a "splitting" result stating that, given an intermediary time t ≤ T , the problem can be decomposed into two independent parts, one corresponding to the past (before t) and one to the future (after t). We point out that no regularity conditions on the reward g are required to obtain the sub and super-optimality principles, which is not the case in the previous literature even in the linear case (see [START_REF] Bouchard | Weak Dynamic Programming Principle for Viscosity Solutions[END_REF], [START_REF] Bouchard | Weak Dynamic Programming for Generalized State Constraints[END_REF] and [START_REF] Bayraktar | A Weak Dynamic Programming Principle for Zero-Sum Stochastic Differential Games with Unbounded Controls[END_REF]). Using this weak dynamic programming principle, we derive that the value function u, which is not necessarily continuous, not even measurable, is a weak viscosity solution of an associated nonlinear Hamilton-Jacobi-Bellman (HJB) equation.

Moreover, in this chapter, we consider the combined problem when there is an additional control in the form of a stopping time. We thus consider mixed generalized optimal control/stopping problems of the form

sup α∈A sup τ ∈T E α 0,τ [ h(τ, X α τ )], (6.1.5)
where T denotes the set of stopping times with values in [0, T ], and h is an irregular reward function.

The chapter is organized as follows: in Section 6.2, we introduce our generalized mixed controloptimal stopping problem. Using results on reflected BSDEs, we express this problem as an optimal control problem for reflected BSDEs. In Section 6.3, we prove a weak dynamic programming principle for our mixed problem with f -expectation. This requires some specific techniques of stochastic analysis and BSDEs to handle measurability and other issues due to the nonlinearity of the expectation and the lack of regularity of the terminal reward.

Using the dynamic programming principle and properties of reflected BSDEs, we prove in Section 6.4 that the value function of our mixed problem is a weak viscosity solution of a nonlinear Hamilton-Jacobi-Bellman (HJB) variational inequality. In Section 6.5, we give illustrating examples in mathematical finance such as optimization problems involving recursive utilities and dynamic risk measures.

Formulation of the problem

We consider the product space Ω := Ω W ⊗ Ω N , where Ω W := C([0, T ]) is the Wiener space, that is the set of continuous functions ω 1 from [0, T ] into R p such that ω 1 (0) = 0, and Ω N := D([0, T ]) is the Skorohod space of right-continuous with left limits (RCLL) functions ω 2 from [0, T ] into R d , such that ω 2 (0) = 0. Recall that Ω is a Polish space for the topology of Skorohod. Here p, d ≥ 1, but for notational simplicity, however, we shall consider only R-valued functions, that is the case

p = d = 1. Let B = (B 1 , B 2 ) be the canonical process defined for each t ∈ [0, T ] and each ω = (ω 1 , ω 2 ) by B i t (ω) = B i t (ω i ) := ω i t , for i = 1, 2.
Let us denote the first coordinate process B 1 by W . Let P W be the probability measure on (Ω W , B(Ω W )) such that W is a Brownian motion. Here B(Ω W ) denotes the Borelian σ-algebra on Ω W .

Set E := R n \{0} equipped with its Borelian σ-algebra B(E), where n ≥ 1. We define the jump random measure N as follows: for each t > 0 and each B ∈ B(E),

N (., [0, t] × B) := 0<s≤t 1 {∆B 2 s ∈B} . (6.2.1)
The measurable set (E, B(E)) is equipped with a σ-finite positive measure ν such that E (1 ∧ |e|)ν(de) < ∞. Let P N be the probability measure on (Ω N , B(Ω N )) such that N is a Poisson random measure with compensator ν(de)dt and such that B 2 t = 0<s≤t ∆B 2 s a.s. (note that the sum of jumps is well defined up to a P N -null set). In the following, we set Ñ (dr, de) = N (dr, de)ν(de)dt.

Let F := (F t ) t≥0 be the completed filtration associated with the canonical process B. We also define the product probability measure P := P W ⊗ P N .

Let T > 0 be fixed. Let H 2 T (denoted also by H 2 ) be the set of real-valued predictable processes (Z t ) such that E T 0 Z 2 s ds < ∞ and let S 2 be the set of real-valued RCLL adapted processes (ϕ s ) with

E[sup 0≤s≤T ϕ 2 s ] < ∞, Let L 2
ν be the set of measurable functions l : (E, B(E)) → (R, B(R)) such that l 2 ν := E l 2 (e)ν(de) < ∞. The set L 2 ν is a Hilbert space equipped with the scalar product l, l ′ ν := E l(e)l ′ (e)ν(de) for all l, l

′ ∈ L 2 ν × L 2 ν . Let H 2 ν denote the set of predictable real-valued processes (k t (•)) with E T 0 k s 2 L 2
ν ds < ∞. Let A be the set of controls, defined as the set of predictable processes α valued in a compact subset A of R p , where p ∈ N * . For each α ∈ A, initial time t ∈ [0, T ] and initial condition x in R, let (X α,t,x s ) t≤s≤T be the unique R-valued solution in S 2 of the stochastic differential equation (SDE):

X α,t,x s = x + s t b(X α,t,x r , α r )dr + s t σ(X α,t,x r , α r )dW r + s t E β(X α,t,x r -, α r , e) Ñ (dr, de),
where b, σ : R × A → R, are Lipschitz continuous with respect to x and α, and β : R × A × E → R is a measurable bounded function such that for some constant C ≥ 0, and for all e ∈ E

|β(x, α, e)| ≤ C Ψ(e), x ∈ R, α ∈ A where Ψ ∈ L 2 ν . |β(x, α, e) -β(x ′ , α ′ , e)| ≤ C(|x -x ′ | + |α -α ′ |)Ψ(e), x, x ′ ∈ R, α, α ′ ∈ A.
The criterion of our mixed control problem, depending on α, is defined via a BSDE with driver function f satisfying the following hypothesis: Remark 6.2.2. Note that if f is of the form f (α, x, y, z, k) := f (α, x, y, z, E k(e)Ψ(e)ν(de)), where Ψ is a non negative function belonging to L 2 ν and f : A × [0, T ] × R 4 → R is Borelian, non-decreasing with respect to its last variable, and Lipschitz continuous with constant C (as in [START_REF] Barles | Backward stochastic differential equations and integralpartial differential equations[END_REF]), then f satisfies condition (iii

Assumption 6.2.1. f : A × [0, T ] × R 3 × L 2 ν → (R, B(R)) is B(A) ⊗ B([0, T ]) ⊗ B(R 3 ) ⊗ B(L 2 ν )- measurable and satisfies (i) |f (α, t, x, 0, 0, 0)| ≤ C(1 + |x| p ), ∀α ∈ A, t ∈ [0, T ], x ∈ R, where p ∈ N * . (ii) |f (α, t, x, y, z, k) -f (α ′ , t, x ′ , y ′ , z ′ , k ′ )| ≤ C(|α -α ′ | + |x -x ′ | + |y -y ′ | + |z -z ′ | + k -k ′ L 2 ν ), ∀t ∈ [0, T ], x, x ′ , y, y ′ , z, z ′ ∈ R, k, k ′ ∈ L 2 ν , α, α ′ ∈ A. (iii) f (α, t, x, y, z, k 2 ) -f (α, t, x, y, z, k 1 ) ≥< γ(α, t, x, y, z, k 1 , k 2 ), k 2 -k 1 > ν , ∀t, x, y, z, k 1 , k 2 , α, where γ : A × [0, T ] × R 3 × (L 2 ν ) 2 → (L 2 ν , B(L 2 ν )) is B(A) ⊗ B([0, T ]) ⊗ B(R 3 ) ⊗ B((L 2 ν ) 2 )-measurable,
) with γ(k 1 , k 2 )(e) := CΨ(e)1 { E (k 2 (e)-k 1 (e))Ψ(e)ν(de) ≤ 0} .
For all (t, x) ∈ [0, T ] × R and all control α ∈ A, let f α,t,x be the driver defined by f α,t,x (r, ω, y, z, k) := f (α r (ω), r, X α,t,x r (ω), y, z, k).

Since f satisfies condition (iii), the driver f α,t,x satisfies Assumption 6.3.9, which ensures that the Comparison Theorem for BSDEs with jumps holds (see Section 6.3.3 or [START_REF] Quenez | BSDEs with jumps, optimization and applications to dynamic risk measures[END_REF]).

We introduce the nonlinear conditional expectation E f α,t,x (denoted more simply by E α,t,x ) associated with f α,t,x , defined for each stopping time S and for each η ∈ L 2 (F S ) as:

E α,t,x r,S [η] := X α,t,x r , t ≤ r ≤ S,
where (X α,t,x r ) t≤r≤S is the solution in S 2 of the BSDE associated with driver f α,t,x , terminal time S and terminal condition η, that is satisfying:

-dX α,t,x r = f (α r , r, X α,t,x r , X α,t,x r , Z α,t,x r , K α,t,x r (•))dr -Z α,t,x r dW r -E K α,t,x r (e) Ñ (dr, de) X α,t,x S = η,
where (Z α,t,x s ), (K α,t,x s ) are the associated processes, which belong respectively to H 2 and H 2 ν . For all (t, x) ∈ [0, T ] × R and all control α ∈ A, we define the reward by h(s, X α,t,x s ) for t ≤ s < T and g(X α,t,x T ) for t = T , where

• g : R → R is Borelian. • h : [0, T ] × R → R
is a function which is continuous with respect to x uniformly in t, and right-continuous and left-limited with respect to t on [0, T ].

• |h(t, x)| + |g(x)| ≤ C(1 + |x| p ), ∀t ∈ [0, T ], x ∈ R, with p ∈ N * .
Let T be the set of stopping times with values in [0, T ]. Suppose the initial time is equal to 0. For each initial condition x ∈ R, we consider the mixed optimal control/stopping problem:

u(0, x) := sup α∈A sup τ ∈T E α,0,x 0,τ [ h(τ, X α,0,x τ )], (6.2.2)
where h(t, x) := h(t, x)1 t<T + g(x)1 t=T .

Note that h is Borelian but not necessarily regular in (t, x).

We now make the problem dynamic. We define, for t ∈ [0, T ] and each ω ∈ Ω the t-translated path ω t = (ω t s ) s≥t := (ω s -ω t ) s≥t . Note that (ω 1,t s ) s≥t := (ω 1 s -ω 1 t ) s≥t corresponds to the realizations of the translated Brownian motion W t := (W s -W t ) s≥t and that the translated Poisson random measure N t := N (]t, s], .) s≥t can be expressed in terms of (ω 2,t s ) s≥t := (ω 2 sω 2 t ) s≥t similarly to (6.2.1). Let F t = (F t s ) t≤s≤T be the completed filtration associated with W t and N t . Let us denote by T t t the set of stopping times with respect to F t with values in [t, T ]. Let P t be the predictable σ-algebra on Ω × [t, T ] equipped with the filtration F t . We now introduce the following spaces of processes. Let t ∈ [0, T ]. Let H 2 t be the

P t -measurable processes Z on Ω × [t, T ] such that Z H 2 t := E[ T t Z 2 u du] < ∞. We define H 2 t,ν as the set of P t - measurable processes K on Ω × [t, T ] such that K H 2 t,ν := E[ T t ||K u || 2 ν du] < ∞.
Also, we denote by S 2 t the set of real-valued RCLL processes ϕ on Ω × [t, T ], adapted to the filtration F t , with E[sup t≤s≤T ϕ 2 s ] < ∞. Let A t t be the set of controls α : Ω × [t, T ] → A, which are P t -measurable. For each initial time t and each initial condition x, the associated value function is defined by: u(t, x) := sup

α∈A t t sup τ ∈T t t E α,t,x t,τ [ h(τ, X α,t,x τ )]. (6.2.3)
Note that since α and τ depend only on ω t , the SDE satisfied by X α,t,x and the BSDE satisfied by

E α,t,x t,τ [ h(τ, X α,0,x τ )] can be solved in S 2 t × H 2 t × H 2 t,ν
, with respect to the translated Brownian motion W t and the translated Poisson random measure N t and the filtration F t . Note that the solution depends on ω only through ω t . Hence the function u is well defined as a deterministic function of t and x.

For each α ∈ A t t , we introduce the function u α defined as

u α (t, x) := sup τ ∈T t t E α,t,x t,τ [ h(τ, X α,t,x τ )].
We thus get u(t, x) = sup

α∈A t t u α (t, x). (6.2.4)
Note that for each α, u α (t, x) ≥ h(t, x), and hence u(t, x) ≥ h(t, x). Moreover, u α (T, x) = u(T, x) = g(x). By Theorem 3.2 in [START_REF] Quenez | Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps[END_REF], for each α, the value function u α corresponds to the solution of the reflected BSDE associated with driver f α,t,x := f (α • , •, X α,t,x

• , y, z, k), (RCLL) obstacle process ξ α,t,x s := h(s, X α,t,x s ) t≤s≤T , and terminal condition g(X α,t,x T ), that is

u α (t, x) = Y α,t,x t , (6.2.5)
where (Y α,t,x , Z α,t,x , K α,t,x ) ∈ S 2 × H 2 × H 2 ν is the solution of the reflected BSDE:

                         Y α,t,x s = g(X α,t,x T ) + T s f (α r , r, X α,t,x r , Y α,t,x r , Z α,t,x r , K α,t,x r (•))dr + A α,t,x T -A α,t,x s - T s Z α,t,x r dW r - T s E K α,t,x (r, e) Ñ (dr, de) Y α,t,x s ≥ ξ α,t,x s = h(s, X α,t,x s ), 0 ≤ s < T a.s. , A α,t,
x is a RCLL nondecreasing predictable process with A α,t,x t = 0 and such that

T 0 (Y α,t,x s -ξ α,t,x s )dA α,t,x,c s = 0 a.s. and ∆A α,t,x,d s = -∆A α,t,x s 1 {Y α,t,x s - =ξ α,t,x s -} a.s. (6.2.6)
Here A α,t,x,c denotes the continuous part of A and A α,t,x,d its discontinuous part. In the particular case when h(T -, x) ≤ g(x), then the obstacle ξ α,t,x satisfies for all predictable stopping time τ , ξ τ -≤ ξ τ a.s. which implies the continuity of the process A α,t,x (see [START_REF] Quenez | Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps[END_REF]).

In the following, for each α ∈ A t t , Y α,t,x s will be also denoted by Y α,t,x s,T [g(X α,t,x T )]. Using this notation, equality (6.2.5) can be written:

u α (t, x) = Y α,t,x t = Y α,t,x t,T [g(X α,t,x T )]. (6.2.7)
The reflected BSDE (6.2.6) can be solved in S 2 t ×H 2 t ×H 2 t,ν with respect to the t-translated Brownian motion and the t-translated Poisson random measure. Note that the solution depends on ω only through ω t .

Our mixed optimal stopping/control problem (6.2.3) can thus be reduced to an optimal control problem for reflected BSDEs:

u(t, x) = sup α∈A t t Y α,t,x t = sup α∈A t t Y α,t,x t,T [g(X α,t,x T )].
This key property will be used to solve our mixed problem. We point out that in the classical case of linear expectations, this approach allows us to provide alternative proofs of the dynamic programming principle to those given in the previous literature.

Remark 6.2.3. Some mixed optimal control/stopping problems with nonlinear expectations have been studied in [START_REF] Bayraktar | Optimal stopping for Non-linear Expectations[END_REF][START_REF] Quenez | Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps[END_REF]. In these papers, the reward process does not depend on the control, which yields the characterization of the value function as the solution of a reflected BSDE. This is not the case here.

Weak dynamic programming 6.3.1 Splitting properties

Let s ∈ [0, T ]. For each ω, let s ω := (ω r∧s ) 0≤r≤T and ω s := (ω rω s ) s≤r≤T . We shall identify the path ω with ( s ω, ω s ), which means that a path can be splitted into two parts: the path before time s and the s-translated path after time s.

Let α be a given control in A. We show below the following: at time s, for fixed past path ω := s ω, the process α(ω, .) which only depends on the future path ω s is an s-admissible control, that is α(ω, .) ∈ A s s ; furthermore, the criterium Y α,0,x (ω, .) from time s coincides with the solution of the reflected BSDE driven by W s and Ñ s , controlled by α(ω, .) and associated with initial time s and initial state condition X α,0,x s (ω). We introduce the following random variables defined on Ω by

S s : ω → s ω ; T s : ω → ω s .
Note that they are independent. For each ω ∈ Ω, we have

ω = S s (ω) + T s (ω)1 ]s,T ] ,
or equivalently ω r = ω r∧s + ω s r 1 ]s,T ] (r), fort all r ∈ [0, T ]. We introduce the following notation : for all paths ω, ω ′ ∈ Ω, ( s ω, T s (ω ′ )) denotes the path such that the past trajectory before s is that of ω, and the s-translated trajectory after s is that of ω ′ . This can also be written as:

( s ω, T s (ω ′ )) := s ω + T s (ω ′ )1 ]s,T ] = s ω + ω ′ s 1 ]s,T ] = (ω r 1 r≤s + (ω s + ω ′ r -ω ′ s )1 r>s ) 0≤r≤T .
Note that for each ω ∈ Ω, we have ( s ω, T s (ω)) = ω.

Lemma 6.3.1. Let s ∈ [0, T ]. Let Z ∈ H 2 .
There exists a P -null set N such that for each ω in the complement N c of N , setting ω := s ω = ω .∧s , the process Z(ω, T s ) (denoted also be Z(ω, .)) defined by

Z(ω, T s ) : Ω × [s, T ] → R ; (ω ′ , r) → Z r (ω, T s (ω ′ )) belongs to H 2 s . Moreover, if Z ∈ A, then Z(ω, T s ) ∈ A s s .
Proof. By a classical property of conditional expectations, we have

E[ T s Z 2 r dr]= E[E[ T s Z 2 r dr|F s ]] < +∞.
Using the independence of T s with respect to F s and the measurability of S s with respect to F s , we then derive that

E[ T s Z 2 r dr| F s ] = E[ T s Z r (S s , T s ) 2 dr| F s ] = F ( S s ) < +∞ P -a.s. ,
where

F (ω) := E[ T s Z r (ω, T s (•)) 2 dr]. It remains to prove that the process Z(ω, T s ) : (ω ′ , r) → Z r (ω, T s (ω ′ )) is P s -measurable.
Suppose we have shown that the map

ψ : (Ω × [s, T ], P s ) → (Ω × [s, T ], P) ; (ω ′ , r) → ((ω, T s (ω ′ )), r) is measurable. Note that we have Z(ω, T s )(ω ′ , r) = Z • ψ(ω ′ , r) for each (ω ′ , r) ∈ Ω × [s, T ].
Since Z is P-measurable, the P s -measurability of Z(ω, T s ) thus follows by composition.

It remains to show that the map ψ is P s -measurable. The proof is based on classical arguments of Integration Theory. Recall that the σ-algebra P is generated by the sets H×]v, T ], where H is a cylindrical set belonging to F v , that is of the following form:

H = {B t i ∈ A i , 1 ≤ i ≤ n}, where A i ∈ B(R 2 ) and t 1 < t 2 < ... ≤ v. It is thus sufficient to show that ψ -1 (H×]v, T ]) ∈ P s . Note that ψ -1 (H×]v, T ]) = H ′ ×]v, T ]
, where H ′ = {ω ′ ∈ Ω , (ω, T s (ω ′ )) ∈ H}. We have:

H ′ = ∅ if ∃ i such that t i ≤ s and ωt i ∈ A i {ω ′ t i -ω ′ s ∈ A i , ∀ i such that t i > s} otherwise.
Hence H ′ ∈ F s v . This implies that ψ -1 (H×]v, T ]) ∈ P s . The map ψ is thus P s -measurable.

Remark 6.3.2. The same proof shows that this property still holds for each initial time t

∈ [0, T ]. More precisely, let s ∈ [t, T ]. Let Z ∈ H 2 t (resp. A t t ). For almost every ω ∈ Ω, the process Z( s ω, .) = (Z r ( s ω, T s )) r≥s belongs to H 2 s (resp. A s s ).
Let Z ∈ H 2 . Let us give an intermediary time s ∈ [0, T ] and a fixed past path s ω. Note that the Lebesgue integral ( u s Z r dr)( s ω, .) is equal a.s. to the integral u s Z r ( s ω, .)dr. This property is not so clear for a stochastic integral. We now show that the stochastic integral ( u s Z r dW r )( s ω, .) coincides with the stochastic integral of the process Z( s ω, .) with respect to the translated Brownian motion W s , that is u s Z r ( s ω, .)dW s r . A similar property holds for the integral with respect to the Poisson random measure. Lemma 6.3.3 (Splitting properties for stochastic integrals).

Let t ∈ [0, T ]. Let s ∈ [t, T ]. Let Z ∈ H 2 t and K ∈ H 2 t,ν
. There exists a P -null set N such that for each ω ∈ N c , and ω := s ω, we

For each n, since Z n is a simple process, it satisfies equality (6.3.1) everywhere, that is

( u s Z n r dW r )(ω, T s ) = u s Z n r (ω, T s )dW s r .
By the convergence property (6.3.2), equalities (6.3.3) and (6.3.4), and the uniqueness property of the limit in L 2 , we derive equality (6.3.1), which ends the proof.

Using the above lemmas, we now show that for each s ≥ t, for almost every ω ∈ Ω, setting ω = s ω, the process Y α,t,x (ω, T s ) coincides with the solution of the reflected BSDE on Ω × [s, T ], associated with driver f α(ω,T s ),s,η(ω) , with obstacle h(r, X α(ω,T s ),s,X α( ω),t,x s (ω) r

) and filtration F s , and driven by W s and Ñ s .

To simplify notation, T s will be replaced by • in the following. In particular Y α,t,x (ω, T s ) will be simply denoted by Y α,t,x (ω, .). Theorem 6.3.5 (Splitting properties for the forward-backward "system"). Let t ∈ [0, T ], α ∈ A t t and s ∈ [t, T ]. There exists a P -null set N such that for each ω ∈ N c , setting ω = s ω, the following properties hold:

• There exists an unique solution (X α(ω,•),s,η(ω) r

) s≤r≤T in S 2
s of the following SDE:

X α(ω,•),s,η(ω) u = η(ω) + u s b(X α(ω,•),s,η(ω) r , α r (ω, •))dr + u s σ(X α(ω,•),s,η(ω) r , α r (ω, •))dW s r + u s E β(X α(ω,•),s,η(ω) r - , α r (ω, •), e) Ñ s (dr, de), (6.3.5)
where η(ω) := X α(ω),t,x s

(ω). We also have X α,t,x r (ω, .) = X α(ω,•),s,η(ω) r

, s ≤ r ≤ T P -a.s.

• There exists an unique solution (Y

α(ω,•),s,η(ω) r , Z α(ω,•),s,η(ω) r , K α(ω,•),s,η(ω) r ) s≤r≤T in S 2 s ×H 2 s ×H 2 s,ν
of the reflected BSDE on Ω × [s, T ] driven by W s and Ñ s and associated with filtration F s , driver f α(ω,.),s,η(ω) , and with obstacle h(r, X α(ω,•),s,η(ω) r

). We have:

Y α,t,x r (ω, .) = Y α(ω,.
),s,η(ω) r , s ≤ r ≤ T, Pa.s. (6.3.6)

Z α,t,x r (ω, .) = Z α(ω,•),s,η(ω) r , s ≤ r ≤ T, dP ⊗ dr -a.s. K α,t,x r (ω, ., e) = K α(ω,•),s,η(ω) r (., e), s ≤ r ≤ T, dP ⊗ dr ⊗ dν(e) -a.s. Y α,t,x s (ω) = Y α(ω,•),s,η(ω) s = u α(ω,•) (s, η(ω)). (6.3.7)
Proof. Recall that by Lemma 6.3.1 the process α(ω, •) := (α r (ω, •)) r≥s belongs to A s s . Let us show the first assertion. To simplify the exposition, we suppose that there is no Poisson random measure. There exists a P -null set N such that for each ω ∈ N c , setting ω = s ω, X α,t,x u (ω, .) = η(ω) + which implies that the process (X α,t,x r (ω, •)) r∈[s,T ] is a solution of SDE (6.3.5), and then, by uniqueness of the solution of this SDE, we have X α,t,x r (ω, .) = X α(ω,.),s,η(ω) r , s ≤ r ≤ T, P -a.s. Let us show the second assertion. First, note that since the filtration F s is generated by W s and Ñ s , we have a martingale representation theorem for F s -martingales with respect to W s and Ñ s .

Hence, there exists an unique solution (Y

α(ω,•),s,η(ω) r , Z α(ω,•),s,η(ω) r , K α(ω,•),s,η(ω) r ) s≤r≤T in S 2 s ×H 2 s ×H 2 s,ν
of the reflected BSDE on Ω × [s, T ] driven by W s and Ñ s and associated with filtration F s and with obstacle h(r, X α(ω,•),s,η(ω) r

). Equalities (6.3.6) then follow from similar arguments as above together with the uniqueness of the solution of a Lipschitz BSDE. Equality (6.3.7) is obtained by taking r = s in equality (6.3.6) and by using the definition of u α(ω,.) .

Existence of weak ε-optimal controls

We first show a measurability property of the function u α (t, x) with respect to control α and initial condition x.

Lemma 6.3.6. Let s ∈ [0, T ]. The map (α, x) → u α (s, x); (A s s × R , B(A s s ) ⊗ B(R)) → (R, B(R)) is measurable. Proof. Recall that u α (s, x) = Y α,s,x s,T [g(X α,s,x T
)] also denoted by Y α,s,x s,T [ h(., X α,s,

)]. Let x 1 , x 2 ∈ R, and α 1 , α 2 ∈ A s s . By classical estimates on diffusion processes and the assumptions made on the coefficients, we get

E[sup r≥s |X α 1 ,s,x 1 r -X α 2 ,s,x 2 r | 2 ] ≤ C( α 1 -α 2 2 H 2 s + |x 1 -x 2 | 2 ). (6.3.8)
We introduce the map Φ :

A s s × R × S 2 s → S 2 s ; (α, x, ξ) → Y α,s,x s,T [ξ . ]
, where Y α,s,x s,T [ξ . ] denotes here the solution at time s of the reflected BSDE associated with driver f α,s,x := (f (α r , r, X α,s,x r , .)1 r≥s ) and obstacle ξ . . By the estimates on reflected BSDEs (see the Appendix in [START_REF] Dumitrescu | Optimal Stopping for Dynamic Risk Measures with Jumps and Obstacle Problems[END_REF]), using the Lipschitz property of f with respect to x, α and estimates (6.3.8), for all x 1 , x 2 ∈ R, α 1 , α 2 ∈ A s s and ξ 1 . , ξ 2 . ∈ S 2 s , we have

|Y α 1 ,s,x 1 s [ξ 1 . ] -Y α 2 ,s,x 2 s [ξ 2 . ]| 2 ≤ C( α 1 -α 2 2 H 2 s + |x 1 -x 2 | 2 + ξ 1 . -ξ 2 . 2 S 2 s ).
The map Φ is thus Lipschitz-continuous with respect to the norm . 2

H 2 s + | . | 2 + . 2 S 2
s . Moreover, the reward map h is Borelian, which implies that the map (α, x) → (α, x, h(., X α,s,x .

)) defined on (A s s ×R, B(A s s )⊗B(R)) and valued in (A s s ×R×S 2 s , B(A s s )⊗B(R)⊗B(S 2 s ) is measurable. By composition, it follows that the map (α, x) → Y α,s,x s,T [ h(., X α,s,x

.

)] = u α (s, x) is measurable.

For each (t, s) with s ≥ t, we introduce the set A t s of restrictions to [s, T ] of the controls in A t t . They can also be identified to the controls α in A t t which are equal to 0 on [t, s]. Let η ∈ L 2 (F t s ). Since η is F s -measurable, it can be written as a measurable map, still denoted by η, of the past trajectory s ω.

For each ω ∈ Ω t , by using the definition of the function u we have:

u(s, η( s ω)) = sup α∈A s s u α (s, η( s ω)). (6.3.9)
Note that for fixed s, the map x → u(s, x) is not necessarily Borelian. We now introduce the map u * defined by u * (t, x) := lim inf

(t ′ ,x ′ )→(t,x) u(t ′ , x ′ ) ∀(t, x) ∈ [0, T [×R and u * (T, x) = g(x) ∀x ∈ R. (6.3.10)
The map u * thus coincides with the classical lower-semicontinuous envelope of u on [0, T [×R. It follows that u * is Borelian. Note also that u * ≤ u.

Using the measurability of the maps (α, x) → u α (s, x) and x → u * (s, x), we derive the existence of nearly optimal controls for (6.3.9) satisfying some specific measurability properties. Theorem 6.3.7. (Existence of weak ε-optimal controls) Let t ∈ [0, T ], s ∈ [t, T ] and η ∈ L 2 (F t s ). Let ε > 0. There exists α ε ∈ A t s such that, for almost every ω ∈ Ω, α ε ( s ω, T s ) is weak ε-optimal for Problem (6.3.9), in the sense that

u * (s, η( s ω)) ≤ u α ε ( s ω,T s ) (s, η( s ω)) + ε.
Proof. Without loss of generality, we may assume that t = 0. We introduce the space s Ω := {(ω r ) 0≤r≤s ; ω ∈ Ω}, equipped with the σ-algebra F s , that is the σ-algebra associated with the coordinate process, and the probability measure s P , which corresponds to the image of P by s S i.e. P • (S s ) -1 .

Let x ∈ R. From the definition of u(s, x) as a supremum (see (6.2.4)), we derive that for each ω ∈ s Ω, there exists ᾱε ∈ A s s such that u(s, η(ω)) ≤ u ᾱε (s, η(ω)) + ε and hence such that

u * (s, η(ω)) ≤ u ᾱε (s, η(ω)) + ε.
Recall that the Hilbert space H 2 s of square-integrable predictable processes on Ω s × [s, T ], equipped with the norm • H 2 s is separable. It follows that A s s is a Polish space because it is a closed subset of H 2 s . Also, recall that the space s Ω of paths (RCLL) before s is Polish for the Skorohod metric. Now, η is F s -measurable and the map u * is Borelian. Moreover, by Lemma 6.3.6, u α is B(R) ⊗ B(A s s )-measurable with respect to (x, α). We thus have that

A := {(ω, α) ∈ s Ω × A s s , u * (s, η(ω)) ≤ u α (s, η(ω)) + ε } ∈ F s ⊗ B(A s s ). (6.3.11)
Now, a measurable selection theorem of [START_REF] Dellacherie | Probabilités et Potentiel[END_REF] in Section 81, Appendix of Ch. III (see also [START_REF] Barbu | Convexity and Optimization in Banach Spaces[END_REF]) ensures that if A is a Borel subset of Ω × E, where Ω is a metrizable space and E a Polish space, and if for each ω ∈ Ω, there exists α ∈ E such that (ω, α) ∈ A, then there exists an universally measurable map h : Ω → E such that for all ω ∈ Ω, (ω, h(ω)) ∈ A. Also, by a result of Measurable Theory (see Remark 6.6.4), if E is a (separable) Hilbert space, for each probability P on B(Ω) there exists a Borelian map ĥ : Ω → E such that h(ω) = ĥ(ω) for P -almost every ω.

Let us apply these properties to our case with Ω replaced by s Ω, P replaced by s P , E = A s s , A = A defined by (6.3.11). We thus obtain that there exists a Borelian map αε :

( s Ω, F s ) → (A s s , B(A s s )) ; ω → αε (ω, •) such that u * (s, η(ω)) ≤ u αε (ω,•) (s, η(ω)) + ε for s P -almost every ω ∈ s Ω.
Since H 2 s is a separable Hilbert space, for each ω, we have αε u (ω, ω) = i β i,ε (ω)e i u (ω) dP (ω) ⊗ dua.s. , where β i,ε (ω) =< αε (ω, •), e i (•) > H 2 s and {e i , i ∈ N} is a countable orthonormal basis of H 2 s .

obstacle (η s ) s<θ , and terminal condition ξ is defined as the unique solution in S 2 × H 2 × H 2 ν of the reflected BSDE with terminal time T , driver f (t, y, z, k)1 {t≤θ} , terminal condition ξ and obstacle η t 1 t<θ + ξ1 t≥θ . Note that Y t,θ (ξ) = ξ, Z t,θ (ξ) = 0, k t,θ (ξ) = 0 for t ≥ θ.

We first prove a continuity property for reflected BSDEs where the limit involves both terminal condition and terminal time. Lemma 6.3.10 (A continuity property for reflected BSDEs). Let T > 0. Let (η t ) be an RCLL process in S 2 . Let f be a given Lipschitz driver. Let (θ n ) n∈N be a non increasing sequence of stopping times in T , converging a.s. to θ ∈ T as n tends to ∞. Let (ξ n ) n∈N be a sequence of random variables such that E[sup n (ξ n ) 2 ] < +∞, and for each n, ξ n is F θ n -measurable. Suppose that ξ n converges a.s. to an F θ -measurable random variable ξ as n tends to ∞. Suppose that η θ ≤ ξ a.s. (6.3.13) Let Y .,θ n (ξ n ); Y .,θ (ξ) be the solutions of the reflected BSDEs associated with driver f , obstacle (η s ) s<θ n (resp. (η s ) s<θ ) , terminal time θ n (resp. θ), terminal condition ξ n (resp. ξ). We have

Y 0,θ (ξ) = lim n→+∞ Y 0,θ n (ξ n ) a.s.
When for each n, θ n = θ a.s. , the result still holds without Assumption (6.3.13).

Remark 6.3.11. Compared with the case of non reflected BSDEs (see Proposition A.6 in [START_REF] Quenez | BSDEs with jumps, optimization and applications to dynamic risk measures[END_REF]), there is an additional difficulty due both to the obstacle and to the variation of the terminal time.

An additional assumption (Assumption (6.3.13)) is here required to obtain the result.

Proof. Let n ∈ N. We apply a classical estimate on reflected BSDEs (see Proposition 5.7.6) with

f 1 = f 1 t≤θ n , f 2 = f 1 t≤θ , ξ 1 = ξ n , ξ 2 = ξ, η
|Y 0,θ n (ξ n )-Y 0,θ (ξ)| 2 ≤ K E[(ξ n -ξ) 2 ] + E[ θ n θ f 2 (s, ξ, 0, 0)ds] +φ sup θ≤s<θ n |η s -η θ | L 2 , (6.3.14)
where the constant K depends only on the Lipschitz constant C of f and the terminal time T , and where the constant φ depends only on C, T , η S 2 , sup n ξ n L 2 and f (s, 0, 0, 0) IH 2 . Since the obstacle (η t ) is right-continuous and θ n ↓ θ a.s. , we have lim n→+∞ sup θ≤s≤θ n |η sη θ | L 2 = 0. The right member of (6.3.14) thus tends to 0 as n tends to +∞. The result follows.

Using this lemma, we derive a Fatou lemma in the reflected case, where the limit involves both terminal condition and terminal time. Theorem 6.3.12 (A Fatou lemma for reflected BSDEs). Let T > 0. Let (η t ) be an RCLL process in S 2 . Let f be a Lipschitz driver satisfying Assumption 6.3.9. Let (θ n ) n∈N be a non increasing sequence of stopping times in T , converging a.s. to θ ∈ T as n tends to ∞. Let (ξ n ) n∈N be a sequence of random variables such that E[sup n (ξ n ) 2 ] < +∞, and for each n, ξ n is F θ n -measurable.

Let Y .,θ n (ξ n ) ; Y .,θ (lim inf n→+∞ ξ n ) and Y .,θ (lim sup n→+∞ ξ n ) be the solution(s) of the reflected BSDE(s) associated with driver f , obstacle (η s ) s<θ n (resp. (η s ) s<θ ) , terminal time θ n (resp. θ), terminal condition ξ n (resp. lim inf n→+∞ ξ n and lim sup n→+∞ ξ n ). Suppose that lim inf n→+∞ ξ n ≥ η θ (resp. lim sup n→+∞ ξ n ≥ η θ ) a.s. (6.3.15) then Y 0,θ (lim inf

n→+∞ ξ n ) ≤ lim inf n→+∞ Y 0,θ n (ξ n ) resp. Y 0,θ (lim sup n→+∞ ξ n ) ≥ lim sup n→+∞ Y 0,θ n (ξ n ) .
When for each n, θ n = θ a.s. , the result still holds without Assumption (6.3.15).

Proof. We present only the proof of the first inequality, since the second one is obtained by similar arguments. For all n, we have by the monotonicity of reflected BSDEs with respect to terminal condition, Y 0,θ n (inf p≥n ξ p ) ≤ Y 0,θ n (ξ n ). We derive that lim inf n→+∞ Y 0,θ n (inf

p≥n ξ p ) ≤ lim inf n→+∞ Y 0,θ n (ξ n ).
By Assumption (6.3.15), since lim n→+∞ inf p≥n ξ p = lim inf n→+∞ ξ n a.s. , Lemma 6.3.10 yields that lim n→+∞ Y 0,θ n (inf

p≥n ξ p ) = Y 0,θ (lim inf n→+∞ ξ n ).
The desired result follows.

A weak dynamic programming principle

Since for fixed s, the value function x → u(s, x) is not necessarily Borelian, we cannot a priori establish a classical dynamic programming principle. We will provide a weak dynamic programming principle involving the map u * (defined above by (6.3.4)) and the map u * defined by u * (t, x) := lim sup

(t ′ ,x ′ )→(t,x) u(t ′ , x ′ ) ∀(t, x) ∈ [0, T [×R and u * (T, x) = g(x) ∀x ∈ R.
The map u * coincides with the classical upper semicontinuous envelope of u on [0, T [×R. It follows that u * is Borelian (as u * ). Note that u * (resp. u * ) is not necessarily upper (resp. lower) semicontinuous on [0, T ]×R, because the terminal reward map g is Borelian but is not supposed to satisfy any regularity assumption. Note also that u * ≤ u ≤ u * and u * (T, .) = u(T, .) = u * (T, .) = g(.).

We now prove that the value function satisfies a weak dynamic programming principle, in the sense that u * (resp.u * ) satisfies a weak sub-(resp. super-) optimality principle of dynamic programming. In order to do this, we will use the splitting properties (Th. 6.3.5 ), the existence of weak ε-optimal controls (Th. 6.3.7) and the above Fatou lemma for reflected BSDEs, where the limit involves both terminal condition and terminal time (Th. 6.3.12). Theorem 6.3.13 (A weak dynamic programming principle). The function u * satisfies the weak sub-optimality principle of dynamic programming, that is for each t ∈ [0, T ] and for each stopping time θ ∈ T We stress that no regularity condition is required on the terminal reward map g to ensure these dynamic programming principles, even the second one, which is the most difficult one to establish. This is not the case in the previous literature even in the case of a classical expectation, where g is supposed to be lower-semicontinuous (see [START_REF] Bouchard | Weak Dynamic Programming Principle for Viscosity Solutions[END_REF], [START_REF] Bouchard | Weak Dynamic Programming for Generalized State Constraints[END_REF] and [START_REF] Bayraktar | A Weak Dynamic Programming Principle for Zero-Sum Stochastic Differential Games with Unbounded Controls[END_REF]).

Before giving the proof, we introduce the following notation. For each θ ∈ T and each ξ in L 2 (F θ ), we denote by (Y α,t,x .,θ (ξ), Z α,t,x .,θ (ξ), k α,t,x .,θ (ξ)) the unique solution in S 2 × H 2 × H 2 ν of the reflected BSDE with driver f α,t,x 1 {s≤θ} , terminal time T , terminal condition ξ and obstacle h(r, X α,t,x Hence, we have lim inf n→+∞ u * (θ n , X α,0,x θ n ) ≥ h(θ, X α,0,x θ ) a.s. Condition (6.3.15) is thus satisfied with ξ n = u * (θ n , X α,0,x θ n ) and ξ t = h(t, X α,0,x t

). We can thus apply the above Fatou lemma for reflected BSDEs (Th. 6.3.12). We thus get:

Y α,0,x 0,θ u * (θ, X α,0,x θ ) ≤ Y α,0,x 0,θ lim inf n→+∞ u * (θ n , X α,0,x θ n ) ≤ lim inf n→∞ Y α,0,x 0,θ n u * (θ n , X α,0,x θ n ) a.s.
Fix n ∈ N. Let A t k be the set of the restrictions to [t k , T ] of the controls α in A. By Theorem 6.3.7, for each ε > 0, for each k, there exists a weak ε-optimal control control α n,ε,k in A 0 t k for the control problem at time t k with initial condition η = X α,0,x t k , that is satisfying the inequality

u * (t k , X α,0,x t k ( t k ω)) ≤ u α n,ε,k ( t k ω,•) (t k , X α,0,x t k ( t k ω)) + ε (6.3.20)
for almost every ω ∈ Ω. By definition of u α n,ε,k ( t k ω,•) , we have

u α n,ε,k ( t k ω,•) (t k , X α,0,x t k ( t k ω)) = Y α n,ε,k ( t k ω,•),t k ,X α,0,x t k ( t k ω) t k ,T = Y α n,ε,k ,t k ,X α,0,x t k t k ,T ( t k ω) Here, Y α n,ε,k ,t k ,X α,0,x t k .,T = Y f α n,ε,k ,t k ,X α,0,x t k .,T [ h(r, X α n,ε,k ,t k ,X α,0,x t k r
)] denotes the solution of the reflected BSDE associated with terminal time T , obstacle ( h(r, X α n,ε,k ,t k ,X α,0,x t k r )) t k ≤r≤T and driver

f α n,ε,k ,t k ,X α,0,x t k (r, y, z, k) := f (α n,ε,k r , r, X α,t k ,X α,0,x t k r , y, z, k). Set α n,ε s := 2 n -1 k=0 α n,ε,k s 1 A k . Since for each k, A k ∈ F t k , we have Y α n,ε,k ,t k ,X α,0,x t k t k ,T 1 A k = Y f α n,ε,k ,t k ,X α,0,x t k 1 A k t k ,T [ h(r, X α n,ε,k ,t k ,X α,0,x t k r )1 A k ] = Y f α n,ε ,θ n ,X α,0,x θ n 1 A k t k ,T [ h(r, X α n,ε ,θ n ,X α,0,x θ n r )1 A k ] = Y α n,ε ,θ n ,X α,0,x θ n θ n ,T 1 A k a.s. ,
where, for a given driver f , Y f 1 A k denotes the solution of the reflected BSDE associated with f 1 A k . Using inequality (6.3.20), we get

u * (θ n , X α,0,x θ n ) = 2 n -1 k=0 u * (t k , X α,0,x t k )1 A k ≤ Y α n,ε ,θ n ,X α,0,x θ n θ n ,T + ε a.s.
We set: αn,ε s := α s 1 s<θ n + α n,ε s 1 θ n ≤s≤T . Note that αn,ε ∈ A. Using the comparison theorem together with the estimates on reflected BSDEs (see [START_REF] Dumitrescu | Optimal Stopping for Dynamic Risk Measures with Jumps and Obstacle Problems[END_REF]), we obtain

Y α,0,x 0,θ n [u * (θ n , X α,0,x θ n )] ≤ Y α,0,x 0,θ n [Y α n,ε ,θ n ,X α,0,x θ n θ n ,T ] + Kε = Y αn,ε ,0,x 0,T + Kε,
where the last equality follows from the flow property. Since Y αn,ε ,0,x 0,T ≤ u(0, x), we have

Y α,0,x 0,θ n [u * (θ n , X α,0,x θ n )] ≤ u(0, x) + Kε a.s.
which holds for all n. Hence, by inequality (6.3.4), we get Y α,0,x 0,θ u * (θ, X α,0,x θ ) ≤ u(0, x) + Kε.

Taking the supremum on α ∈ A and letting ε tend to 0, we obtained inequality (6.3.18). It remains to show the first assertion, that is, u * satisfies the sub-optimality principle of dynamic programming (which is the easiest one). It is sufficient to show that for each θ ∈ T , u(0, x) ≤ sup α∈A Y α,0,x 0,θ u * (θ, X α,0,x θ ) .

(6.3.21)

Let θ ∈ T . As in the proof of the super-optimality principle, we approximate θ by the sequence of stopping times (θ n ) n∈N . Let n ∈ N. By applying the flow property for reflected BSDEs, we get

Y α,0,x 0,T = Y α,0,x 0,θ n [Y α,θ n ,X α,0,x θ n θ n ,T
]. By similar arguments as in the proof of the super-optimality principle (but without using the existence of weak-optimal controls), we derive that:

Y α,θ n ,X α,0,x θ n θ n ,T ≤ u * (θ n , X α,0,x θ n ) a.s.
By the comparison theorem for reflected BSDEs, it follows that

Y α,0,x 0,T = Y α,0,x 0,θ n [Y α,θ n ,X α,0,x θ n θ n ,T ] ≤ Y α,0,x 0,θ n [u * (θ n , X α,0,x θ n )] a.s.
By taking the limit in n in the above relation and using the Fatou lemma for reflected BSDEs with respect to both terminal time and terminal condition (Th. 6.3.12), we get:

Y α,0,x 0,T ≤ lim sup n→∞ Y α,0,x 0,θ n [u * (θ n , X α,0,x θ n )] ≤ Y α,0,x 0,θ [lim sup n→∞ u * (θ n , X α,0,x θ n )] a.s.
By the upper semicontinuity property of u * on [0, T [×IR and the fact that u * (T, x) = g(x), we finally obtain

Y α,0,x 0,T ≤ Y α,0,x 0,θ [lim sup n→∞ u * (θ n , X α,0,x θ n )] ≤ Y α,0,x 0,θ [u * (θ, X α,0,x θ )] a.s.
Since α ∈ A is arbitrary, we get inequality (6.3.21), which completes the proof.

Remark 6.3.15. The above proof also shows that the weak dynamic programming principle of Theorem 6.3.13 still holds with θ replaced by θ α in inequalities (6.3.16) and (6.3.17), given a family of stopping times indexed by controls {θ α , α ∈ A t t }.

6.4 Nonlinear HJB variational inequalities 6.4.1 Some extensions of comparison theorems for BSDEs and reflected BSDEs

We provide two results which will be used to prove that the value function u, defined by (6.2.3), is a weak viscosity solution of some nonlinear Hamilton Jacobi Bellman variational inequalities (see Theorem 6.4.5). We first show a slight extension of the comparison theorem for BSDEs given in [START_REF] Quenez | BSDEs with jumps, optimization and applications to dynamic risk measures[END_REF] which formally states that if two terminal conditions ξ 1 , ξ 2 satisfy ξ 1 ≥ ξ 2 + ε, then the associated solutions X 1 and X 2 satisfy X 1 ≥ X

f 1 (t, X 1 t , π 1 t , l 1 t ) ≥ f 2 (t, X 1 t , π 1 t , l 1 t ), t 0 ≤ t ≤ θ, dt ⊗ dP a.s. X 1 t ≥ ξ 2 t + ε, t 0 ≤ t ≤ θ a.s. (6.4.1)
Then for each t ∈ [t 0 , θ], we have X ) + e -CT ε a.s. By taking the supremum over τ ∈ T [t,θ] , the result follows. Remark 6.4.3. We stress that unlike to the comparison theorem for two reflected BSDEs where condition f 1 (t, y, z, k) ≥ f 2 (t, y, z, k) is required for all y, z, k, in the above Proposition, this condition is required to be satisfied only along the solution of the BSDE. This point will be used in the proof of Theorem 6.4.5.

The value function, weak solution of a nonlinear HJBVI

We introduce the following Hamilton Jacobi Bellman variational inequality (HJBVI):

             min(u(t, x) -h(t, x), inf α∈A (- ∂u ∂t (t, x) -L α u(t, x) -f (α, t, x, u(t, x), (σ ∂u ∂x )(t, x), B α u(t, x))) = 0, (t, x) ∈ [0, T ) × R u(T, x) = g(x), x ∈ R (6.4.2)
where L α := A α + K α , and for φ ∈ C 2 (R),

• A α φ(x) := 1 2 σ 2 (x, α) ∂ 2 φ ∂x 2 (x) + b(x, α) ∂φ ∂x (x) and B α φ(x) := φ(x + β(x, α, •)) -φ(x).
• K α φ(x) := E φ(x + β(x, α, e))φ(x) -∂φ ∂x (x)β(x, α, e) ν(de). Definition 6.4.4. • A function u is said to be a viscosity subsolution of (6.4.2) if it is upper semicontinuous on [0, T [×R, if u(T, x) ≤ g(x), x ∈ R, and if for any point (t 0 , x 0 ) ∈ [0, T [×R and for any φ ∈ C 1,2 ([0, T ]×R) such that φ(t 0 , x 0 ) = u(t 0 , x 0 ) and φ-u attains its minimum at (t 0 , x 0 ), we have

min(u(t 0 , x 0 ) -h(t 0 , x 0 ), inf α∈A (- ∂φ ∂t (t 0 , x 0 ) -L α φ(t 0 , x 0 ) -f (α, t 0 , x 0 , u(t 0 , x 0 ), (σ ∂φ ∂x )(t 0 , x 0 ), B α φ(t 0 , x 0 ))) ≤ 0.
In other words, if u(t 0 , x 0 ) > h(t 0 , x 0 ), inf α∈A (-∂φ ∂t (t 0 , x 0 ) -L α φ(t 0 , x 0 )f (α, t 0 , x 0 , u(t 0 , x 0 ), (σ ∂φ ∂x )(t 0 , x 0 ), B α φ(t 0 , x 0 ))) ≤ 0.

• A function u is said to be a viscosity supersolution of (6.4.2) if it is lower semicontinuous on [0, T [×R, u(T, x) ≥ g(x), x ∈ R, and if for any point (t 0 , x 0 ) ∈ [0, T [×R and any φ ∈ C 1,2 ([0, T ]×R) such that φ(t 0 , x 0 ) = u(t 0 , x 0 ) and φu attains its maximum at (t 0 , x 0 ), we have

min(u(t 0 , x 0 ) -h(t 0 , x 0 ), inf α∈A (- ∂ ∂t φ(t 0 , x 0 ) -L α φ(t 0 , x 0 ) -f (α, t 0 , x 0 , u(t 0 , x 0 ), (σ ∂φ ∂x )(t 0 , x 0 ), B α φ(t 0 , x 0 ))) ≥ 0.
In other words, we have both u(t 0 , x 0 ) ≥ h(t 0 , x 0 ) and inf α∈A (-∂φ ∂t (t 0 , x 0 ) -L α φ(t 0 , x 0 )f (α, t 0 , x 0 , u(t 0 , x 0 ), (σ ∂φ ∂x )(t 0 , x 0 ), B α φ(t 0 , x 0 ))) ≥ 0.

Note that if a map is both a viscosity subsolution and a viscosity supersolution, then it is continuous and a viscosity solution in the classical sense. Here, since the value function u is not regular, it is not in general a viscosity solution in the classical sense.

Using the weak dynamic programming principle (Theorem 6.3.13) and the comparison theorem between a BSDE and a reflected BSDE (Proposition 6.4.2), we now prove that the value function of our problem is a weak viscosity solution of the above HJBVI. More precisely, without additional assumptions, the following theorem holds. Theorem 6.4.5. Under the same assumptions as those of Theorem 6.3.13, the function u, defined by (6.2.3), is a weak viscosity solution of the HJBVI (6.4.2), in the sense that u * is a viscosity subsolution of (6.4.2) and u * is a viscosity supersolution of (6.4.2).

Proof. • We first prove that u * is a subsolution of (6.4.2). Let (t 0 , x 0 ) ∈ [0, T [×R and φ ∈ C 1,2 ([0, T ] × R) be such that φ(t 0 , x 0 ) = u * (t 0 , x 0 ) and φ(t, x) ≥ u * (t, x), ∀(t, x) ∈ [0, T ] × R. Without loss of generality, we can suppose that the minimum of φu * attained at (t 0 , x 0 ) is strict. We suppose that u * (t 0 , x 0 ) > h(t 0 , x 0 ) and that inf α∈A (-∂ ∂t φ(t 0 , x 0 ) -L α φ(t 0 , x 0 )f (α, t 0 , x 0 , φ(t 0 , x 0 ), (σ ∂φ ∂x )(t 0 , x 0 ), B α φ(t 0 , x 0 ))) > 0.

By uniform continuity of K α φ and B α φ : [0, T ] × R → L 2 ν with respect to α, we can suppose that there exists ǫ > 0 , η ǫ > 0 such that: ∀(t, x) such that t 0 ≤ t ≤ t 0 + η ǫ < T and |xx 0 | ≤ η ǫ , we have: φ(t, x) ≥ h(t, x) + ǫ and

- ∂ ∂t φ(t, x) -L α φ(t, x) -f (α, t, x, φ(t, x), (σ ∂φ ∂x )(t, x), B α φ(t, x)) ≥ ǫ, ∀α ∈ A. (6.4.3)
We denote by B ηε (t 0 , x 0 ) the ball of radius η ε and center (t 0 , x 0 ). By definition of u * , there exists a sequence (t n , x n ) n in B ηε (t 0 , x 0 ), such that (t n , x n , u(t n , x n )) → (t 0 , x 0 , u * (t 0 , x 0 )).

Fix n ∈ N. Let α be an arbitrary control of A tn tn and X α,tn,xn the associated state process. We define the stopping time θ α,n as:

θ α,n := (t 0 + η ǫ ) ∧ inf{s ≥ t n , |X α,tn,xn s -x 0 | ≥ η ǫ }.
Applying Itô's lemma to φ(t, X α,tn,xn t ), we obtain: ), terminal time θ α,n and terminal value φ(θ α,n , X α,tn,xn θ α,n

φ(t n , X α,tn,xn tn ) = φ(θ α,n , X α,tn,xn θ α,n ) - θ α,n tn ψ αs (s, X α,
). By (6.4.3) and by definition of θ α,n , we get that for each s ∈ [t n , θ α,n ]:

-ψ αs (s, X α,tn,xn s ) ≥ f (α s , s, X α,tn,xn s , φ(s, X ) and f (α s , •) of two BSDEs. Now, since the minimum (t 0 , x 0 ) is strict, there exists γ ǫ such that:

u * (t, x) -φ(t, x) ≤ -γ ǫ on [0, T ] × R \ B ηǫ (t 0 , x 0 ). (6.4.5)
We have

φ(θ α,n ∧ t, X α,tn,xn θ α,n ∧t ) = φ(t, X α,tn,xn t )1 t<θ α,n + φ(θ α,n , X α,tn,xn θ α,n )1 t≥θ α,n , t n ≤ t ≤ T.
To simplify notation, set δ ε := min(ǫ, γ ǫ ). Using (6.4.5) together with the definition of θ α,n , we derive that for each t ∈ [t n , θ α,n ]:

φ(t, X α,tn,xn t ) ≥ (h(t, X α,tn,xn t ) + δ ε )1 t<θ α,n + (u * (θ α,n , X α,tn,xn θ α,n ) + δ ε )1 t=θ α,n a.s.
This, together with inequality (6.4.4) on the drivers and the above comparison theorem between a BSDE and a reflected BSDE (see Proposition 6.4.2) lead to:

φ(t n , x n ) ≥ Y α,tn,xn tn,θ α,n [h(t, X α,tn,xn t )1 t<θ α,n + u * (θ α,n , X α,tn,xn θ α,n )1 t=θ α ] + δ ε K,
where K is a positive constant which only depends on T and the Lipschitz constant of f . Now, recall (t n , x n , u(t n , x n )) → (t 0 , x 0 , u * (t 0 , x 0 )) and φ is continuous with φ(t 0 , x 0 ) = u * (t 0 , x 0 ).

We can thus assume that n is sufficiently large so that |φ(t

n , x n ) -u(t n , x n )| ≤ δ ε K 2 .Hence, u(t n , x n ) ≥ Y α,tn,xn tn,θ α,n [h(t, X α,tn,xn t )1 t<θ α,n + u * (θ α,n , X α,tn,xn θ α,n )1 t=θ α ] + δ ε K 2 .
As this inequality holds for all α ∈ A tn tn , we get a contradiction of the sub-optimality principle of dynamic programming principle (6.3.16) satisfied by u * (see also Remark 6.3.15).

• We now prove that u * is a viscosity supersolution of (6.4.2). Let (t 0 , x 0 ) ∈ [0, T [×R and φ ∈ C 1,2 ([0, T ] × R) be such that φ(t 0 , x 0 ) = u * (t 0 , x 0 ) and φ(t, x) ≤ u * (t, x), ∀(t, x) ∈ [0, T ] × R. Without loss of generality, we can suppose that the maximum is strict in (t 0 , x 0 ). Since the solution (Y α,t 0 ,x 0 s ) stays above the obstacle, for each α ∈ A, we have u * (t 0 , x 0 ) ≥ h(t 0 , x 0 ). Our aim is to show that:

inf α∈A (- ∂ ∂t φ(t 0 , x 0 ) -L α φ(t 0 , x 0 ) -f (α, t 0 , x 0 , φ(t 0 , x 0 ), (σ ∂φ ∂x )(t 0 , x 0 ), B α φ(t 0 , x 0 ))) ≥ 0.
Suppose for contradiction that this inequality does not hold. By continuity, we can suppose that there exists α ∈ A, ǫ > 0 and η ǫ > 0 such that: ∀(t, x) with t 0 ≤ t ≤ t 0 + η ǫ < T and |xx 0 | ≤ η ǫ , we have:

- ∂ ∂t φ(t, x) -L α φ(t, x) -f (α, t, x, φ(t, x), (σ ∂φ ∂x )(t, x), B α φ(t, x)) ≤ -ǫ. (6.4.6)
We denote by B ηε (t 0 , x 0 ) the ball of radius η ε and center (t 0 , x 0 ). Let (t n , x n ) n be a sequence in B ηε (t 0 , x 0 ) such that (t n , x n , u(t n , x n )) → (t 0 , x 0 , u * (t 0 , x 0 )). We introduce the state process X α,tn,xn associated with the above constant control α and define the stopping time θ n as:

θ n := (t 0 + η ǫ ) ∧ inf{s ≥ t n , |X α,tn,xn s -x 0 | ≥ η ǫ }.
By Itô's lemma applied to φ(s, X )]. This inequality with (6.4.8) lead to a contradiction. Remark 6.4.6. We mention the paper [START_REF] Peng | A generalized dynamic programming principle and Hamilton-Jacobi-Bellman-Equation[END_REF] which studies stochastic control with nonlinear expectation in the regular case (and when there is no stopping time optimization). The approach is different and relies on the continuity assumption of the reward function. We also mention [START_REF] Soner | Wellposedness of second order backward SDEs[END_REF] where relations between some nonlinear HJB equations and second order BSDEs in the Brownian case are studied.

Examples in mathematical finance

Maximization of recursive utility of terminal wealth. We consider a portfolio optimization problem for an agent with recursive utility. His wealth process X α,t,x is controlled by α, which represents a portfolio-strategy. The recursive utility process is defined via a BSDE associated with a driver f :

[0, T ] × R 2 × L 2 ν → R; (t, y, z, k) → f (t, y, z, k) satisfying Assumption 6.
2.1 and concave with respect to y, z, k. The terminal reward is given by h(X α,t,x T ), where h is a concave non decreasing map. Recall that the recursive utility generalizes the standard additive utilities but in this case, the utility depends on the future utility through the dependance of the driver f on y (see [START_REF] Epstein | Stochastic Differential Utility[END_REF]). Also, the recursive utility may depend on the future utility "variability" or "volatility" through the dependance of f with respect to z and k.

If x is the initial wealth, for each strategy α, the associated recursive utility function at initial time t is equal to E t,T [h(X α,t,x T )], where E is the f -conditional expectation associated with the driver f . The aim of the investor is to maximize his recursive utility of wealth over all portfolio-strategies α ∈ A t t . By Theorem 6.4.5, the value function u(t, x) defined by u(t, x) := sup

α∈A t t E t,x t,T [h(X α,t,x T )]
is a weak viscosity solution of the nonlinear HJB equation:

sup α∈A ( ∂u ∂t (t, x) + L α u(t, x) + f (t, x, u(t, x), (σ ∂u ∂x )(t, x), B α u(t, x))) = 0, (6.5.1)
with u(T, x) = h(x), where the operators are defined by (6.4.2). An example given in [START_REF] Epstein | Stochastic Differential Utility[END_REF] in a Brownian framework is f (t, x, y, z, k) := -C|z|. We can generalize this example to the case of jumps by setting f := f 1 with

f 1 (t, z, k) := -C 1 |z| -C 2 E
|k(e)|Ψ(e)ν(de).

(6.5.2)

The constants C 1 , C 2 are here positive constants which can be interpreted as risk-aversion coefficients. Note that this driver allows us to model asymmetry in risk-aversion, depending on whether the risk comes from the Brownian random source or from the jumps random source (Poisson random measure). If C 2 ≤ 1, then, f 1 satisfies Assumption 6.2.1, in particular condition (iii), which ensures the monotonicity of the recursive utility with respect to terminal reward, and thus to terminal wealth because h is non decreasing. We can also consider an extension of this example to the case of a seller of a European option with payoff G(S T ), where G is an irregular function, for example

G(x) = 1 B (x),
where B is a Borelian, and S is a Markovian jump-diffusion process representing the price of the underlying asset. He wants to maximize his recursive utility of terminal wealth over all portfoliostrategies. In this case, the value function at time t is then given by u(t, x) := sup

α∈A t t E t,x t,T [h(X α,t,x T -G(S T ))].
Note that u is not necessarily continuous, not even measurable. By Theorem 6.4.5, one can derive that u is a weak viscosity solution of an associated nonlinear HJB equation with terminal condition

u(T, x) = h(x 1 -G(x 2 )) for x = (x 1 , x 2 ).
Minimization of the risk of terminal wealth. We consider the same model as in the above example and a dynamic risk-measure ρ defined for each position ξ ∈ L 2 (F T ) by

ρ t,T (ξ) := -E t,T [g(ξ)], 0 ≤ t ≤ T,
where g is a Borelian non decreasing function with polynomial growth. At time t, for a given initial wealth x, the aim of the investor is to minimize his risk-measure of terminal wealth over all portfolio-strategies α ∈ A t t . The value function v(t, x) is given by v(t, x) := inf

α∈A t t ρ t,T [g(X α,t,x T )] = -u(t, x),
where u(t, x) := sup α∈A t t E t,T [g(X α,t,x T )]. As in the previous example, u is a weak viscosity solution of HJB equation (6.5.1) with u(T, x) = g(x). An example is given by f 1 defined by (6.5.2), where the coefficients C 1 and C 2 can also be interpreted as risk-aversion coefficients when the risk comes from the Brownian random source, respectively from the jumps random source.

We can also consider the minimization problem of shortfall risk for the seller of a European option with payoff G(S T ), where G is an irregular function such as an indicator function. More precisely, at time t, for an initial wealth x, the aim of the seller is to minimize the risk measure associated with the negative part (shortfall) of his terminal position given by -(X α,t,x T -G(S T )) -. The value function v(t, x) is then given by v(t, x) := inf

α∈A t t ρ t,T [-(X α,t,x T -G(S T )) -].
The extension to the case when the agent or the seller also acts on stopping times leads to a mixed optimal control/stopping problem. Lemma 6.6.3 (A result of Measure Theory). Let (X, F, Q) be a probability space. Let F Q be the completion σ-algebra of F with respect to Q, that is the class of sets of the form B ∪ M , with B ∈ F and M being a subset of a set N belonging to F with Q-measure 0. Let E be a separable Hilbert space, equipped with its scalar product < . , . >, and its Borel σ-algebra B(E). Then, for each F-measurable map f : X → E, there exists an F-measurable map f Q such that f Q (x) = f (x) for Q-almost every x, in the sense that the set {x ∈ X , f Q (x) = f (x)} is included in a set belonging to F with Q-measure 0.

Proof. For completeness, we give the proof of this lemma. Since E is a separable Hilbert space, its admits a countable orthonormal basis {e i , i ∈ N}. Hence, for each x ∈ X, we have f

(x) = i f i (x)e i with f i (x) =< f (x), e i >. Note that f i : (X, F Q ) → (R, B(R)) is measurable. It is thus sufficient to show that the result holds for f valued in E = R.
For r ∈ Q, let Br := {x ∈ X, f (x) ≤ r}. For each x ∈ X, we have f (x) = inf{r ∈ Q, x ∈ Br }. Since Br ∈ F Q , there exists B r ∈ F such that B r ⊂ Br and Br \B r ⊂ N r , where N r ∈ F with Q-measure 0. For each x ∈ N c r , x ∈ B r if and only if x ∈ Br , which ensures that

f (x) = inf{r ∈ Q, x ∈ B r }. Define f Q for each each x ∈ X by f Q (x) := inf{r ∈ Q, x ∈ B r }. The map f Q is F-measurable because for all x ∈ X, we have f Q (x) = inf r∈Q ϕ r (x)
, where ϕ r is the F-measurable map defined by ϕ r (x) := r if x ∈ B r and ϕ r (x) := +∞ otherwise. Also, {x ∈ X , f

Q (x) = f (x)} ⊂ ∪ r∈Q N r . Hence, f Q (x) = f (x) for Q-almost every x.
Remark 6.6.4. One can immediately derive the following result, used in the proof of Theorem 6.3.7. Let X be a topological space. Let P(X) be the set of all probability measures on B(X), the Borelian σ-algebra of X. For each Q ∈ P(X), B Q (X) denotes the completion of B(X) with respect to Q. The universal σ-algebra on X is then defined by U (X) := ∩ Q∈P(X) B Q (X). Let now E be a separable Hilbert space. Let f : X → E be an universally measurable map, that is U (X)-measurable. By Lemma 6.6.3, for each probability Q on B(X), since U (X) ⊂ B Q (X), there exists a Borelian map f Q : X → E such that f (x) = f Q (x) for Q-almost every x.

Part II

Numerical methods for Doubly Reflected BSDEs with Jumps and RCLL obstacles Chapter 1

Numerical approximation of doubly reflected BSDEs with jumps and RCLL obstacles

Abstract. We study a discrete time approximation scheme for the solution of a doubly reflected Backward Stochastic Differential Equation (DRBSDE in short) with jumps, driven by a Brownian motion and an independent compensated Poisson process. Moreover, we suppose that the obstacles are right continuous and left limited (RCLL) processes with predictable and totally inaccessible jumps and satisfy Mokobodski's condition. Our main contribution consists in the construction of an implementable numerical sheme, based on two random binomial trees and the penalization method, which is shown to converge to the solution of the DRBSDE. Finally, we illustrate the theoretical results with some numerical examples in the case of general jumps.

Introduction

In this chapter, we study in the non-markovian case a discrete time approximation scheme for the solution of a doubly reflected Backward Stochastic Differential Equation (DRBSDE in short) when the noise is given by a Brownian motion and a Poisson random process mutually independent. Moreover, the barriers are supposed to be right-continuous and left-limited (RCLL in short) processes, whose jumps are arbitrary, they can be either predictable or inaccessible. The DRBSDE we solve numerically has the following form:

                               (i) Y t = ξ T + T t g(s, Y s , Z s , U s )ds + (A T -A t ) -(K T -K t ) - T t Z s dW s - T t U s d Ñs , (ii) ∀t ∈ [0, T ], ξ t ≤ Y t ≤ ζ t a.s., (iii) T 0 (Y t --ξ t -)dA c t = 0a.s. and T 0 (ζ t --Y t -)dK c t = 0a.s. (iv) ∀τ predictable stopping time , ∆A d τ = ∆A d τ 1 Y τ -=ξ τ -and ∆K d τ = ∆K d τ 1 Y τ -=ζ τ -.
(1.1.1)

Here, A c (resp. K c ) denotes the continuous part of A (resp. K) and A d (resp. K d ) its discontinuous part, {W t : 0 ≤ t ≤ T } is a one dimensional standard Brownian motion and { Ñt := N tλt, 0 ≤ t ≤ T } is a compensated Poisson process. Both processes are independent and they are defined on the probability space (Ω, F T , F = {F t } 0≤t≤T , P ). The processes A and K have the role to keep the solution between the two obstacles ξ and ζ. Since we consider the general setting when the jumps of the obstacles can be either predictable or totally inaccessible, A and K are also discontinuous.

In the case of a Brownian filtration, non-linear backward stochastic differential equations (BS-DEs in short) were introduced by Pardoux and Peng [START_REF] Pardoux | Adapted solution of a backward stochastic differential equation[END_REF]. One barrier reflected BSDEs have been firstly studied by El Karoui et al in [START_REF] El Karoui | Reflected solutions of backward SDE's, and related obstacle problems for PDE's[END_REF]. In their setting, one of the components of the solution is forced to stay above a given barrier which is a continuous adapted stochastic process. The main motivation is the pricing of American options especially in constrained markets. The generalization to the case of two reflecting barriers has been carried out by Cvitanic and Karatzas in [START_REF] Cvitanic | Backward stochastic differential equations with reflection and Dynkin games[END_REF]. It is also well known that doubly reflected BSDEs are related to Dynkin games and in finance to the pricing of Israeli options (or Game options, see [START_REF] Kifer | Dynkin games and Israeli options[END_REF]). The case of standard BSDEs with jump processes driven by a compensated Poisson random measure was first considered by Tang and Li in [START_REF] Tang | Necessary conditions for optimal control of stochastic systems with random jumps[END_REF]. The extension to the case of reflected BSDEs and one reflecting barrier with only inaccessible jumps has been established by Hamadène and Ouknine [START_REF] Hamadène | Reflected backward stochastic differential equation with jumps and random obstacle[END_REF]. Later on, Essaky in [START_REF] Essaky | Reflected backward stochastic differential equation with jumps and RCLL obstacle[END_REF] and Hamadène and Ouknine in [START_REF] Hamadène | Reflected backward SDEs with general jumps[END_REF] have extended these results to a RCLL obstacle with predictable and inaccessible jumps. Results concerning existence and uniqueness of the solution for doubly reflected BSDEs with jumps can be found in [START_REF] Crépey | Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison[END_REF], [START_REF] Dumitrescu | Generalized Dynkin games and Doubly reflected BSDEs with Jumps[END_REF], [START_REF] Hamadène | BSDEs with two reacting barriers driven by a Brownian motion and an independent Poisson noise and related Dynkin game[END_REF], [START_REF] Hamadène | BSDEs with two RCLL Reflecting Obstacles driven by a Brownian Motion and Poisson Measure and related Mixed Zero-Sum Games[END_REF] and [START_REF] Essaky | Backward stochastic differential equation with two reflecting barriers and jumps[END_REF].

Numerical shemes for DRBSDEs driven by the Brownian motion and based on a random tree method have been proposed by Xu in [START_REF] Xu | Numerical algorithms and Simulations for Reflected Backward Stochastic Differential Equations with Two Continuous Barriers[END_REF] (see also [START_REF] Memin | Convergence of solutions of discrete Reflected backward SDE's and Simulations[END_REF] and [START_REF] Peng | Numerical algorithms for BSDEs with 1-d Brownian motion: convergence and simulation[END_REF]) and, in the Markovian framework, by Chassagneux in [START_REF] Chassagneux | A discrete-time approximation for doubly reflected BSDEs[END_REF]. In the case of a filtration driven also by a Poisson process, some results have been provided only in the non-reflected case. In [START_REF] Bouchard | Discrete-time approximation of decoupled Forward-Backward SDE with jumps[END_REF], the authors propose a scheme for Forward-Backward SDEs based on the dynamic programming equation and in [109] the authors propose a fully implementable scheme based on a random binomial tree. This work extends the paper [START_REF] Briand | Donsker-type theorem for BSDEs[END_REF], where the authors prove a Donsker type theorem for BSDEs in the Brownian case.

Our aim is to propose an implementable numerical method to approximate the solution of DRBSDEs with jumps and RCLL obstacles (2.1.1). As for standard BSDEs, the computation of conditional expectations is an important issue. Since we consider reflected BSDEs, we also have to model the constraints. To do this, we consider the following approximations

• we approximate the Brownian motion and the Poisson process by two independent random walks,

• we introduce a sequence of penalized BSDEs to approximate the reflected BSDE.

These approximations enable us to provide a fully implementable scheme, called explicit penalized discrete scheme in the following. We prove in Theorem 2.4.3 that the scheme weakly converges to the solution of (2.1.1). Moreover, in order to prove the convergence of our sheme, we prove, in the case of jump processes driven by a general Poisson random measure, that the solutions of the penalized equations converge to the solution of the doubly reflected BSDE in the case of a driver depending on the solution, which was not the case in the previous literature (see [START_REF] Essaky | Backward stochastic differential equation with two reflecting barriers and jumps[END_REF], [START_REF] Hamadène | BSDEs with two reacting barriers driven by a Brownian motion and an independent Poisson noise and related Dynkin game[END_REF], [START_REF] Hamadène | BSDEs with two RCLL Reflecting Obstacles driven by a Brownian Motion and Poisson Measure and related Mixed Zero-Sum Games[END_REF]). This gives another proof for the existence of a solution of DRBSDEs with jumps and RCLL barriers. Our method is based on a combination of penalization, Snell envelope theory, stochastic games, comparison theorem for BSDEs with jumps (see [START_REF] Quenez | BSDEs with jumps, optimization and applications to dynamic risk measures[END_REF], [START_REF] Quenez | Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps[END_REF]) and a generalized monotonic theorem under the Mokobodski's condition. It extends [START_REF] Lepeltier | Reflected backward stochastic differential equations with two rcll barriers[END_REF] to the case when the solution of the DRBSDE also admits totally inaccessible jumps. Finally, we illustrate our theoretical results with some numerical simulations in the case of general jumps. We point out that the practical use of our scheme is restricted to low dimensional cases. Indeed, since we use a random walk to approximate the Brownian motion and the Poisson process, the complexity of the algorithm grows very fast in the number of time steps n (more precisely, in n d , d being the dimension) and, as we will see in the numerical part, the penalization method requires many time steps to be stable.

The chapter is organized as follows: in Section 2 we introduce notation and assumptions. In Section 3, we precise the discrete framework and give the numerical scheme. In Section 4 we provide the convergence by splitting the error : the error due to the approximation by penalization and the error due to the time discretization. Finally, Section 5 presents some numerical examples, where the barriers contain predictable and totally inaccessible jumps. In Appendix, we extend the generalized monotonic theorem and prove some technical results for discrete BSDEs to the case of jumps. For the self-containment of the chapter, we also recall some recent results on BSDEs with jumps and reflected BSDEs.

Notations and assumptions

Although we propose a numerical scheme for reflected BSDEs driven by a Brownian motion and a Poisson process, one part of the proof of the convergence of our scheme is done in the general setting of jumps driven by a Poisson random measure. Then, we first introduce the general framework, in which we prove the convergence of a sequence of penalized BSDEs to the solution of (2.1.1).

General framework Notation

As said in Introduction, let (Ω, IF, P ) be a probability space, and P be the predictable σ-algebra on [0, T ] × Ω. W is a one-dimensional Brownian motion and N (dt, de) is a Poisson random measure, independent of W , with compensator ν(de)dt such that ν is a σ-finite measure on IR * , equipped with its Borel field B(IR * ). Let Ñ (dt, du) be its compensated process. Let IF = {F t , 0 ≤ t ≤ T } be the natural filtration associated with W and N .

For each T > 0, we use the following notations:

• L 2 (F T ) is the set of random variables ξ which are F T -measurable and square integrable.

• IH 2 is the set of real-valued predictable processes φ such that φ 2

IH 2 := E T 0 φ 2 t dt < ∞. with θ :Ω × [0, T ] × IR 2 × (L 2 ν ) 2 -→ L 2 ν ; (ω, t, y, z, k 1 , k 2 ) -→ θ y,z,k 1 ,k 2 t (ω, •) P⊗B(IR 2 )⊗B((L 2 ν )
2 )-measurable, bounded, and satisfying dP ⊗dt⊗ν(du)-a.s., for each (y, z, k

1 , k 2 ) ∈ IR 2 × (L 2 ν ) 2 , θ y,z,k 1 ,k 2 t (u) ≥ -1 and |θ y,z,k 1 ,k 2 t (u)| ≤ ψ(u),
where ψ ∈ L 2 ν . We now recall the "Mokobodski's condition" which is essential in the case of doubly reflected BSDEs, since it ensures the existence of a solution. This condition essentially postulates the existence of a quasimartingale between the barriers.

                         (i) Y ∈ S 2 , Z ∈ IH 2 , U ∈ IH 2 ν and α ∈ S 2 , where α = A -K with A, K in A 2 (ii) Y t = ξ T + T t g(s, Y s , Z s , U s )ds + (A T -A t ) -(K T -K t ) - T t Z s dW s - T t IR * U s (e) Ñ (ds, de), (iii) ∀t ∈ [0, T ], ξ t ≤ Y t ≤ ζ t a.s., (iv)
T 0 (Y t --ξ t -)dA t = 0a.s. and T 0 (ζ t --Y t -)dK t = 0a.s. ( 1

Framework for our numerical scheme

In order to propose an implementable numerical scheme we consider that the Poisson random measure is simply generated by the jumps of a Poisson process. We consider a Poisson process {N t : 0 ≤ t ≤ T } with intensity λ and jumps times {τ k : k = 0, 1, ...}. The random measure is then

Ñ (dt, de) = Nt k=1 δ τ k ,1 (dt, de) -λdtδ 1 (de)
where δ a denotes the Dirac measure at the point a. In the following, Ñt := N tλt. Then, the unknown fonction U s (e) does not depend on the magnitude e anymore, and we write U s := U s (1). In this particular case, (1.2.1) becomes:

                         (i) Y ∈ S 2 , Z ∈ IH 2 , U ∈ IH 2 and α ∈ S 2 , where α = A -K with A, K in A 2 (ii) Y t = ξ + T t g(s, Y s , Z s , U s )ds + (A T -A t ) -(K T -K t ) - T t Z s dW s - T t U s d Ñs , (iii) ∀t ∈ [0, T ], ξ t ≤ Y t ≤ ζ t a.s., (iv)
T 0 (Y t --ξ t -)dA t = 0a.s. and T 0 (ζ t --Y t -)dK t = 0 a.s. (1.2.2)
In view of the proof of the convergence of the numerical scheme, we also introduce the penalized version of (1.2.2):

Y p t =ξ + T t g(s, Y p s , Z p s , U p s )ds + A p T -A p t -(K p T -K p t ) - T t Z p s dW s - T t U p s d Ñs , ( 1

Convergence result

The following result states the convergence of Θ converges weakly to α t in L 2 (F T ) as n → ∞ and p → ∞.

In order to prove this result, we split the error in three terms, by introducing Θ p,n t := (Y p,n t , Z p,n t , U p,n t , α p,n t ), the solution of the implicit penalized discrete scheme (2.4.3) and Θ p t := (Y p t , Z p t , U p t , α p t ), the penalized version of (1.2.2), defined by (1.2.3). For the error on Y , we get

E[ T 0 |Y p,n s -Y s | 2 ds] ≤ 3 E[ T 0 |Y p,n s -Y p,n s | 2 ds] + E[ T 0 |Y p,n s -Y p s | 2 ds] + E[ T 0 |Y p s -Y s | 2 ds] ,
and the same splitting holds for |Z p,n -Z| r and |U p,n -U | r . For the increasing processes, we have:

E[|α p,n ψ n (t) -α t | 2 ] ≤ 3 E[|α p,n ψ n (t) -α p,n ψ n (t) | 2 ] + E[|α p,n ψ n (t) -α p t | 2 ] + E[|α p t -α t | 2 ] . (1.4.2)
The proof of Theorem 1.4.1 ensues from Proposition 1.4.2, Corollary 1.4.4 and Proposition 1.4.5. Proposition 1.4.2 states the convergence of the error between Θ p,n , the explicit penalization scheme defined in (1.3.10), and Θ p,n , the implicit penalization scheme. It generalizes the results of [START_REF] Peng | Numerical algorithms for BSDEs with 1-d Brownian motion: convergence and simulation[END_REF]. We refer to Section 1.4. The rest of the Section is devoted to the proof of these results.

Error between explicit and implicit penalization schemes

We prove the convergence of the error between the explicit penalization scheme and the implicit one. The scheme of the proof is inspired from [START_REF] Peng | Numerical algorithms for BSDEs with 1-d Brownian motion: convergence and simulation[END_REF]Proposition 5].

Proposition 1.4.2. Assume Assumption 1.3.3 (i) and g is a Lipschitz driver. We have

lim n→∞ sup 0≤t≤T E[|Y p,n t -Y p,n t | 2 ] + E[ T 0 |Z p,n s -Z p,n s | 2 ds] + E[ T 0 |U p,n s -U p,n s | 2 ds] = 0.
Moreover, lim n→∞ (α p,n tα p,n t ) = 0 in L 2 (F t ), for t ∈ [0, T ].

Proof. By using the definitions of the implicit and explicit schemes (1.3.4) and (1.3.7), we obtain that:

y p,n j+1y p,n j+1 =(y p,n jy p,n j ) + (g p (t j , E[y p,n j+1 |F n j ], y p,n j , z p,n j , u p,n j )g(t j , y p,n j , y p,n j , z p,n j , u p,n j ))

δ n + (z p,n j -z p,n j )e n j+1 √ δ n + (u p,n j -u p,n j )η n j+1 + (v p,n j -v p,n j )µ n j+1 where g p (t, y 1 , y 2 , z, u) = g(t, y 1 , z, u) + p(y 2 -ξ n t ) --p(ζ n t -y 2 ) -. It implies that: E[(y p,n j -y p,n j ) 2 ] =E[(y p,n j+1 -y p,n j+1 ) 2 ] -E[(g p (t j , E[y p,n j+1 |F n j ], y p,n j , z p,n j , u p,n j ) -g p (t j , y p,n j , y p,n j , z p,n j , u p,n j )) 2 ]δ 2 n -E[(z p,n j -z p,n j ) 2 ]δ n -E[(u p,n j -u p,n j ) 2 ](1 -κ n )κ n -E[(v p,n j -v p,n j ) 2 ](1 -κ n )κ n + 2E[(g p (t j , y p,n
j , y p,n j , z p,n j , u p,n j ) g p (t j , E[y p,n j+1 |F n j ], y p,n j , z p,n j , u p,n j ))(y p,n jy p,n j )]δ n .

In the above relation, we take the sum over j from i to n -1. We have:

E[(y p,n i -y p,n i ) 2 ] + δ n n-1 j=i E[(z p,n j -z p,n j ) 2 ] + (1 -κ n )κ n n-1 j=i E[(u p,n j -u p,n j ) 2 ] ≤ 2δ n n-1 j=i E[(g p (t j , y p,n j , y p,n j , z p,n j , u p,n j ) -g p (t j , E[y p,n j+1 |F n j ], y p,n j , z p,n j , u p,n j ))(y p,n j -y p,n j )]. Let us introduce f : y -→ (y -ξ n t ) --(ζ n t -y) -. We have g p (t, y 1 , y 2 , z, u) = g(t, y 1 , z, u) + pf (y 2 )
. The last expectation of the previous inequality can be written E[(g(t j , y p,n j , z p,n j , u p,n j )g(t j , E[y p,n j+1 |F n j ], z p,n j , u p,n j ))(y p,n jy p,n j ) + p(f (y p,n j )f (y p,n j ))(y p,n jy p,n j )]

Since f is decreasing and g is Lipschitz, we obtain:

E[(y p,n i -y p,n i ) 2 ] + δ n n-1 j=i E[(z p,n j -z p,n j ) 2 ] + (1 -κ n )κ n n-1 j=i E[(u p,n j -u p,n j ) 2 ] ≤ 2δ n n-1 j=i E (C g |y p,n j -E[y p,n j+1 |F n j ]| + C g |z p,n j -z p,n j | + C g |u p,n j -u p,n j |)|y p,n j -y p,n j | .
Consequently, by applying the inequality 2ab ≤ a 2 +b 2 for a = C g |y p,n j -y

p,n j | √ 2δ n ; b = δ n 2 |z p,n j - z p,n j | and a = C g |y p,n j -y p,n j | √ 2 δ n κ n (1 -κ n ) ; b = κ n (1 -κ n ) 2 |u p,n j -u p,n j | we get that: E[(y p,n i -y p,n i ) 2 ] + δ n n-1 j=i E[(z p,n j -z p,n j ) 2 ] + (1 -κ n )κ n n-1 j=i E[(u p,n j -u p,n j ) 2 ] ≤ 2δ n C 2 g n-1 j=i E[(y p,n j -y p,n j ) 2 ] + δ n 2 n-1 j=i E[(z p,n j -z p,n j ) 2 ] + 2C 2 g δ 2 n κ n (1 -κ n ) n-1 j=i E[(y p,n j -y p,n j ) 2 ] + (1 -κ n )κ n 2 n-1 j=i E[(u p,n j -u p,n j ) 2 ] + 2C g δ n E[ n-1 j=i |y p,n j -y p,n j ||y p,n j -E[y p,n j+1 |F n j ]|].
Now, since y p,n j -E[y p,n j+1 |F n j ] = g p (t j , E[y p,n j+1 |F n j ], z p,n j , u p,n j )δ n , the last term is dominated by

δ n n-1 j=i (2C g + 1)E[(y p,n j -y p,n j ) 2 ] + C 2 g δ 3 n n-1 j=i E[g p (t j , E[y p,n j+1 |F n j ], y p,n j , z p,n j , u p,n j ) 2 ].
Using the definition of g p yields

g p (t j , E[y p,n j+1 |F n j ], y p,n j , z p,n j , u p,n j ) ≤ |g(t j , E[y p,n j+1 |F n j ], z p,n j , u p,n j )| + p(|y p,n j | + |ξ n j | + |ζ n j |), ≤ |g(t j , 0, 0, 0)| + C g (|E[y p,n j+1 |F n j ]| + |z p,n j | + |u p,n j |) + p(|y p,n j | + |ξ n j | + |ζ n j |).
We get

δ 3 n n-1 j=i E[g p (t j , E[y p,n j+1 |F n j ], y p,n j , z p,n j , u p,n j ) 2 ] ≤ C 0 δ 2 n (δ n n-1 j=i |g(t j , 0, 0, 0)| 2 + δ n n-1 j=i |z p,n j | 2 + δ n n-1 j=i |u p,n j | 2 ) + C 0 (pδ n ) 2 (max j E(|ξ n j | 2 ) + max j E(|ζ n j | 2 )) + C 0 δ 2 n (1 + p 2 ) max j E(|y p,n j | 2 )
where C 0 denotes a generic constant depending on C g . Since

δ n (1 -κ n )κ n = 1 λ λδ n (1 -e -λδn )
E[(y p,n i -y p,n i ) 2 ]+ δ n 2 E[ n-1 j=i (z p,n j -z p,n j ) 2 ] + 1 2 (1 -κ n )κ n E[ n-1 j=i (u p,n j -u p,n j ) 2 ] ≤ 1 + 2C g + 2C 2 g + 2C 2 g δ n (1 -κ n )κ n δ n E[ n-1 j=i (y p,n j -y p,n j ) 2 ] + C 1 (p)δ 2 n , (1.4.3)
where C 1 (p) = C 0 ( g(•, 0, 0, 0)

lim n→∞ ||Y p,n -Y p || 2 J1-L 2 + E[ T 0 |Z p,n s -Z p s | 2 ds + T 0 |U p,n s -U p s | 2 ds] = 0. (1.4.4)
Proof. For a fixed p, we have the following:

Y p,n -Y p = (Y p,n -Y p,n,q ) + (Y p,n,q -Y p,∞,q ) + (Y p,∞,q -Y p ). (1.4.5)
where (Y p,∞,q , Z p,∞,q , U p,∞,q ) is the Picard approximation of (Y p , Z p , U p ) and (Y p,n,q , Z p,n,q , U p,n,q ) represents the continuous time version of the discrete Picard approximation of (y p,n k , z p,n k , u p,n k ), denoted by (y p,n,q k , z p,n,q k , u p,n,q k ). Note that (y p,n,q+1 k , z p,n,q+1 k , u p,n,q+1 k ) is defined inductively as the solution of the backward recursion given by [109, Eq. (3.16)], for the penalized driver g n (ω, t, y, z, u) := g(ω, t, y, z, u) + p(yξ n t (ω))

lim n→∞ E[ T 0 |Y p,n s -Y p s | 2 ds + T 0 |Z p,n s -Z p s | 2 ds + T 0 |U p,n s -U p s | 2 ds] = 0,
Moreover, A p,n (resp. K p,n ) converges to A p (resp. K p ) when n tends to infinity in L 2 for the J1-Skorohod topology.

Proof. Note that:

T 0 |Y p,n s -Y p s | 2 ds ≤ 2 T 0 |Y p,n s -Y p η n (s) | 2 dt + 2 T 0 |Y p η n (s) -Y p s | 2 ds
|A p,n ψ n (t) -A p t | = sup t∈[0,T ] |A p,n t -A p η n (t) | = sup k∈{0,••• ,n} |A p,n t k -A p t k | + sup k∈{0,••• ,n} sup t∈[t k ,t k+1 ] |A p t k -A p η n (t) |.
since ξ and Y p belong to S 2 , we get that the second term in the right hand side tends to 0 in L 2 when n → ∞. The proof of Proposition 1.4.5 is postponed to Section 1.4.3.

sup k∈{0,••• ,n} |A p,n t k -A p t k | ≤ p T 0 |Y p,n s -Y p s | + |ξ n s -ξ s |ds. Since lim n→∞ E[ T 0 |Y p,n s -Y p s | 2 ds] = 0, lim n→∞ E|ξ n s -ξ η(s) | 2 = 0 (

Intermediate result

For each p, q in IN , since the driver g(s, y, z, u) + q(yξ s ) -p(ζ sy) -is Lipschitz in (y, z, u), the following classical BSDE with jumps admits a unique solution (Y p,q , Z p,q , U p,q ) (see

[?]) Y p,q t =ξ T + T t g(s, Y p,q s , Z p,q s , U p,q s )ds + q T t (Y p,q s -ξ s ) -ds -p T t (ζ s -Y p,q s ) -ds - T t Z p,q s dW s - T t IR *
U p,q s (e) Ñ (ds, de).

(1.4.9)

We set A p,q t = q t 0 (Y p,q s ξ s ) -ds and K p,q t = p t 0 (ζ s -Y p,q s ) -ds.

Theorem 1.4.6. Let us assume that Assumption 1.2.4 holds. The quadruple (Y p,q , Z p,q , U p,q , α p,q ), where α p,q = A p,q -K p,q , converges to (Y, Z, U, α), the solution of (1.2.1), as p → ∞ then q → ∞ (or equivalently as q → ∞ then p → ∞) in the following sense : Y p,q converges to Y in IH 2 , Z p,q weakly converges to Z in IH 2 , U p,q weakly converges to U in IH 2 ν , α p,q t weakly converges to α t in L 2 (F t ). Moreover, for each r ∈ [1, 2[, the following strong convergence holds

lim p→∞ lim q→∞ E T 0 |Y p,q s -Y s | 2 ds + E T 0 |Z p,q s -Z s | r ds + T 0 IR * |U p,q s -U s | 2 ν(de) r 2 ds = 0. (1.4.10)
The proof of Theorem 1.4.6 is divided in several steps. We prove 1. the quadruple (Y p,q , Z p,q , U p,q , α p,q ) converges as q → ∞ then p → ∞ 2. the quadruple (Y p,q , Z p,q , U p,q , α p,q ) converges as p → ∞ then q → ∞ Consequently, (1.4.11) and (1.4.12) imply that: Then, there exists a constant C, independent of p and q such that we have :

Y * t =ξ T + T t g(s, Y * s , Z * s , U * s )ds - T t g(s, Y * s , Z * s , U * s )ds + (V T -V t ) -(V ′ T -V ′ t ) - T t Z * s dW s - T t R * U * s (e) Ñ (
E sup 0≤t≤T (Y p,q t ) 2 + E T 0 |Z p,q t | 2 dt + E T 0 IR * |U p,q t (e)| 2 ν(de)dt + E[(A p,q T ) 2 ] + E[(K p,q T ) 2 ] ≤ C. (1.4.13)
Proof. This proof generalizes the proof of [START_REF] Lepeltier | Reflected backward stochastic differential equations with two rcll barriers[END_REF]Proposition 4.1] to the case of jumps. Since p and q play symmetric roles, the calculations over p and q are uniform throughout this proof. From Lemma 1.4.7, we know that there exists (Y

* , Z * , U * , A * , K * ) in S 2 × IH 2 × IH 2 ν × A 2 × A 2 such that Y * t = ξ T + T t g(s, θ * s )ds + (A * T -A * t ) -(K * T -K * t ) - T t Z * s dW s - T t IR * U * s (e) Ñ (
Y * t =ξ T + T t g(s, θ * s )ds + (A * T -A * t ) -(K * T -K * t ) + q T t (ξ s -Y * s ) + ds -p T t (Y * s -ζ s ) + ds - T t Z * s dW s - T t IR * U * s (e) Ñ (ds, de).
Let θ p,q := (Y p,q , Z p,q , U p,q ) and θp,q = ( Ỹ p,q , Zp,q , Ũ p,q ) be the solutions of the following equations

Y p,q t =ξ T + T t g(s, θ p,q s )ds + (A * T -A * t ) + q T t (ξ s -Y p,q s ) + ds -p T t (Y p,q s -ζ s ) + ds (1.4.14) - T t Z p,q s dW s - T t IR * U p,q
s (e) Ñ (ds, de).

(1.4.15)

Ỹ p,q t =ξ T + T t g(s, θp,q s )ds -(K * T -K * t ) + q T t (ξ s -Ỹ p,q s ) + ds -p T t ( Ỹ p,q s -ζ s ) + ds (1.4.16) - T t Zp,q s dW s - T t IR *
Ũ p,q s (e) Ñ (ds, de).

(1.4.17)

By the comparison theorem for BSDEs with jumps (see Theorem 1.9.1), we get that for all p, q in

IN , Ỹ p,q t ≤ Y p,q t ≤ Y p,q t , ξ t ≤ Y * t ≤ Y p,q t and Ỹ p,q t ≤ Y * t ≤ ζ t .
Applying this result to (1.4.14) gives that (Y p,q , Z p,q , U p,q ) is also solution to

Y p,q t = ξ T + T t g(s, θ p,q s )ds + (A * T -A * t ) -p T t (Y p,q s -ζ s ) + ds - T t Z p,q s dW s - T t IR * U p,q
s (e) Ñ (ds, de).

(1.4.18)

Doing the same with (1.4.16) gives that ( Ỹ p,q , Zp,q , Ũ p,q ) is also solution to

Ỹ p,q t = ξ T + T t g(s, θp,q s )ds -(K * T -K * t ) + q T t (ξ s -Ỹ p,q s ) + ds - T t Zp,q s dW s - T t IR *
Ũ p,q s (e) Ñ (ds, de).

(1.4.19)

Let us consider the following BSDEs

Y + t = ξ T + T t g(s, θ + s )ds + (A * T -A * t ) - T t Z + s dW s - T t IR * Ũ + s (e) Ñ (ds, de), (1.4.20)
Y - t = ξ T + T t g(s, θ - s )ds -(K * T -K * t ) - T t Z - s dW s - T t IR * Ũ - s (
≤ Y p,q t ≤ Y p,q t leads to ∀(p, q) ∈ IN 2 , ∀t ∈ [0, T ], Y - t ≤ Ỹ p,q t ≤ Y p,q t ≤ Y p,q t ≤ Y + t . (1.4.22)
Then we have Let us now prove that E[(A p,q T

E[ sup 0≤t≤T (Y p,q t ) 2 ] ≤ max{E[ sup 0≤t≤T (Y + t ) 2 ], E[ sup 0≤t≤T (Y - t ) 2 ]}. ( 1
) 2 ] + E[(K p,q T ) 2 ] ≤ C. Since for all p, q in IN , Ỹ p,q t ≤ Y p,q t ≤ Y p,q t , then Ãp,q t ≥ A p,q t ≥ 0 and K p,q t ≥ K p,q t ≥ 0 . It boils down to prove E[( Ãp,q T ) 2 ] + E[(K p,q T ) 2 ] ≤ C. Let us first prove that E[( Ãp,q T ) 2 ] ≤ C.
To do so, we apply [START_REF] Essaky | Reflected backward stochastic differential equation with jumps and RCLL obstacle[END_REF]Equation (17)] to (1.4.19) (as a sequence in q). In the same way, we apply [START_REF] Essaky | Reflected backward stochastic differential equation with jumps and RCLL obstacle[END_REF]Equation (17)] to (1.4.18) (as a sequence in p). We get

E[(K p,q T ) 2 ] ≤ C. It remains to prove E T 0 |Z p,q t | 2 dt + E T 0 IR * |U p,q t (e)| 2 ν(de)dt ≤ C. By applying Itô's formula to |Y p,q t | 2 , we get E |Y p,q t | 2 + E T t |Z p,q s | 2 ds + E T t IR * |U p,q s (e)| 2 ν(de)ds = E(ξ 2 T ) + 2E T t
Y p,q s g(s, Y p,q s , Z p,q s , U p,q s )ds

+ 2E T t Y p,q s q(Y p,q s -ξ s ) -ds -2E T t Y p,q s p(ζ s -Y p,q s ) -ds .
The third term of the right hand side is null if Y p,q s ≥ ξ s . Then we can bound it by 2E sup 0≤t≤T |ξ t |(A p,q T -A p,q t ) . The last term of the right hand side is bounded in the same way. We bound it by 2E sup 0≤t≤T |ζ t |(K p,q T -K p,q t ) . By using that g is Lipschitz, we bound the second term of the right hand side 2E T t Y p,q s g(s, Y p,q s , Z p,q s , U p,q s )ds

≤2E T t |Y p,q s |( g(•, 0, 0, 0) ∞ + C g (|Y p,q s | + |Z p,q s | + |U p,q s |))ds .
By applying Young's inequality, we get

E |Y p,q t | 2 + E T t |Z p,q s | 2 ds + E T t IR * |U p,q s (e)| 2 ν(de)ds ≤ g(•, 0, 0, 0) 2 ∞ + (1 + 2C g + 4C 2 g )E T t |Y p,q s | 2 ds + 1 2 E T t |Z p,q s | 2 ds + 1 2 E T t IR * |U p,q s (e)| 2 ν(de)ds (1.4.25) + E[ sup 0≤t≤T ξ 2 t ] + E[ sup 0≤t≤T ζ 2 t ] + E[(A p,q T ) 2 ] + E[(K p,q T ) 2 ].
By combining the assumptions on ξ, ζ, (1.4.24) and the previous result bounding E

[(A p,q T ) 2 ] + E[(K p,q T ) 2 ], we get E[ T t |Z p,q s | 2 ds] + E[ T t IR * |U p,q
s (e)| 2 ν(de)ds] ≤ C. In (1.4.9), for fixed p we set g p (s, y, z, u) = g(s, y, z, u)p(ζ sy) -. g p is Lipschitz and

E T 0 (g p (s, 0, 0, 0)) 2 ds ≤ 2E T 0 (g(s, 0, 0, 0)) 2 ds + 2p 2 T E( sup 0≤t≤T (ζ t ) 2 ) < ∞.
By Theorem 1.9.1, we know that (Y p,q ) is increasing in q for all p. Thanks to Theorem 1.9.4, we know that (Y p,q , Z p,q , U p,q ) q∈IN has a limit (Y p,∞ , Z p,∞ , U p,∞ ) := θ p,∞ such that (Y p,q ) q converges increasingly to Y p,∞ ∈ S 2 , and thanks to Theorem 1.9.3, we know that there exists Z p,∞ ∈ IH 2 , U p,∞ ∈ IH 2 ν and A p,∞ ∈ A 2 such that (Y p,∞ , Z p,∞ , U p,∞ , A p,∞ ) satisfies the following equation

Y p,∞ t =ξ T + T t g(s, θ p,∞ s )ds + (A p,∞ T -A p,∞ t ) -p T t (ζ s -Y p,∞ s ) -ds - T t Z p,∞ s dW s - T t IR * U p,∞ s (e) Ñ (ds, de) (1.4.26)
Z p,∞ is the weak limit of (Z p,q ) q in IH 2 , U p,∞ is the weak limit of (U p,q ) q in IH 2 ν and A p,∞ t is the weak limit of (A p,q t ) q in L 2 (F t ). Moreover, for each r ∈ [1, 2[, the following strong convergence holds

lim q→∞ E T 0 |Y p,q s -Y p,∞ s | 2 ds + E T 0 |Z p,q s -Z p,∞ s | r ds + T 0 IR * |U p,q s -U p,∞ s | 2 ν(de) r 2 ds = 0. (1.4.27)
From [78, Theorem 5.1], we also get that ∀t ∈

[0, T ], Y p,∞ t ≥ ξ t and T 0 (Y p,∞ t - -ξ t -)dA p,∞ t = 0 a.s. Set K p,∞ t = p t 0 (ζ s -Y p,∞ s ) -ds. Since Y p,q ր Y p,∞ when q → ∞, K p,q ր K p,∞ when q → ∞.
By the monotone convergence theorem and (1.4.13), we get that E((K p,∞ T ) 2 ) ≤ C. Then we get the following Lemma Lemma 1.4.9. There exists a constant C independent of p such that

E sup 0≤t≤T (Y p,∞ t ) 2 + E T 0 |Z p,∞ t | 2 dt + E T 0 IR * |U p,∞ t (e)| 2 ν(de)dt + E[(A p,∞ T ) 2 ] + E[(K p,∞ T ) 2 ] ≤ C. From Theorem 1.9.2, we have Y p,∞ t ≥ Y p+1,∞ t
, then there exists a process Y such that Y p,∞ ց Y . By using Fatou's lemma, we get

E sup 0≤t≤T (Y t ) 2 ≤ C,
and the dominated convergence theorem gives us that lim p→∞ Y p,∞ = Y in IH 2 . Since (Y p,q ) p is a decreasing sequence, (A p,q ) p is an increasing sequence, and by passing to the limit ((A p,q t ) q weakly converges to A p,∞ t ), we get A p,∞ t ≤ A p+1,∞ t . Then, we deduce from Lemma 1.4.9 that there exists a process A such that A p,∞ ր A and

E(A 2 T ) < ∞. Since A p,q t -A p,q s = t s q(ξ r -Y p,q r ) + dr ≤ t s q(ξ r -Y p+1,q r ) + dr = A p+1,q t -A p+1,q s , we get that A p,∞ t -A p,∞ s ≤ A p+1,∞ t -A p+1,∞ s ∀ 0 ≤ s ≤ t ≤ T.
Thanks to Lemma 1.4.9, we can apply the "generalized monotonic Theorem" 1.6.1: there exist

Z ∈ IH 2 , U ∈ IH 2 ν and K ∈ A 2 such that Y t = ξ T + T t g(s, Y s , Z s , U s )ds + A T -A t -(K T -K t ) - T t Z s dW s - T t IR *
U s (e) Ñ (ds, de), (1.4.28) K t is the weak limit of K p,∞ t in L 2 (F t ), Z is the weak limit of Z p,∞ in IH 2 and U is the weak limit of U p,∞ in IH 2 ν . Moreover, A p,∞ t strongly converges to A t in L 2 (F t ) and A ∈ A 2 , and we have for

each r ∈ [1, 2[, lim p→∞ E T 0 |Y p,∞ s -Y s | 2 ds + E T 0 |Z p,∞ s -Z s | r ds + T 0 IR * |U p,∞ s -U s | 2 ν(de) r 2 ds = 0. (1.4.29) Proof of point 2.
Similarly, (Y p,q ) p is decreasing for any fixed q. The same arguments as before give that (Y p,q , Z p,q , U p,q ) p∈IN has a limit (Y ∞,q , Z ∞,q , U ∞,q ) := θ ∞,q such that (Y p,q ) p converges decreasingly to Y ∞,q ∈ S 2 , and thanks to Theorem 1.9.3, we know that there exists Z ∞,q ∈ IH 2 , U ∞,q ∈ IH 2 Z ∞,q is the weak limit of (Z p,q ) p in IH 2 , U ∞,q is the weak limit of (U p,q ) p in IH 2 ν and K ∞,q t is the weak limit of (K p,q t ) p in L 2 (F t ). From [78, Theorem 5.1], we also get that ∀t ∈

[0, T ], Y ∞,q t ≤ ζ t and T 0 (Y ∞,q t - -ζ t -)dK ∞,q t = 0 a.s. Set A ∞,q t = q t 0 (Y ∞,q s -ξ s ) -ds. Since Y p,
q ց Y ∞,q when p → ∞, A p,q ր A ∞,q when p → ∞. By the monotone convergence theorem and (1.4.13), we get that E((A ∞,q T ) 2 ) ≤ C. We get the following result, equivalent to Lemma 1.4.9 Lemma 1.4.10. There exists a constant C independent of q such that

E sup 0≤t≤T (Y ∞,q t ) 2 + E T 0 |Z ∞,q t | 2 dt + E T 0 IR * |U ∞,q
t (e)| 2 ν(de)dt

+ E[(A ∞,q T ) 2 ] + E[(K ∞,q T ) 2 ] ≤ C.
From Theorem 1.9.2, we have Y ∞,q t ≤ Y ∞,q+1 t , then there exists a process Y ′ such that Y ∞,q ր Y ′ . By using Fatou's lemma, we get that Y ′ belongs to S 2 , and the convergence also holds in IH 2 . By using the same proof as before, we can apply Theorem 1.6.1: there exist Z ′ ∈ IH 2 , U ′ ∈ IH 2 ν and A ′ ∈ A 2 such that

Y ′ t = ξ T + T t g(s, Y ′ s , Z ′ s , U ′ s )ds + A ′ T -A ′ t -(K ′ T -K ′ t ) - T t Z ′ s dW s - T t IR *
U ′ s (e) Ñ (ds, de), A ′ t is the weak limit of A ∞,q t in L 2 (F t ), Z ′ is the weak limit of Z ∞,q in IH 2 and U ′ is the weak limit of U ∞,q in IH 2 ν . Moreover, K ∞,q t strongly converges to K ′ t in L 2 (F t ) and K ′ ∈ A 2 . We will now prove that the two limits are equal.

Proof of point 3. Proof. Since Y p,q ր Y p,∞ and Y p,q ց Y ∞,q , we get that for all p, q ∈ IN , Y ∞,q ≤ Y p,q ≤ Y p,∞ . Then, since Y p,∞ ց Y and Y ∞,q ր Y ′ , we get Y ′ ≤ Y . On the other hand, since Y ∞,q ≤ Y p,q , we get that for all 0 ≤ s ≤ t ≤ T A p,q t -A p,q s ≤ A ∞,q t -A ∞,q s . Since (A p,q t ) q weakly converges to A p,∞ t in L 2 (F t ), (A ∞,q t ) q weakly converges to A ′ t in L 2 (F t ), and (A p,∞ t ) p strongly converges to A t in L 2 (F t ), taking limit in q and then limit in p gives

A t -A s ≤ A ′ t -A ′ s .
(1.4.31)

Since Y p,q ≤ Y p,∞ , we get that for all 0 ≤ s ≤ t ≤ T K p,q t -K p,q s ≤ K p,∞ t -K p,∞ s .

Letting p → ∞ and q → ∞ leads to

K ′ t -K ′ s ≤ K t -K s .
(1.4.32)

Combining (1.4.31) and (1.4.32) gives that for all 0 ≤ s ≤ t ≤ T , A t -A s -(K t -K s ) ≤ A ′ t -A ′ s -(K ′ t -K ′ s ). Thanks to Theorem 1.9.1, we get that Y ′ ≥ Y . Then Y ′ = Y , and we get Z ′ = Z, U ′ = U , and A ′ -K ′ = A -K.

Proof of point 4.

It remains to prove that the limit (Y, Z, U, A -K) of the penalized BSDE is the solution of the reflected BSDE with two RCLL barriers ξ and ζ. To do so, we use stochastic game theory (see Proposition 1.9.5) and Snell envelope theory (see Appendix 1.7).

Theorem 1.4.12. Let α := A -K. The quartuple (Y, Z, U, α) solving (1.4.28) is the unique solution to (1.2.1).

Proof. We know from Theorem 1.2.7 that (1.2.1) has a unique solution. We already know that (Y, Z, U, A, K) belongs to S 2 × IH 2 × IH 2 ν × A 2 × A 2 and satisfies (ii). It remains to check (iii) and (iv). We first check (iii). From (1.4.26), we know that (Y p,∞ , Z p,∞ , U p,∞ , A p,∞ ) is the solution of a reflected BSDE (RBSDE in the following) with one lower barrier ξ. Let α p,∞ := A p,∞ -K p,∞ . Then, (Y p,∞ , Z p,∞ , U p,∞ , α p,∞ ) can be considered as the solution of a RBSDE with two barriers Since Y p,∞ → Y in IH 2 , Z p,∞ → Z in IH r for r < 2, and U p,∞ → U in IH r ν for r < 2, there exists a subsequence p j such that the last conditional expectation converges to 0 a.s. Taking the limit in p in the last inequality gives (1.4.33)

Y t ≥ essinf
In the same way, we know that (Y ∞,q , Z ∞,q , U ∞,q , K ∞,q ) is the solution of a RBSDE with one upper barrier ζ. Let α ∞,q := A ∞,q -K ∞,q . Then (Y ∞,q , Z ∞,q , U ∞,q , α ∞,q ) is the solution of a RBSDE with two barriers ξ -(Y ∞,qξ) -and ζ. By Proposition 1.9.5 we know that

Y ∞,q t ≤ esssup τ ∈Tt essinf σ∈Tt E σ∧τ t g(s, θ s )ds + ξ τ 1 τ ≤σ + ζ σ 1 σ<τ F t + C g E T 0 |Y ∞,q s -Y s | + |Z ∞,q s -Z s | + U ∞,q s -U s ν ds|F t .
Since Y ∞,q → Y in IH 2 , Z ∞,q → Z in IH r for r < 2, and U ∞,q → U in IH r ν for r < 2, there exists a subsequence q j such that the last conditional expectation converges to 0 a.s. Taking the limit in q in the last inequality gives Since ξT = ζT = 0 and ξ and ζ satisfy Mokobodski'condition, we can apply [START_REF] Lepeltier | Reflected backward stochastic differential equations with two rcll barriers[END_REF]Theorem 5.1]: there exists a pair of non-negative RCLL supermatingales (X + , X -) in S 2 such that

Y t ≤
X + t = R t (X -+ ξ), X - t = R t (X + -ζ),
where R t (φ) denotes the Snell enveloppe of φ (see Appendix 1.7). Thanks to [112, Theorem 5.2], we know that Y t -M t = X + t -X - t . Moreover, by the Doob-Meyer decomposition theorem, we get and then Z t = Z 1 t , U t = U 1 t and K t -A t = K 1 t -A 1 t . By using the properties of the Snell envelope in (1.4.35), we get the X + ≥ X -+ ξ and X -≥ X + -ζ, which leads to ξ = M + ξ ≤ Y = M + X + -X -≤ M + ζ = ζ and (iii) follows. It remains to check (iv). From the theory of the Snell envelope (see Section 1.7), we get that 0

X + t = E(A 1 T |F t ) -A 1 t , X - t = E(K 1 T |F t ) -K 1 t ,
= T 0 (X + t --( ξt -+ X - t -))dA 1 t = T 0 (X + t --X - t --ξ t -+ M t -)dA 1 t = T 0 (Y t --ξ t -)dA 1 t , 0 = T 0 (X - t --(X + t --ζt -))dK 1 t = T 0 (X - t --X + t -+ ζ t --M t -)dK 1 t = T 0 (ζ t --Y t -)dK 1 t ,
which ends the proof.

Proof of Proposition 1.4.5

In order to prove the convergence of (Y p , Z p , U p , α p ), we rewrite (1. By using the Cauchy-Schwarz inequality, the convergence of Y p to Y in IH 2 , and the fact that g(s, θ p s ) and g(s, θ s ) are bounded in L 2 (Ω × [0, T ]), we get that the second term of the r.h.s. tends to zero when p tends to ∞. From the dominated convergence theorem the last two terms of the r.h.s. also tend to zero. Since 2 σ≤s≤τ ∆ s A∆ s K ≤ σ≤s≤τ (∆ s A) 2 + σ≤s≤τ (∆ s K) 2 , we are back to Theorem 1.9.3, which ends the proof of (1.4.8).

It remains to prove that Z p weakly converges to Z in IH ). Then, we can extract subsequences which weakly converge in the related spaces. Let us denote Z ′ , U ′ , A ′ and K ′ the respective limits. Since (Z p , U p ) strongly converge to (Z, U ) for any q < 2 (see (1.4.8)), we get that Z = Z ′ and U = U ′ .

Let us prove that A ′ -K ′ = A -K. We have U s (e) Ñ (ds, de).

A p t -K p t = Y p 0 -
Taking the limit in p in the first equation, we get A ′ t -K ′ t = A t -K t .

Numerical simulations

In this section, we illustrate the convergence of our scheme with two examples. The difficulty in the choice of examples is given by the hypothesis we assume, in particular the Mokobodsi's condition which is difficult to check in practice.

Example 1 : inaccessible jumps

We consider the simulation of the solution of a DRBSDE with obstacles having only totally inaccessible jumps. More precisely, we take the barriers and driver of the following form: ξ t := (W t ) 2 + Ñt + (Tt), ζ t := (W t ) 2 + Ñt + 3(Tt), g(t, ω, y, z, u) := -5|y + z| + 6u -1.

Our example satisfies the assumptions assumed in the theoretical part, in particular Hypotheses 2.2.5 and 1.3.3 (see Remark 1.3.5, point 2.). Assumption (1.2.4), which represents the Mokobodski's condition, is fulfilled, since H t := (W t ) 2 + Ñt +2(T -t) satisfies ξ t ≤ H t ≤ ζ t and H t = M t +A t , where M t := (W t ) 2 + Ñt +T -t is a martingale and A t := T -t is a decreasing finite variation process. Table 2.1 gives the values of Y 0 with respect to parameters n and p of our explicit sheme. We notice that the algorithm converges quite fast in p and n. However, when n is too small (n = 20 and n = 50), the result for p = 20000 is quite far from the "reference" result (n = 600 and p = 20000). Concerning the computational time, we notice that it is low, even for big values of p and n.

Example 2 : predictable and totally inaccessible jumps

We consider now the simulation of the DRBSDE with obstacles having general jumps (totally inaccessible and predictable). More precisely, we take the barriers and driver of the fol-lowing form: ξ t := (W t ) 2 + Ñt + (Tt)(1 -1 Wt≥a ), ζ t := (W t ) 2 + Ñt + (Tt)(2 + 1 Wt≥a ), g(t, ω, y, z, u) := -5|y + z| + 6u -1. We first give the numerical results for two different values of a, in order to show the influence of the predictable jumps given by 1 Wt≥a on the solution Y and also the convergence in n and p of the numerical explicit scheme (see Tables 1.2 and 1.3).

Then, Figures 1.2, 1.3 and 1.4 allow to distinguish the predictable jumps of totally inaccesible ones and their influence on the barriers (for e.g. the first jump of the barriers is totally inaccessible, the second and third ones are predictable). Moreover, we remark, as in the previous example, that the solution Y stays between the two obstacles ξ and ζ.

Generalized monotonic limit theorem

The following Theorem generalizes [133, Theorem 3.1] and Theorem 1.9.3 to the case of doubly reflected BSDEs with jumps.

Theorem 1.6.1 (Monotonic limit theorem). Assume that g satisfies Assumption 1.2.2, and ξ belongs to L 2 (F T ). We consider the following sequence (in n) of BSDEs : U s (e) Ñ (ds, de).

Y n t =
Z is the weak limit of (Z n ) n in IH 2 , K t is the strong limit of (K n t ) n in L 2 (F t ), A t is the weak limit of (A n t ) n in L 2 (F t ) and U is the weak limit of (U n ) n in IH Then E(A 2 T ) < ∞. Since the process (A n t ) t is increasing, predictable and such that A n 0 = 0, the limit process A remains an increasing predictable process with A 0 = 0. We deduce from [

Y t = ξ + T 0 f (s, Y s , Z s , V s )ds + K T -K t - T t Z s dW s - T t U
V s (u) Ñ (ds, du), t ≤ T.

Here Z is the weak limit of (Z n ) n in H 2 , K t is the weak limit of (K n t ) n in L 2 (F t ) and V is the weak limit of (V n ) n in H . The sequence (Y n , Z n , V n ) n has a limit (Y, Z, V ) such that Y n converges to Y in S 2 and Z is the weak limit in H 2 , K t is the weak limit of (K n t ) n in L 2 (F t ) and V is the weak limit in H 2 ν .

Stochastic game for DRBSDE

Let us now give the characterization of the solution of the DRBSDE as the value function of a stochastic game we introduce. For more details on stochastic games applied to DRBSDE, we refer to [START_REF] Quenez | Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps[END_REF]. (1.9.7)

The upper and lower value functions at time S associated to the stochastic game are defined respectively by V (S) := essinf This game has a value V , given by the state-process Y solution of DRBSDE, i.e.

Y S = V (S) = V (S).

(1.9.10)

Proof. For each S ∈ T 0 and for each ε > 0, let Chapter 2

τ ε S := inf{t ≥ S,
Reflected scheme for doubly reflected BSDEs with jumps and RCLL obstacles Abstract. We introduce a discrete time reflected scheme to solve doubly reflected Backward Stochastic Differential Equations with jumps (in short DRBSDEs), driven by a Brownian motion and an independent compensated Poisson process. As in [START_REF] Dumitrescu | Numerical approximation for Doubly reflected BSDEs with Jumps and RCLL obstacles[END_REF], we approximate the Brownian motion and the Poisson process by two random walks, but contrary to this chapter, we discretize directly the DRBSDE, without using a penalization step. This gives us a fully implementable scheme, which only depends on one parameter of approximation: the number of time steps n (contrary to the scheme proposed in [START_REF] Dumitrescu | Numerical approximation for Doubly reflected BSDEs with Jumps and RCLL obstacles[END_REF], which also depends on the penalization parameter). We prove the convergence of the scheme, and give some numerical examples.

Introduction

Non-linear backward stochastic differential equations (BSDEs in short) have been introduced by Pardoux and Peng in the Brownian framework in their seminal chapter [START_REF] Pardoux | Adapted solution of a backward stochastic differential equation[END_REF] and then extended to the case of jumps by Tang and Li [START_REF] Tang | Necessary conditions for optimal control of stochastic systems with random jumps[END_REF]. BSDEs appear as a useful mathematical tool in finance (hedging problems) and in stochastic control. Moreover, these stochastic equations provide a probabilistic representation for the solution of semilinear partial differential equations. BSDEs have been extended to the reflected case by El Karoui et al in [START_REF] El Karoui | Reflected solutions of backward SDE's, and related obstacle problems for PDE's[END_REF]. In their setting, one of the components of the solution is forced to stay above a given barrier which is a continuous adapted stochastic process. The main motivation is the pricing of American options especially in constrained markets. The generalization to the case of two reflecting barriers has been carried out by Cvitanic and Karatzas in [START_REF] Cvitanic | Backward stochastic differential equations with reflection and Dynkin games[END_REF]. It is well known that doubly reflected BSDEs (DRBSDEs in the following) are related to Dynkin games and to the pricing of Israeli options (or Game options). The extension to the case of reflected BSDEs with jumps and one reflecting barrier with only inaccessible jumps has been established by Hamadène and Ouknine [START_REF] Hamadène | Reflected backward stochastic differential equation with jumps and random obstacle[END_REF]. Later on, Essaky in [START_REF] Essaky | Reflected backward stochastic differential equation with jumps and RCLL obstacle[END_REF] and Hamadène and Ouknine in [START_REF] Hamadène | Reflected backward SDEs with general jumps[END_REF] have extended these results to a right-continuous left limited (RCLL) obstacle with predictable and inaccessible jumps. Results concerning existence and uniqueness of the solution for doubly reflected BSDEs with jumps can be found in [START_REF] Crépey | Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison[END_REF], [START_REF] Dumitrescu | Generalized Dynkin games and Doubly reflected BSDEs with Jumps[END_REF], [START_REF] Hamadène | BSDEs with two reacting barriers driven by a Brownian motion and an independent Poisson noise and related Dynkin game[END_REF], [START_REF] Hamadène | BSDEs with two RCLL Reflecting Obstacles driven by a Brownian Motion and Poisson Measure and related Mixed Zero-Sum Games[END_REF] and [START_REF] Essaky | Backward stochastic differential equation with two reflecting barriers and jumps[END_REF].

The chapter is organized as follows: in Section 2 we introduce notations and assumptions. In Section 3, we precise the discrete time framework and present the numerical schemes. In Section 4 we provide the convergence of the schemes. Numerical examples are given in Section 5 .

Notations and assumptions

In this Section we introduce notations and assumptions. We recall the result on existence and uniqueness of solution to (2.1.1). We also introduce some assumptions on the obstacles ξ and ζ specific to this chapter (Assumption 2.2.5).

Let (Ω, IF, P) be a probability space, and P be the predictable σ-algebra on [0, T ] × Ω. Let W be a one-dimensional Brownian motion and N be a Poisson process with intensity λ > 0. Let IF = {F t , 0 ≤ t ≤ T } be the natural filtration associated with W and N .

For each T > 0, we use the following notations:

• L 2 (F T ) is the set of F T -measurable and square integrable random variables.

• IH 2 is the set of real-valued predictable processes φ such that φ 2 IH 2 := E T 0 φ 2 t dt < ∞.

• B(IR 2 ) is the Borelian σ-algebra on IR 2 .

• S 2 is the set of real-valued RCLL adapted processes φ such that φ 2 S 2 := E(sup 0≤t≤T |φ t | 2 ) < ∞.

• A 2 is the set of real-valued non decreasing RCLL predictable processes A with A 0 = 0 and E(A 2 T ) < ∞.

Definition 2.2.1 (Driver, Lipschitz driver). A function g is said to be a driver if

• g : Ω × [0, T ] × IR 3 → IR (ω, t, y, z, u) → g(ω, t, y, z, u) is P ⊗ B(IR 3 )-measurable,

• g(., 0, 0, 0) ∞ < ∞.

A driver g is called a Lipschitz driver if moreover there exists a constant C g ≥ 0 and a bounded, non-decreasing continuous function Λ with Λ(0) = 0 such that dP ⊗ dt-a.s. , for each (s 1 , y Let us now introduce an additional assumption on g, which ensures the comparison theorem for BSDEs with jumps (see [START_REF] Quenez | BSDEs with jumps, optimization and applications to dynamic risk measures[END_REF]Theorem 4.2]). The comparison theorem plays a key role in the proof of the convergence of the penalized scheme (see [START_REF] Dumitrescu | Numerical approximation for Doubly reflected BSDEs with Jumps and RCLL obstacles[END_REF]), which is useful to prove the convergence of the reflected scheme (see Section 2.4). Assumption 2.2.4. A lipschitz driver g is said to satisfy Assumption 2.2.4 if the following holds : dP ⊗ dt a.s. for each (y, z, u 1 , u 2 ) ∈ IR 4 , we have g(t, y, z, u 1 )g(t, y, z, u 2 ) ≥ θ(u 1u 2 ), with -1 ≤ θ ≤ θ 0 .

We also assume the following hypothesis on the barriers.

Discrete time framework and numerical scheme 2.3.1 Discrete time framework

For the numerical part of the chapter, we adopt the framework of [109] and [START_REF] Dumitrescu | Numerical approximation for Doubly reflected BSDEs with Jumps and RCLL obstacles[END_REF], presented below.

Random walk approximation of (W, Ñ )

For n ∈ N, we introduce δ := T n and the regular grid (t j ) j=0,...,n with step size δ (i.e. t j := jδ) to discretize [0, T ]. In order to approximate W , we introduce the following random walk

W n 0 = 0 W n t = √ δ [t/
+ 2 δ n-1 i=j E (y p,n i -ζ n i ) + 2 1 2 1 δ n-1 i=j E[(k n i ) 2 ] 1 2 ≤ 2C g + 2C 2 g + 2C 2 g δ κ n (1 -κ n ) δ n-1 i=j E[(y n i -y p,n i ) 2 ] + 2 √ p 1 pδ n-1 i=j E[(a p,n i ) 2 ] 1 2 1 δ n-1 i=j E[(a n i ) 2 ] 1 2 + 2 √ p 1 pδ n-1 i=j E[(k p,n i ) 2 ] 1 2 1 δ n-1 i=j E[(k n i ) 2 ] 1 2
.

Since n ≥ N 0 , Lemma 2.4.

Technical result for the implicit penalized scheme

In this Section, we use N 0 and c introduced in Definition 2.4.

  In a recent paper, Bouchard, Elie and Reveillac[START_REF] Bouchard | BSDEs with weak terminal condition[END_REF] have introduced a new class of Backward Stochastic Differential Equations with weak terminal condition, in which the Tterminal value Y T of the solution (Y, Z) is not fixed as a random variable, but only satisfies a constraint of the form E[Ψ(Y T )] ≥ m. The aim of this paper is to study a more general class of BSDEs, with nonlinear expectation constraints on the terminal condition, induced by the solution of a Backward Stochastic Differential Equation. More precisely, the constraint takes the form

Assumption 3 . 3 . 2 .

 332 The map m ∈ [0, 1] → Φ(ω, m) is continuous for a.e. ω ∈ Ω.

Theorem 3 . 3 . 3 .

 333 Under the Assumption 3.3.2, for each α ∈ A 0 , there exists a right-continuous left limited process (Y α t ) t≤T which aggregates the family {Y α (τ ), τ ∈ T }.

  τ n,k such that: τ n,k → τ when n tends to +∞, τ n,k > τ a.s. on {τ < T } for all n ∈ N and M αk T ≤ Mk,α ′ T . The proof is postponed to Step 1.b. Thanks to the above assertion, we can appeal to (3.3.1) and obtain:

T

  and M μ2 ,α n T belong to [0, 1] and by construction M μ1 , αn T ≥ M μ2 ,α n T a.s., we obtain:

1 2

 1 .5.23) From (3.5.21), (3.5.22) and (3.5.23), by letting ε tend to 0, we get that lim ε→0 µ ε N = μN a.s. This implies that ∆(μ Nµ ε N ) = C|μ Nµ ε N | → 0. Since Φ satisfies Assumption 3.5.3, we get by Theorem 3.5.1, the desired result.

  .5.25) where the last inequality follows by definition of the sequence (µ n , Y n ) n and (1.4.4). The desired result (4.4.18) is obtained by combining (3.5.20) with Step a and (3.5.25).

∇= 1 0∇

 1 x f s, Ms + rδM ε s , αs dr; B α,ε s :π f s, Ms , αs + rδα ε s dr.

Theorem 4 . 3 . 2 .

 432 The function u, defined by (4.2.8), is a viscosity solution (i.e. both a viscosity sub-and supersolution) of the obstacle problem (4.3.1).

  x + β(x, e)) -Φ(t, x)πβ(x, e) ν(de). (4.4.10) B δ [t, x, φ] := B δ φ(t, x + β(x, e))φ(t, x) γ(x, e)ν(de) (4.4.11) Bδ [t, x, Φ] := B c δ Φ(t, x + β(x, e)) -Φ(t, x) γ(x, e)ν(de) (4.4.12)

.4. 18 )

 18 Lemma 4.4.4. Let

.4. 24 )

 24 By Lemma 4.4.2, letting successively α, δ, ε and η tend to 0 in (4.4.24) we obtain that 0 < r 2 M ≤ 0. Hence, the assumption M > 0 made above (see (4.4.21)) is wrong. This ends the proof of Theorem 4.4.1. Corollary 4.4.5 (Uniqueness). Under the additional Assumption (H 2 ), the value function is the unique solution of the obstacle problem (4.3.1) in the class of bounded continuous functions.

  4.29), (4.4.30), (4.4.31) and using the hypothesis on β and γ, we get:

Now, by ( 4 . 4 . 5 )

 445 , we have |x ε,η | ≤ C η and |y ε,η | ≤ C η . Hence, using the hypothesis on β, γ and integrating on B c δ , we get

  [START_REF] Alario-Nazaret | Dynkin games[END_REF], Kobylanski et al.[START_REF] Kobylanski | Dynkin games in a general framework[END_REF]. Let ξ, ζ be two Right Continuous Left-Limited (RCLL) adapted processes with ξ ≤ ζ and ξ T = ζ T a.s. The criterium is given, for each pair (τ, σ) of stopping times valued in [0, T ], by

.3. 1 )

 1 For any S ∈ T 0 , the upper and lower value functions at time S are defined respectively byV (S) := essinfσ∈T S ess sup τ ∈T S E[I S (τ, σ)|F S ] (5.3.2) V (S) := ess sup τ ∈T S essinf σ∈T S E[I S (τ, σ)|F S ]. (5.3.3)

) 2 .

 2 If ξ and ζ are l.u.s.c. along stopping times, then the processes A and A ′ are continuous. The proof is based on classical arguments and is given in the Appendix. Remark 5.4.2. Note that the solution Y of the DRBSDE (5.2.2) coincides with the value function of the classical Dynkin game (5.3.2) and (5.3.3) with the gain:

Definition 5 . 4 . 4 .

 544 ∧σ (I(τ, σ)).(5.4.5) We clearly have the inequality V (S) ≤ V (S) a.s. By definition, we say that there exists a value function at time S for the generalized Dynkin game if V (S) = V (S) a.s.We now introduce the definition of an S-saddle point for this game problem. Let S ∈ T 0 . A pair (τ * , σ * ) ∈ T 2 S is called an S-saddle point for the generalized Dynkin game if for each (τ, σ) ∈ T 2 S we have E g S,τ ∧σ * (I(τ, σ * )) ≤ E g S,τ * ∧σ * (I(τ * , σ * )) ≤ E g S,τ * ∧σ (I(τ * , σ)) a.s.

  s. Similarly, one can show that Y S ≤ E g S,τ ε S ∧σ (I(τ ε S , σ)) + Kε a.s. , which ends the proof of Lemma 5.4.11.

  4.10) is an ε ′ -saddle point at time S with ε ′ = Kε. Remark 5.4.13. Note that contrary to the classical Dynkin game with payoff (5.4.1) (see Remark 5.4.2), the generalized Dynkin game is well-posed in the sense that the criterium does not depend on the value function. The characterization of the solution Y of the DRBSDE (5.2.2) in terms of the value function of the generalized Dynkin game is thus more interesting and exploitable than the one given in Remark 5.4.2.

  v∈V,σ∈T S ess sup u∈U ,τ ∈T S E u,v S,τ ∧σ (I(τ, σ)); (5.4.16) V (S) := ess sup u∈U ,τ ∈T S essinf v∈V,σ∈T S E u,v S,τ ∧σ (I(τ, σ)).

.5. 2 )

 2 Moreover, equality (5.5.1) holds on [S, τ1 ∧ τ2 ∧ σ1 ∧ σ2 ].

  consider now the Markovian case, and we study the links between Markovian generalized Dynkin games (or equivalently DRBSDEs) and obstacle problems. Let b : IR → IR , σ : IR → IR be continuous mappings, globally Lipschitz and β : IR × E → IR a measurable function such that for some nonnegative real C, and for all e ∈ E |β(x, e)| ≤ Cϕ(e), |β(x, e)β(x ′ , e)| ≤ C|xx ′ |ϕ(e), x, x ′ ∈ IR,

  Let us prove recursively that for each n, J and nonnegative. Then J

  and ξ T otherwise. Remark 5.7.4. The proof of Th. 5.3.4 together with Lemma 5.3.

  that A and A ′ are continuous. Since Y and ξ are right-continuous, we have Y σ * S = ξ σ * S and Y τ * S = ζ τ * S a.s. By definition of τ * S , on [S, τ * S [, we have Y t > ξ t a.s. Since (Y, Z, k(.), A, A ′ ) is the solution of the DRBSDE, A is constant on [S, τ * S [ a.s. and even on [S, τ * S ] because A is continuous. Similarly, A ′ is constant on [S, σ * S ] a.s. The process (Y t + t 0 g(s)ds, S ≤ t ≤ τ * S ∧ σ * S ) is thus a martingale. Hence, we have Y S = E[I S (τ * S , σ * S ) | F S ] a.s. By similar arguments as above, one can show that for each τ, σ ∈ T S , E[I S (τ, σ * S ) | F S ] ≤ Y S and Y S ≤ E[I S (τ * S , σ) | F S ] a.s. , which yields that (τ * S , σ * S ) is an S-saddle point.

0 e βs l s 2 ν

 2 condition (5.7.8) holds. Proof of Theorem 5.4.1: For β > 0, φ ∈ IH 2 , and l ∈ IH 2 ν , we introduce the norms φ 2 β := E[ T 0 e βs φ 2 s ds], and l 2 ν,β := E[ T ds]. Let IH 2 β,ν (below simply denoted by IH 2 β ) the space

Chapter 6 A

 6 Weak Dynammic Programming Principle for Stochastic Control/Optimal Stopping with f -Expectations.

  |γ(.)(e)| ≤ Ψ(e) and γ(.)(e) ≥ -1 dν(e)a.s. where Ψ ∈ L 2 ν .

  .), α r (ω, .))du + ( u s σ(X α,t,x r , α r )dW s u )(ω, .), on [s, T ] P -a.s. Now, by the first equality in Lemma 6.3.3, there exists a P -null set N such that for each ω ∈ N c , setting ω = s ω, we have ( u s σ(X α,t,x r , α r )dW s u )(ω, .) = u s σ(X α,t,x r (ω, .), α r (ω, •))dW s u Pa.s. ,

Lemma 1 . 4 . 11 .

 1411 The two limits Y and Y ′ are equal. Moreover Z = Z ′ , U = U ′ and A -K = A ′ -K ′ .

--)

  ξ and ζ + (ζ -Y p,∞ ) -, since we have ξ ≤ Y p,∞ ≤ ζ + (ζ -Y p,∞ ) -, ζ t -(ζ -Y p,∞ ) - t )dK p,∞ t ζ t ) -(ζ t -Y p,∞ t ξ τ 1 τ ≤σ + ζ σ 1 σ<τ + (ζ σ -Y p,∞ σ ) -1 σ<τ F t ξ τ 1 τ ≤σ + ζ σ 1 σ<τ F t , θ s )ds + ξ τ 1 τ ≤σ + ζ σ 1 σ<τ F t -C g E T 0 |Y p,∞ s -Y s | + |Z p,∞ s -Z s | + U p,∞ s -U s ν ds|F t .

  σ∈Tt

  , θ s )ds + ξ τ 1 τ ≤σ + ζ σ 1 σ<τ F t .

|Y s | 2 + 2 ν

 22 |Z s | 2 + U s ds) < ∞.

YY-T-

  4.26), the solution of the reflected BSDE with one lower obstacle ξ Ñ (ds, de), and (1.4.30), the solution of the reflected BSDE with one upper obstacle ζ ξ s ) -ds -(K ∞,p Ñ (ds, de).

Figure 1 . 1 :

 11 Figure 1.1: Trajectories of the solution y p,n and the barriers ξ n and ζ n for λ = 5, N = 200, p = 20000.

Figure 1 . 2 :

 12 Figure 1.2: Trajectories of the Brownian motion for a = -0.2, N = 200.

Figure 1 . 3 :

 13 Figure 1.3: Trajectories of the Compensated Poisson process for λ = 5, N = 200.

Figure 1 . 4 :

 14 Figure 1.4: Trajectories of the solution Y and the barriers ξ and ζ for a=-0.2, λ = 5, N = 200.

2 ν 2 ds = 0 . 0 Z s dW s + τ 0

 22000 . Moreover, for all r ∈ [1, 2[, the following strong convergence holds lim n→∞ ) -U s (e)| 2 ν(de) r Proof of Theorem 1.6.1. This proof follows the proofs of Theorem 1.9.3 and [133, Theorem 3.1]. From the hypotheses, the sequences (Z n ) n , (U n ) n and (g(•, Y n , Z n , U n )) n are bounded in IH 2 , IH 2 νand L 2 ([0, T ] × Ω), then we can extract subsequences which weakly converge in the related spaces. Let Z, U and g 0 denote the respective weak limits. Thus, for each stopping time τ ≤ T , the following weak convergence holds in L 2 (F τ ) n t ) n ր K t in L 2 (F t ). ) Ñ (ds, de)we also have the following weak convergence in L 2 (F τ )A n τ ⇀ A τ := Y 0 -Y τ -τ 0 g 0 (s)ds + K τ + τ IR *U s (e) Ñ (ds, de).

Proposition 1 . 9 . 5 .

 195 Let (Y, Z, U, α) ∈ S 2 × H 2 × H 2 ν × A 2 be a solution of the DRBSDE (1.2.1).For any S ∈ T 0 and any stopping times τ, σ ∈ T S , consider the payoff:I S (τ, σ) = τ ∧σ S g(s, Y s , Z s , U s (•))ds + ξ τ 1 {τ ≤σ} + ζ σ 1 {σ<τ } .

U

  , Y s , Z s , U s (•))ds + Y σ ε S + ε -(K σ ε S -K S ) + (A σ ε S -A S )s (e) Ñ (ds, de) + ε.(1.9.12)On the set {τ ≤ σ ε S }, we obtain:I S (τ, σ ε S ) ≤ τ S g(s, Y s , Z s , U s (•))ds + Y τ -(K τ -K S ) + (A τ -A S ) ≤ Y S + τ S Z s dW s + τ S R *U s (e) Ñ (ds, de).The two above inequalities imply:E[I S (τ, σ ε S )|F S ] ≤ Y S + ε.Similarly, one can show that:E[I S (τ ε S , σ)|F S ] ≥ Y Sε. Consequently, we get that for each ε > 0 esssup τ ∈Ts E[I S (τ, σ ε S )|F S ]ε ≤ Y S ≤ essinf σ∈T S E[I S (τ ε S , σ)|F S ] + ε a.s.,that is V (S)ε ≤ Y S ≤ V (S) + ε a.s. Since V (S) ≤ V (S) a.s., the result follows.

Assumption 2 . 2 )

 22 2.5. ξ and ζ are It processes of the following form ξ t = ξ 0 + where b ξ , b ζ , σ ξ , σ ζ , β ξ and β ζ are adapted RCLL processes such that there exists r > 2 and a constant C ξ,ζ such that E(sup s≤T |b ξ s | r ) + E(sup s≤T |b ζ s | r ) + E(sup s≤T |σ ξ s | r ) + E(sup s≤T |σ ζ s | r ) + E(sup s≤T |β ξ s | r ) + E(sup s≤T |β ζ s | r ) ≤ C ξ,ζ . We also assume ξ T = ζ T a.s., ξ t ≤ ζ t for all t ∈ [0, T ] and the Mokobodski's condition holds.

Figure 2 . 2 :

 22 Figure 2.2: One path of the compensated Poisson process for λ = 5 and n = 400.

Figure 2 . 3 : 2 and 2 1 i=j k p,n i 2 .| 2 ]

 2322122 Figure 2.3: Trajectories of the solution y n and the barriers ξ n and ζ n for λ = 5 and n = 400.

  à la filtration générée par le mouvement brownien. Une définition plus précise est donnée ci-dessous. Definition 2.1.1. La solution d'une EDSR est un couple de processus (Y, Z) à valeurs dans R×R tel que Y est continu et adapté, Z est prévisible et P

.1.1) où {W t } 0≤t≤T est un mouvement brownien défini sur un espace de probabilité équipé de la filtration naturelle complète notée {F t } 0≤t≤T . Les données d'une telle équation sont la condition terminale ξ, qui est une variable aléatoire F T -mesurable à valeurs dans R et un générateur g qui est une fonction aléatoire définie sur [0, T ] × Ω × R × R, à valeurs dans R, mesurable par rapport à la tribu P ⊗ B(R) ⊗ B(R) et B(R) où P représente la tribu prévisible. Résoudre cette équation signifie trouver un couple de processus {(Y t , Z t )} 0≤t≤T qui satisfont l'équation (2.1.1) et qui sont adaptés

  Le développement des EDSR réfléchies a été motivé par deux importantes applications: l'évaluation et la couverture d'options américaines, particulièrement dans les marchés avec contraintes, et la représentation probabiliste des solutions des problèmes d'obstacles pour les EDP non linéaires.Concernant l'application en mathématiques financières, El Karoui, Pardoux et Quenez[START_REF] Karoui | Backward Stochastic Differential Equations in Finance[END_REF] montrent qu'en marché complet, le prix d'une option américaine avec actif sous-jacent (ξ t ) t≤T et prix d'exercice γ est donné par Y 0 où (Y t , π t , A t ) t≤T est la solution de l'EDSR réfléchie suivante:

	.2.2)
	où A est un processus continu et croissant dont le rôle est de repousser la solution Y afin qu'elle
	reste au-dessus de la barrière ξ. La condition de manière minimale. Plus précisément, A croît seulement sur l'ensemble {Y = ξ}. T 0 (Y t -ξ t )dA t = 0 assure que le processus A agit

  .2.6) où X t (η, T ′ ) désigne la solution de l'EDSR avec générateur f et condition terminale (T ′ , η). Si T ′ représente une maturité et η une position financière à la date T ′ , alors ρ(η, T ′ ) est interprété comme le risque de η à la date t. La fonctionnelle ρ : (η, T ′ ) → ρ • (η, T ′ ) représente alors une mesure de risque dynamique induite par l'EDSR avec générateur g.Soit (ξ t , 0 ≤ t ≤ T ) un processus càdlàg et adapté dans S 2 qui représente une position financière dynamique. Soit S ∈ T 0 . Le problème est minimiser la mesure de risque à la date S.

			Soit v(S) la
	fonction valeur associée, égale à la variable aléatoire F S -mesurable (unique pour l'égalité au sens presque sûr) définie par
	v(S) := essinf τ ∈T S	ρ S (ξ τ , τ ),	(2.2.7)

  Cela fournit un résultat d'existence pour le problème d'obstacle sous des hypothèses relativement faibles. Dans le cas brownien, ce résultat est obtenu en utilisant une méthode de pénalisation par les EDSR non réfléchies. Cette méhode pourrait aussi être adaptée dans notre cas avec sauts, mais impliquerait de lourds calculs afin de prouver la convergence des solutions des EDSR pénalisées vers la solution de l'EDSR réfléchie. Cela demanderait également des résultats de convergence des solutions de viscosité dans le cas intégro-différentiel. Nous obtenons une preuve directe et plus courte.

	où (t, x) est une condition initiale fixée et X t,x est un processus d'état qui a la dynamique suivante:
	X t,x s = x +	t	s	b(X t,x r )dr +	t	s	σ(X t,x r )dW r +	t	s	R *	β(X t,x r -, e) Ñ (dr, de).	(2.2.11)
	Les fonctions f, h, g, b, σ, β sont déterministes et satisfont les conditions de Lipschitz habituelles (le
	lecteur pourra se référer au chapitre correspondant). Dans le cadre markovien, pour chaque (t, x),
	la mesure de risque minimale v(t, x) est définie par:
							v(t, x) = -Y t,x t ,			(2.2.12)
	où Y t,x est l'EDSR réfléchie dont les paramètres sont donnés par (1.2.10).
	Notre contribution principale est l'établissement d'un lien entre la fonction valeur de notre
	problème d'arrêt optimal et les inégalités variationnelles des EDP intégro-différentielles et paraboliques.
	Nous prouvons que la mesure de risque minimale est une solution de viscosité d'une EDP intégro-
	différentielle.
												(2.2.10)
							t,x s (ω), y, z, k), s ≤ T,

  Le cadre associé au problème est très simple. Il y a deux joueurs, nommés Joueur 1 et Joueur 2, qui observent deux processus de gain ξ et ζ définis sur un espace de probabilité (Ω, F, P). Joueur 1 (respectivement Joueur 2) choisit un temps d'arrêt τ (respectivement σ) comme contrôle pour ce problème d'arrêt optimal. Au temps (d'arrêt) σ ∧ τ le jeu est terminé et Joueur 2 paie le montant ζ σ 1 τ >σ + ξ τ 1 τ ≤σ à Joueur 1. Par conséquent, l'objectif de Joueur 1 est de maximiser son paiement pendant que Joueur 2 essaie de le minimiser. Il est alors naturel d'introduire les valeurs inférieures et supérieures du jeu.

	et le gain dépend de qui l'a arrêté. Ce jeu d'arrêt stochastique, de nos jours appelé jeu de Dynkin,
	a été introduit pour la première fois par Dynkin [66] comme une généralisation des problèmes
	d'arrêt optimal. Depuis, une quantité considérable de travaux de recherche a été réalisée sur les
	jeux de Dynkin et les problèmes associés. Quelques exemples: Dynkin et Yushkevich (1968) [67],
	Bensoussan et Friedman (1974) [21], Neveu (1975) [121], Bismut (1977) [23], Stettner (1982) [146],
	Alario, Lepeltier et Marechal (1982) [1], Morimoto (1984) [119], Lepeltier et Maingueneau (1984)
	[111], Cvitanic et Karatzas (1996) [52], Karatzas et Wang (2001) [101], Ekstrom et Peskir [77],
	Laraki et Solan [114], Peskir [135], Rosenberg et al. [139], Touzi et Vieille (2002) [149] etc. La
	plupart de la littérature s'intéresse à établir l'existence d'arrêts optimaux ainsi que la valeur sous
	différents modèles et hypothèses de gain. En temps discret, il est facile de montrer l'existence
	d'arrêt optimaux ainsi que la valeur en utilisant des arguments d'induction rétrograde. En temps
	continu, un résultat important est dû à Lepeltier et Maingueneau [111] qui prouvent l'existence de
	temps d'arrêts ε-optimaux ainsi que la valeur.
	Rappelons la formulation mathématique d'un jeu de Dynkin classique.
	Ce chapitre repose sur l'article ≪Generalized Dynkin games and Doubly Reflected BSDEs with jumps≫ [62], travail en collaboration avec M.C. Quenez et A. Sulem et soumis pour publication.
	2.3.1 Préliminaires et vue d'ensemble de la littérature
	Le jeu de Dynkin est un jeu à somme nulle d'arrêt optimal entre deux joueurs. Chaque joueur
	peut soit arrêter le jeu soit le continuer. Le jeu est arrêté dès lors que l'un des deux joueurs arrête

  (Principe de Programmation Dynamique Faible). 1. Soit {θ α , α ∈ U t } une famille de temps d'arrêt finis indépendants de F t à valeurs dans [t, T ]. Alors:

	.4.5)
	Rappelons le Principe de Programmation Dynamique faible.
	Theorem 2.4.1

  Soner, Touzi et Zhang ([145]) ont récemment introduit la notion d'EDSR du second ordre (EDSR2) dont l'idée de base est d'exiger que la solution vérifie les équations P α p.s. pour chaque mesure de probabilité dans une classe non dominée de mesures mutuellement singulières. Leur théorie est étroitement liée à la notion

de G-espérance de Peng ([129]) et permet d'obtenir une représentation probabiliste différente des solutions d'équations HJB complètement non linéaires.

  for all τ ∈ T . Hence, the following limits:

	lim s∈(t,T ]↓t	Y α s and	lim s∈(t,T ]↑t

  Step 2. Let us prove now the converse inequality Y

	α τ ≥ Y α τ a.s. We apply on [τ, τ n ] the stability result for BSDEs with parameters (Y	α τ , 0) and (Y α τn , g1 [0,τn) ), we
	obtain:

.3.11) Since σ k = τ on A k and since (3.3.10), (3.3.11) hold, we finally obtain: M α τn = E f τn,T [ Mk,α ′ T ] = M αk τn a.s. on A k . (3.3.12) By (3.3.6), (3.3.8) and (3.3.12), we deduce that M α τ n,k = M αk τ n,k a.s. and hence that αk ∈ A α τ n,k .

  .3.13) Definition 3.3.1, together with the assumptions on the driver g, the convergence of τ n to τ , the observation on the integrability of Y α (see 3.2.8), and Lebesgue's Theorem imply that E[

	τ n τ |g(s, Y we let n tend to ∞ in (3.3.13), and obtain E g α τ , 0)| 2 ds] → 0. By the same arguments and (3.3.1), we get ||Y τ,τn [Y α α τ a.s., up to a subsequence. α τ -Y α τn || L 2 → 0. Now, τn ] → Y Moreover, Lemma 3.3.1 implies that E g τ,τn [Y α τn ] ≥ Y α τ

  by definition of the value function Y α τ ′ and the comparison theorem. As above, there exists (α n

  By 3.2.8, each Y n of the above family is bounded below by -η t and hence this also holds for Y c t (µ). The converse inequality Y c t ≤ η t is clear since Remark 3.2.8 holds and, by construction

  s. As Assumption 3.5.3 holds, we can apply Theorem 3.3.3 and get Y t+ε (M µn,αn t+ε ) → Y t (µ n ) a.s. This convergence together with inequality 3.2.8 allow us to apply Lebesgue's Theorem and to obtain the desired result.

  is a Hilbert space equipped with the scalar product δ, ℓ ν := R * δ(e)ℓ(e)ν(de) for all δ, ℓ ∈ L 2 ν × L 2 ν . We also introduce the set H 2 (resp. H 2 ν ) of predictable processes (π t ) (resp. (l t (•))) such that E the set S 2 of real-valued RCLL adapted processes (ϕ s ) with E[sup s ϕ 2 s ] < ∞, and the set L 2 (F T ) of F T -measurable and square-integrable random variables. Let (t, x) be a fixed intial condition. For each maturity S in [t, T ] and each position ζ in L 2 (F S ), the associated risk measure at time s ∈ [t, S] is defined by

	T 0 π 2 s ds<∞ (resp. E	T 0 l s	2 L 2 ν ds<∞);

  Here T t denotes the set of stopping times with values in [t, T ].By Th. 3.2 in[START_REF] Quenez | Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps[END_REF], the minimal risk measure is characterized via the solution Y t,x in S 2 of the following reflected BSDE (RBSDE) associated with driver f and obstacle ξ:

	x t,τ (ξ t,x τ ).	(4.2.5)

  x is a nondecreasing, continuous predictable process in S 2 with We can thus define a deterministic function u called value function of our optimal stopping problem by setting for each t, x By Lemma 4.6.4 and Lemma 4.6.5 given in Appendix, the function u is continuous and has at most polynomial growth.

					(4.2.6)
			A t,x t = 0 and such that
	T
	t	(Y t,x s	-ξ t,x s )dA t,x s = 0 a.s. ,
	3.2 in [138] ensures that
			Y t,x t	= ess sup τ ∈Tt	E t,x t,τ (ξ t,x τ ) a.s.	(4.2.7)
	The SDE (4.2.1) and the RBSDE (4.2.6) can be solved with respect to the translated Brownian
	motion (W u(t, x) := Y t,x t .	(4.2.8)

with Z t,x , K t,x ∈ H 2 (resp. H 2 ν ). Note that by the assumptions made on h and g, the obstacle (ξ ,t,x s ) s≥t is continuous except at the inaccessible jump times of the Poisson measure, and at time T with ∆ξ t,x T ≤ 0 a.s., and this implies the continuity of A t,x by Th. 2.6 in

[START_REF] Quenez | Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps[END_REF]

. Moreover, Th. s -W t ) s≥t . Hence Y t,x t is constant for each t, x.

  Theorem 5.4.1 (Existence and uniqueness for DRBSDEs). Suppose ξ and ζ are RCLL adapted process in S 2 such that ξ

T = ζ T a.s. and ξ t ≤ ζ t , 0 ≤ t ≤ T a.s. Suppose that J 0 ∈ S 2 (or equivalently suppose that Mokobodski's condition is satisfied). Then, DRBSDE (5.2.2) admits a unique solution (Y, Z, k(.

), A, A ′ ) ∈ S 2 × IH 2 × IH 2 ν × (A 2

  τ ∧σ (I(τ , σ)) a.s.Let τ ∈ T S . We want to show that for each τ ∈ T S

	Y S ≥ E g S,τ ∧σ (I(τ, σ)) a.s.	(5.4.6)
	Since the process (Y

t , S ≤ t ≤ τ ∧ σ) is a strong E g -supermartingale, we get Y S ≥ E g S,τ ∧σ (Y τ ∧σ ) a.s. (5.4.7)

Since Y ≥ ξ and Y σ = ζ σ a.s. , we also have

  Proof. Let S ∈ T 0 . Since Y and ξ are right-continuous processes, we have Y σ * S = ζ σ * S and Y τ * S = ξ τ * S a.s. By definition of τ * S , for almost every ω, we have Y t (ω) > ξ t (ω) for each t ∈ [S(ω), τ * S (ω)[. Hence, since Y is solution of the DRBSDE, the continuous process A is constant on [S, τ * S ] a.s. because A is continuous. Hence, A τ * S = A S a.s. Similarly, A ′

	σ * S = A ′ S a.s. By Lemma 5.4.5, (τ * S , σ * S

.s. The same properties hold for τ S , σ S . Remark 5.4.7. Note that σ * S ≤ σ S and τ * S ≤ τ S a.s. Moreover, by Proposition 5.7.8 in the Appendix, (Y t , S ≤ t ≤ τ S ) is a strong E g -submartingale and (Y t , S ≤ t ≤ σ S ) is a strong E g -supermartingale.

  .s. Remark 5.4.10. By the second point and Proposition 5.7.8 in the Appendix, the process (Y t , S ≤ t ≤ τ ε S ) is a strong E g -submartingale and the process (Y t , S ≤ t ≤ σ ε S ) is a strong E g -supermartingale. Lemma 5.4.11. Let ε > 0. For all S ∈ T 0 and (τ, σ) ∈ T 2 S , we have where K is a positive constant which only depends on T and the Lipschitz constant C of f . Proof. Let τ ∈ T S . By Remark 5.4.7, the process (Y t , S ≤ t ≤ σ ε S ) is a strong E g -supermartingale.

	The first point follows from the definitions of τ ε S and σ ε S and the right-continuity of ξ, ζ and Y . Let us show the second point. Note that τ ε S ∈ T S and σ ε S (I(τ, σ ε S )) -Kε ≤ Y S ≤ E g S,τ ε S ∧σ (I(τ ε S , σ)) + Kε a.s. , (5.4.13) Hence, Y S ≥ E g S,τ ∧σ ε S (Y τ ∧σ ε S ) a.s. (5.4.14) S ∈ T E S,τ ∧σ ε Since Y ≥ ξ and Y σ ε S ≥ ζ σ ε S -ε a.s. (see Lemma 5.4.9), we have:

S . Fix ε > 0. For a.e. ω, if t ∈ [S(ω), τ ε S (ω)[, then Y t (ω) > ξ t (ω) + ε and hence Y t (ω) > ξ t (ω). It follows that almost surely, A c is constant on [S, τ ε S ] and A d is constant on [S, τ ε S [. Also, Y (τ ε S ) -≥ ξ (τ ε S ) -+ ε a.s. Since ε > 0, it follows that Y (τ ε S ) -> ξ (τ ε S ) - a.s., which implies that ∆A d τ ε S = 0 a.s. Hence, almost surely, A is constant on [S, τ ε S ]. Similarly, A ′ is a.s. constant on [S, σ ε S ].

  S . Since Y ≥ ξ and Y σ * S = ζ σ * S a.s. , we have Y τ ∧σ * S = Y τ 1 τ ≤σ * S + Y σ * S 1 σ * S <τ ≥ ξ τ 1 τ ≤σ * S + ζ σ * S 1 σ * S <τ = I(τ, σ * S ) a.s. -dY t = g u,v (t, Y t , Z t , k t )dt + dA t -Z t dW t -S≤t≤τ ∧σ *S is the solution of the BSDE associated with generalized driver g u,v (•)dt+dA t and terminal condition Y τ ∧σ * S . By using Assumption(5.4.18), the inequality Y τ ∧σ * S ≥ I(τ, σ * S ) and the comparison theorem for BSDEs with jumps, we obtain that for each u ∈ U:

	Moreover, by Theorem 5.4.6, A ′ σ * S = A ′ s a.s., which implies that:

τ * S ∧σ * S (I(τ * S , σ * S )) a.s. Let τ ∈ T E k t (e) Ñ (dt, de); S ≤ t ≤ σ * S , dt ⊗ dP a.s. Hence, (Y t )

  (Comparison theorem for DRBSDEs.). Let ξ 1 , ξ 2 , ζ 1 , ζ 2 be processes in S 2 such that ξ i T = ζ i T a.s. and ξ i t ≤ ζ i t , 0 ≤ t ≤ T a.s. for i = 1, 2. Suppose that for i = 1, 2, ξ i , ζ i satisfies Mokobodski's condition. Let g 1 and g 2 be Lipschitz drivers satisfying Assumption 5.4.3. Suppose that

	5.5 Comparison theorems for DRBSDEs with jumps and
	a priori estimates
	5.5.1 Comparison theorems
	Theorem 5.5.1

  1 t = η t 1 t<θ n + ξ n 1 θ n ≤t<T and η 2 t = η t 1 t<θ + η θ 1 θ≤t<θ n + ξ1 θ n ≤t<T . Using the notation of Proposition 5.7.6, (Y i , Z i , K i ) denotes the solution of the reflected BSDE associated with terminal time T , driver f i , obstacle (η i t ) and terminal condition ξ i . We have Y 1 . = Y .,θ n (ξ n ) a.s. Moreover, since by assumption η θ ≤ ξ a.s. , we have Y 2 .

	= Y .,θ (ξ) a.s. Note that
	(Y 2 t , Z 2 t , k 2 t ) = (ξ, 0, 0) a.s. on {t ≥ θ}. We thus obtain

  The function u * satisfies the weak super-optimality principle of dynamic programming, that is for each t ∈ [0, T ] and for each stopping time θ ∈ T t t , that is

	u(t, x) ≥ sup α∈A t t	sup τ ∈T t t	E α,t,x t,θ∧τ h(τ, X α,t,x τ	)1 τ <θ + u * (θ, X α,t,x θ	)1 τ ≥θ .	(6.3.17)
	Remark 6.3.14.
	t t , that is
	u(t, x) ≤ sup α∈A t t	sup τ ∈T t t	E α,t,x t,θ∧τ h(τ, X α,t,x τ	)1 τ <θ + u * (θ, X α,t,x θ	)1 τ ≥θ ,	(6.3.16)

  Lemma 6.4.1. Let t 0 ∈ [0, T ] and let θ ∈ T t 0 . Let ξ 1 and ξ 2 ∈ L 2 (F θ ). Let f 1 be a driver. Let f 2 be a Lipschitz driver with Lipschitz constant C > 0, satisfying Assumption 6.3.9. For i = 1, 2, let (X i t , π i t , l i t ) be a solution in S 2 × IH 2 × IH 2 ν of the BSDE associated with driver f i , terminal time θ and terminal condition ξ i . Suppose thatξ 1 ≥ ξ 2 + ε a.s. and f 1 (t, X 1 t , π 1 t , l 1 t ) ≥ f 2 (t, X 1 t , π 1 t , l 1 t ) t 0 ≤ t ≤ θ, dt ⊗ dP a.s.where ε is a real constant. Then, for each t ∈ [t 0 , θ], we have X 1 t -X 2 t ≥ ε e -CT a.s. Proof. From inequality(4.22) in the proof of the Comparison Theorem in[START_REF] Quenez | BSDEs with jumps, optimization and applications to dynamic risk measures[END_REF], we derive thatX 1 t 0 -X 2 t 0 ≥ e -CT E [H t 0 ,θ ε |F t 0 ] a.s., where C is the Lipschitz constant of f 2 , and (H t 0 ,s ) s∈[t 0 ,T ] is the non negative martingale satisfying dH t 0 ,s = H t 0 ,s -[β s dW s + E γ s (u) Ñ (ds, du)] with H t 0 ,t 0 = 1, (β s ) being a predictable process bounded by C. The result follows.From this property, we derive the following comparison result. Proposition 6.4.2 (A comparison theorem between a BSDE and a reflected BSDE). Let t 0 ∈ [0, T ] and let θ ∈ T t 0 . Let ξ 1 ∈ L 2 (F θ ). Let f 1 be a driver. Let f 2 be a Lipschitz driver with Lipschitz constant C > 0 which satisfies Assumption 6.3.9. Let (ξ 2 t ) ∈ S 2 . Let (X 1 t , π 1 t , l 1 t ) be a solution of the BSDE associated with f 1 , terminal time θ and terminal condition ξ 1 . Let (Y 2 t ) be the solution of the reflected BSDE associated with f 2 , terminal time θ and obstacle (ξ 2 t ). Suppose that

2

+ εK.

  1 t ≥ Y 2 t + εe -CT a.s. Proof. Let t ∈ [t 0 , θ]. By the characterization of the solution of the reflected BSDE as the value function of an optimal stopping problem (see Theorem 3.2 in [137]), Y 2 t = ess sup τ ∈T [t,θ] E 2

	t,τ (ξ 2 τ ).
	Now, by Lemma 6.4.1, for each τ ∈ T [t,θ] , X 1 t ≥ E 2 t,τ (ξ 2 τ

  ∈ [t n , θ α,n ]. This inequality gives a relation between the drivers -ψ αs (s, X α,tn,xn s

	α,tn,xn s	), (σ	∂φ ∂x	)(s, X α,tn,xn s	), Bφ(s, X α,tn,xn s	)) + ǫ (6.4.4)
	for each s

  X α,tn,xnwhere the second inequality is obtained by using the above extension of the comparison theorem (Lemma 6.4.1). Now, we can assume that n is sufficient large so that |φ(tn , x n )u(t n , x n )| ≤ δ ε K 2 .Hence, we get:u(t n , x n ) ≤ E α,tn,xn tn,θ n [u * (θ n , X α,tn,xn

	α,tn,xn s ), (σ ∂φ ∂x )(s, θ n ), we have that Since u * satisfies the super-optimality principle of dynamic programming (Th. 6.3.13), we have (φ(s, X α,tn,xn s )] -γ ǫ K 2 . (6.4.8) u(t n , x n ) ≥ E α,tn,xn tn,θ n [u * (θ n , X α,tn,xn θ n

  Definition 1.2.3 (Mokobodski's condition). Let ξ, ζ be in S 2 . There exist two nonnegative RCLL supermartingales H and H ′ in S 2 such that ∀t ∈ [0, T ], ξ t 1 t≤T ≤ H t -H ′ t ≤ ζ t 1 t≤T a.s. Assumption 1.2.4. ξ and ζ are two adapted RCLL processes with ξT = ζ T a.s., ξ ∈ S 2 , ζ ∈ S 2 , ξ t ≤ ζ t for all t ∈ [0, T ],the Mokobodski's condition holds and g is a Lipschitz driver satisfying Assumption 2.2.4. We introduce the following general reflected BSDE with jumps and two RCLL obstacles Definition 1.2.5. Let T > 0 be a fixed terminal time and g be a Lipschitz driver. Let ξ and ζ be two adapted RCLL processes with ξ T = ζ T a.s., ξ ∈ S 2 , ζ ∈ S 2 , ξ t ≤ ζ t for all t ∈ [0, T ] a.s. A process (Y, Z, U, α) is said to be a solution of the double barrier reflected BSDE (DRBSDE) associated with driver g and barriers ξ, ζ if

  .2.1) Remark 1.2.6. Condition (iv) is equivalent to the following condition : ifK = K c + K d and A = A c + A d ,where K c (resp. K d ) represents the continuous (resp. the discontinous) part of K (the same notation holds for A), then Theorem 1.2.7 ([62], Theorem 4.1). Suppose ξ and ζ are RCLL adapted processes in S 2 such that for all t ∈ [0, T ], ξ t ≤ ζ t and Mokobodski's condition holds (see Definition 1.2.3). Then, DRBSDE (1.2.1) admits a unique solution (Y, Z, U, α) in S 2 × IH 2 × IH 2 ν × A 2 . Remark 1.2.8. As said in [62, Remark 4.3], if for all t ∈]0, T ] ξ t -< ζ t -a.s., [62, Proposition 4.2]gives the uniqueness of A, K ∈ (A 2 ) 2 .Definition 1.2.9 (convergence in J1-Skorokhod topology). ξ n is said to converge in probability (resp. in L 2 ) to ξ for the J1-Skorokhod topology, if there exists a family (ψ n ) n∈IN of one-to-one random time changes from [0, T ] to [0, T ] such that sup t∈[0,T ] |ψ n (t) -t| ---→ in probability (resp. in L 2 ). Throughout the chapter, we denote this convergence ||ξ n -ξ|| J1-P → 0 (resp. ||ξ n -ξ|| J1-L 2 → 0).

	sup t∈[0,T ] |ξ n ψ n (t) -ξ t | ---→	n→∞	0 almost surely and

T 0 (Y tξ t )dA c t = 0 a.s., T 0 (ζ t -Y t )dK c t = 0 a.s. and ∀τ ∈ T 0 predictable, ∆A d τ = ∆A d τ 1 Y τ -=ξ τ -and ∆K d τ = ∆K d τ 1 Y τ -=ζ τ -. n→∞ 0

  As for the implicit scheme, we define the continuous time version (Y

												p,n t , Z	p,n t , U	p,n t , A p,n t , K	p,n t ) 0≤t≤T
	of the solution to (1.3.9):
										[t/δn]				[t/δn]
	Y	p,n t	= y p,n [t/δn] , Z	p,n t	= z p,n [t/δn] , U	p,n t	= u p,n [t/δn] , A p,n t	=	a p,n j	K	p,n t	=	k	p,n j .	(1.3.10)
										j=0				j=0
	We also introduce α p,n t	:= A p,n t -K	p,n
																.2.3)
	with A p t := p

t 0 (Y p sξ s ) -ds and K p t := p t 0 (ζ s -Y p s ) -ds, and α p t := A p t -K p t for all t ∈ [0, T ]. t , for all t ∈ [0, T ].

  1. Corollary 1.4.4 states the convergence (in n) of Θ p,n to Θ p . This is based on the convergence of a standard BSDE with jumps in discrete time setting to the associated BSDE with jumps in continuous time setting, which is proved in[109]. We refer to

Section 1.4.2. Finally, Proposition 1.4.5 proves the convergence (in p) of the penalized BSDE with jumps Θ p to Θ, the solution of the DRBSDE (1.2.2). In fact, we prove a more general result in Section 1.4.3, since we show the convergence of penalized BSDEs to (1.2.1) in the case of jumps driven by a Poisson random measure.

  e -λδn and

	e x ≤ 2C g + 2C 2 xe 2x e x -1 g (1 + 1 ≤ e 2x , we get λ e 2λT ))δ n < 1, Lemma 1.8.1 enables to write: δ n (1 -κ n )κ n 1 λ e 2λT . Hence, for δ n small enough such that (3 + 2p + ≤

  2 ∞ + p 2 (sup n max j E|ξ n j | 2 + sup n max j E|ζ n j | 2 ) + (1 + p 2 )K Lem.1.8.1 ), K Lem.1.8.1 denotes the constant appearing in Lemma 1.8.1. Discrete Gronwall's Lemma (see [134, Since δ n ≤ T , (1κ n )κ n ≥ λδ n e -2λT , and Equation (1.4.3) gives

	Lemma 3]) gives
				sup i≤n	E[(y p,n i	-y p,n i ) 2 ] ≤ C 1 (p)δ 2 n e (1+2Cg+2C 2 g (1+ 1 λ e 2λT ))T .
			E[	0	T	|Z	p,n s -Z p,n s | 2 ds] + E[	0	T	|U	p,n s -U p,n s | 2 ds] ≤ C ′ 1 (p)δ 2 n ,
	where C ′ 1 (p) is another constant depending on C g , λ, T and C 1 (p). It remains to prove the
	convergence for the increasing processes. We have
	A p,n t -K	p,n t	= Y	p,n 0 -Y	p,n t -	0	t	g(s, Y	p,n s , Z	p,n s , U	p,n s )ds +	0	t	Z	p,n s dW n s +	0	t	U	p,n s d Ñ n s ,
	A p,n t -K p,n t	= Y p,n 0	-Y p,n t	-	0	t	g(s, Y p,n s , Z p,n s , U p,n s )ds +	0	t	Z p,n s dW n s +	0	t	U p,n s d Ñ n s .
	Using the Lispchitz property of g and the convergence of (Y get the result.	p,n s -Y p,n s , Z	p,n s -Z p,n s , U	p,n s -U p,n s ), we
	1.4.2 Convergence of the discrete time setting to the continuous time
	setting
								p,n t , Z p,n t , U p,n t ) converges to (Y p t , Z p t , U p t ) in the following sense:

The following Proposition ensues from [109].

Proposition 1.4.3. Let g be a Lipschitz driver and assume that Assumption 1.3.3 (ii) holds. For any p ∈ N * , the sequence (Y

  Corollary 1.4.4. Let g be a Lipschitz driver, ξ and ζ belong to S 2 , ψ n is the random mapping introduced in Proposition 1.4.3 and assume that Assumption 1.3.3 holds. For any p ∈ N

	The following Corollary ensues from Proposition 1.4.3.
	, the
	sequence (Y p,n t , Z p,n t , U p,n t ) converges to (Y p t , Z p t , U p t ) in the following sense:

-p(ζ n t (ω)y) -. Since ξ n and ζ n satisfy Assumption 1.3.3 (ii), (g n (ω, •, •, •, •)) n converges uniformly to g(ω, •, •, •, •) + p(yξ t (ω)) -p(ζ t (ω)y) -almost surely up to a subsequence (i.e. g n satisfies [109, Assumption (A')]). Now, by using (1.4.5), [109, Proposition 1], [109, Proposition 3] and [109, Eq. (3.17)], one can easily show that (1.4.4) holds. *

  , where η n (s) represents the inverse of ψ n (s). Proposition 1.4.3 gives that the first term in the right-hand side converges to 0. Concerning the second term, s → Y p s is continuous except at the times at which the Poisson process jumps. Consequently, Y p η n (s) converges to Y p s for almost every s and as Y p belongs to S 2 , we get that E[

	T 0 |Y p η Now, remark that we can rewrite A p,n t	and A p t as follows:
	A p,n t	= p	0	t	(Y p,n s	-ξ	n s ) -ds	A p t = p	0	t	(Y p s -ξ s ) -ds.	(1.4.6)
	Then
	sup
	t∈[0,T ]

n (s) -Y p s | 2 ds] → 0 when n → ∞.

  see Remark 1.3.4) and lim n→∞ E[ T 0 |ξ η n (s) -ξ s | 2 ds] = 0 (ξ is RCLL, its jumps are countable), we get that sup k∈{0,••• ,n} |A p,n t k -A p t k | converges to 0 in L 2 in n, which ends the proof. 1.4.3 Convergence of the penalized BSDE to the reflected BSDE As said in the Introduction, this part of the proof deals with the general case of jumps driven by a random Poisson measure. We state in Proposition 1.4.5 that a sequence of penalized BSDEs converges to the solution to (1.2.1). To do so, we give in Section 1.4.3 an other proof of existence of solutions to reflected BSDEs with jumps and RCLL barriers based on the penalization method.We extend the proof of[START_REF] Lepeltier | Reflected backward stochastic differential equations with two rcll barriers[END_REF] Section 4] to the case of totally inacessible jumps. We are able to generalize their proof thanks to Mokobodski's condition (which in particular enables to get Lemma 1.4.7, generalizing[START_REF] Lepeltier | Reflected backward stochastic differential equations with two rcll barriers[END_REF] Lemma 4.1]), to the comparison Theorem for BSDEs with jumps (see Theorem 1.9.1 and Theorem 1.9.2) and to the caracterization of the solution of the DRBSDE as the value function of a stochastic game (proved in Proposition 1.9.5).

	We introduce the penalization scheme, generalizing (1.2.3) to the case of random Poisson
	measure :
	Y p t =ξ T +	t	T	g(s, Y p s , Z p s , U p s )ds + p	t	T	(Y p s -ξ s ) -ds -p	t	T	(ζ s -Y p s ) -ds -	t	T	Z p s dW s
			-	t	T	IR *	U p s (e) Ñ (ds, de)				(1.4.7)
	with A p t = p
	Moreover, for all r ∈ [1, 2[, the following strong convergence holds
	lim p→∞	E	0	T	|Y p s -Y s | 2 ds + E	0	T	|Z p s -Z s | r ds +	0	T	IR *	|U p s -U s | 2 ν(de)	r 2	ds = 0. (1.4.8)

t 0 (Y p sξ s ) -ds and K p t = p t 0 (ζ s -Y p s ) -ds. Proposition 1.4.5. Under Hypothesis 1.2.4, Y p converges to Y in IH 2 , Z p weakly converges in IH 2 to Z, U p weakly converges in IH 2 ν to U , and α p t := A p t -K p t weakly converges to α t in L 2 (F t ).

  Let us first state the following preliminary result. Lemma 1.4.7. Suppose that H, H ′ ∈ S 2 are two supermartingales such that Assumption 1.2.4 holds. Let Y * be the RCLL adapted process defined by Y * Proof. By assumption, H and H ′ are square integrable supermartingales. The process Y * is thus well defined. By the Doob-Meyer decomposition of supermartingales, there exist two square integrable martingales M and M ′ , two square integrable nondecreasing predictable RCLL processes V and V By the above relation and (1.4.11), we derive dY * t = dM t -dV t + dV Now, by the martingale representation theorem, there exist Z * ∈ H 2 , U * ∈ H 2 ν such that:

	3. the two limits are equal (see Lemma 1.4.11)
	4. the limit of the penalized BSDE is the solution of the reflected BSDE (1.2.1) (see Theorem
	1.4.3)
	5. Equation (1.4.10) ensues from (1.4.27) and (1.4.29).
	Proof of point 1.
		′ t .	(1.4.11)
	Define
	M dM t = Z * t dW t +	U * t (e) Ñ (de, dt).	(1.4.12)
	R *

t := (H t -H ′ t )1 t<T + ξ T 1 t=T . There exists (Z * , U * , A * , K * ) ∈ H 2 × H 2 ν × A 2 × A 2 such that (Y * , Z * , U * , A * , K * ) solves (i), (ii), (iii) of (1.2.1). ′ with V 0 = V ′ 0 = 0 such that: dH t = dM t -dV t ; dH ′ t = dM ′ t -dV t := M t -M ′ t . ′ t .

  Now let g + (resp. g -) denote the positive (resp. negative) part of the function g. By setting A * t := V t + Proposition 1.4.8. Suppose Assumption 1.2.4 holds.

	t 0 g + (s, Y * s , Z * s , U * s )ds and K * t := V ′ t +	t 0 g -(s, Y * s , Z * s , U * s )ds, the result follows.

ds, de).

  esssup , θ s )ds + ξ τ 1 τ ≤σ + ζ σ 1 σ<τ F t .(1.4.34) Comparing(1.4.33) and(1.4.34) and since esssup essinf ≤ essinf esssup, we deduceY t = esssup )ds + ξ τ 1 τ ≤σ + ζ σ 1 σ<τ F t , θ s )ds + ξ τ 1 τ ≤σ + ζ σ 1 σ<τ F t . Let M t := E(ξ T + T 0 g(s, θ s )ds|F t ) -t 0 g(s, θ s )ds, ξt = ξ t -M t and ζt = ζ t -M t .We can rewrite Y in the following formY t = esssup τ ∈Tt essinf σ∈Tt E ξτ 1 τ ≤σ + ζσ 1 σ<τ F t + M t ξτ 1 τ ≤σ + ζσ 1 σ<τ F t + M t Then Y t -M t isthe value of a stochastic game problem with payoff I t (τ, σ) = ξτ 1 τ ≤σ + ζσ 1 σ<τ . Let us check that ξ and ζ are in S 2 . Since ξ and ζ are in S 2 , we only have to check that M ∈ S 2 .

	σ∧τ
	τ ∈Tt g(sτ ∈Tt essinf σ∈Tt E t σ∧τ essinf σ∈Tt E t σ∧τ g(s, θ s = essinf σ∈Tt esssup τ ∈Tt E t g(s= essinf esssup σ∈Tt τ ∈Tt
	Using Doob's inequality
	E( sup

E 0≤t≤T (M t ) 2 ) ≤ 2E sup

  where A 1 , K 1 are predictable increasing processes belonging to A 2 . With the representation theorem for the martingale part we know that there existsZ 1 ∈ H 2 and U 1 ∈ H 2 ν such that Y t = M t + X + t -X -Then, we compare the forward form of (1.4.28) and the previous equality, we get(A t -K t ) -(A 1 t -K 1 t ) =

	t		t
	0	(Z s -Z 1 s )dW s +	0	IR

t = E(ξ + T 0 g(s, θ s )ds + A 1 T -K 1 T |F t ) -t 0 g(s, θ s )ds -A 1 t + K 1 t , = Y 0 + t 0 Z 1 s dW s + t 0 IR * U 1 s (e) Ñ (ds, de) -t 0 g(s, θ s )ds -A 1 t + K 1 t . * (U s (e) -U

1

s (e)) Ñ (ds, de)

  -ds to the first BSDE and we can addp T t (ζ s -Y ∞,p s ) -ds to the second BSDE. By the comparison theorem we get Y ∞,p . Since Y p,∞ ց Y and Y ∞,p ր Y when p → ∞,we get that Y p t → Y t almost surely, for all t ∈ [0, T ]. From (1.4.29) and the corresponding result for Y ∞,p , we get that lim p→∞ E( T 0 |Y p s -Y s | 2 ds) = 0. Applying Itô's formula to E(|Y p t -Y t | 2 ) between [σ, τ ], a pair of stopping times such that t ≤ σ ≤ τ ≤ T , we get E |Y p σ -Y σ | 2 +

		Since Y p,∞
	t	≤ Y p t ≤ Y p,∞ t
		τ		τ
		σ	|Z p s -Z s | 2 ds +	σ	IR *	|U p s (e) -U s (e)| 2 ν(de)ds = E(|Y p τ -Y τ | 2 )
		+ 2E(

t ≥ ξ t and Y ∞,p ≤ ζ t , we can substract p T t (Y p,∞ s ξ s ) τ σ (Y p s -Y s )(g(s, θ p s )g(s, θ s ))ds) + σ≤s≤τ (∆ s A) 2 + σ≤s≤τ (∆ s K) 2 + 2 σ≤s≤τ ∆ s A∆ s K + 2 τ σ (Y p s -Y s )d(A p -A) s -2 τ σ (Y p s -Y s )d(K p -K) s .

  2 , U p weakly converges to U in IH 2 ν and α p t weakly converges to α in L 2 (F t ). Since Y ∞,p Lemmas 1.4.9 and 1.4.10, we obtain E((A p T ) 2 ) + E((K p T ) 2 ) ≤ C, where C does not depend on p. By applying Itô's formula to |Y p t | 2 and by using Young's inequality as in (1.4.25) we get E( IR * |U p s (e)| 2 ν(de)ds)) ≤ C, where C does not depend on p. The sequences (Z p ) p≥0 , (U p ) p≥0 , (A p t ) p≥0 and (K p t ) p≥0 are bounded in the respective spaces IH 2 , IH 2 ν , L 2 (F t ) and L 2 (F t

	Then, by using T 0 |Z p t | 2 dt +	T 0 (	t	≤ Y p t ≤ Y p,∞ t	, we get A p t ≤ A ∞,p t	and K p t ≤ K p,∞ t	.

Table 1 . 1 :

 11 The solution y p,n at time t = 0 Y p,n

	0	n=20	n=50 n=100 n=200 n=400 n=500 n=600
	p=20	1.1736 1.2051 1.2181 1.2245 1.2277 1.2283 1.2288
	p=50	1.2077 1.2482 1.2648 1.2728 1.2767 1.2775 1.2780
	p=100	1.2214 1.2634 1.2808 1.2894 1.2936 1.2945 1.2950
	p=500	1.2350 1.2753 1.2939 1.3033 1.3079 1.3088 1.3094
	p=1000	1.2365 1.2767 1.2957 1.3051 1.3098 1.3107 1.3113
	p=5000	1.2376 1.2778 1.2971 1.3066 1.3113 1.3122 1.3129
	p=20000	1.2377 1.2780 1.2974 1.3069 1.3116 1.3125 1.3132
	CPU time for p=20000 0.00071 0.0084 0.0644 0.6622 6.3560 12.5970 20.0062
	Figure 2.1 represents one path of (y p,n t , ξ	n t , ζ n t ) t≥0 . We notice that for all t, y p,n t	stays between
	the two obstacles.

Table 1 . 2 :

 12 The solution Y at time t = 0 for a=-1 Y p,n .1353 1.1448 1.1419 1.1431 p=1000 1.1387 1.1369 1.1465 1.1437 1.1449 p=5000 1.1399 1.1382 1.1481 1.1453 1.1466 p=20000 1.1401 1.1385 1.1484 1.1456 1.1469

	0	n=100 n=200 n=400 n=500 n=600
	p=20	1.0745 1.0698 1.0782 1.0748 1.0759
	p=50	1.1138 1.1103 1.1191 1.1159 1.1170
	p=100	1.1266 1.1238 1.1328 1.1297 1.1308
	p=500	1.1373 1

Table 1 . 3 :

 13 The solution Y at time t = 0 for a=1 Y p,n

	0	n=100 n=200 n=400 n=500 n=600
	p=20	1.2125 1.2177 1.2203 1.2208 1.2212
	p=50	1.2582 1.2647 1.2680 1.2686 1.2690
	p=100	1.2738 1.2808 1.2843 1.2850 1.2855
	p=500	1.2866 1.2944 1.2982 1.2990 1.2995
	p=1000 1.2884 1.2962 1.3001 1.3008 1.3013
	p=5000 1.2898 1.2976 1.3016 1.3023 1.3029
	p=20000 1.2900 1.2979 1.3018 1.3026 1.3032

  ξ + such that Y n ∈ S 2 , A n and K n are in A 2 , and sup n E(IR * |U n s (e)| 2 ν(de)ds) < ∞.We also assume that for each n ∈ IN 1. (A n ) n is continuous and increasing and such that A n 0 = 0 and sup n E((A n T )Then K ∈ A 2 and there exist Z ∈ IH 2 , A ∈ A 2 and U ∈ IH 2 ν such thatY t = ξ + T t g(s, Y s , Z s , U s )ds + A T -A t -(K T -K t ) -

					T	T
					t	Z s dW s -	t	IR *
				T		T
				t	g(s, Y n s , Z n s , U n s )ds + (A n T -A n t ) -(K n T -K n t ) -	t	Z n s dW s
	-	t	T	IR *	U n s (e) Ñ (ds, de)
					T 0 |Z n s | 2 ds)+sup n E(	T 0

2 ) < ∞ 2. K j t -K j s ≥ K i t -K i s , for all 0 ≤ s ≤ t ≤ T and for all i ≤ j 3. for all t ∈ [0, T ], (K n t ) n ր K t and E(K 2 T ) < ∞ 4. (Y n t ) n increasingly converges to Y t with E(sup 0≤t≤T |Y t | 2 ) < ∞.

  133, Lemma 3.2] that K is a RCLL process, and from [133, Lemma 3.1] that A and Y are RCLL processes. Then Y has the formY t = ξ + (s)ds + A T -A t -(K T -K t ) -, Y s , Z s , U s )ds. U s (e) Ñ (ds, de) and N n t = t 0 IR * U n s (e) Ñ (ds, de). We have ∆ s (Y n -Y ) = ∆ s (N n -N + K n -K + A).We appply Itô's formula to (Y n t -Y t ) 2 on each subinterval ]σ, τ ], where σ and τ are two predictable stopping times such that 0 ≤ σ ≤ τ ≤ T . Let θ n It comes down to[START_REF] Essaky | Reflected backward stochastic differential equation with jumps and RCLL obstacle[END_REF] Equation (10)], we refer to this paper for the end of the proof.1.7 Snell envelope theoryDefinition 1.7.1. Any F t -adapted RCLL process η = (η t ) 0≤t≤T is of class D[0, T ] if the family {η(τ )} τ ∈T 0 is uniformly integrable. Definition 1.7.2. Let η = (η t ) t≤T be a F t -adapted RCLL process of class D[0, T ]. Its Snell envelope R t (η) is defined as R ψ ∈ L 2 ν and such that f 1 (t, X 2 t , π 2 t , l 1 t )f 1 (t, X 2 t , π 2 t , l 2 t ) ≥ γ t , l 1 tl 2 t ν , t ∈ [0, T ], dt ⊗ dP a.s. (1.9.2)Assume thatξ 1 ≥ ξ 2 a.s. and f 1 (t, X 2 t , π 2 t , l 2 t ) ≥ f 2 (t, X 2 t , π 2 t , l 2 t ) t ∈ [0, T ],dt ⊗ dP a.s.(1.9.3)Moreover, if inequality (1.9.3) is satisfied for (X 1 t , π 1 t , l 1 t ) instead of (X 2 t , π 2 t , l 2 t ) and if f 2 (instead of f 1 )is Lipschitz and satisfies (1.9.2), then (1.9.4) still holds.Theorem 1.9.2 (Comparison Theorem for reflected BSDEs with jumps ([START_REF] Quenez | Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps[END_REF], Theorem 5.1)). Let ξ 1 , ξ 2 be two RCLL obstacle processes in S 2 . Let f 1 and f 2 be Lipschitz drivers satisfying Assumption 2.2.4. Suppose thatξ 2 t ≤ ξ 1 t , 0 ≤ t ≤ T a.s. f 2 (t, y, z, k) ≤ f 1 (t, y, z, k), for all (y, z, k) ∈ R 2 × L 2 ν , dP ⊗ dt a.s. Let (Y i , Z i , k i , A i ) be a solution in S 2 × H 2 × H 2 ν × S 2 of the reflected BSDE -dY i t = f i (t, Y i t , Z i t , k i t (•))dt + dA i t -Z it dW t -.9.2 Convergence results on reflected BSDEs with jumps Theorem 1.9.3 (Monotonic limit theorem for reflected BSDEs with jumps ([78], Theorem 3.1)). Assume that f satisfies [78, Assumption A.2], ξ ∈ L 2 and K n is a continuous and increasing process such that sup n∈NE(K n T ) 2 < ∞ and K n 0 = 0 for any n ∈ N. Let (Y n , Z n , V n) be the solution of the following BSDE (u)| 2 ν(du)ds < ∞. If Y n converges increasingly to Y with E(sup 0≤t≤T Y 2 t ) < ∞, then there exists Z ∈ H 2 , K ∈ A 2 and V ∈ H 2 ν such that the triple (Z, K, V ) satisfies the following equation

	and for all t ∈ [0, T ]
				t	t
	0 g(sLet N t = g 0 (s)ds = 0
	s denotes (Y n s , Z n s , U n s ) s -Y s )dA n (Y n s -2 τ σ (Y n s --Y s -)dA s s -Z s )dW s s ) -g 0 (s))ds + 2 σ≤s≤τ ∆ s (Y n -Y ) 2 s -Y s )(g(s, θ n s -Z s | 2 ds + |Z n τ -Y τ ) 2 + 2 σ -Y σ ) 2 + (Y n τ σ = (Y n τ σ (Y n τ σ -2 τ σ (Y n s --Y s -)d(K n s -K s ) -2 τ σ (Y n s --Y s -)(Z n -2 τ σ (Y n s Since τ σ (Y n s -Y s )dA n s ≤ 0, -2 τ σ (Y n s Then we have X 1 t ≥ X 2 t a.s. for all t ∈ [0, T ]. (1.9.4)
					k i t (u) Ñ (dt, du); Y i T = ξ i T ,	(1.9.5)
					R *
	Y i t ≥ ξ i t , 0 ≤ t ≤ T a.s.	(1.9.6)
	and A i is a non decreasing RCLL predictable process with A i 0 = 0 and such that
		0	T	(Y i t -ξ i t )dA i,c t = 0 a.s. and ∆A i,d t = -∆Y i t 1 Y i t -=ξ i t -a.s.
	Then Y 2
	Y n t = ξ + where sup n∈N E T t	T f (s, Y n t g 0 T t s , Z n s , V n s )ds + K n T -K n t -T t Z n s dW s -Z s dW s -T t T 0 |Z n s | 2 ds < ∞ and sup n∈N E T 0 U |V n s	T U t	V n s (u) Ñ (ds, du), t ≤ T, IR r
					2
					ds = 0

*

U s (e) Ñ (ds, de).

It remains to prove that for all

r ∈ [1, 2[ lim n→∞ E T 0 |Z n s -Z s | r ds + T 0 IR * |U n s (e) -U s (e)| 2 ν(de) t 0 IR * --Y s -)(U n s (e) -U s (e)) Ñ (ds, de). --Y s -)d(K n s -K s ) ≤ 0 and σ≤s≤τ ∆ s (Y n -Y ) 2 = σ≤s≤τ ∆ s (N n -N ) 2 + σ≤s≤τ ∆ s (K n -K) 2 + σ≤s≤τ (∆ s A) 2 + 2 σ≤s≤τ ∆ s A∆ s (K n -K).

By taking expectation and using

Y n s --Y s -= (Y n s -Y s ) -∆ s (Y n -Y ), we get E(Y n σ -Y σ ) 2 + E τ σ |Z n s -Z s | 2 ds + E τ σ IR * |U n s (e) -U s (e)| 2 ν(de)ds + E σ≤s≤τ ∆ s (K n -K) 2 ≤ E(Y n τ -Y τ ) 2 + 2E τ σ (Y n s -Y s )(g(s, θ n s )g 0 (s))ds -2E τ σ (Y n s -Y s )dA s + E σ≤s≤τ (∆ s A)

2 . t (η) = esssup ν∈Tt E(η ν |F t ). Proposition 1.7.3. R t (η) is the lowest RCLL F t -supermartingale of class D[0, T ] which dominates η, i.e. P-a.s., for all t ∈ [0, T ], R(η) t ≥ η t . where t ≤ Y 1 t for all t in [0, T ] a.s. 1

  2 ν . Moreover, for every p ∈ [1, 2[, the following strong convergence holds -V s (u)| 2 ν(du) S s ) -ds. We have Theorem 1.9.4 ([78], Theorem 4.2)

	lim n→∞	E	0	T	|Y n s -Y s | 2 ds + E	0	T	|Z n s -Z s | p ds +	0	T	U	|V n s (u) p 2	ds = 0.
	Now we introduce the following penalized equation
	Y n t = ξ +	t	T	f (s, Y n s , Z n s , V n s )ds + K n T -K n t -	t	T	Z n s dW s -	t	T	U	V n s (u) Ñ (ds, du), t ≤ T,
	where K n t = n	t 0 (Y n s -

  Y t ≤ ξ t + ε} σ ε S := inf{t ≥ S, Y t ≥ ζ t -ε}.(1.9.11)Remark that σ ε S and τ ε S ∈ T S . Fix ε > 0. We have that almost surely, if t ∈ [S, τ ε S [, then Y t > ξ t + ε and hence Y t > ξ t . It follows that the function t → A c t is constant a.s. on [S, τ ε S ] and t → A d t is constant a.s. on [S, τ ε S [. Also, Y (τ ε S ) ε S ]. Furthermore, by the right-continuity of (ξ t ) and (Y t ), we clearly have Y τ ε S ≤ ξ τ ε S + ε a.s. Similarly, one can show that the process K is constant on [S, σ ε S ] and that Y σ ε S ≥ ζ σ ε Sε a.s. Let us now consider two cases. First, on the set {σ ε S < τ }, by using the definition of the stopping times and the fact that K is constant on [S, σ ε S ], we have:

	I S (τ, σ ε S ) ≤

-≥ ξ (τ ε S ) -+ ε a.s. Since ε > 0, it follows that Y (τ ε S ) -> ξ (τ ε S ) - a.

s. , which implies that ∆A d τ ε S = 0 a.s. (see Remark 1.2.6). Hence, the process A is constant on [S, τ

  1 , z 1 , u 1 ), (s 2 , y 2 , z 2 , u 2 ), |g(ω, s 1 , y 1 , z 1 , u 1 )g(ω, s 2 , y 2 , z 2 , u 2 )| ≤ Λ(|s 2s 1 |) + C g (|y 1y 2 | + |z 1z 2 | + |u 1u 2 |). Definition 2.2.2 (Mokobodski's condition). Let ξ,ζ be in S 2 . There exist two nonnegative RCLL supermartingales H and H ′ in S 2 such that Theorem 2.2.3 ([62],Theorem 4.1). Suppose ξ and ζ are RCLL adapted processes in S 2 such that for all t ∈ [0, T ], ξ t ≤ ζ t , Mokobodski's condition holds and g is a Lipschitz driver. Then, DRBSDE (2.1.1) admits a unique solution (Y, Z, U, α) in S 2 × IH 2 × IH 2 × S 2 , where α := A -K, A and K in A 2 .

∀t ∈ [0, T ], ξ t 1 t≤T ≤ H t -H ′ t ≤ ζ t 1 t≤T a.s.

The following Theorem states existence and uniqueness of solutions to (2.1.1).

  Theorem 2.4.3. Assume that Assumption 2.2.5 holds and g is a Lipschitz driver satisfying Assumption 2.2.4. The sequence (Y p,n , Z p,n , U p,n ) defined by (2.4.3) converges to (Y, Z, U ), the solution of the DRBSDE (2.1.1), in the following sense: ∀r ∈ [1, 2[Moreover, Z p,n (resp. U p,n ) weakly converges in IH 2 to Z (resp. to U ) and for 0 ≤ t ≤ T , α p,n ψ n (t) converges weakly to α t in L 2 (F T ) as n → ∞ and p → ∞, where (ψ n ) n∈IN is a one-to-one random map from [0, T ] to [0, T ] such that sup t∈[0,T ] |ψ n (t) -t| ---→In this Section we state two intermediate results useful for Section 2.4.3. Lemma 2.4.4. Under Assumption 2.2.5 we have sup Proof. Since ξ n j ≤ y n j ≤ ζ n j , Assumption 2.2.5 gives sup j E(|y n j | 2 ) ≤ c. Let us deal with z n j and u n j . We apply the discrete Itô's formula and we get: Proof. Let us first prove (2.4.9). From (2.3.6), (2.4.2) and the discrete Itô's formula applied to (y n jy p,n j ) 2 , we get

					n-1											n-1	n-1
	E|y n j -y p,n j | 2 + δ	lim p→∞ i=j E|z n lim n→∞ i -z p,n E i | 2 + (1 -κ n )κ n T 0 |Y p,n s -Y s | 2 ds + E i=j E[|u n i -u p,n T 0 i | 2 ] + (1 -κ n )κ n |Z p,n s -Z s | r ds n-1	i=j	E[|v n i -v p,n i | 2 ]
								+E	0	T	|U p,n s = 2 i=j -U s | r ds E[(y n i -y p,n = 0. i )(g(t i , y n i , z n i , u n i ) -g(t i , y p,n i , z p,n i , u p,n i ))δ] (2.4.4)
														n-1		n-1
											+ 2	i=j	E[(y n i -y p,n i )(a n i -a p,n i )] -2	i=j	E[(y n i -y p,n i )(k n i -k p,n i )].
	n→∞ i -a p,n i )(a n i -y p,n Let us deal with the last two terms (y n 2.4.2 Intermediate results i ) = (y n 0 a.s.. i -ξ n i )a n i -(y p,n i ξ n i )a p,n i + (y p,n i -ξ n i )a p,n i ≤ (y p,n i -ξ n i ) -a n i . By using same computations, we derive (y n -ξ n i )a n i -(y n i -i -y p,n i )(k n i -k p,n i ) ≥ -(y p,n i i ) + k n i . -ζ n By using the Lipschitz property of g, we get
	E[|y n j -y p,n j | 2 ] +	1 2	δE[|z n j -z p,n j | 2 ] +		κ n (1 -κ n ) 2	E[|u n j -u p,n j | 2 ]
	≤ 2C g + 2C 2 g +	2C 2 g δ κ n (1 -κ n )		δ	n-1 i=j	n-1 j=0 i -y p,n E[(y n i ) 2 ] + 2 |u n j | 2 + n-1 1 δ i=j E[(y p,n n-1 j=0 |a n j | 2 + i -ξ n i ) -a n 1 δ n-1 j=0 i + (y p,n |k n j | 2 ≤ c. i -ζ n i ) + k n i ].
	Using Cauchy-Schwarz inequality gives
	E[|y n j | 2 ] + δ E[|y n j -y p,n j | 2 ] + n-1 i=j ≤E[|ξ n n | 2 ] + 2δ n-1 E[|z n 1 2 δE[|z n j -z p,n j | 2 ] + i=j E[y n i g(t i , y n i , z n i , u n i )] + 2 κ n (1 -κ n ) 2 n-1 i=j E[|u n E[|u n j -u p,n j | 2 ] i | 2 ] n-1 i=j E[y n i a n i ] -2 ≤ 2C g + 2C 2 g + 2C 2 n-1 g δ δ i=j E[(y n i -y p,n i ) 2 ] κ n (1 -κ n ) n-1 i=j ≤E[|ξ n n | 2 ] + δ n-1 i=j g(t i , 0, 0, 0) 2 + δ 1 + 2C g + 2C 2 g + 2C 2 κ n (1 -κ n ) E[y n i k n i ] i=j g δ n-1 + 2 δ n-1 i=j E (y p,n i -ξ n i ) -2 1 2 i=j δ E[(a n i ) 2 ] 1 1 n-1 2	E[|y n i | 2 ]
	+	δ 2	n-1 i=j	E[|z n i | 2 ] +	κ n (1 -κ n ) 2	n-1 i=j	E[|u n i | 2 ] +	2δ α	n-1 i=j	E(|y n i | 2 ) +	α δ	n-1 i=j	E(|a n i | 2 ) +	α δ	n-1 i=j	E(|k n i | 2 ).
	Since ξ n i ≤ y n i ≤ ζ n i , we get
	δ] i=1 e n i -= δ(b ξ t i + g(t i , ξ n i , z n i , u n i )) -, i ) -ξ n i , u n i , z n i ) + δg(t i , ξ n i+1 |G n i ≤ E(ξ n a n k n i ≤ E(ζ n i+1 |G n i ) + δg(t i , ζ n i , z n i , u n i + = δ(b ζ t i + g(t i , ζ n i , z n i , u n i )) + . i ) -ζ n i Then, using the Lipschitz property of g gives where e n 1 , e n 2 , ..., e n n are independent identically distributed random variables with the following (2.3.1) (2.4.5) symmetric Bernoulli law: P(e n 1 = 1) = P(e n 1 = -1) = 1 2 . δ i=j E(|a n i | 2 ) ≤ 5αδ α n-1 n-1

j E[|y n j | 2 ] + E δ n-1 j=0 |z n j | 2 + κ n (1κ n ) i | 2 ] + κ n (1κ n ) i=j E[|b ξ i | 2 + |g(t i , 0, 0, 0)| 2 + C 2 g (|ξ n i | 2 + |z n i | 2 + |u n i | 2 )] (2.4.6)

  4, Lemma 2.6.1 and Gronwall inequality give (2.4.9).

								Concerning
	α n t -α p,n t α n t -α p,n we have t =(Y n t -Y p,n t ) -(Y n 0 -Y p,n 0 ) -	0	t	g(s, Y n s , Z n s , U n s ) -g(s, Y p,n s , Z p,n s , U p,n s )ds
	+	0	t	(Z n s -Z p,n s )dW n s +	0	t	(U n s -U p,n s )d Ñ n s .

2 .

 2 Lemma 2.6.1. Suppose Assumption 2.2.5 holds and g is a Lipschitz driver. For each p ∈ N and n ≥ N 0 we have

	sup j	E[|y p,n j | 2 ] + δ	n-1 j=0	E[|z p,n					n-1 j=0	E[|u p,n j | 2 ] +	1 pδ	n-1 j=0	E[|a p,n j | 2 ] +	1 pδ	n-1 j=0	E[|k p,n j | 2 ] ≤ c
	Proof. By applying the discrete Itô's formula, we get
								n-1						n-1
		E[|y p,n j | 2 ] + δ	i=j	E[|z p,n				i=j	E[|v p,n i | 2 ]
									n-1				n-1
		≤ E[|ξ n n | 2 ] + 2	i=j	E[|y p,n i ||g(t i , y p,n i , z p,n i , u p,n i )δ|] + 2E[	i=j	(y p,n i a p,n i	-y p,n i k p,n i )].
	Note that y p,n i a p,n i	= -1 pδ (a p,n i ) 2 + ξ n i a p,n i	and y p,n i k p,n i	= 1 pδ (k p,n i ) 2 + ζ n i k p,n i . We have that:
	E[|y p,n j | 2 ] +	δ 2	n-1 i=j	E[|z p,n i | 2 ] +	κ n (1 -κ n ) 2	n-1 i=j	E[|u p,n i | 2 ] +	1 pδ	n-1 i=j	E[|a p,n i | 2 ] +	1 pδ	n-1 i=j	E[|k p,n i | 2 ]
		≤ E[|ξ n n | 2 ] + δE[	n-1 i=j	|g(t i , 0, 0, 0)| 2 ] + 2δ 1 + 2C g + 2C 2 g +	2C 2 g δ κ n (1 -κ n )	n-1 i=j	E[|y p,n i | 2 ]
				n-1						n-1
		+ 2	i=j		E[(ξ n i )a p,n i ] -2	i=j	E[(ζ n i )k p,n i ].

j | 2 ] + κ n (1κ n ) i | 2 ] + κ n (1κ n ) n-1 i=j E[|u p,n i | 2 ] + κ n (1κ n )

I also express my gratitude to Romuald Elie for all the challenging and very useful discussions he has initiated.

J'exprime aussi ma gratitude envers Prof. Romuald Elie pour toutes les discussions très fructueuses qu'il a initiées.

(Y, B, µ), if µ is σ-finite and B is countably generated, then L 2 (Y, B, µ) is separable. Applying this property to Y = E (where E = R * ), B = B(E) and µ = ν, since B(E) is countably generated, it follows that L 2 ν = L 2 (E, B(E), ν) is separable.

Proposition 5.7.2. Let (X, A) be a measurable space. Let f : (X × L 2 ν , A ⊗ B(L 2 ν ) → (R, B(R)); (α, k) → f (α, k). Suppose that f satisfies one of the three following conditions: 1. f is of class C 1 with respect to k such that for all (α, k) ∈ X × L 2 ν ,

|∇ k f (α, k)(e)| ≤ ψ(e) and ∇ k f (α, k)(e) ≥ -1 dν(e)a.s.

(5.7.1)

where ψ ∈ L 2 ν . 2. f is convex (resp. concave) with respect to k and Gâteaux-differentiable with respect to k such that the Gâteaux-gradiant ∇ g k f (α, k), which is also the sub-(resp. super-) differential with respect to k, satisfies (5.7.1). 3. f of the form f (α, k) := f (α, E k(e)ψ(e)ν(de)), where ψ is a nonnegative function in L 2 ν and f : X × R → R is a measurable map, supposed to be non-decreasing with respect to its second variable and Lipschitz continuous with Lipschitz constant denoted by C.

Then, there exists a measurable map γ :

) such that |γ(.)(e)| ≤ ψ(e), where ψ ∈ L 2 ν ; γ(.)(e) ≥ -1 ν(de)a.s. and

Proof. 1. Since L 2 ν is a separable Hilbert space, it admits a countable orthonormal basis {e i , i ∈ N}. Let (α, k) ∈ X ×L 2 ν . Since f is differentiable at k, for each h in V we have: f (α, k+h) = f (α, k)+ < ∇ k f (α, k), h > ν +||h|| ν ε(||h|| ν ), where lim x→0 ε(x) = 0. By taking h = te i , t ∈ R, i ∈ N we obtain that

Hence, the map δ i defined for each (α, k) ∈ X × V by δ i (α, k) :=< ∇ k f (α, k), e i > is A ⊗ B(L 2 ν ) -measurable. We thus obtain that ∇ k f (., .) :

) is C 1 . Hence, by the mean theorem, we have that

where for each l ∈ L 2 ν , we have denoted its coordinates in the basis (e i ) i∈N by (l i ) i∈N . Now, by (5.7.1), ||∇ k f (.)|| ν is uniformly bounded. Using this property and Fubini's theorem, one can show that

have (Z r (ω, T s )) r≥s ∈ H 2 s and (K r (ω, T s )) r≥s ∈ H 2 s,ν , and thus for each u ∈ [s, T ], K r (T s , •, e) Ñ s (dr, de) Pa.s.. Remark 6.3.4. Equality (6.3.1) is equivalent to ( u s Z r dW r )(ω, T s (ω ′ )) = ( u s Z r (ω, T s )dW s r )(ω ′ ) for P -almost every ω ′ ∈ Ω. A similar property holds for the second equality.

Proof. We shall only prove the first equality with the Brownian motion. The second one with the Poisson random measure can be shown by similar arguments. Let us first show that equality (6.3.1) holds for a simple process. Let a < T and let H ∈ L 2 (F a ). For each ω ≡ ( s ω, ω s ) = (S s (ω), T s (ω)) ∈ Ω, we have

Let now Z ∈ H 2 . Let us show that Z satisfies equality (6.3.1). The idea is to approximate Z by an appropriate sequence of simple processes (Z n ) n∈N so that the sequence (Z n ) n∈N converges in H 2 to Z, and that, for almost every past path s ω, the sequence (Z n ( s ω, T s )) n∈N converges to Z( s ω, T s ) in H 2 s .

For each n ∈ N * , define Z n r := n n-1 i=1 ( iT n (i-1)T n

] (r). By inequality (6.6.1) in the Appendix, we have

excepted on a P -null set N . Since S s is F s -measurable and T s is independant of F s , there exists a P -null set including the previous one, such that for each ω ∈ N c , setting ω = s ω, we have

The second equality follows by the classical isometry property. Now, for each square integrable martingale M , M 2 -M is a martingale. Hence, for each ω ∈ N c , where N is a P -null set included the previous one, setting ω = s ω, we have

It is a measurable map. We now define a process α ε on [0, T ] × Ω by α ε r (ω) := i β i,ε (S s (ω))e i (ω). It remains to prove that it is predictable, that is P-measurable. Note that β i,ε • S s is F s -measurable by composition. Since the process (e i u ) s≤u≤T is P s -measurable, the process (β i,ε • S s ) e i u is P-measurable. Indeed, if we take e i of the form e i u = H1 ]r,T ] (u) with r ≥ s and H a random variable F s r -measurable, then the random variable (β i,ε •S s ) H is F r -measurable and hence the process (β i,ε • S s ) H1 ]r,T ] is P-measurable. The process α ε is thus P-measurable. Note also that α ε (ω, T s (ω)) = i β i,ε (ω)e i (ω, ω). Now, we have e i (ω, T s (ω)) = e i (ω) because e i (ω) depends on ω only through T s (ω). Hence, α ε (ω, T s (ω)) = ᾱε (ω, ω), which completes the proof. Remark 6.3.8. Recall that in control theory, selection theorems are closely related to the existence of optimal or nearly optimal controls and their regularity or measurability properties (see, among others, [START_REF] Filippov | On certain questions in the theory of optimal control[END_REF] in a deterministic framework, and [START_REF] Bertsekas | Stochastic Optimal Control: The Discrete Time Case[END_REF] in a discrete time Markovian stochastic framework).

The above result will be used to prove that the map u * satisfies a super-optimality principle of dynamic programming (see Theorem 6.3.13). In the next section, we provide a Fatou lemma for reflected BSDEs which will be also used to prove this super-optimality principle.

A Fatou lemma for reflected BSDEs

In this section, we establish a Fatou lemma for reflected BSDEs, where the limit involves both terminal condition and terminal time. We first introduce some notation.

A function f is said to be a Lipschitz driver if

ν )-measurable, uniformly Lipschitz with respect to y, z, k(•) and such that f (., 0, 0, 0) ∈ IH 2 .

A Lipschitz driver f is said to satisfy Assumption 6.3.9 if the following holds:

, where ψ ∈ L 2 ν . (6.3.12)

Recall that this assumption ensures the comparison theorem for BSDEs with jumps (see [START_REF] Quenez | BSDEs with jumps, optimization and applications to dynamic risk measures[END_REF] Th 4.2).

Let (η t ) be a given obstacle RCLL process in S 2 and let f be a given Lipschitz driver. In the following, we will consider the case when the terminal time is a stopping time θ ∈ T and the terminal condition is a random variable ξ in L 2 (F θ ). In this case, the solution, denoted (Y .,θ (ξ), Z .,θ (ξ), k .,θ (ξ)), of the reflected BSDEs associated with terminal stopping time θ, driver f , 6.6 Appendix Lemma 6.6.1. The function u has at most polynomial growth at infinity.

Proof. Applying some estimates on the solution of a reflected BSDE (see Prop. 5.1 in [?]), we obtain:

where K is a real constant which depends only on C and T . Using now the hypothesis of polynomial growth on f, h, g and the standard estimate A result of classical analysis. We state a result of classical analysis concerning the approximation of a real-valued function in L 2 ([0, T ]), dt) equipped with the norm ||f || 2 2 dr by a specific sequence of step functions as well as useful inequalities used in the chapter. For each n ∈ N, we consider the linear operator

By Cauchy-Schwartz's inequality, we have that for each

Indeed, the above convergence clearly holds when f is continuous, and the general case follows by using the uniform continuity of f and the density of

A result of Measure Theory (see Lemma 1.2 in [START_REF] Crauel | Random Probability Measures on Polish Spaces[END_REF]) ensures the following property.

• L 2 ν is the set of Borelian functions ℓ :

ν is a Hilbert space equipped with the scalar product δ, ℓ ν := IR * δ(u)ℓ(u)ν(du) for all δ, ℓ ∈ L 2 ν × L 2 ν , and the norm ℓ 2 ν := IR * |ℓ(u)| 2 ν(du).

• B(IR 2 ) (resp B(L 2 ν )) is the Borelian σ-algebra on IR 2 (resp. on L 2 ν ).

• IH 2 ν is the set of processes l which are predictable, that is, measurable

• S 2 is the set of real-valued RCLL adapted processes φ such that φ 2

• A 2 is the set of real-valued non decreasing RCLL predictable processes A with A 0 = 0 and

• T 0 is the set of stopping times τ such that τ ∈ [0, T ] a.s

• For S in T 0 , T S is the set of stopping times τ such that S ≤ τ ≤ T a.s.

Definitions and assumptions

We start this section by recalling the definition of a driver and a Lipschitz driver. We also introduce DRBSDEs and our working assumptions.

Definition 1.2.1 (Driver, Lipschitz driver). A function g is said to be a driver if

A driver g is called a Lipschitz driver if moreover there exists a constant C g ≥ 0 and a bounded, non-decreasing continuous function Λ with Λ(0) = 0 such that dP ⊗ dt-a.s. , for each (s

In the case of BSDEs with jumps, the coefficient g must satisfy an additional assumption, which allows to apply the comparison theorem for BSDEs with jumps (see Theorem 1.9.1), which extends the result of [START_REF] Royer | Backward stochastic differential equations with jumps and related non-linear expectations[END_REF]. More precisely, the driver g satisfies the following assumption:

Numerical scheme

The basic idea is to approximate the Brownian motion and the Poisson process by random walks based on the binomial tree model. As explained in Section 1.3.1, these approximations enable to get a martingale representation whose coefficients, involving conditional expectations, can be easily computed. Then, we approximate (W, Ñ ) in the penalized version of our DRBSDE (i.e. in (1.2.3)) by using these random walks. Taking conditional expectation and using the martingale representation leads to the explicit penalized discrete scheme (1.3.9). In view of the proof of the convergence of this explicit scheme, we introduce an implicit intermediate scheme (1.3.5).

Discrete time approximation

We adopt the framework of [109], presented below.

Random walk approximation of (W, Ñ )

For n ∈ N, we introduce δ n := T n and the regular grid (t j ) j=0,...,n with step size δ n (i.e. t j := jδ n ) to discretize [0, T ]. In order to approximate W , we introduce the following random walk

where e n 1 , e n 2 , ..., e n n are independent identically distributed random variables with the following symmetric Bernoulli law:

To approximate Ñ , we introduce a second random walk

where η n 1 , η n 2 , ..., η n n are independent and identically distributed random variables with law

where κ n = e -λ n . We assume that both sequences e n 1 , ..., e n n and η n 1 , η n 2 , ..., η n n are defined on the original probability space (Ω, F, P ). The (discrete) filtration in the probability space is F n = {F n j : j = 0, ..., n} with F n 0 = {Ω, ∅} and F n j = σ{e n 1 , ..., e n j , η n 1 , ..., η n j } for j = 1, ..., n. The following result states the convergence of (W n , Ñ n ) to (W, Ñ ) for the J 1 -Skorokhod topology, and the convergence of W n to W in any L p , p ≥ 1, for the topology of uniform convergence on [0, T ]. We refer to [109, Section 3] for more results on the convergence in probability of F nmartingales.

Lemma 1.3.1 ([109], Lemma3, (III), and [START_REF] Briand | Donsker-type theorem for BSDEs[END_REF],Proof of Corollary 2.2). The couple (W n , Ñ n ) converges in probability to (W, Ñ ) for the J 1 -Skorokhod topology, and

in probability and in L p , for any 1 ≤ p < ∞.

Martingale representation

Let y j+1 denote a F n j+1 -measurable random variable. As said in [109], we need a set of three strongly orthogonal martingales to represent the martingale difference m j+1 := y j+1 -E(y j+1 |F n j ). We introduce a third martingale increments sequence {µ n j = e n j η n j , j = 0, • • • , n}. In this context there exists a unique triplet (z j , u j , v j ) of F n j -random variables such that

and

Remark 1.3.2 (Computing the conditional expectations). Let Φ denote a function from IR 2j+2 to IR. We use the following formula to compute the conditional expectations

Fully implementable numerical scheme

In this Section we present two numerical schemes to approximate the solution of the penalized equation (1.2.3): the first one, (1.3.5), is an implicit intermediate scheme, useful for the proof of convergence. We also introduce the main scheme (1.3.9), which is explicit. The implicit scheme (1.3.5) is not easy to solve numerically, since it involves to inverse a function, as we will see below. However, it plays an important role in the proof of the convergence of the explicit scheme, that's why we introduce it.

In both schemes, we approximate the barrier (ξ

). We also introduce their continuous time versions:

where b X , σ X and c X are Lipschitz functions. We approximate it by

Since (W n , Ñ n ) converges in probability to (W, Ñ ) for the J1-topology, [141, Corollary 1] gives that X n converges to X in probability for the J1-topology (for more details on the convergence of sequences of stochastic integrals on the space of RCLL functions endowed with the J1-Skorokhod topology, we refer to [START_REF] Jakubowski | Convergence en loi des suites d'intégrales stochastiques sur l'espace D 1 de Skorokhod[END_REF]). Then, X n satisfies Assumption 1.3.3 (ii). We deduce from Doob and Burkhölder-Davis-Gundy inequalities that X and X n satisfy Assumption 1.3.3 (i) and that X belongs to S 2 .

2. X is defined by X t := Φ(t, W t , Ñt ), where Φ satisfies the following assumptions (a) Φ(t, x, y) is uniformly continuous in (t, y) uniformly in x, i.e. there exist two continuous non decreasing functions g 0 (•) and g 1 (•) from IR + to IR + with linear growth and satisfying g 0 (0) = g 1 (0) = 0 such that

We denote a 0 (resp. a 1 ) the constant of linear growth for g 0 (resp. g 1 ) i.e. ∀ (t, y) ∈

Φ(t, x, y) is "strongly" locally Lispchitz in x uniformly in (t, y), i.e. there exists a constant K 0 and an integer p 0 such that

Then, ∀(t, x, y) we have |Φ(t, x, y)| ≤ a 0 |t|+a 1 |y|+K 0 (1+|x| p 0 )|x|+|Φ(0, 0, 0)|+a 0 +a 1 . From this inequality, we prove that X satisfies Assumption 1.3.3 (i) by standard computations. Since ( Ñ n ) converges in probability to ( Ñ ) for the J1-topology and lim n→∞ sup t |W n t -W t | = 0 in L p for any p (see Lemma 1.3.1), we get that (X n t ) t := (Φ(δ n [t/δ n ], W n t , Ñ n t )) t converges in probability to X for the J1-topology.

Intermediate penalized implicit discrete scheme

After the discretization of the penalized equation (1.2.3) on time intervals [t j , t j+1 ] 0≤j≤n-1 , we get the following discrete backward equation. For all

(1.3.4) Following (1.3.3), the triplet (z p,n j , u p,n j , v p,n j ) can be computed as follows

where we refer to Remark 1.3.2 for the computation of conditional expectations. By taking the conditional expectation w.r.t. F n j in (1.3.4), we get the following scheme, called implicit penalized discrete scheme: y p,n n := ξ n n and for j

where Θ p,n (y) = yg(jδ n , y, z p,n j , u p,n j )δ npδ n (yξ n j ) -+ pδ n (ζ n jy) -. We also introduce the continuous time version (Y p,n t , Z p,n t , U p,n t , A p,n t , K We also introduce α p,n t := A p,n t -K p,n t , for all t ∈ [0, T ].

Main scheme

As said before, the numerical inversion of the operator Θ p,n is not easy and is time consuming. If we replace y p,n j by E[y p,n j+1 |F n j ] in g, (1.3.4) becomes

(1.3.7)

Now, by taking the conditional expectation in the above equation, we obtain:

Solving this equation, we get the following scheme, called explicit penalized scheme: y p,n n := ξ n n and for

(1.3.9)

Remark 1.3.6 (Explanations on the derivation of the main scheme). We give below some explanations concerning the derivation of the values of a p,n j and k p,n j . We consider the following cases:

• If ξ n j ≥ y p,n j , then by (1.3.7) we have a p,n j = pδ n (ξ n j -y p,n j ) and k p,n j = 0; we then replace a p,n j and k p,n j in (1.3.8) and we get a p,n j

We also have

• The case ζ n j ≤ y p,n j is symmetric to the one studied above: ξ n j ≥ y p,n j .

Proposition 1.7.4 (Doob-Meyer decomposition of Snell envelopes). Let η := (η t ) t≤T be of class D([0, T ]). There exists a unique decomposition of the Snell envelope

where M t is a RCLL F t -martingale, K c is a continuous integrable increasing process with K c 0 = 0, and K d is a pure jump integrable increasing predictable RCLL process with K d 0 = 0. Moreover, we have

Proof. The first part of the proposition corresponds to the Doob-Meyer decomposition of supermartingales of class D[0, T ]. To prove the second part of the proof, we write

The first term of the right hand side is null, since

). Let us prove that the second term of the r.h.s. is also null. We know that (R

1.8 Technical result for standard BSDEs with jumps Lemma 1.8.1. We assume that δ n is small enough such that (3+2p+2C g +2C 2 g (1+ 1 λ e 2λT ))δ n < 1. Then we have:

where

Proof. From the explicit scheme, we derive that:

Taking the sum for j = i, ..., n -1 yields

Hence, we get that:

, the assumption on δ n enables to apply Gronwall's Lemma, and the result follows.

Some recent results on BSDEs and reflected BSDEs with jumps

For the self-containment of the chapter, we recall in this Section some recent results used several times in the chapter.

Comparison theorem for BSDEs and reflected BSDEs with jumps

Theorem 1.9.1 (Comparison Theorem for BSDEs with jumps ([137], Theorem 4.2)). Let ξ 1 and ξ 2 be in L 2 (F T ). Let f 1 be a Lipschitz driver and f 2 be a driver.

(1.9.1)

Assume that there exists a bounded predictable process (γ t ) such that dt ⊗ dP ⊗ ν(du)-a.s.

Numerical shemes for DRBSDEs driven by the Brownian motion have been proposed by Xu in [START_REF] Xu | Numerical algorithms and Simulations for Reflected Backward Stochastic Differential Equations with Two Continuous Barriers[END_REF] (see also [START_REF] Memin | Convergence of solutions of discrete Reflected backward SDE's and Simulations[END_REF] and [START_REF] Peng | Numerical algorithms for BSDEs with 1-d Brownian motion: convergence and simulation[END_REF]) and, in the Markovian framework, by Chassagneux in [START_REF] Chassagneux | A discrete-time approximation for doubly reflected BSDEs[END_REF]. In this chapter, we are interested in numerically solving DRBSDEs driven by a Brownian motion and an independent Poisson process in the case of RCLL obstacles with only totally inacessible jumps. More precisely, we consider equations of the following form:

{W t : 0 ≤ t ≤ T } is a one dimensional standard Brownian motion and { Ñt := N tλt, 0 ≤ t ≤ T } is a compensated Poisson process. Both processes are independent and they are defined on the probability space (Ω, F T , F = {F t } 0≤t≤T , P). The processes A and K have the role to keep the solution between the two obstacles ξ and ζ. Since we consider that the jumps of the obstacles are totally inaccessible, A and K are continuous processes.

In the non-reflected case, some numerical methods have been provided: in [START_REF] Bouchard | Discrete-time approximation of decoupled Forward-Backward SDE with jumps[END_REF], the authors propose a scheme for Forward-Backward SDEs based on the dynamic programming equation and in [109], the authors propose a fully implementable scheme based on a random binomial tree. In the reflected case, a fully implementable numerical scheme has been recently provided by Dumitrescu and Labart in [START_REF] Dumitrescu | Numerical approximation for Doubly reflected BSDEs with Jumps and RCLL obstacles[END_REF]. Their method is based on the approximation of the Brownian motion and the Poisson process by two random walks and on the approximation of the reflected BSDE by a sequence of penalized BSDEs.

The aim of this chapter is to propose an alternative scheme to [START_REF] Dumitrescu | Numerical approximation for Doubly reflected BSDEs with Jumps and RCLL obstacles[END_REF] to solve (2.1.1). The scheme proposed here takes the following form:

It generalizes the scheme proposed by [START_REF] Xu | Numerical algorithms and Simulations for Reflected Backward Stochastic Differential Equations with Two Continuous Barriers[END_REF] to the case of jumps. Compared to the scheme proposed in [START_REF] Dumitrescu | Numerical approximation for Doubly reflected BSDEs with Jumps and RCLL obstacles[END_REF], the scheme proposed here -called reflected scheme in the following -is based on the direct discretization of (2.1.1). In particular, there is no penalization step. Then, this method only depends on one parameter of approximation (the number of time steps n), contrary to the scheme proposed in [START_REF] Dumitrescu | Numerical approximation for Doubly reflected BSDEs with Jumps and RCLL obstacles[END_REF] (which also depends on the penalization parameter). We provide here an explicit reflected scheme and an implicit reflected scheme and we show the convergence of both schemes. We illustrate numerically the theoretical results and show they coincide with the ones obtained by using the penalized scheme presented in [START_REF] Dumitrescu | Numerical approximation for Doubly reflected BSDEs with Jumps and RCLL obstacles[END_REF], for large values of the penalization parameter.

To approximate Ñ , we introduce a second random walk

where η n 1 , η n 2 , ..., η n n are independent and identically distributed random variables with law P(η n 1 = κ n -1) = 1 -P(η n 1 = κ n ) = κ n where κ n = e -λδ . We assume that both sequences e n 1 , ..., e n n and η n 1 , η n 2 , ..., η n n are defined on the original probability space (Ω, F, P). The (discrete) filtration in the probability space is F n := {F n j : j = 0, ..., n} with F n 0 = {Ω, ∅} and F n j = σ{e n 1 , ..., e n j , η n 1 , ..., η n j } for j = 1, ..., n. The following result states the convergence of (W n , Ñ n ) in the J 1 -Skorokhod topology. We refer to [109, Section 3] for more results on the convergence in probability of F n -martingales .

Lemma 2.3.1 ([109],Lemma3, (III)). The couple (W n , Ñ n ) converges in probability to (W, Ñ ) for the J 1 -Skorokhod topology.

We recall that the process ξ n converges in probability to ξ in the J1-Skorokhod topology if there exists a family (ψ n ) n∈IN of one-to-one random time changes from

Martingale representation

Let y j+1 denote a F n j+1 -measurable random variable. As said in [109], we need a set of three strongly orthogonal martingales to represent the martingale difference m j+1 := y j+1 -E(y j+1 |F n j ). We introduce a third martingale increment sequence {µ n j = e n j η n j , j = 0, • • • , n}. In this context there exists a unique triplet (z j , u j , v j ) of F n j -random variables such that m j+1 := y j+1 -E(y j+1 |F n j ) = √ δz j e n j+1 + u j η n j+1 + v j µ n j+1 , and

The computation of conditional expectations is done in the following way:

Remark 2.3.2 (Computing the conditional expectations). Let Φ denote a function from IR 2j+2 to IR. We use the following formula

Reflected schemes

The barriers ξ and ζ given in Assumption 2.2.5 are approximated in the following way: for all k ∈ {1, • • • , n}

Proof. (i) ensues from Burkholder-Davis-Gundy and Rosenthal inequalities, and (ii) ensues from [96, Theorem 6.22 and Corollary 6.29].

Implicit reflected scheme

After the discretization of the time interval, our discrete reflected BSDEs with two RCLL barriers on small interval [t j , t j+1 [, for 0 ≤ j ≤ n -1 is

with terminal condition y n n = ξ n n . By taking the conditional expectation in (2.3.6) w.r.t. F n j , we get

Lemma 2.3.4. For δ small enough, (S 1 ) is equivalent to

where Ψ(y) := yg(t j , y, z n j , u n j )δ.

Proof. For δ small enough, Ψ is invertible because the Lipschitz property of g leads to (Ψ(y) -

We first prove that (S 1 ) implies (S 2 ). Let us firstly assume that ∀j ≤ n -1 ξ n j < ζ n j . On the set {y n j = ξ n j } we have k n j = 0, then

The same type of proof leads to the fourth line of (S 2 ). If there exists j ≤ n -1 such that ξ n j = ζ n j , we get ξ n j = ζ n j = y n j . Then, we have a n j = 0 or k n j = 0. If both are null, we get Ψ(y

). This coincides with the definitions of a n j and k n j given in (S 2 ). If a n j > 0, k n j = 0 and we get

Since Ψ is a one to one map, we get y n j = ξ n j . The same argument holds to prove (y n jζ n j )k n j = 0. Let us prove that ξ n j ≤ y n j . To do so, assume that y n j < ξ n j . In this case a n j = k n j = 0, which gives Ψ(ξ n j ) ≤ E[y n j+1 |F n j ], by definition of a n j . Then Ψ(y n j ) = E[y n j+1 |F n j ] ≥ Ψ(ξ n j ). Ψ being a non decreasing function, this leads to absurdity. We also introduce the continuous time version (Y n t , Z n t , U n t , A n t , K n t ) 0≤t≤T of (y n j , z n j , u n j , a n j , k n j ) j≤n :

(2.3.7)

In the following Θ n := (Y n , Z n , U n , A n -K n ).

Explicit reflected scheme

The explicit reflected scheme is introduced by replacing y n j by E[y n j+1 |F n j ] in g. We obtain

with terminal condition y n n = ξ n n . By taking the conditional expectation in (2.3.8) with respect to F n j , we derive that:

As for the implicit reflected scheme, we get that (S 1 ) is equivalent to (S 2 )

We also introduce the continuous time version (Y n t , Z n t , U n t , A n t , K n t ) 0≤t≤T of (y n j , z n j , u n j , a n j , k n j ) j≤n :

(2.3.9)

In the following Θ

Convergence result

We prove in this Section that Θ n to converges to Θ := (Y t , Z t , U t , A t -K t ) 0≤t≤T , the solution to the DRBSDE (2.1.1). The main result is stated in the following Theorem.

Theorem 2.4.1. Suppose that Assumption 2.2.5 holds and g is a Lipschitz driver satisfying Assumption 2.2.4. Then we have

Moreover, α n ψ n (t) converges weakly to α t in L 2 (F T ).

Proof. To prove this result, we split the error in three terms. The first one is the error Θ n -Θ n , the second one is Θ n -Θ p,n , where Θ p,n := (Y p,n , Z p,n , U p,n , A p,n -K p,n ) represents the solution given by the implicit penalization scheme (see (2.4.3)), and the third error term is Θ p,n -Θ, whose convergence has already been proved in [START_REF] Dumitrescu | Numerical approximation for Doubly reflected BSDEs with Jumps and RCLL obstacles[END_REF]. The result on the convergence of Θ p,n to Θ is recalled in Theorem 2.4.3.

We have the following inequality for the error on Y (the same inequality holds for the errors on Z and U )

For the increasing processes, we have:

λ ).

The rest of the Section is organized as follows: Section 2.4.1 recalls the implicit penalization scheme introduced in [START_REF] Dumitrescu | Numerical approximation for Doubly reflected BSDEs with Jumps and RCLL obstacles[END_REF] and the convergence of Θ

Implicit penalization scheme

In this Section we recall the implicit penalization scheme introduced in [START_REF] Dumitrescu | Numerical approximation for Doubly reflected BSDEs with Jumps and RCLL obstacles[END_REF]. For all j in {0, • • • , n-1} we have

(2.4.2) Following (2.3.3), the triplet (z p,n j , u p,n j , v p,n j ) can be computed as follows

Taking the conditional expectation w.r.t. F n j in (2.4.2), we get

where Ψ p,n (y) = yg(jδ, y, z p,n j , u p,n j )δpδ(yξ n j ) -+ pδ(ζ n jy) -. We also introduce the continuous time version (Y p,n t , Z p,n t , U p,n t , A p,n t , K

Plugging this result in (2.4.6) ends the proof.

The same type of proof gives the following Lemma Lemma 2.4.5. Under Assumption 2.2.5, we have

Proposition 2.4.6. Assume that Assumption 2.2.5 holds and g is a Lipschitz driver. We have

Proof. Let us consider y n j , the solution of the discrete implicit reflected sheme (2.3.6) and y n j , the solution of the explicit reflected scheme (2.3.8). We compute |y n jy n j | 2 , we take the expectation and we get:

-E δ(g(t j , y n j , z n j , u The last inequality comes from (y n jy n j )(a n ja n j ) ≤ 0 and (y n jy n j )(k n jk n j ) ≥ 0 (this ensues from the third and fourth lines of (S 1 ) and (S 1 )). Taking the sum from j = i to n -1 we get Plugging the previous inequality in (2.4.8) and using Lemma 2.4.5 gives

Let n be bigger than N 0 , then δ(1 + 2C g + 2C It remains to take the square of both sides, then the expectation, and to use the Lipschitz property of g combining with (2.4.9) to get the result.

Numerical simulations

We consider the simulation of the solution of a DRBSDE with obstacles and driver of the following form: ξ t := (W t ) 2 +2(1-t T ) Ñt + 1 2 (T -t), ζ t := (W t ) 2 +(1-t T )(( Ñt ) 2 +1)+ 1 2 (T -t), g(t, ω, y, z, u) := -5|y + z| + 6u.

Table 2.1 gives the values of Y 0 with respect to n. We notice that the algorithm converges quite fast in n. Moreover, the computational time is low.