Hélène Halconruy

Malliavin calculus and Dirichlet structures for independent random variables

English. NNT : 2020IPPAT016. tel-03099427

Si c'est seule en scène qu'il faut me défendre ce travail, nombreux sont ceux qui m'ont aidée au montage, ou qui des coulisses continuent à me donner du courage. Aussi c'est de bonne grâce que je me plie à l'exercice des remerciements.

Les premiers que j'adresse vont à mon directeur de thèse, et metteur en scène d'un jour, Laurent Decreusefond. Un immense merci Laurent de n'avoir eu égard à l'atypicité de mon profil pour me suggérer la définition d'un gradient discrètement efficace, point de départ de l'expérience mathématique enthousiasmante que tu m'as donné la chance de vivre. Depuis les après-midis de début de thèse à noircir tableaux et feuilles de papier, jusqu'à cette soutenance finale, merci d'avoir distribué pendant ces années traits d'humour et pléthore d'encouragements, et de m'avoir fait visiter un monde pas toujours chaotique mais souvent intégrable par parties, pour qu'à la question de Leonard Gross :

How far can you go with integration by parts?

Je réponde finalement -et presque en chanson -Far far away.

À ses côtés le jour de la soutenance, cinq membres du jury que je tiens à remercier très chaleureusement pour le privilège qu'ils m'offrent d'en faire partie. I would like to thank Elisa Alòs for kindly agreeing to review this thesis. I am both pleased and honoured that you accepted to read my modest contributions. Un grand merci à Raphäel Lachièze-Rey d'avoir également accepté de rapporter cette thèse et de m'avoir suggéré au détour d'une conversation d'après séminaire à Descartes, de mettre en regard des points de vue "Hoeffding" et "Malliavin". Thank you both for writing a report on the current difficult working conditions of confinement; all the more generous! Merci à François Roueff pour son suivi en tant que directeur adjoint de l'EDMH, dont j'ai apprécié conseils avisés et bienveillants et les agréables discussions que nous avons notamment eues à ces moments-là. Je tiens aussi à remercier Laure Coutin pour son chaleureux accueil à l'université Paul Sabatier lors d'un séjour très agréablement logistique à Toulouse. Je suis également honorée et très reconnaissante à Francesco Russo d'avoir gracieusement accepté de prendre part à mon jury. Enfin un très grand merci à Ivan Nourdin dont les travaux ont été particulièrement inspirants tout au long de cette thèse et dont je me réjouis d'honorer la généreuse invitation à exposer à l'Université du Luxembourg.

La grande modestie de mes travaux ne reflète pas la fascination que les mathématiques exercent en moi. J'y associe l'esthétique, pour ne pas dire l'artistique. Abstraites et distillées comme une statue de Giacometti, elles exhortent à une création dynamique et un remodelage infini. Symboliques aussi, si bien que la recherche en mathématique est une Invitation au voyage que 5

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m'évoquent les deux derniers vers du poème éponyme de Charles Baudelaire Les Fleurs du Mal (1857) :

Là, tout n'est qu'ordre et beauté, Calme, luxe et volupté.

Je suis d'ailleurs infiniment reconnaissante à mon ami et collègue Nicolas Marie d'avoir été le véritable instigateur de ce projet (de voyage). Depuis l'idée de thèse que nos discussions a fait germer et tu t'es employé à semer en organisant la rencontre avec Laurent, jusqu'à ces invitations à participer à quelques de tes activités (RMR en 2018, cours dans la majeure finance etc.) et cette collaboration que tu me proposes aujourd'hui, un immense merci Nicolas ! Ma seule sortie mathématique hors du périphérique m'a conduite à Toulouse. En dépit d'une information pas encore tout à fait mutuelle et de quelques problèmes logistiques persistants et prometteurs de biens des tableaux encore à noircir, mille mercis à Anthony Réveillac et Lorick Huang pour leur accueil incomparable et un séjour exquis à bien des égards ! Mener cette thèse en parallèle de mon activité salariée de responsable pédagogique et enseignante à l'ESME Sudria fut un défi ; je remercie chaleureusement Marc Sellam, PDG de IONIS Education Group et Véronique Bonnet, directrice générale déléguée de l'ESME Sudria, d'avoir accompagné ce projet par leur confiance et leur soutien financier. J'en profite pour adresser un clin d'oeil à mes collègues de la Team Montparnasse, à nos discussions enflammées en "salle de réu", nos apéros du QG d'origine de la BAC à Falguière et qui se déclinent temporairement en virtuel ! Un grand merci également à son coach Hervé Coum, directeur des études du cycle prépa, qui par un cocktail détonant de confiance, de bienveillance, d'écoute attentive et de questions subtilement posées donne à chacun les clefs pour qu'il dessine ses objectifs et y apporte ses propres solutions ; ceci en toute agilité et sans mais.

Anaïs Nin écrit dans son Journal (1947-1955) :

Les moments de courage et de divination sont tous nés de la passion.

Les déserts qui suivent, je ne m'y attarde pas.

J'ai eu la chance de croiser le chemin d'enseignants qui au-delà de l'expertise disciplinaire et des qualités de pédagogues que je leur ai toujours prêtées, m'ont aidée à ne pas trop m'attarder dans le désert. Je pense en particulier à Claire Laurent, avec qui les moments partagés, depuis les discussions passionnantes d'après-cours de géopolitique jusqu'à une inoubliable visite au musée Granet, en passant par les soirées mémorables en terre lettreuse ont scellé une amitié fidèle et complice. À Guillermo Moreno-Socías et Otared Kavian pour leur soutien constant et aussi précieux. À Rachel Ollivier, à nos discussions inspirantes d'il y a quelques années et au souvenir indélébile qu'il en reste aujourd'hui. À Martin Andler et Mourad Besbes, qui ont accompagné mes années d'étudiante, et ont aujourd'hui délicatement substitué à leur casquette de mentor celle d'ami fidèle cher à mes yeux.

Je n'oublie pas ceux et celles qui ont accompagné ces années (et celles d'avant!) par leur humour, sourires, diatribes et affection, et avec qui j'ai eu la joie de partager : soirées au Parc des Princes ou à l'Opéra, balades en forêt de Chantilly ou dans la cache à Luce, verres de Graves ou de Coup Franc, expositions, dîners confidentiels ou après-midis à refaire le monde,

Introduction (in English)

Malliavin's tree

Malliavin calculus is at the heart of our work. Or rather, it is the trunk of the tree by which we choose to draw the outlines of this thesis. The anatomy of Malliavin's tree is described in the following sections; the ground on which it flourishes is made of analysis and probability theories. Malliavin calculus takes its roots in processes characterized by (semi/normal) martingales, sample-paths and/or independence of increments properties. Its formalism sits on what embodies its basement /cornerstone: the integration by parts formula. The meeting with other branches of analysis or probability has produced much fruit: the applications of Malliavin calculus.

Roots and basement: Malliavin's framework and integration by parts

Historically, its initial development was not really to provide an infinite-dimensional differential calculus on the Wiener space, but rather to elaborate a probabilistic toolbox from an application perspective. The first ones were to prove the ellipticity of Hörmander operators and to rule on the existence and regularity of the density function of random vectors. The criterions stated by Paul Malliavin (see [START_REF] Malliavin | Stochastic calculus of variations and hypoelliptic operators[END_REF]) in both cases were based on what would become the cornerstone of his eponymous calculus: the integration by parts formula. Many works leaning on the seminal paper of Paul Malliavin were led around the hypoellipticity of second order degenerate elliptic differential operators; see for instance that of Jean-Michel Bismut [START_REF] Bismut | Martingales, the Malliavin calculus and hypoellipticity under general Hormander's condition[END_REF], Hiroshi Kunita [START_REF] Kunita | Stochastic flows and stochastic differential equations[END_REF], Daniel W. Stroock [START_REF] Stroock | The malliavin calculus, a functional analytic approach[END_REF] or Shinzo Watanabe [START_REF] Watanabe | Lectures on stochastic differential equations and Malliavin calculus[END_REF]. In the same time, other probabilists focused on another perspective that it suggested: the possibility to formalize a differentiation on the Wiener space W and to connect it with the preexisting integration notion.

Two main approaches lead to Malliavin calculus on the classical Wiener space supported by the Banach space C 0 pT ; Rq (where T "r 0,Ts or T " R `) and equipped with the Wiener measure P. Apossiblevariational approach consists in coupling the Wiener space with as e p a r a b l eH i l b e r t space H " L 2 pT, B, q, where is the Lebesgue measure on T. As explained in the reference books of David Nualart [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF] or Svante Janson [START_REF] Janson | Gaussian Hilbert Spaces[END_REF], there exists thus aG a u s s i a nr a n d o m measure tWpAq, A P B, pAq † 8u independent on any family of disjoint subsets of T,s u c h that WpAq has variance pAq, and that the paths of the Brownian motion can be reconstructed via the Centsov representation:

B t " Wpr0,tsq " W `1r0,ts ˘; t P T.

9

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In fact, Wphq coincides with the Wiener integral of the function h P H with respect to B. The Wiener space is thus entirely characterized by the so-called associated isonormal Gaussian process tWphq,hP Hu. Even if the Wiener integral naturally connects the Wiener space and a notion of integration, the challenge remains to construct a differentiation operator applicable to a wide class of Wiener functionals. A definition as a Fréchet derivative, if it seems natural at first, is not relevant: usual Wiener functionals such as Wiener integrals or solutions of stochastic differential equations with smooth coefficients are not even continuous with respect to the norm of the Wiener space. The suitable definition of a directional derivative on L p pPq derives from the Cameron-Martin theorem that indicates -implicitly -that it makes sense only if the increases generating the derivative are performed in the directions of the eponymous vector space. The derivative operator D is thus defined on the space S of cylindrical random variables composed of smooth random variables F of the form F " f pWph 1 q, ¨¨¨, Wph n qq ; h 1 , ¨¨¨,h n P H, (0.0.1)

where f belongs to the Schwartz space S pR n q. It is then extended to its domain D 1,2 ,b y density of the class S in the space of square integrable Wiener functionals. The adjoint of the operator D, called divergence and denoted by , coincides with the Wiener (resp. Skorohod) integral when applied to adapted (resp. non-adapted) processes. The operators D and are connected by the relationship:

E rxDF, Uy H s"E rF Us ;FP D 1,2 , U P Dom .

As detailed for instance in the books of Paul-André Meyer [START_REF] Meyer | Quantum probability for probabilists[END_REF] or Nobuaki Obata [START_REF] Obata | White noise calculus and Fo ck space[END_REF], an alternative approach, lying on the Wiener-Itô chaotic representation property, can also lead to Malliavin calculus. It is based on the development of an infinite dimensional analogue of the Schwartz theory where the Gaussian measure µ on the dual E ˚of a nuclear space E " S pRq would supersede the Lebesgue measure on R n . The space E (resp. E ˚)o ftest (resp. generalized) white noise functionals is constructed by a continuous and dense embedding in L 2 pE ˚,µq (resp. by duality):

E Ä L 2 pE ˚,µq Ä E ˚.
The space L 2 pE ˚,µq is in fact canonically isomorphic to the Fock space over L 2 pRq through the Wiener-Itô-Sega isomorphism. Thus, any square integrable Wiener functionals admits a unique expansion in terms of multiple Wiener integrals. The so-called chaotic decomposition characterizes this alternative approach. In the general context of a Fock space the operators D and coincide respectively with the annihilation operator and the creation operator met in quantum probability and satisfy a generalized canonical commutation relation. The equivalence of the approaches lies on the intrinsic relation between the standard Brownian motion and Hermite polynomials. Paul-André Meyer stated in [START_REF] Meyer | Transformations de riesz pour les lois gaussiennes[END_REF] the equivalence of the norms associated to the different Sobolev spaces thus defined.

Even if the Malliavin calculus was associated with the Wiener space in a first place, it has since been extended to other classes of processes; to name just a few, Gaussian processes in general (see Nualart [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF],Nourdin and Peccati [START_REF] Nourdin | Normal Approximations with Malliavin Calculus: From Stein's Method to Universality[END_REF]), Poisson processes (see Bichteler et al. [START_REF] Bichteler | Malliavin calculus for processes with jumps[END_REF] for a variational approach, Nualart and Vives [START_REF] Nualart | Anticipative calculus for the Poisson process based on the Fock space[END_REF] or Privault [START_REF] Privault | Chaotic and variational calculus in discrete and continuous time for the Poisson process[END_REF] for a chaotic approach), Lévy processes (see Nualart and Schoutens [START_REF] Nualart | Chaotic and predictable representations for lévy processes[END_REF]) and Rademacher processes (see Privault [START_REF] Privault | Stochastic Analysis in Discrete and Continuous Settings[END_REF]). The terminology of classical differential calculus in Banach spaces can be declined in the paradigm of Malliavin calculus: the paths of the canonical process replace vectors, the functionals operating on the space of the paths take the place of functions, while the two key operators are called by the same names of gradient and divergence. The fundamental relationship between these latter appears as the basement of this calculus and conceptualizes the notion of integration by parts in Malliavin's sense:

E rxDF, Uy R s"E rF Us , (0.0.2)

where pΩ, A, Pq is the canonical probability space and R is a generic Hilbert space seen as the perturbations space when DF is interpreted as a directional derivative (for instance R is the Cameron-Martin's space for Brownian motion).

During the preparation of this thesis work, we found two frameworks equipped with a Malliavin structure and of particular interest: the Poisson space and the Rademacher space. Their Malliavin formalism will be described and compared to our contributions later.

Branches and fruits: applications of Mallavin calculus

The use of Malliavin's powerful toolbox in other branches of analysis or probability has been particularly fruitful for years.

Partial/Stochastic differential equations and anticipative calculus

Having in mind the results of the seminal work of Paul Malliavin many authors re-used the non-degeneracy condition to study the regularity of solutions of partial differential equations and stochastic differential equations. This led to many applications reviewed for instance by Martha Sanz-Solé in [START_REF] Sanz-Solé | Malliavin calculus: With applications to stochastic partial differential equations[END_REF] (resp. by Hu, Huang, Lê, Nualart and Tindel [START_REF] Hu | Stochastic heat equation with rough dependence in space[END_REF]) for partial differential equations driven by coloured noises (resp. by a white noise in time/rough noise in space with Hurst index H Pp 1{4, 1{2q). We can cite the works of Vlad Bally and Denis Talay [START_REF] Bally | The Euler scheme for stochastic differential equations: Error analysis with Malliavin calculus[END_REF], Yaozhong Hu, David Nualart and Jian Song [START_REF] Hu | Malliavin calculus for backward stochastic differential equations and application to numerical solutions[END_REF], or Shigeo Kusuoka and Daniel W. Stroock [START_REF] Kusuoka | Applications of the Malliavin calculus, part i[END_REF] for stochastic differential equations driven by a Brownian motion and that of David Nualart and Bruno Sausserau [START_REF] Nualart | Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion[END_REF] for SDEs driven by a fractional Brownian motion with Hurst index H Pp1{2, 1q.

Malliavin calculus results were also exploited to tackle with non-causal stochastic differential equations formulated in terms of anticipating stochastic integrals. Indeed the divergence operator allows to define an extension of Itô's stochastic integral to anticipating integrands in Brownian motion case (see Skorohod [START_REF] Skorohod | On a generalization of a stochastic integral[END_REF]). An anticipating formula is presented by David Nualart and Pardoux in [START_REF] Nualart | Stochastic calculus with anticipating integrands[END_REF] or in its earlier (and very unknown) version by Masayuki Hitsuda in [START_REF] Hitsuda | Formula for brownian partial derivatives, the second japan-ussr symp[END_REF]. Elisa Alòs and David Nualart improved it in [START_REF] Alòs | An extension of Itô's formula for anticipating processes[END_REF]. An anticipative calculus was thus first designed through this approach for Brownian motion (see Berger [START_REF] Berger | A malliavin-type anticipative stochastic calculus[END_REF] or the chapter 3 of Nualart [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF]) and has since been extended to Poisson processes (see Decreusefond and Savy [START_REF] Decreusefond | Anticipative calculus with respect to filtered Poisson processes[END_REF], Nualart and Vives [START_REF] Nualart | Anticipative calculus for the Poisson process based on the Fock space[END_REF]) and Lévy processes (see [START_REF] Alòs | An anticipating Itô formula for Lévy processes[END_REF], [START_REF] Di Nunno | Malliavin calculus and anticipative Itô formulae for Lévy processes[END_REF]). Elisa Alòs, Olivier Mazet and David Nualart used also Malliavin calculus (in particular the divergence) to give a sense to stochastic integrals with respect to the fractional Brownian motion with Hurst index H Pp0, 1{2q and then to design a stochastic calculus in that framework.

CONTENTS

Statistics

A wide range of Malliavin tools were also used for statistical purposes. We give a quick overview through some examples. For instance, Ciprian A. Tudor and Frederi G. Viens [START_REF] Tudor | Variations and estimators for self-similarity parameters via malliavin calculus[END_REF] used the chaotic decomposition to provide an estimator of the self-similarity parameter H of the Rosenblatt process. Fabienne Comte and Nicolas Marie [START_REF] Comte | Nonparametric Estimation in Fractional SDE[END_REF] showed that Skorohod's integral (defined from the divergence) was the only suitable extension of Itô's integral to study nonparametric estimation in SDEs driven by a fractional Brownian motion of Hurst index H Pp 1{2, 1q. José M. Corcuera and Arturo Kohatsu-Higa [START_REF] Corcuera | Statistical inference and malliavin calculus[END_REF] wielded Malliavin integration by parts to study asymptotic inference of stochastic (jump) processes. Nicolas Privault and Anthony Réveillac also used it with another aim: to provide estimators of Stein-type estimators of the drift of some Gaussian processes [START_REF] Privault | Stein estimation for the drift of gaussian processes using the malliavin calculus[END_REF].

Finance

A wide range of the fruits born of the intersection of Malliavin calculus and finance can be found in the monograph of Paul Malliavin and Anton Thalmaier (see [START_REF] Malliavin | Stochastic Calculus of Variations in Mathematical Finance[END_REF]). Among them, three have sparked special interest throughout our work: the calculation of Greeks, portfolios hedging and insider's trading.

In order to frame our point in the vast universe of financial mathematics, we restrict first our frame to a simple financial market model composed of two assets: one risk-free asset A "p A t q tPT and one risky asset S "p S t q tPT .W ea s s u m et h a tA is defined by an interest rate and that S satisfies a stochastic differential equation (in the Black-Scholes model for instance) or an equation in differences (in discrete models) on the trading interval T. The derivatives, namely the financial products which value derives from the performance of S,w e are focused on are called vanilla options. An option is a contract which gives the buyer (the owner or holder of the option) the right, but not the obligation, to buy or sell the risky asset at a specified strike price K and at a specified date T . The value of an option at expiry, i.e. what the holder will receive, is called payoff ; this is given for a vanilla option by a random variable ΦpS T q. On this market act buyers and sellers who are considered as regular agents (resp. insiders) if decisions as to the composition of their portfolio (with both types of assets at their disposal) are made based on public information (resp. with extra information). The Greeks, so-called in reminiscence of the Greek letters by which they are denoted, designate the sensitivities of the price of an option with respect to a change in the underlying parameters on which its value depends. They are of capital use for the trader giving him the power to control his risk exposure. For instance, Delta and Rho are the first-order Greeks measuring respectively the sensitivity to the underlying asset price and to the interest rate whereas Gamma designates the second-order Greek quantifying the rate of change in the Delta with respect to changes in the underlying price. For a long time, these computations were the only prerogative of analysts who were treating them with finite difference methods in a Monte-Carlo frame. Without precisely introducing the formalism (that will be done in chapter 4) consider an option of payoff K " ΦpS T q where pS t q tPT is the price of a risky asset defined by the Black-Scholes model:

S t " 1 `ª t 0 rS u du `ª t 0 S u dB u ,tP T, (0.0.3)
where the parameters r and stand respectively for the risk-free interest rate and the volatility of returns of S. The value of the option is given by V t at time t P T. Delta can be expressed (in the Black-Scholes model) as

∆ " BV 0 BS 0 " e ´rT E Q " Φ 1 pS T q BS T BS 0 ⇢ , (0.0.4)
where Q denotes the risk-neutral probability measure. Generally, the function Φ is not derivable; for instance the payoff of a european call is ΦpS T q"p S T ´Kq `. Therefore, it would be wise to write ∆ " e ´rT E Q rΦpS T q ⇡s, where the weight ⇡ does not depend of the payoff. The basic idea introduced first by Éric Fournié, Jean-Michel Lasry, Jérôme Lebuchoux, Pierre-Louis Lions and Nizar Touzi in [START_REF] Fournié | Applications of malliavin calculus to monte carlo methods in finance[END_REF] and [START_REF] Fournié | Applications of Malliavin calculus to Monte Carlo methods in finance[END_REF] making use of the integration by parts and the chain rule formulas leads to

∆ " e ´rT E Q rΦpS T q puqs , so that the weight can be expressed as the Skorohod integral of a generator u, here deterministic and constant equal to p T q ´1. Éric Benamou gave in [START_REF] Benhamou | A Generalisation of Malliavin Weighted Scheme for Fast Computation of the Greeks[END_REF] necessary and sufficient conditions on u so that it generates a weight for the Greeks simulation. The pioneer work [START_REF] Fournié | Applications of malliavin calculus to monte carlo methods in finance[END_REF] was restated and transposed to financial jump models driven by Poisson processes by Nicolas Privault et al. (see [START_REF] El-Khatib | Computations of greeks in a market with jumps via the malliavin calculus[END_REF], [START_REF] Privault | A Malliavin calculus approach to sensitivity analysis in insurance[END_REF]) and by Lévy processes by (in particular) Marie-Pierre Bavouzet-Morel and Marouen Messaoud in [START_REF] Bavouzet-Morel | Computation of greeks using malliavin's calculus in jump type market models[END_REF]. Strongly connected to the semi-martingales theory, the valuation of options and in particular hedging problems have also benefited from the contributions of Malliavin calculus. Hedging is crucial since it gives an understanding of how sellers or buyers can manage dynamically and compose their portfolio to replicate the payoff of the option, i.e. to attain its value at maturity. Mathematically, this boils down to determine a R 2 -valued process satisfying the self-financing condition (see chapter 4 for its expression in the Black-Scholes and the Cox-Ross-Rubinstein models) such that V T p q"ΦpS T q.

In discrete time models, the additional condition V t p q • 0, ; @t P N T , P ´a.s., is required. In complete markets, where all claims are reachable, and in which the underlying model is equipped with a Malliavin structure, the process can be elegantly described in terms of Malliavin derivative; this was suggested by the contemporary works (both published in 1991) of Ioannis Karatzas and Daniel Ocone [START_REF] Karatzas | An extension of Clark's formula[END_REF] on the one hand and with Jinlu Li [START_REF] Ocone | A generalized clark representation formula, with application to optimal portfolios[END_REF] on the other hand. The so-called Karatzas-Ocone hedging formula is actually derived from the Clark-Ocone one. Analogue formulas were provided for Poisson processes by Günter Last and Matthew Penrose [START_REF] Last | Martingale representation for poisson processes with applications to minimal variance hedging[END_REF] and for Lévy processes by Giulia Di Nunno [START_REF] Nunno | Stochastic integral representations, stochastic derivatives and minimal variance hedging[END_REF]. Insider trading is a related topic to portfolio optimization. The frame is that of a financial market where trade two agents with different levels of information: an ordinary agent whose decisions are made in light of public flow and an insider enjoying an additional information from the start of the trading period. Two questions arise: how to quantify insider's additional expected utility? Does the additional information produce an arbitrage? This latter can be traduced by an enrichment of the filtration on which the insider can base his portfolio decisions and leads then to the theory of enlargement of filtrations. Following the pioneer work of Igor Pikovsky and Ioannis Karatzas [START_REF] Pikovsky | Anticipative portfolio optimization[END_REF], Jürgen Amendinger et al. [START_REF] Amendinger | Additional logarithmic utility of an insider[END_REF], [START_REF] Amendinger | A monetary value for initial information in portfolio optimization[END_REF], [START_REF] Amendinger | Martingale representation theorems for initially enlarged filtrations[END_REF], Axel Grorud and

CONTENTS

Monique Pontier [START_REF] Grorud | Insider trading in a continuous time market model[END_REF] precised criterions for optimization and computed the additional utility of the insider in their respective works. Peter Imkeller connected it to Malliavin calculus in [START_REF] Imkeller | Malliavin's calculus in insider models: Additional utility and free lunches[END_REF] by expressing the information drift as the logarithmic Malliavin trace of a conditional density characterizing insider's advantage. Jorge A. León, Reyla Navarro and David Nualart use in [START_REF] León | An anticipating calculus approach to the utility maximization of an insider[END_REF] some techniques of Malliavin calculus to analyze the properties the forward integral (introduced by Russo and Vallois in [START_REF] Russo | Forward, backward and symmetric stochastic integration[END_REF]) and to maximize the expected logarithmic utility of the insider. In some extensions of the Black-Scholes model allowing the volatility to be a stochastic process (see for instance Hull and White [START_REF] Hull | The pricing of options on assets with stochastic volatilities[END_REF], or Heston [START_REF] Heston | A closed-form solution for options with stochastic volatility with applications to bond and currency options[END_REF]), in particular in some stochastic volatility diffusion models, where the volatility also follows a diffusion process, it can be observed that the implied volatility (forward-looking measure to estimate the future fluctuations of the underlying asset) behaves roughly at any reasonable timescale. Malliavin's toolbox whose effectiveness in anticipative calculus was mentioned, is also efficient to analyze future volatilities, which are non-adapted processes. In [START_REF] Alòs | On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility[END_REF], Elisa Alòs, Jorge León and Josep Vives, on a seminal work on the nowadays named rough volatilities, give an expression for the short-time behavior of the implied volatility in a jump-diffusion models in terms of the Malliavin derivative of it.

Stein's method

In a seminal paper ( [START_REF] Stein | A bound for the error in the normal approximation to the distribution of a sum of dependent random variables[END_REF]), Charles Stein described a process to quantify the errors in the Normal approximation by sums of random variables having a stationary dependence structure. By the same, he laid the foundations of a very powerful method whose perspectives of application very quickly exceeded the framework of its birth; Stein's method notably stood out as one (not to say the one) efficient way to compute distance between two probability measures of the form dist H pP ‹ , Qq"sup hPH ˇˇˇª F h dP ‹ ´ªF h dQ ˇˇˇ, where P ‹ stands for the target measure (to approximate) and Q the initial measure, both defined on the same metric space pF,cq and H is a set of test functions. Even if Stein's method keeps a special bond with the Normal approximation for which it was born (refer to the reference book of Louis H.Y. Chen, Larry Goldstein and Qi-Man Shao [START_REF] Chen | Normal approximation by Stein's method[END_REF]), its principles could be to transposed to other target distributions. It first gave birth to Stein-Chen method for Poisson convergence (see Chen [START_REF] Chen | On the convergence of Poisson binomial to Poisson distributions[END_REF]). Many approximation criterions of more general random variables by varied distributions followed: Gamma (Arras and Swan [START_REF] Arras | A stroll along the gamma[END_REF], Döbler and Peccati [START_REF] Döbler | The gamma stein equation and non-central de jong theorems[END_REF], Peköz, Röllin and Ross [START_REF] Peköz | Generalized gamma approximation with rates for urns, walks and trees[END_REF]), Exponential (Chatterjee, Fulman and Röllin [START_REF] Chatterjee | Exponential approximation by stein's method and spectral graph theory[END_REF], Peköz and Röllin [START_REF] Peköz | New rates for exponential approximation and the theorems of rényi and yaglom[END_REF]), Geometric (Peköz, Röllin and Ross [START_REF] Peköz | Total variation error bounds for geometric approximation[END_REF])... Besides, in the Section Taxonomy of the webpage Stein's method he has dedicated to it, Yvik Swan reviews all distributions which approximation can be quantified through this method. Many surveys were published to teach the proselyte and initiate the layman with the effectiveness of the concept. If it remains impossible to give an exhaustive list, we recommend the reference and pedagogical works of Benjamin Arras and Yvik Swan, [START_REF] Arras | A stroll along the gamma[END_REF], Andrew D. Barbour [14] or Nathan Ross [START_REF] Ross | Fundamentals of Stein's method[END_REF]. Stein's method seems to be split in two stages; the first one consists in converting the problem of bounding the error in the approximation of the measure P ‹ by Q into a problem of bounding an expression of the form E Q rL'pXqs " E rL 1 'pXqs `E rL 2 'pXqs , (0.0.5)

where L (resp. the class of ') is determined to characterize the target measure P ‹ (resp. P ‹ and H), and X is a random variable of law Q. The second component of Stein's method gathers techniques to bound (0.0.13); by taking into account how X is defined this consists into transforming L 1 'pXq into ´L2 'pXq`remainder. This remainder is what gives the bound of the distance and in a problem of convergence, provides its rate. To make the transformation of L 1 'pXq, several approaches appeared along the years. One of the most popular approach (see for instance Barbour and Chen [START_REF] Barbour | An introduction to Stein's method[END_REF]) is to use exchangeable pairs. Alternative to it, are the size-biased (see Chen, Goldstein and Shao [START_REF] Chen | Normal approximation by Stein's method[END_REF]) or zero biased (see Goldstein and Reinert [START_REF] Goldstein | Stein's method and the zero bias transformation with application to simple random sampling[END_REF]) couplings, which again conveniently transform L 1 '. In a path-breaking work, Ivan Nourdin and Giovanni Peccati (see [START_REF] Nourdin | Stein's method on wiener chaos[END_REF], [START_REF] Nourdin | Normal Approximations with Malliavin Calculus: From Stein's Method to Universality[END_REF]) showed that the transformation step can be advantageously made simple using integration by parts in the sense of Malliavin calculus, and by the same gave an intersection to the two theories. This approach is efficient provided there exists a Malliavin gradient on the space on which X is defined. It was in particular applied to functionals of Rademacher (see Nourdin, Peccati and Reinert [START_REF] Nourdin | Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos[END_REF], Zheng [START_REF] Zheng | Normal approximation and almost sure central limit theorem for nonsymmetric rademacher functionals[END_REF], Poisson (see for instance Decreusefond, Schulte and Thäle [START_REF] Decreusefond | Functional Poisson approximation in Kantorovich-Rubinstein distance with applications to U-statistics and stochastic geometry[END_REF], Lachièze-Rey and Peccati [START_REF] Lachièze-Rey | Fine Gaussian fluctuations on the Poisson space II: rescaled kernels, marked processes and geometric U-statistics[END_REF], Peccati, Solé, Taqqu and Utzet [START_REF] Peccati | Stein's method and Normal approximation of Poisson functionals[END_REF]) or Gaussian random variables (see Nualart and Peccati [START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF]) or processes (see Coutin and Decreusefond [32,[START_REF] Coutin | Higher order expansions via Stein's method[END_REF]). A review of the works resulting from the Malliavin-Stein approach can be retrieved in the webpage designed by Ivan Nourdin.

Dirichlet forms

Parallel to its connection with Stein's method, Malliavin calculus meets profitably Dirichlet forms in the frame of error calculus (see Bouleau and Hirsch [START_REF] Bouleau | Dirichlet forms and analysis on Wiener space[END_REF]). The related topics have generated considerable interest (among the analyst community before probabilists take it) since the beginning of XX-th century. The use of Dirichlet forms is particularly efficient to perform those computations when errors are supposed to be infinitesimal and probabilistic. If Y stands for a scalar erroneous quantity, the error propagates by the function f according to the formulas: " bias of error on f pYq"p bias of error on Yq f 1 pYq`1 2 pvar of error on Yq f 2 pYqp E 1 q var of error on f pYq"p var of error on Yq

f 12 pYqp E 2 q
where, if the variance is of the same order of magnitude as the bias or if the bias is negligible with respect to the variance, the calculus has to be • for the bias of error: a second order differential equation with variance and bias (E 1 ),

• for the variance of error: a first order differential equation for the calculus of the variance which does not involve the bias (E 2 ).

By considering up to now a quantity F " FpX A q (A Ä N ˚) function of the erroneous quantities tX k ,kP Au (supposed to be known, small and independent) the potential quadratic error to expect on F denoted 2 F can be written as: The use of these tools makes most sense when tackling with the estimation of the convergence rate of a sequence pX n q nPN ˚towards a random variable X, all assumed to be defined on a probability space pΩ, A, Pq. Indeed, if the form Γ is closed, the error calculus theory deploys in this context by lying on the following principle: if the sequence of pairs pX n , error on X n q converges suitably, it converges necessarily to a pair pX, error on Xq. When the underlying space is provided with a Malliavin structure the operator L appearing in (0.0.15) coincides with the Laplacian/number operator L "´ D. This key fact will be explained in Section 1.3.2 and illustrated in the case of the Wiener space in Example 1.3.6. This suggests that Dirichlet forms implicitly operate in the Stein-Malliavin combination to lead to a Stein-Dirichlet-Malliavin method (see Decreusefond [START_REF] Decreusefond | The Stein-Dirichlet-Malliavin method[END_REF]). Besides, Ehsan Azmoodeh, Simon Campese and Guillaume Poly exploited the connection between the three theories to revisit the fourth moment theorem in [START_REF] Azmoodeh | Fourth moment theorems for Markov diffusion generators[END_REF].

We conclude this state of the art by drawing Malliavin's tree.

Malliavin's tree

IBP Normal/Semi Martingales

Sample paths

Brownian Motion

Contributions

By grabbing the tree metaphor again, we could sum up the contributions of this thesis to the addition of two roots: the construction of a Malliavin calculus for independent random variables on the one hand, and for compound geometric processes on the other one. The fruits resulting from these two formalisms are displayed in colour in the above figure: in purple for those relating to independent random variables framework, in blue for those relating to compound geometric processes framework.

Malliavin calculus and Dirichlet structures for independent random variables

The motivation to develop a Malliavin calculus for independent random variables was twofold.

As mentioned above, after some years of development, the Malliavin calculus has reached a certain maturity. The most complete and fruitful theories are for Wiener (see for instance Nualart [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF]) and Poisson point processes (see for instance Albeverio, Kondratiev and Röckner [START_REF] Albeverio | Analysis and geometry on configuration spaces[END_REF], Privault [START_REF] Privault | Stochastic Analysis in Discrete and Continuous Settings[END_REF]). The only Malliavin's foray in discrete settings had been so far for Rademacher processes (see Privault [START_REF] Privault | Stochastic Analysis in Discrete and Continuous Settings[END_REF]); our initial aim was to generalize it. Malliavin calculus can be constructed in the three aforementioned frameworks (Gaussian, Poisson, Rademacher) via a chaotic approach. In those, the existence of normal martingales as solutions of structure equations (see Émery [START_REF] Émery | On the Azéma martingales[END_REF], Privault, Solé and Vives [START_REF] Privault | Chaotic kabanov formula for the azéma martingales[END_REF] for the continuous time, Privault [START_REF] Privault | Stochastic Analysis in Discrete and Continuous Settings[END_REF] for Rademacher processes) entails directly a chaotic decomposition by induction of the martingale representation property (see Émery [START_REF] Émery | On the Azéma martingales[END_REF]). The existence of structure equations is not necessary to state a chaotic decomposition (see for instance Dasgupt and Kallianpur [START_REF] Dasgupta | Multiple fractional integrals[END_REF], Eddhabi and Vives [START_REF] Eddahbi | Chaotic expansion and smoothness of some functionals of the fractional brownian motion[END_REF] for chaotic decomposition and Clark formula for the fractional Brownian motion with index H Pp1{2, 1q); but it considerably facilitates it. The independence and stationarity of increments seems to play a major role in the resolution of the so-called structure equations and thus in the effectiveness of the concepts. This motivated to investigate and explore what we can inherit in the simplest situation of all with independence (without requiring stationarity): that of a family of independent, non necessarily identically distributed, random variables. The closest situation to our investigations is that of the Rademacher space, namely t´1, 1u N , equipped with the product probability b kPN µ k where µ k is a Bernoulli probability on t´1, 1u.

The chaotic approach that leads to a Malliavin calculus on it is intrinsically linked to the existence of a normal martingale or novation (in Emery's terminology [START_REF] Émery | A discrete approach to the chaotic representation property[END_REF]) coupled with the canonical process X. The existence of a normal martingale in a discrete setting also requires that the law of the random variables be only supported by two points (see Émery [START_REF] Émery | On the Azéma martingales[END_REF], [START_REF] Émery | A discrete approach to the chaotic representation property[END_REF], Privault and Schoutens [START_REF] Privault | Discrete chaotic calculus and covariance identities[END_REF]). This is also reflected in the very definition of the gradient (see Nourdin, Peccati and Reinert [START_REF] Nourdin | Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos[END_REF] or Privault [START_REF] Privault | Stochastic Analysis in Discrete and Continuous Settings[END_REF]), usually defined as

p D k FpX 1 , ¨¨¨, X n q " E rX k FpX 1 , ¨¨¨, X n q|X `, `‰ ks (0.0.8) " PpX k " 1q FpX 1 , ¨¨¨, `1 lo omo on k´th , ¨¨¨, X n q´PpX k "´1q FpX 1 , ¨¨¨, ´1 lo omo on k´th , ¨¨¨, X n q,
and that implies to be meaningful, either that the random variables are real valued or that they only have two possible outcomes. Our initial aim was to provide any countable product of probability spaces with a Malliavin structure; it must be clear that in the present work all the random variables may leave on different spaces, which are then only supposed to be Polish spaces. That means that in the definition of the gradient, we can not use any algebraic property of the underlying spaces. Although some of our applications do concern random variables with finite number of outcomes, it did not seem straightforward to devise what should be the weights, replacing PpX k " 1q and ´PpX k "1q. In that respect, we introduce a discrete gradient D as a difference operator which can be interpreted as the measure of the "influence" of the k-th component of the process X on F. The definition of the divergence operator , as the adjoint of D, satisfies a discrete version of integration by parts formula (1.3.1). Last the introduction of the operator number L "´ D, also generator of a Markovian semi-group pP t q tPR `, completes the definition of the modicum of Malliavin operators. We bring kind of legitimacy to our formalism from several perspectives. We connect our formalism to its elder, the Rademacher framework; not with the gradient but with the number operator. Indeed, many applications, notably those revolving around functional identities, rely not directly on the gradient D but rather on the operator number L. It turns out that for the Rademacher space, the operators p L "´p p D defined according to (0.0.8) and L defined in our construction do coincide. Besides, we attach our construction to preexisting continuous time theories. Actually, we bring to light that the usual Poisson and Brownian Dirichlet structures associated to their respective gradient, could be retrieved as limits of the structures induced by our formalism. We obtained versions of almost all the classical functional inequalities in discrete settings, and revisit some of them to which we give a new point of view. To mention just a few, we show that the Efron-Stein inequality can be interpreted as a Poincaré inequality or that the Hoeffding decomposition of U -statistics can be viewed as an avatar of the Clark representation formula. We transpose the classical covariance identity, log-Sobolev inequality, and deviation inequality in our framework. Then our formalism finds its place within Malliavin's landscape; it fully generalizes what is known about Rademacher spaces, and connects with Brownian and Poisson frameworks with both limit procedure and similar functional identities. The second main motivation of our first work came from a branch of the tree with which such formalism would be likely to combine advantageously: Stein's method. We provide here a "Stein-Malliavin criterion" analogue to the eponymous results in the Gaussian and Poisson cases (see Nourdin and Peccati [START_REF] Nourdin | Stein's method on wiener chaos[END_REF], Peccati, Solé, Taqqu and Utzet [START_REF] Peccati | Stein's method and Normal approximation of Poisson functionals[END_REF]), or in the Rademacher space (see Nourdin, Peccati and Reinert [START_REF] Nourdin | Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos[END_REF]) giving a "Berry-Esseen" general bound for the Kolmogorov-Rubinstein distance between the law of a functional of independent random variables (without any additional hypothesis on the initial laws) and the Normal distribution, and this in terms of Malliavin operators. We establish a similar criterion for the Gamma approximation. Applied to the particular case of degenerate U -statistics of order two, it allows to provide a de Jong type estimate in the Gamma approximation of those U -statistics (see Döbler and Peccati [START_REF] Döbler | The gamma stein equation and noncentral de jong theorems[END_REF]). To the best of our knowledge, there does not yet exist a Stein criterion for Gaussian or Gamma approximation which does not rely on exchangeable pairs or any other sort of coupling. Our results appear as a natural continuation of those stated by Ivan Nourdin, Giovanni Peccati and Gesine Reinert [START_REF] Nourdin | Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos[END_REF] for Rademacher functionals; we generalize thus the use of the Nourdin-Peccati approach (Nourdin and Peccati [START_REF] Nourdin | Normal Approximations with Malliavin Calculus: From Stein's Method to Universality[END_REF]) to general discrete settings. For all these reasons, it seems that our Dirichlet-Malliavin structure gives a unified framework for many results scattered in the literature so that we hope to bring new insights on why these apparently disjoint results (Efron-Stein, exchangeable pairs, etc.) are in fact multiple sides of the same coin... with the "effigy" of the integration by parts formula.

CONTENTS

Malliavin calculus for compound geometric processes and insider trading in a ternary model

The initial aim of the second work was not to design a stochastic calculus for geometric compound processes but rather to use our discrete Malliavin calculus to address insider's trading issues. This is also the reason why the title of the thesis does not mention it. Indeed, we first tried to deploy our artillery to hit an identified target: the computation of the additional logarithmic expected utility of an insider in the trinomial model. This problem is of particular interest in several respects; first, and as mentioned above, financial mathematics is the flagship scope of Malliavin calculus. If option hedging has already been studied in the Cox-Ross-Rubinstein model (see chapter 1 in Privault [START_REF] Privault | Stochastic Analysis in Discrete and Continuous Settings[END_REF]) using the Malliavin equipment in the Rademacher space, no similar project has been carried out so far in the framework of multinomial trees (with at least three branches at each stage). Moreover, while many works deal with enlargement of filtrations and insider issues in continuous time (see the works of Amendinger, Imkeller and Schweizer ( [START_REF] Amendinger | Additional logarithmic utility of an insider[END_REF]), very few were carried on discrete settings. Some results already exist for the enlargement of filtrations in discrete time. Catherine Blanchet-Scalliet, Monique Jeanblanc and Roméo Romo Romero showed in [START_REF] Blanchet-Scalliet | Enlargement of filtration in discrete time[END_REF] that the results known in continuous time extend immediately in a discrete time setting. Most of them are easily obtained as a result of Doob's decomposition. In this respect, we wanted to discuss the possibility to interpret the increasing predictable process of Doob's decomposition as the information drift and to express it in terms of Malliavin derivative, as done by Peter Imkeller in continuous time [START_REF] Imkeller | Malliavin's calculus and applications in stochastic control and finance[END_REF].

The results of our investigations follow; while our formalism lends itself very well to the computation of Greeks in the trinomial model, it turns out to be impossible to state a Karatzas-Ocone hedging formula in this context. This is a direct consequence of the absence of a martingale representation theorem which requires that the law of the random variables be only supported by two points (see Émery [START_REF] Émery | A discrete approach to the chaotic representation property[END_REF]). Without completely losing sight of our original goal, we chose thus to substitute what we have called aternarymo delto the trinomial one. Equivalent in law to the initial one, it is supported by a compound geometric process. This is based on a jump structure as the compound Poisson process, and thereby inherits number of properties from this latter. In particular, the definition of the measurable space pΩ, Aq on which it is defined is a crucial point to characterize the compound geometric process. Following the frame designed by Laurent Decreusefond and Nicolas Savy for filtered Poisson processes [START_REF] Decreusefond | Anticipative calculus with respect to filtered Poisson processes[END_REF], we describe Ω as the set composed of the couple pt, kq where t stands for the jump time and k the height of the corresponding jump. This definition plays a major role in the effectiveness of the concepts; the so-called ternary model lying on this new structure, yet equivalent in law to the trinomial model, best suits for hedging problems. As expected, it remains impossible to state a chaotic decomposition from a normal martingale (for the reasons evoked above); nevertheless this new paradigm enables to establish a modified chaotic decomposition in terms of multiple integrals with respect to an non-orthogonal family of random variables. The gradient is thus introduced as the annihilation operator acting on this pseudo-chaotic decomposition and such that the stochastic integral defined appears -up to a linear transformation -as the "inverse" operation of it and as its adjoint. These latters are linked via an integration by parts formula so that we equip compound geometric processes with a Malliavin calculus. Besides, the gradient coincides with a difference operator more prone to state functional identities from which derives the expected Clark-Ocone-Karatzas formula.

The second part of the work consists in making use of this new formalism to compute the additional utility of an insider in the new ternary model. The insider benefits from an extra information hidden in a Γ-valued random variable G. The insider's filtration G does not coincide with the initial one, so that we need to use the techniques of enlargement of filtrations in a discrete setting. Using the toolbox of Catherine Blanchet-Scalliet, Monique Jeanblanc and Roméo Romo Romero [START_REF] Blanchet-Scalliet | Enlargement of filtration in discrete time[END_REF], we define an analogue of the information drift. We express it by means of the Malliavin gradient (for compound geometric processes) applied to the conditional density processes p c (c P Γ). Besides, we focus on a particular process 1{p G from wich we define the martingale preserving measure pQ t q tPT . Following the method of Freddy Delbaen et Walter Schachermayer [START_REF] Delbaen | The Mathematics of Arbitrage[END_REF], we compute and express the insider's optimal portfolio in terms of the G-martingale pQ t q tPT . We show that the insider's additional expected logarithmic utility can be expressed as the relative entropy of the initial measure P with respect to Q t ;w e retrieve exactly the result of Jürgen Amendinger, Peter Imkeller and Martin Schweizer [START_REF] Amendinger | Additional logarithmic utility of an insider[END_REF] stated in the continuous case. We end the work by doing explicit computations in the specific case where the insider gets an extra information on the terminal value of the risky asset.

Manuscript

The manuscript is organized as follows; it consists of two parts, each of which refers to one of the above-mentioned works. The first one, entitled Malliavin and Dirichlet structures for independent random variables is a collaboration with Laurent Decreusefond and was published in Stochastic Processes and their Applications in 2019 (see [START_REF] Decreusefond | Malliavin and dirichlet structures for independent random variables[END_REF]). The second one will be submitted soon. Both parts are deployed according to the same scheme; they start with an introductory chapter where is giving the state of the art and presenting the formalism that need to be developed to respond to an identified problematic. The necessary tools are built in the chapters immediatly following them whereas the fruits are harvested in the last chapter of each part, as the response provided to the initial problem.

Before ending this introduction, let me mention that any new contribution (original definition or result) is flagged with a symbol p‹q. Despite my poor English (I apologize for that), I really enjoyed preparing this manuscript; I wish you a pleasant reading! Introduction (en français) b L'arbre de Malliavin Au coeur de notre travail, se situe le calcul de Malliavin. Ou plutôt, il constitue le tronc de l'arbre par lequel nous avons choisi de dessiner les contours de cette thèse. L'anatomie de cet arbre de Malliavin se décline au travers des sections suivantes ; la composition du sol sur lequel il s'érige a en héritage les théories analytique et probabiliste. Il prend ses racines dans les familles de processus caractérisés par des propriétés de (semi/normales) martingales, trajectorielles ou d'indépendance des incréments. Son formalisme repose sur un(e) pierre angulaire/socle qu'incarne la formule d'intégration par parties. Enfin, sa rencontre avec d'autres branches de l'analyse ou des probabilités a donné de nombreux fruits, les applications du calcul de Malliavin.

Racines et socle : le cadre du calcul de Malliavin et la formule d'intégration par parties

Historiquement, le développement du calcul de Malliavin était davantage motivé par la conception d'outils probabilistes à des fins applicatives que par l'élaboration -à proprement parler -d'un calcul différentiel en dimension infinie sur l'espace de Wiener. Dans cette perspective d'applications, les premiers résultats de Malliavin furent utilisés pour fournir une preuve probabiliste du théorème de Hörmander et statuer sur l'existence et la régularité de la densité de vecteurs aléatoires. Les critères établis par Paul Malliavin (voir [START_REF] Malliavin | Stochastic calculus of variations and hypoelliptic operators[END_REF]) dans le cadre de ces deux études reposaient sur ce qui deviendrait la pierre angulaire de sa théorie éponyme : la formule d'intégration par parties. Ce travail pionner de Paul Malliavin inspira peu après de nombreux travaux autour de l'hypoellipticité d'opérateurs différentiels elliptiques dégénérés d'ordre 2 ; on peut notamment citer les travaux de Jean-Michel Bismut Jean-Michel Bismut [START_REF] Bismut | Martingales, the Malliavin calculus and hypoellipticity under general Hormander's condition[END_REF], Hiroshi Kunita [START_REF] Kunita | Stochastic flows and stochastic differential equations[END_REF], Daniel W. Stroock [START_REF] Stroock | The malliavin calculus, a functional analytic approach[END_REF] or Shinzo Watanabe [START_REF] Watanabe | Lectures on stochastic differential equations and Malliavin calculus[END_REF]. En parallèle, d'autres probabilistes s'intéressèrent alors à ce qu'impliquait notamment ce nouveau formalisme : la possibilité de construire une différentiation sur l'espace de Wiener W et de la connecter avec la précédente notion d'intégration.

Deux approches principales menant au calcul de Malliavin classique sur l'espace de Wiener (supporté par l'espace de Banach C 0 pT ; Rq avec T "r 0,Ts ou T " R `, et équipé de la mesure de Wiener P) peuvent être identifiées. Une première approche, dite variationnelle, consiste à coupler l'espace de Wiener à un espace de Hilbert séparable du type H " L 2 pT, B, q,o ù est la mesure de Lebesgue sur T. Comme 23 expliqué dans les ouvrages de référence de David Nualart [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF] ou Svante Janson [START_REF] Janson | Gaussian Hilbert Spaces[END_REF], il existe alors une mesure Gaussienne tWpAq, A P B, pAq † 8u indépendante sur toute famille de sous-ensembles disjoints de T, telle que la variance de WpAq est égale à pAq et que les trajectoires du mouvement brownien peuvent être reconstruites via la formule de représentation de Centsov :

B t " Wpr0,tsq " W `1r0,ts ˘; t P T.

E Ä L 2 pE ˚,µq Ä E ˚.
L'espace L 2 pE ˚,µq est en fait canoniquement isomorphe à l'espace de Fock sur L 2 pRq à travers l'isomorphisme de Wiener-Itô-Sega. Ainsi toute fonctionnelle de Wiener de carré intégrable admet une unique décomposition en termes d'intégrales de Wiener, de sorte à ce que cette décomposition chaotique caractérise cette approche alternative. Dans le contexte général des espaces de Fock, D et coïncident respectivement avec l'opérateur d'annihilation et l'opérateur de création que l'on rencontre en théorie des probabilités quantique et satisfont une relation canonique de commutation généralisée. L'équivalence de ces deux approches repose sur la relation intrinsèque entre le mouvement Brownien standard et les polynômes de Hermite.

Paul-André Meyer met en lumière dans [START_REF] Meyer | Transformations de riesz pour les lois gaussiennes[END_REF] l'équivalence des normes associées aux différents espaces de Sobolev ainsi définis.

Si le formalisme de Malliavin était circonscrit à l'espace Wiener à ses débuts il fut par la suite étendu à d'autres types de processus ; pour n'en citer que quelques uns, les processus Gaussiens en général (voir Nualart [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF], Nourdin et Peccati [START_REF] Nourdin | Normal Approximations with Malliavin Calculus: From Stein's Method to Universality[END_REF]), les processus de Poisson (voir Bichteler et al. [START_REF] Bichteler | Malliavin calculus for processes with jumps[END_REF] pour un point de vue variationnel, Nualart and Vives [START_REF] Nualart | Anticipative calculus for the Poisson process based on the Fock space[END_REF] ou Privault [START_REF] Privault | Chaotic and variational calculus in discrete and continuous time for the Poisson process[END_REF] dans le cadre d'une approche chaotique), les processus de Lévy (voir Nualart and Schoutens [START_REF] Nualart | Chaotic and predictable representations for lévy processes[END_REF]) et les processus de Rademacher (voir Privault [START_REF] Privault | Stochastic Analysis in Discrete and Continuous Settings[END_REF]). V T p q"ΦpS T q.

À noter que dans les modèles à temps discret l'hypothèse supplémentaire, V t p q • 0, ; @t P N T , P ´a.s., est requise. Dans les marchés complets où toutes les options sont réplicables, et dans lesquels l'espace probabilisé sous-jacent est muni d'une structure de Malliavin, le processus s'exprime élégamment à l'aide de la dérivée de Malliavin ; cela fut suggéré dans les deux travaux indépendants et contemporains (tous deux publiés en 1991) de Ioannis Karatzas et Daniel Ocone [START_REF] Karatzas | An extension of Clark's formula[END_REF] d'une part et avec Jinlu Li [START_REF] Ocone | A generalized clark representation formula, with application to optimal portfolios[END_REF] d'autre part. La formule de couverture de Karatzas-Ocone hedging est en effet dérivée de celle de Clark-Ocone. Des formules analogues furent établies pour les processus de Poisson par Günter Last et Matthew Penrose [START_REF] Last | Martingale representation for poisson processes with applications to minimal variance hedging[END_REF] et pour les processus de Lévy par Giulia Di Nunno [START_REF] Nunno | Stochastic integral representations, stochastic derivatives and minimal variance hedging[END_REF]. Le délit d'initié est un sujet intrinsèquement lié à l'optimisation de portefeuille. Le cadre est celui d'un marché financier où deux agents ayant des niveaux différents d'information coexistent : un agent ordinaire dont les décisions sont prises à la lumière des informations publiques et un initié bénéficiant d'un surplus d'information (confidentielle) dès le début de la période d'échanges. Deux questions se posent naturellement : comment quantifier l'utilité supplémentaire espérée par l'initié ? L'information additionnelle dont il bénéficie produit-elle un arbitrage ? Ce surplus d'information peut se traduire mathématiquement par un élargissement de la filtration sur lequel l'initié fonde ses décisions de gestion de portefeuille et conduit donc alors à la théorie de grossissement de filtrations. Suite aux travaux pionniers d'Igor Pikovsky et de Ioannis Karatzas [START_REF] Pikovsky | Anticipative portfolio optimization[END_REF] dans le domaine, Jürgen Amendinger et al. [START_REF] Amendinger | Additional logarithmic utility of an insider[END_REF], [START_REF] Amendinger | A monetary value for initial information in portfolio optimization[END_REF], [START_REF] Amendinger | Martingale representation theorems for initially enlarged filtrations[END_REF], Axel Grorud et Monique Pontier [START_REF] Grorud | Insider trading in a continuous time market model[END_REF] précisèrent les critères pour l'optimisation de portefeuille et calculèrent l'utilité additionnelle de l'initié dans leur travaux respectifs. Peter Imkeller connecta cette utilité au calcul de Malliavin dans [START_REF] Imkeller | Malliavin's calculus in insider models: Additional utility and free lunches[END_REF] en exprimant le drift d'information comme la trace de Malliavin logarithmique de densités conditionelles caractérisant le bénéfice de l'initié. Jorge A. León, Reyla Navarro et David Nualart utilisèrent dans [START_REF] León | An anticipating calculus approach to the utility maximization of an insider[END_REF] certaines techniques du calcul de Malliavin pour analyser les propriétés de l'intégrale anticipative (introduite par Russo et Vallois dans [START_REF] Russo | Forward, backward and symmetric stochastic integration[END_REF]) et pour maximiser l'utilité logarithmique espérée de l'initié. Dans certaines extensions du modèle de Black-Scholes où la volatilité peut être un processus stochastique (voir par exemple Hull et White [START_REF] Hull | The pricing of options on assets with stochastic volatilities[END_REF], ou Heston [START_REF] Heston | A closed-form solution for options with stochastic volatility with applications to bond and currency options[END_REF]), en particulier dans certains modèles de diffusion à volatilité stochastique, où la volatilité suit également un processus de diffusion, on peut observer que le volatilité implicite (mesure prospective pour estimer les fluctuations futures de l'actif sous-jacent) se comporte rugueusement à toute échelle de temps raisonnable. La boîte à outils de Malliavin, dont l'efficacité en matière de calcul anticipatif a déjà été mentionnée, est alors également efficace pour analyser les volatilités futures, qui sont des processus non adaptés. Dans un travail précurseur sur les volatilités rugueuses [START_REF] Alòs | On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility[END_REF], Elisa Alòs, Jorge León et Josep Vives, donnent une expression pour le comportement à court terme de la volatilité implicite dans un modèle de diffusion à sauts en fonction de sa dérivée de Malliavin.

La méthode de Stein

Dans son article fondateur ( [START_REF] Stein | A bound for the error in the normal approximation to the distribution of a sum of dependent random variables[END_REF]), Charles Stein décrivit une procédure visant à quantifier les erreurs commises dans l'approximation Normale par des sommes de variables aléatoires ayant une structure de dépendance stationnaire. Par la même, il posa les bases d'une méthode très puissante dont les perspectives d'application dépassèrent rapidement le cadre de sa naissance ; la méthode de Stein est devenue l'une des (pour ne pas dire la) façons les plus efficaces de calculer la distance entre deux mesures de probabilité s'écrivant sous la forme : [START_REF] Peköz | Generalized gamma approximation with rates for urns, walks and trees[END_REF]), Exponentiel (Chatterjee, Fulman et Röllin [START_REF] Chatterjee | Exponential approximation by stein's method and spectral graph theory[END_REF], Peköz et Röllin [START_REF] Peköz | New rates for exponential approximation and the theorems of rényi and yaglom[END_REF]), Géométrique (Peköz, Röllin et Ross [START_REF] Peköz | Total variation error bounds for geometric approximation[END_REF]) ... En outre, dans la section Taxonomie de la page web Malliavin-Stein approach qu'il lui a consacré, Yvik Swan passe en revue toutes les distributions dont l'approximation peut être quantifiée par cette méthode. De nombreuses études ont été publiées pour enseigner au prosélyte et initier le profane à l'efficacité du concept. S'il reste impossible d'en donner une liste exhaustive, nous recommandons les ouvrages de référence et particulièrement pédagogiques de Benjamin Arras et Yvik Swan, [START_REF] Arras | A stroll along the gamma[END_REF], Andrew D. Barbour [14] ou Nathan Ross [START_REF] Ross | Fundamentals of Stein's method[END_REF]. La méthode de Stein semble se déployer en deux étapes ; la première consiste à convertir le problème initial (difficile) de borne de l'erreur dans l'approximation de la mesure P ‹ par Q en celui du contôle d'une expression de la forme : [START_REF] Barbour | An introduction to Stein's method[END_REF]) consiste à utiliser des paires échangeables. Des solutions cousines, appelées size-biased (coupling) (voir Chen, Goldstein et Shao [START_REF] Chen | Normal approximation by Stein's method[END_REF]) ou zero-biased (coupling) (voir Chen, Goldstein et Shao [START_REF] Chen | Normal approximation by Stein's method[END_REF]) basées également sur des couplages tirent bénéfice des propriétés de la mesure initiale (voir Goldstein et Reinert [START_REF] Goldstein | Stein's method and the zero bias transformation with application to simple random sampling[END_REF]), pour, là encore, transformer commodément L 1 '. Dans un travail inédit, Ivan Nourdin et Giovanni Peccati (voir [START_REF] Nourdin | Stein's method on wiener chaos[END_REF], [START_REF] Nourdin | Normal Approximations with Malliavin Calculus: From Stein's Method to Universality[END_REF] ont montré que l'étape de transformation pouvait être avantageusement simplifiée en utilisant l'intégration par parties au sens de Malliavin ; ils ont donné par la même occasion une intersection aux deux théories. Cette approche est efficace sous couvert de l'existence d'un gradient de Malliavin sur l'espace sur lequel X est défini. Elle a notamment été appliquée aux fonctionnelles de Rademacher (voir Nourdin, Peccati and Reinert [START_REF] Nourdin | Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos[END_REF], Zheng [START_REF] Zheng | Normal approximation and almost sure central limit theorem for nonsymmetric rademacher functionals[END_REF], Poisson (voir par exemple Decreusefond, Schulte et Thäle [START_REF] Decreusefond | Functional Poisson approximation in Kantorovich-Rubinstein distance with applications to U-statistics and stochastic geometry[END_REF], Lachièze-Rey et Peccati [START_REF] Lachièze-Rey | Fine Gaussian fluctuations on the Poisson space II: rescaled kernels, marked processes and geometric U-statistics[END_REF], Peccati, Solé, Taqqu et Utzet [START_REF] Peccati | Stein's method and Normal approximation of Poisson functionals[END_REF]) ou Gaussienne variables aléatoires (voir Nualart et Peccati [START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF]) ou processus (voir Coutin et Decreusefond [START_REF] Coutin | Stein's method for Brownian approximations[END_REF][START_REF] Coutin | Higher order expansions via Stein's method[END_REF]). Une liste des travaux se rapportant à l' approche de Stein-Malliavin est disponible sur la page web entretenue par Ivan Nourdin.

dist H pP ‹ ,
E Q rL'pXqs " E rL 1 '

Les formes de Dirichlet

En parallèle de sa connexion avec la méthode de Stein, le langage du calcul Malliavin se mêle avantageusement à celui des formes de Dirichlet le cadre du calcul d'erreur (voir Bouleau et Hirsch [START_REF] Bouleau | Dirichlet forms and analysis on Wiener space[END_REF]). Les sujets connexes ont généré un intérêt croissant (au sein de la communauté des analystes avant que les probabilistes ne s'en emparent) depuis le début du XXe siècle. L'utilisation des formes Dirichlet s'avère particulièrement efficace pour effectuer des calculs d'erreurs lorsque celles-ci sont supposées être infinitésimales et probabilistes. Si Y représente une quantité scalaire erronée, la propagation de l'erreur par la fonction f est caractérisée par les formules : " biais de l 1 erreur sur f pYq"p biais de l 1 erreur sur Yq f 1 pYq`1 2 pvar de l 1 erreur sur Yq f 2 pYqp E 1 q var de l 1 erreur sur f pYq"p var de l 1 erreur sur Yq f 12 pYqp E 2 q , qui deviennent, si la variance est de même ordre de magnitude que le biais ou si le biais est négligeable par rapport à la variance :

• pour le biais de l'erreur : une équation différentielle du second ordre avec biais et variance (E 1 ), • pour la variance de l'erreur : une équation différentielle du premier ordre pour la variance n'incluant pour de terme de biais (E 2 ).

Considérant [START_REF] Decreusefond | Malliavin and dirichlet structures for independent random variables[END_REF]). La deuxième sera soumise prochainement. Les deux parties se déploient selon le même schéma ; elles commencent par un chapitre où est présenté l'état de l'art et où l'on identifie le formalisme à développer pour répondre à une problématique donnée. Les outils nécessaires sont construits dans les chapitres qui suivent immédiatement alors que les fruits sont récoltés dans le dernier chapitre de chaque partie, comme la réponse apportée au problème initial.

F " FpX A q (A Ä N ˚)
Avant de clore cette introduction, je signale que toute nouvelle contribution (définition ou résultat original) sera signalée dans le manuscrit par le symbole p‹q. Malgré la piètre qualité de l'anglais dont souffre la rédaction (et pour laquelle je présent par avance mes excuses aux lecteurs), j'ai beaucoup apprécié la préparation de ce manuscrit de thèse et vous en souhaite une agréable lecture ! Part I

Malliavin calculus and Dirichlet structures for independent random variables

Chapter 1

From Stein's to Stein-Dirichlet-Malliavin method

Stein's method, initially developed to quantify the rate of convergence in the Central Limit Theorem [START_REF] Stein | A bound for the error in the normal approximation to the distribution of a sum of dependent random variables[END_REF] and then for Poisson convergence [START_REF] Chen | On the convergence of Poisson binomial to Poisson distributions[END_REF], has become a very popular not to say the most famous procedure to assess distances between two probability measures. Associated to a collection of probability metrics, it seems to deploy by two main stages: the first one consists in converting the initial problem into a more tractable expression before developping in the second one tools to derive from it the bound on the distance.

As announced by the title of the chapter, we are to start from Stein's method to get to the Stein-Dirichlet-Malliavin structure. The chapter is organized as follows; the first leg of our journey, by stating the principle of Stein's method, will be an opportunity to underscore its two identified main steps. We stop at Dirichlet theory in the second section and explain why the conversion of the initial problem means the existence of an underlying Dirichlet structure on both initial and target spaces. In the third section, we make a detour via Malliavin calculus.

We show then its relevance to tract the second step in Stein's method (fourth section). The last section is devoted to detail one of the abundant results stated by following this path: a Stein-Malliavin criterion on the Gaussian space.

Stein's method principle

This first section is charged with bringing to light the efficiency of Stein's method in the estimation of distances between two probability measures.

Taxonomy on probability metrics

In order to provide a mathematical sense to the notion "of proximity" between two probability measures, we give in this first part some elements of classification of probability metrics. The framework is that of two probability measures P ‹ and Q, respectively the target and the initial measures, defined on a same metric space pF,cq. The associated Borel -field is denoted BpFq.

It appears that we can basically pigeonhole the numerous existing metrics between probability measures in three classes.

The first distance we introduce, which is capital since its associated topology is precisely the one of the convergence in distribution is called Prokhorov distance:

dist Pro pP ‹ , Qq"inft" °0:P ‹ pAq § QpA " q`" and QpAq § P ‹ pA " q`", @A P BpFqu,

where A " stands for the "-neighborhood of A and is defined by A " "ty P F : Dx P A,d px, yq § "u. The set M 1 pFq of probability measures on F is usually equipped with the weak convergence generated by the semi-norms

p f pP ‹ q" ˇˇˇª F f dP ‹ ˇˇˇ,
for any function f bounded and continuous from F to R. The separability of F enables to find a countable family of bounded continuous functions tf n ,nP N ˚u which generates the Borel -field on F and the topology of the weak convergence is metrizable by considering the distance:

⇢pP ‹ , Qq" 8 ÿ n"1 2 ´n pp fn pP ‹ ´Qqq " dist Pro pP ‹ , Qq,
where the function is defined on R by pxq"x{p1 `xq. This definition is not prone to calculations in practice such that the possibility to propose alternatives has been investigated.

The second category of probability metrics are expressed as u´divergences which definition follows:

Definition 1.1.1. Let u be a convex function on R such that up1q"0,a n dP ‹ and Q be two measures defined on the same Polish space F. The u-divergence of Q with respect to P ‹ is defined by:

D u pP ‹ ||Qq" $ & % ª F u ´dP ‹ dQ ¯dQ if P ‹ ! Q , 8 otherwise.
Total variation between absolutely continuous measures is obtained when u : x fi Ñ| x ´1|, whereas by letting u : x fi Ñ x logpxq we get the Kullback-Leibler distance which is of key importance in information theory. Besides, we will meet once more the latter distance when estimating the additional utility of an insider who holds extra... information (see part II).

The last class we deal with is related to the optimal transportation theory. The Monge-Kantorovitch problem between P and Q translates into the determination of

inf PΓpP ‹ ,Qq ª FˆF cpx, yq d px, yq, (1.1.1)
where ΓpP ‹ , Qq denotes the space of probability measures on F ˆF with first marginal P ‹ and second marginal Q,a n dc is a cost function, semi-continuous from F ˆF to R `Yt 8u.

To connect it with the notion of probabilistic metric, we introduce, by letting c " c p , where c is a distance on F and p a positive real number, the p-th Wasserstein distance dist Wp between the measures P ‹ and Q by

dist Wp pP ‹ , Qq" ˆinf PΓpP ‹ ,Qq ª FˆF c p px, yq d px, yq ˙1{p
.

By taking p " 1, the underlying metric c on space F can be viewed through the optimal transportation problem as a cost function, which argument of the minimum is realized by the best "coupling" between the measures P ‹ and Q or, from this point of view, their distance. The compactness of pF,cq ensures, via the duality theorem of Kantorovitch and Rubinstein (1958), or more precisely its extension to separable metric spaces by Dudley (see [START_REF] Dudley | Real analysis and probability[END_REF]) that

dist W 1 pP ‹ , Qq" inf PΓpP ‹ ,Qq ª FˆF cpx, yqd px, yq" sup hPLip 1 " ˇˇª F h dP ‹ ´ªF h dQ ˇˇ* ,
where Lip 1 is the set Lip 1 " h P R F : |hpxq´hpyq| § cpx, yq, @px, yqPF2 ( . That makes the 1-Wasserstein distance the dual representation of (1.1.1). Besides Theorem 11.3.1 of [START_REF] Dudley | Real analysis and probability[END_REF] states that the distances dist W 1 and ⇢ yield the same topology, and legitimizes the choice to retain this definition. In the case where F " R n , for technical reasons, it is often assumed that the test functions are more regular than simply Lipschitz continuous and we are led to compute

d k pP ‹ , Qq" sup hPLip k ˇˇˇª F h dP ‹ ´ªF h dQ ˇˇˇ,
where Lip k is a space included in Lip 1 like the set of k-times differentiable functions with derivatives up to order k bounded by 1.N o t et h a td 1 coincides with the 1-Wasserstein distance.

More broadly, we can define many other probabilistic metrics in the similar fashion as to say by letting

dist H pP ‹ , Qq"sup hPH ˇˇª F h dP ‹ ´ªF h dQ ˇˇ, (1.1.2)
where H is a class of real-valued test functions which is separating, in the sense that ≥ F h dP ‹ " ≥ F h dQ for all h P H if and only if Q " P ‹ . To recite nobody else but them : if the set H "t1 A , A P BpFqu, dist H is the total-variation distance,ifF " R and H "t1 p´8,zs ,zP Ru, dist H coincides with the Kolmogorov distance.

Stein's method principle

Stein's method turns out to be particularly suitable to give upper bounds to distances between probability measures of the form (1.1.2). In particular, when F " R,i ti so n ee fficient way to compute the distance between a real-valued measure and the Gaussian distribution.

In addition to the objects introduced above, we may consider two random variables W and Z respectively distributed by Q and P ‹ . When specified, the measure P ‹ could designate the normal distribution on R. Basically, we can identify two main steps through which Stein's method seems to deploy:

1. The conversion of the difficult initial problem (1.1.2) of the approximation of a target measure P ‹ by another one Q into a simpler one with a more tractable expression.

1.2 The Stein-Dirichlet structure

Conversion of the initial problem

In its very essence, Stein's method carries the idea that the laws of two random variables are close if they verify similar identities. The firts step, which aims at the conversion of the initial problem is based on this idea. It comes in three stages : the characterization of the target distribution, the resolution of the so-called Stein's equation, and the reduction of the expression (1.1.2). By identities we mean functional identities characterizing the target laws. For instance it is well-known that the random variable Z is a standard Gaussian on R if and only if E rZhpZqs " E rh 1 pZqs, for any function h of a class of real-valued test functions H. By this, we introduce the operator L, the so-called Stein's operator which acts on H.F o r t h e s t a n d a r d N o r m a l distribution, Stein's lemma yields such an operator:

Lhpxq"Lhpxq"xhpxq´h 1 pxq ; @x P R.

In order to avoid any future confusion, when it designates the operator associated to the standard Gaussian law on R, the Stein operator will be noted L. In other cases, we keep the notation L.

The underlying idea is the following: if the laws of the variables Y and Z are "near" each other, the quantity E rhpYqs ´E rhpZqs may be small for a wide range of functions h. Then, if Z " N p0, 1q, we can expect the quantity

E " YhpYq´h 1 pYq ‰ " E rLhpYqs
to be small. We come then to solve the Stein equation. i.e. to rule on the existence and the regularity of the functions ' satisfying :

L'pyq"hpyq´E rhpZqs ; @y P F.

(1.2.1)

The class of test functions solutions of (1.2.1) satisfying h P H ñ ' P T , is called the Stein class. For instance, the Stein class associated to the Normal distribution and for the Wasserstein distance, denoted by T W , is the class of twice differentiable functions, whose first derivative is bounded by 1 and whose second derivative is bounded by 2.

By integrating then equation (1.2.1) with respect to Q and taking the supremum over the class T we get:

sup hPH ˇˇˇª F h dP ‹ ´ªF h dQ ˇˇˇ" sup 'PT ˇˇE rL'pYqs ˇˇwhere Y " Q. (1.2.2)
Thus, the initial problem of approximation of a target measure by an initial one has been converted into a simpler expression: that of the expectation of a certain functional on the initial space. This is in fact easier to handle ; various identities and couplings can be applied advantageously.

The Dirichlet structure point of view

Actually, we can address the problem from a distinct point of view. The computation of distpP ‹ , Qq can be interpreted as the measure of the error made by approximating P ‹ by Q.

It turns out that the difficult steps of the functional characterization of P ‹ and the resolution of the Stein equation can be got around by following a smart-paths approach. In fact, the target measure P ‹ can be viewed as the stationary ditribution of a Markovian semi-group pP t q tPR `and of generator L.

Language elements of Dirichlet structures

We give some elements of terminology of Dirichlet structures by introducing some basic notions on Markov semigroups. A substantiated presentation on the subject can be find in the books of Bouleau and Hirsch [START_REF] Bouleau | Dirichlet forms and analysis on Wiener space[END_REF] or Fukushima, Oshima and Takeda [START_REF] Fukushima | Dirichlet Forms and Symmetric Markov Processes[END_REF]. In the first part of this subsection, we successively provide the definitions of Markov semi-group, Markov process and infinitesimal generator before focusing on the fundamental relations that connect them to each other. Consider a measurable space pE, Aq. Let pP t q tPR `be the family of operators defined on some set of real-valued measurable functions on pE, Aq and satisfying the following conditions:

1. For any t P R `, P t is a linear operator; it sends bounded measurable functions on pE, Aq to bounded measurable real functions.

2. P 0 " Id where Id is the identity operator (initial condition).

3. For every s, t P R `Pt`s " P t ˝Ps (semi-group property).

4. For any t P R `, P t conserves the mass and preserves positivity (Markov property): P t p1q"1 and, for any positive function f , P t f is positive.

Definition 1.2.1 (Invariant measure). Let a familly pP t q tPR `of operators defined on pE, Aq and satisfying the properties (1)-( 4). A positive -finite measure µ on pE, Aq is invariant for pP t q tPR `, if for every bounded positive measurable function F:EÑ R and t P R `,

ª E P t Fdµ " ª E Fdµ.
Definition 1.2.2 (Markov semi-group). A family pP t q tPR `of operators defined on the set of bounded measurable functions on pE, Aq with invariant positive -finite measure µ satisfying the properties ( 1)-( 4) as well as the continuity property:

5. For any F P L 2 pµq"L 2 pE,µq, P t F converges to F in L 2 pEq as t tends to 0 (continuity property).

The semi-group is symmetric with respect to the invariant measure µ, if for all F, G elements of E and t P R `, ª

E F pP t Gq dµ " ª E pP t Fq Gdµ.
Remark 1.2.3. Thus defined, the Markov semi-group pP t q tPR `is a semi-group of contractions on L 2 pΩq (for more details on the subject refer to the Hille-Yosida theory) and satisfies the property of contraction and strong continuity:

}P t F} L 2 pµq § }F} L 2 pµq (contraction) and lim tÑ0 P t F " F (strong continuity).
Up to now, consider a probability space pΩ, F, P ‹ q. Let E " L 2 pΩ, P ‹ q and µ " P ‹ .

Definition 1.2.4 (Markov process). Consider a measurable process pX x t q tPR `on pΩ, F, P ‹ q starting from x P E at time t " 0. Denote by pF X t q tPR `the filtration generated by the process pX x t q tPR `. The process pX x t q tPR `is a Markov process if it sastifies the Markov property: for any ps, tqPpR `q2 such that t °s, the lew of X x t given F X s is the same as the law of X x t given X x s as well as the law of X x t´s given X x 0 .

In view of remark 1.2.3, the Markov semi-group pP t q tPR `can be seen as a contraction semigoup; so that its infinitesimal generator can be introduced in the setting of Hille-Yosida theory.

Definition 1.2.5 (Infinitesimal generator). The generator of a contraction semi-group pP t q tPR òn L 2 pΩq is defined by

D L " " F:l i m tÑ0 P t F ´F t exists * and LF " lim tÑ0 P t F ´F t ; @F P D L ,
and such that D L is dense in L 2 pΩq.

Remark 1.2.6. There exists a correspondence between the Markov semi-group pP t q tPR `(definition 1.2.2), the Markov process pX x t q tPR `(definition 1.2.4) and the infinitesimal generator (definition 1.2.5) in the sense that the knowledge of one of pP t q tPR `, pX x t q tPR `or L implies the existence and gives the expression of the two others.

PtF"ErFpXtq|X 0"¨s

›› › › › › › › › › › › ›Ñ LF" dP t F dt ˇˇt"0 ››››››››Ñ pX ¨tq tPR `pP t q tPR `L -››››››››››››› X t P"pPtq ˚⌫ -› › › › › › › › ›› PtF"e tL
In fact, by duality the semi-goup pP t q tPR `acts on the set of measures ⌫ on A via the formula

ª E P t Fd⌫ " ª E FdppP t q ˚⌫q,
so that if ⌫ denotes the law of X 0 , pP t q ˚⌫ is the law of the variable X t .

In order to introduce the carré du champ operator as well as the Dirichlet forms, assume the existence of a vector subspace K of D L such that for any pF, GqPK ˆK, the product FG is an element of D L .

Definition 1.2.7 (Carré du champ operator). The bilinear map defined on

D Γ " K ˆK by ΓpF, Gq" 1 2 ´LpFGq´G LF ´F LG ¯,
is the carré du champ operator associated to the Markov generator L.

We can now transpose all this framework into the Dirichlet forms language. Consider a Markov semi-group pP t q tPR `, symmetric with respect to a finite -measure µ " P ‹ , with inifinitesimal generator L and carré du champ operator Γ.

ª E F p´LGq dµ " ª E p´LFq Gdµ. (1.2.4)
and the relations connecting the Markov semi-group pP t q tPR `and its generator L (see remark 1.2.6). Nevertheless, the domain of the Dirichlet form E is bigger than D Γ . The identity EpFq"EpF, Fq ensures that E can be defined on a set

D E such that D Γ Ä D L Ä D E Ä L 2 pΩq"E.
Within this definition, we can verify that E thus defined is a non-negative definite symmetric form on L 2 pΩq which domain D E is dense in L 2 pΩq. It is moreover closed in the sense that D E equipped with the norm

}F} D E " ´}F} 2 L 2 pΩq `EpFq ¯1 2 ,
is a Hilbert space.

Theorem 1.2.10. The generator of a closed form E is defined by D L "tF P D E : DG P L 2 pΩq, @H P D E , EpF, Hq"´xG, Hy L 2 pΩq u and LF " G.

Moreover, if L is a negative self-adjoint operator and

D E " D ? ´L,t h e n EpFq"} ? ´LF} 2 L 2 pΩq ; @F P D E . (1.2.5) Remark 1.2.11.
There is a one-to-one correspondence between the family of closed symmetric forms on L 2 pΩq and the family of non-positive definite self-adjoint operators on L 2 pEq and which is determined through Theorem 1.2.10.

Definition 1.2.12 (Dirichlet form and Dirichlet structure). A Dirichlet form on pΩ, F, Pq is a symmetric closed form E, Markovian in the following sense :

F P D E ñ F ^1 P D E and EpF ^1, F ^1q § EpF, Fq.
The quintuplet pΩ, F, P, D E , Eq is a Dirichlet structure.

Example 1.2.13. The quintuplet pR, BpRq, P, D E , Eq where P denotes the standard Gaussian measure on R, D E "tu P L 2 pΩq : u 1 w P L 2 pΩqu and EpFq"E

" |F 1 | 2 ‰ ,
(u 1 w stands for the derivative taken in the sense of distributions) defines the 1-dimensional Ornstein-Uhlenbeck structure. Definition 1.2.14 (Carré du champ operator). Let pP t q tPR `a symmetric semi-group of generator L and E a Dirichlet form defined as in (1.2.5). The carré du champ operator Γ associated to E is defined on

D E ˆDE by ΓpF, Gq" 1 2 `LpFGq´G LF ´F LG ˘.
The quintuplet pΩ, F, P, D E , Γq is an error structure.

Example 1.2.15. Within the notations of example (1.2.13), ΓpF, Gq"F 1 G 1 .

Dirichlet structures: a solution to the initial problem

We can now address the initial problem in terms of Dirichlet structures. As evokated above, there exists a strong ergodic Markov process pX x t q tPR `of invariant measure P ‹ and generator L. These ones are connected by the relations

L'pxq" dP t 'pxq dt ˇˇt "0 and P t 'pxq"E r'pX x t q|X x 0 " xs . (1.2.6)
Example 1.2.16 (Gaussian measure on R). Let µ denote the standard Gaussian on E " R, then X "pX t q tPR `is the Ornstein-Uhlenbeck process defined by dX t "´X t `?2dB t and X 0 " x, where pB t q tPR `is a standard one-dimensional Brownian motion. It can be shown, via the Mehler representation formula, that the semi-group pP t q tPR `is defined by

P t 'pxq" ª R ' ´e´t x `a1 ´e´2t y ¯dµpyq. (1.2.7)
Its generator, the operator L, satifies the reminding relation: for ' P C 2 b pR; Rq, L'pxq"x' 1 pxq´' 2 pxq ; @x P R.

In the general case, using (1.2.6) and noting that P 8 ' " E P ‹ r's, P 0 ' " ',a n dt a k i n gt h e expectation with respect to Q of the two members of the equality

P 8 'p!q´P 0 'p!q" ª 8 0 dP t dt 'p!q dt,
we can state the so-called Stein representation formula or smart-path formula

E P ‹ r's´E Q r's" ª F ª 8 0 LpP t 'q dt dQ (1.2.8)
It turns out that the reduction of the problem in the fashion of (1.2.2) which implies to characterize the target measure through a functional identity and to solve the Stein equation can be done here without all this work. In fact we are to bound

E Q "ª 8 0 LpP t 'q dt ⇢ . (1.2.9)
In other words, this "smart-path"-like method gives another interpretation to the computation of distances between measures via Stein's method: that of the estimation of the distance between the underlying ergodic Markov process to its stationary measure. This will be illustrated in Subsection 1.3.2 through the example of the Normal approximation by a Poisson distribution (in dimension 1).

Remark 1.2.17. By reminiscence of Dirichlet forms vocabulary, we can interperate P 8 ' Ṕ0 ' as the error made in the approximation of the measure P ‹ by Q. In this same setting (see Bouleau and Hirsch [START_REF] Bouleau | Dirichlet forms and analysis on Wiener space[END_REF]) the carré du champ operator Γ (resp. the symmetric operator L) represents the variance (resp. the bias) of the error. Thus, some convergence results proved by Stein's method can be interpreted in the language of error calculus. As the matter of fact, one of the exisiting proofs of the famous Fourth Moment Theorem stated by David Nualart and Giovanni Peccati (see [START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF]) lies on the existence of a deterministic constant c such that

varpΓpFqq § c `E " F 4 ‰ ´3E " F 2 ‰ 2 ˘,
where Γ is the carré du champ operator associated with the generator of the Ornstein-Uhlenbeck semi-group and F a random variable in a Wiener chaos. The use of the Stein-Malliavin method (see Nourdin and Peccati [START_REF] Nourdin | Stein's method on wiener chaos[END_REF]) yields another proof ot the theorem and provides a estimation in the total variation distance w.r.t. the Normal distribution P ‹ dist TV pF,

P ‹ q § 2 ? 3 a E rF 4 s´3.
The reader can refer to the work of Louis H.Y. Chen and Guillaume Poly (see [START_REF] Chen | Stein's method, malliavin calculus, dirichlet forms and the fourth moment theorem[END_REF]) for a statement of the links between Stein's method, Dirichlet forms and Malliavin's calculus through the Fourth Moment theorem.

Dirichlet-Malliavin structures

Some words about Malliavin calculus

As recalled in the introduction, the Malliavin calculus is a finite-dimensional stochastic variational calculus initially developped on the Wiener space. One way to introduce the tools of Malliavin calculus is to transpose the terminology of classical differential calculus in Banach spaces to Malliavin's one. We choose to give the analogues for three processes families for which the theories are the most complete: the standard Brownian motion pB t q tPr0,1s , the standard Poisson process pN t q tPR `and a Rademacher process pX n q nPN ˚.

In any case, we assume the existence of a probability space pΩ, A, Pq on which each process is defined. We denote the Hilbert space H " L 2 pTq where T is the parameter space: r0, 1s for the Brownian motion B, R `for the Poisson point process N and N ˚for the Rademacher process X. . . .

Wiener
x n ‹ ' t fi Ñ B t p!q, t fi Ñ N t p!q, n fi Ñ X n p!q Functions defined on R n
Random variables = Functionals of the paths Gradient Malliavin derivative We briefly present the main facts about Malliavin calculus on the Wiener space before giving some complementary elements in the more general case of Gaussian spaces. A more detailed discussion can be found in Nualart [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF], or Üstünel and Zakai [START_REF] Üstünel | Transformations of measure on Wiener spaces[END_REF]. We hold the construction of Malliavin calculus for the jump processes in the second part of the manuscript as a motivation for our second work.

rf " ¨Bf Bx 1 . . . Bf Bx n ‹ ‹ ‹ ‹ ' r h F " n ÿ i"1 Bf Bx i pXph 1 q,...,Xph n qq h i D t FpNq"FpN ` t q´FpNq D k FpX 1 , ¨¨¨, X n q" ? p k q k FpX 1 ,..,

The historical Malliavin calculus: the Wiener space case

As mentioned in the introduction, two approaches were developped to lead to a differential calculus on the Wiener space: a variationnal approach and a chaotic approach. In the first one, the derivative is defined for a class of smooth variables and then extended to the space of square integrable Wiener functionals by density. This can be interpreted as a directional derivative but only in certain directions: that of Cameron-Martin's space. In an alternative construction, the Malliavin derivative is obtained as an annihilation operator acting on square integrable random variables, for which a preliminary chaotic decomposition has been stated. The operators defined through these two different approaches coincide in the Gaussian space; this equivalence is due in particular to the intrinsec relation between the standard Brownian motion and the Hermite polynomials. Fp! `"! 1 q´Fp!q " .

Consider the stochastic differential equation defined for any t Pr0,Ts by

X t " x `ª t 0 bpX s qds `ª t 0 pX s qdB s , (1.3.3)
where b, P C 0 pRq,a n dx P R. The application I:R ˆC0 pR 2 , RqˆΩ Ñ C 0 pr0,Ts, Rq which associates px, pb, q, !q with -if it exists -the unique solution X of (1.3.3) is called the Itô map. This is not continuous on W B and a fortiori not Fréchet differentiable. The Cameron-Martin theorem which states that for two random variables F, G:Ω Ñ R F " G a.s. ùñ FpB `hq"GpB `hq ; @h P H 1 , is prone to explore the possibility to narrow the set of admissible differentiation directions to a smaller one, the Cameron-Martin's space H 1 .L e tS denote the class of smooth random variables F of the form

F " f pWph 1 q, ¨¨¨, Wph n qq
where f belongs to the Schwartz space S pT n q and h 1 , ¨¨¨,h n are elements of H 1 . The derivative of F is the element of H 1 defined by

rF " n ÿ k"1 Bf Bx k pWph 1 q, ¨¨¨, Wph n qq h k , (1.3.4)
which can be interpreted as a directional derivative:

xrF,hy H 1 " lim "Ñ0 f `Wph 1 q`"xh 1 ,hy H 1 , ¨¨¨, Wph n q`"xh n ,hy H 1 ˘´f pWph 1 q, ¨¨¨, Wph n qq "
The map r is closable from L 2 pW B q to L 2 pW B ; H 1 q"L 2 pW B ˆTq. Thus, it is meaningful to define D 1,2 B as the closure of cylindrical functions for the norm

}F} 1,2 " ´}F} 2 L 2 pW B q `}rF} 2 L 2 pW B ; H 1 q ¯1 2 .
The derivative operator satisfies the useful chain rule.

Theorem 1.3.1 (Chain rule). Let ' Pp R n q,a n dar a n d o mv e c t o rF "p F 1 , ¨¨¨, F n q which components belong to D 1,q for some q P N ˚.T h e n'pFq belongs to D 1,q and rp'Fq"

n ÿ i"1 B' Bx i DF i . (1.3.5)
In the same way, it is possible to define for p • 1, F P S,t h ep-th Malliavin derivative of F as the element of L 2 pW B ; pH 1 q dp q (pH 1 q dp is the p-th symmetric tensor product of H 1 ) defined by

r p F " ÿ pi 1 ,¨¨¨,ipqPrns p B p f Bx i 1 ¨¨¨Bx ip `Wph 1 q, ¨¨¨, Wph n q ˘hi 1 b¨¨¨bh ip ,
where rns"t1, ¨¨¨,nu.F o ra n yq Pr1, 8q and p P N ˚, the set D p,q B denotes the closure of S with respect to the norm }F} p,q " ´E r|F| q s `E " }rF} L q pHq ‰ `¨¨¨`E " }r p F} q L q pH bp q ı¯1 q Theorem 1.3.2 (Integration by parts formula). For any cylindrical random variable F and

h element of H 1 , E rxrF,hy H 1 s"E rFWphqs (1.3.6)
The second operator in stake called divergence operator is defined as the adjoint of the derivative operator.

Definition 1.3.3. The adjoint of the derivative operator r, denoted is the unbounded operator on L 2 pW B ; H 1 q which domain is the set of H 1 -valued square integrable random variables u such that |E rxrF,uy

H 1 s| § c u }F} L 2 pW B q ,
for all F element of D The so-called Wiener chaos are designed in the following fashion; H 0 is the set of constants and for any n • 1,t h en-th Wiener chaos, denoted H n linear subspace of L 2 pW B , A B , Pq generated by the random variables tH n pWphqq ,h P H 1 u such that the remarkable chaotic decomposition can be stated:

L 2 pW B , A B , Pq" 8 à n"0 H n
Note in particular that H 1 "t Wphq,hP H 1 u"W. Consequence of this decomposition, any random variable F P L 2 pW B q, admits an unique expansion of the form

F " E rFs`ÿ nPN J n pf n q, (1.3.8)
where

f n P L 2 pT n , B n ,µ bn q (equal to pn!q ´1E " r pnq F ‰ if F P D n,2 )andthemultiplestochastic integral of a function f P L 2 pT n , B n ,µ bn q of the form f " ÿ pi 1 ,¨¨¨,inqPrms n a i 1 ¨¨¨in 1 A i 1 ˆ¨¨¨ˆA in pt 1 , ¨¨¨,t n q,
with A 1 , A 2 , ¨¨¨, A n pairwise-disjoint sets belonging to B 0 , and the coefficients a i 1 ¨¨¨in are zero if any two of the indices i 1 , ¨¨¨,i n are equal, is defined by

J n pf q" ÿ pi 1 ,¨¨¨,inqPrms n a i 1 ¨¨¨im WpA i 1 q¨¨¨WpA in q
The derivative operator r coincides on S with the annihilation operator (in reference to its usual name in the Fock space theory met in quantum probability) r r such that r r t J n pf n q"nJ n´1 pf n p‹,tqq, for any f n element of L 2 pr0, 1sq ˝n and t Pr0, 1s. Its adjoint ˜ , called creation operator is such that H n " ˜ n p1q.

Remark 1.3.5. The very close connection between the standard Brownian motion and the Hermite polynomials is at the heart of the equivalence of variational and chaotic approaches. Indeed, the Wiener chaos are intrinsically produced by Hermite polynomials; on the other hand, the first Hermite polynomial subtly appears in the Cameron-Martin theorem

E rFpB `thqs " E " Fe tWphq´1 2 t 2 }h} 2 H 1 ı " E " Fe tH 1 pWphqq´1 2 t 2 }h} 2 H 1 ı ,
and gives rise to the integration by parts formula.

The Malliavin calculus on infinite-dimensional Gaussian fields

A Malliavin calculus can be constructed in the same fashion on infinite-dimensional Gaussian fields i.e. for families Y "t Y t ,tP Tu (T is a parameter set ) of real random variables defined on a probability space pΩ, A, Pq such that any linear combination of the variables is a centered Gaussian variable. To encrypt the properties of a given Gaussian field Y, we consider its associated Gaussian subspace H Ä L 2 pΩ, A, Pq generated by the equivalence classes of Y. Then, Kolmogorov's theorem ensures the existence of an isonormal Gaussian process over H, denoted X; that is a centered Gaussian family X "t Xphq,hP Hu defined on pΩ, A, Pq and which covariance structure is encoded by the inner product of H via the identity E rXphq Xpgqs " xg, hy H ,( g, h P H). We set F " pXq and assume that A " F.

In this framework, the space S of cylindrical random variables is composed of the variables F of the form

F " f pXph 1 q, ¨¨¨, Xph n qq,
where f belongs to the Schwartz space S pR n q and h 1 , ¨¨¨,h n are elements of H. The space S is dense in L q pΩq for any q P N ˚.

For any p P N ˚and F element of S,t h ep-th Malliavin derivative of F (with respect to X)i s the element of L 2 pΩ, H dp q (H dp is the p-th symmetric tensor product of H) defined by

D p F " ÿ pi 1 ,¨¨¨,ipqPrns p B p f Bx i 1 ¨¨¨Bx ip `Xph 1 q, ¨¨¨, Xph n q ˘hi 1 b¨¨¨bh ip .
For any q Pr1, 8q and p P N ˚, the set D p,q denotes the closure of S with respect to the norm

}F} p,q " ´E r|F| q s`E " }DF} q H ‰ `¨¨¨`E " }D p F} q H bp ı¯1 q
The analogues of the Wiener chaos in this setting are designed in the following fashion; H 0 is the set of constants and for any n P N ˚,t hen-th Wiener chaos, denoted H n linear subspace of L 2 pΩ, A, Pq generated by the random variables tH n pXphqq ,h P Hu. Thus any random variable F P L 2 pΩq can be expanded in a unique way

F " E rFs`ÿ n•1 Proj Hn pFq,
where Proj Hn pFq designates the projection of F on the n-th chaos. If additionally F P D p,2 for some p P N ˚, this can be equivalently written as (via Stroock's formula) The integration by parts formula is retrieved through

F " E rFs`ÿ n•1 1 n! n ´E" r pnq F ‰ ¯.
E rΓpF, Gqs " E rxDF, DGy H s"E rF pDGqs " ´E rF LGs ; @F, G P D 1,2 .
The "reverse" construction as to say the elaboration of a Malliavin calculus from a given Dirichlet structure pΩ, A, P, D E , Eq is due to Gabriel Mokobodzki (see [START_REF] Mokobodzki | Sur l'algebre contenue dans le domaine etendu d'un generateur infinitesimal[END_REF]).

Example 1.3.6. (Dirichlet structure on the Wiener space) We now consider P as the Wiener measure on W B " C 0 pr0, 1s; Rq. The aim is to construct a Dirichlet structure of the form pW B , A B , P, D E , Eq . As a remainder (see Subsection 1.3.1), for any F P S of the form

F " f pWph 1 q, ¨¨¨, Wph n qq, (1.3.10)
where f belongs to the Schwartz space SpR n q and h 1 , ¨¨¨,h n belong to

H 1 , rF " n ÿ k"1 Bf Bx k pWph 1 q, ¨¨¨, Wph n qq h k ,
and D 1,2 B as the closure of cylindrical functions for the norm

}F} 1,2 "}F} L 2 pW B q `}rF} L 2 pW B ;H 1 q .
The Wiener space is endowed with the Dirichlet structure

S "pW B , A B , P, D E , Eq.
It is constructed as the infinite product ± 8 n"1 S n ;foranyn P N ˚where S n is the product structure ± n i"1 S i "pR n , BpR n q, b n i"1 P i , D n E , E n q and S i is the Dirichlet form pR, BpRq, P i , D E i , E i q with P i the standard Gaussian measure on R,

D E i " " F P S,f P SpR n q : Bf Bx i P L 2 pRq * and E i pFq"E « ˇˇˇB f Bx i ˇˇˇ2 .
Then we can define for F of the form (1.3.10)

Γ n pF, Fq" n ÿ i"1 Γ k pF, Fq" n ÿ j"1 n ÿ k"1 Bf Bx j Bf Bx k xv j ,v k y L 2 pr0
D E " D 1,2 B
and EpF, Gq"E rΓpF, Gqs " E r´LF Gs ;F ,

G P D 1,2 B ,
where Γ is the associated carré du champ operator. Thus, the association of a Malliavin apparatus and a Dirichlet structure provides what we call a Dirichlet-Malliavin structure on the Wiener space described by the quintuplet pD, , L, pP t q tPR `, Eq and which the operator L and the integration by parts are the keystones via the identity

EpF, Gq"E r´LF Gs"E rrFrGs ;F , G P D 1,2 B .
Last, note in this frame, that the semi-group pP t q tPR `satifies the usual commutation property: for any F P D 1,2 B , rP t F " e ´tP t rF.

(1.3.11)

Examples and applications of Dirichlet-Malliavin structures

This close connection of construction is found at the level of the applications which it generates.

In other words, most probability theorems have their analogues in terms of Dirichlet structures.

In particular we can revisit some results initially established using Malliavin calculus tools, from a Dirichlet structures point of view. Pick two examples as an illustration. The first one (example 1.3.3), is due to Paul Malliavin and remains the famous result of his seminal paper. It provides a criteria for the absolute continuity of a random vector defined on a Gaussian space.

The second one illustrates the combination of Stein's method and Dirichlet-Malliavin calculus in the (one-dimensional) example of the Normal approximation by a Poisson distribution.

Absolute continuity of a random vector

Consider a random vector F "pF 1 , ¨¨¨, F N q measurable with respect to an underlying isonormal Gaussian process tWphq,h P Hu.

Theorem 1.3.9 (Malliavin). Assume that F belongs to D 1,1 loc componentwise and satisfies the following conditions:

1.

F i P D 2,p loc for all i " 1, ¨¨¨, N,f o rs o m ep °0, 2. The Malliavin covariance matrix F " `xF i , F j y H ˘1 §i,j §N is invertible a.s.
Then, the law of F is absolutely continuous with respect to the Lebesgue measure.

The analogue of theorem 1.3.9 in terms of Dirichlet structures lies on the notion of admissible elements; by definition an element ⇠ P Ω is admissible if ⇠ ‰ 0 and, for any t P R, p⌧ t⇠ q ˚P,t h e pullback measure of P by the translation application ⌧ ⇠ : ! fi Ñ ! `⇠ is equivalent to P. The element ! is additionally said to be strictly admissible, if ⇠ is admissible and, for almost every ! P Ω, the function t fi Ñ k ⇠ p! `t⇠q, where k ⇠ denotes a strictly positive Borel representative of the density of the measure p⌧ t⇠ q ˚P with respect to P, is locally integrable on R for the Lebesgue measure. It is thus possible to define (see Bouleau and Hirsch section 4.2 chapter II for more details) the Fréchet derivative r ⇠ F for certain functions F defined on Ω and to set, given a sequence p⇠ n q n•0 such that where L 1 and L 2 are two operators acting on F and such that L " L 1 `L2 . For instance, if the target distribution is the Gaussian law on R, L 1 'pxq"L 1 'pxq"x' 1 pxq and L 2 'pxq"L 2 'pxq"´' 2 pxq.

@ P Ω 1 , ÿ n•0 | p⇠ n q| 2 †
In the next step, we have to take into account how Y is defined and transform L 1 'pYq such that it can be written as ´L2 'pYq`remainder. This remainder is what gives the rate of convergence. To make the transformation of L 1 'pYq, several approaches appeared along the years. One of the most popular approach (see for instance [START_REF] Barbour | An introduction to Stein's method[END_REF]) is to use exchangeable pairs: construct a copy Y 1 of Y with good properties which gives another expression of L 1 'pYq, suitable to a comparison with L 2 'pYq. To be more specific, for the proof of the CLT, it is necessary to create an exchangeable pair pS, S 1 q with S " ∞ n i"1 Y i . This is usually done by first, choosing uniformly an index I Pt 1, ¨¨¨,nu and then, replacing Y I with Y 1 an independent copy of X I , so that the couple pS, S 1 " S ´YI `Y1 q is an exchangeable pair. This means that

E " FpS 1 q|I " a;Y b ,b " a ‰ " E rFpSq|Y b ,b " as . (1.3.13)
Actually, it is the right-hand-side of (1.3.13) which gave us some clue on how to proceed when dealing with functionals more general than the sum of random variables. An alternative to exchangeable pairs, is the size-biased [START_REF] Chen | Normal approximation by Stein's method[END_REF] or zero biased [START_REF] Goldstein | Stein's method and the zero bias transformation with application to simple random sampling[END_REF] couplings, which again conveniently transform L 1 'pYq"L 1 'pYq. For Gaussian approximation, it amounts to find a distribution

Y ˚such that E rL 1 'pYqs " E " ' 2 pY ˚q‰ .
Note that for S as above, one can choose S ˚" S 1 .I ft h ed i s t r i b u t i o no fW ˚is absolutely continuous with respect to that of W , with Radon derivative Λ,w eo b t a i n

E rL 1 'pYqs " E " ' 2 pYq ΛpYq ‰ ,
which means that we are reduced to estimate how far Λ is from the constant random variable equal to 1. This kind of identity, where the second order derivative is multiplied by a weight factor, is reminiscent to what can be obtained via integration by parts. Actually, Nourdin and Peccati (see [START_REF] Nourdin | Normal Approximations with Malliavin Calculus: From Stein's Method to Universality[END_REF]) showed that the transformation step can be advantageously made simple using integration by parts in the sense of Malliavin calculus. This works well only if there exists a Malliavin gradient on the space on which W is defined (see for instance [START_REF] Decreusefond | Functional Poisson approximation in Kantorovich-Rubinstein distance with applications to U-statistics and stochastic geometry[END_REF]). That is to say, that up to now, this approach is restricted to functionals of Rademacher [START_REF] Nourdin | Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos[END_REF], Poisson [START_REF] Decreusefond | Functional Poisson approximation in Kantorovich-Rubinstein distance with applications to U-statistics and stochastic geometry[END_REF][START_REF] Peccati | Stein's method and Normal approximation of Poisson functionals[END_REF] or Gaussian random variables [START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF] or processes [START_REF] Coutin | Stein's method for Brownian approximations[END_REF][START_REF] Coutin | Higher order expansions via Stein's method[END_REF]. Then, strangely enough, the first example of applications of the Stein's method which was the CLT, cannot be handled through this approach. On the one hand, exchangeable pairs or size-biased coupling have the main drawback to have to be adapted to each particular version of Y. On the other hand, Malliavin integration by parts are in some sense more automatic but we need to be provided with a Malliavin structure.

The setting in which we need to compute a KR distance is very often the situation in which we have another Polish space E with a probability measure Q and a random variable T with value in F. The objective is then to compare some measure P ‹ on F and T ˚Q the distribution of T, i.e. the push-forward of Q by the application T. This means that we have to compute

sup 'PT ˇˇˇª E ' ˝Td Q ´ªF ' dP ‹ ˇˇˇ. (1.3.14) E " p Z 'p p Z q ı I.P.P. " E " pD p Z q 'p p Z q ı " E rDTpZ q Dp' ˝TqpZ qs " E rp' ˝TqpZ `1q´p' ˝TqpZ qs " ? E " ' ´p Z `1 ? ¯´'p p Z q ⇢ " E " ' 1 p p Z q`O `1 ? ˘ı " E " L 2 'p p Z q ı `remainder.
We conclude that the rate of convergence in (1.3.15) is of order p ? q ´1.

It is possible to retrieve by using the smart-path formula (1.2.8). With the notations of Example 1.2.16, as explained in [START_REF] Decreusefond | The Stein-Dirichlet-Malliavin method[END_REF], we get

dist T p Q, P ‹ q"sup 'PT ˇˇE "ª 8 0 Z pP t 'q 1 pZ q´pP t 'q 2 pZ q dt ⇢ ˇ" ? sup 'PT ˇˇE "ª 8 0 pP t 'q 1 ´p Z `1 ? ¯´pP t 'q 1 p p Z q´pP t 'q 2 pZ q dt ⇢ ˇ" 1 ? sup 'PT ˇˇE "ª 8 0 ª 1 0 p1 ´rqpP t 'q p3q ´p Z `1 ? ¯dr dt ⇢ ˇˇ,
where we use the integration by parts formula in the second line, and the Taylor expansion of P t ' which is thrice differentiable for any t °0 by the regularizing properties of P t . Then by choosing

T "t' P C 2 b : }'} C 2 b § 1u,
and using the Mehler formula representation (1.2.7) of P t ', the commutation property, we get for any t P R `,

ˇˇpP t 'q p3q ˇˇ8 § e ´3t ? 1 ´e´2t ª R |x| dP ‹ pxq so that dist T p Q, P ‹ q § ? ⇡ 4 ? 2 ? .
Last, note that the ditance can be expressed in the language of Dirichlet forms

dist T p Q, P ‹ q" ª 8 0 E rLP t 's dt 1.

As a conclusion: Stein-Malliavin criterion on a Gaussian space

As an illustration of the path taken from Stein's method to Malliavin calculus we choose to present in the last section of this chapter one significant and particularly inspiring infinitedimensional result: a Stein-Malliavin criterion on a Gaussian space. It provides a "Berry-Esseen" general bound for the Wasserstein distance between the law of a functional of independent random variables (without any hypothesis on the laws of the sequence of initial variables) and the Normal distribution, and this in terms of Malliavin operators (see Nourdin and Peccati [START_REF] Nourdin | Stein's method on wiener chaos[END_REF]). Throuhghout this section, we consider a Gaussian field embedded in an isonormal process X "t Xphq,hP Hu where H Ä L 2 pΩ, A, Pq is its associated Gaussian subspace. We assume that A " pXq and denote Q its distribution. The elaboration of Malliavin calculus within this framework is detailed in subsection 1.3.1 and we keep the notations introduced there.

Theorem 1.4.1 (Nourdin, Peccati -2009). Let P ‹ denote the standard Gaussian distribution on R.F o ra n yF P D 1,4 such that E rFs"0,

d 1 `F˚Q , P ‹ ˘ § c E " |1 ´xrF , ´rL ´1Fy H | 2 ı . (1.4.1)
A look to the proof allows to extract the ingredients to gather to establish the result. As explained in the section devoted to Stein's method, the reduction of problem leads to tackle with E rL 1 'pFqs"E rF'pFqs , that we want to transform into ´E rL 2 'pFqs`remainder. Using the relations between the main Malliavin-type operators D, and L, we get:

E rF'pFqs " E " LpL ´1Fq'pFq ‰ " E " ´ pDL ´1Fq 'pFq ‰ " E " xD'pFq , ´DL ´1Fy H ‰ pIPP formulaq " E " ' 1 pFqxDF , ´DL ´1Fy H ‰ pchain ruleq " E " ´L2 'pFq hk kik kj ' 1 pFq remainder hkkkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkkkj ´'1 pFq `1 `xDF , ´DL ´1Fy H ˘ı
The last step which consists in using Cauchy-Schwarz inequality and requires that F P D 1,4 to use the regularizing property of the semi-group pP t q tPR `, i.e.

E " }DL ´1F} 4 H ‰ " E « › › › › ª 8 0 e ´tP t DF dt › › › › 4 H § E " }DF} 4 H ‰ .
As suggested in blue, the statement of this result requires to gather the following ingredients:

• A characterization of P in terms of 1 st -order differential operators.

• An underlying Dirichlet-Malliavin structure pD, , L, pP t q t•0 , Eq.

• An integration by parts formula including .

One of the challenges of the next chapter is to develop the aforementioned tools to state an analogue of the Stein-Malliavin criterion in any countable product of probability spaces.

Chapter 2

Malliavin calculus and Dirichlet structures for independent random variables

In the first chapter we notably investigated the possibility to derive probabilistic approximations by regular functionals of processes for which there exists a Malliavin calculus. This means exploiting the underlying Dirichlet-Malliavin structure. The motivation of this second chapter is to shape the suitable tools to be able -in the third one -to provide an analogue of the "Stein-Malliavin criterion" for functionals of independent random variables. As suggested earlier, we need to define a modicum of Malliavin-type operators -such as derivative, divergence and Ornstein-Uhlenbeck operators -and the yielded Dirichlet strucure in the general discrete context we choose to investigate: that of a family of independent, non necessarily identically distributed, random variables. The design of our stochastic calculus of variations on any countable product of probability spaces enables to generalize to a certain extent what is known about Rademacher spaces (see Privault [START_REF] Privault | Stochastic Analysis in Discrete and Continuous Settings[END_REF]), i.e. t´1, 1u N endowed by the probability product b kPN µ k where µ k is a Bernoulli distribution on t´1, 1u. Despite apparent similarities in their patterns (countable product of probability spaces in either case), the two approaches of constructing a Malliavin calculus nevertheless remain very different from each other. The Rademacher space, as product of two-components state spaces, shows a good range of algebraic specifical properties which enable to define a normal martingale pY n q nPN satisfying a structure equation of the form

Y 2 n " 1 `'n Y n . (2.0.1)
where p' n q nPN is F-predictable process. This one satisfies a "discrete" predictable representation property such that any square integrable Rademacher functional can be expanded into chaos. The gradient operator expression comes thus from the definition of the annihilation operator as it is introduced. The gradient on the Rademacher space (see Nourdin, Peccati and Reinert [START_REF] Nourdin | Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos[END_REF], Privault [START_REF] Privault | Stochastic Analysis in Discrete and Continuous Settings[END_REF]) is usually defined as

p D k FpX 1 , ¨¨¨, X n q"E rX k FpX 1 , ¨¨¨, X n q|X `, `‰ ks (2.0.2) " PpX k " 1qFpX 1 , ¨¨¨, `1, ¨¨¨, X n q´PpX k "´1qFpX 1 , ¨¨¨, ´1, ¨¨¨, X n q,
where the ˘1 are put in the k-th coordinate. It requires, for its very definition to be meaningful, either that the random variables are real valued or that they only have two possible outcomes.

59

Moreover, as highlighted by Privault and Schoutens there exists a discrete structure equation if and only if the law of X k is supported by two points (see proposition 3 in Privault and Schoutens [START_REF] Privault | Discrete chaotic calculus and covariance identities[END_REF]). This leads, in a general context of unspecified Polish spaces, to the obvious impossibility of stating a chaotic decomposition and drives us to define an ad hoc gradient, which does not coincide with the usual one on the Rademacher space (2.0.2). We proceed as follows. In Section 2.1, we define the gradient D and its adjoint , which we call divergence as it appears as the sum of the partial derivatives,a si nR n . It is defined thanks to what may arise as the Malliavin calculus cornerstone: the integration by parts formula.

The following section is devoted to the exploitation of this new formalism to the statement of representation formulas and functional identities. We establish a Clark representation formula of square integrable random variables and an Helmholtz decomposition of vector fields. We establish a log-Sobolev inequality, strongly reminding that obtained for Poisson processes (see [START_REF] Wu | A new modified logarithmic Sobolev inequality for Poisson point processes and several applications[END_REF]), together with a concentration inequality. Then, we define the number operator L "´ D. It coincides with the generator of a Markov process whose stationary distribution is the tensor probability we started with. We show in Section 2.3 that we can retrieve the classical Dirichlet-Malliavin structures for Poisson processes and Brownian motion as limits of our structures. In that respect, our theory seems to find its place in the pre-existing Malliavin's landscape.

Malliavin calculus for independent random variables

Let A be an at most countable set equipped with the counting measure :

L 2 pAq" # u :AÑ R, ÿ aPA |u a | 2 † 8 + and xu, vy L 2 pAq " ÿ aPA u a v a .
Let (E a ,a P A) be a family of Polish spaces. For any a P A, let E a and P a be respectively a -field and a probability measure defined on E a . We consider the probability space E A " ± aPA E a equipped with the product -field E A "_ aPA E a and the tensor product measure

P "b aPA P a .
The coordinate random variables are denoted by pX a ,a P Aq.F o ra n yB Ä A, X B denotes the random vector pX a ,a P Bq, defined on E B " ± aPB E a equipped with the probability P B "b aPB P a . For any a P A, let G a and F a be the -files defined by G a " pX b ,b‰ aq, F a " pX b ,b § aq, and assume that E A "pF a q aPA . A process U is a measurable random variable defined on pE A ˆA, PpAqbE A q. We denote by L 2 pE A ˆAq the Hilbert space of processes which are square integrable with respect to the measure ∞ aPA " a b P A (where " a is the Dirac measure at point a):

}U} 2 L 2 pE A ˆAq " ÿ aPA E " U 2 a ‰ and xU, Vy L 2 pE A ˆAq " ÿ aPA E rU a V a s .
Our presentation follows closely the usual construction of Malliavin calculus from the class of cylindrical functionals, denoted in the following by S. This means that F only depends on the finite set of random variables pX a ,aP Bq. It is clear that S is dense in L 2 pE A q.

The very first tool to be considered is the discrete gradient, whose form is motivated by what follows ; throughout this work, it must be made clear that all the random variables may leave on different spaces, which are only supposed to be Polish spaces. That means that in the definition of the gradient, we cannot use any algebraic property of the underlying spaces. Though some of our applications does concern random variables with finite number of outcomes, it does not seem obvious to exhibit what should be the weights, replacing PpX k " 1q and ´PpX k "´1q appearing in (2.0.2). We offer an alternative definition. We first define the gradient of cylindrical functionals, for there is no question of integrability and then extend the domain of the gradient to a larger set of functionals by a limit procedure. In functional analysis terminology, we need to verify the closability of the gradient : If a sequence of functionals converges to 0 and the sequence of their gradients converges, then it should also converges to 0. This is the only way to guarantee in the limit procedure that the limit does not depend on the chosen sequence. Definition 2.1.2 (‹ Discrete gradient). For F P S, DF is the process of L 2 pE A ˆAq defined by one of the following equivalent formulations : for all a P A,

D a FpX A q"FpX A q´E rFpX A q|G a s " FpX A q´ª Ea FpX Ara , X a qdP a pX a q " FpX A q´E 1 " FpX Ara , X 1 a q ‰ ,
where X 1 a is an independent copy of X a . Remark 2.1.3. During the preparation of this work, we found strong reminiscences of our gradient with the map ∆, introduced by Stéphane Boucheron, Gábor Lugosi and Pascal Massart in [START_REF] Boucheron | Concentration inequalities[END_REF], or WanSoo T. Rhee and Michel Talagrand in [START_REF] Rhee | Martingale inequalities and the jackknife estimate of variance[END_REF] for the proof of the Efron-Stein inequality, defined by

∆ k FpX 1 , ¨¨¨, X n q"E rF | X 1 , ¨¨¨, X k s ´E rF | X 1 , ¨¨¨, X k´1 s .
Actually, our point of view diverges from that of these works as we do not focus on a particular inequality but rather on the intrinsic properties of our newly defined gradient. Its expression an be interpreted as the measure of the "influence" of the a-th component of the process X on F. Remark 2.1.4. A straightforward calculation shows that for any F, G P S,a n ya P A,w e have

D a pFGq"FD a G `GD a F ´Da FD a G ´E rFG | G a s`E rF | G a s E rG | G a s .
This formula has to be compared with the formula DpFGq"F pDGq`G pDFq for the Gaussian Malliavin gradient (see (2.3.1) below) and DpFGq"F pDGq`G pDFq`pDFqpDGq for the Poisson gradient (see (2.3.3) below).

For F P S, there exists a finite subset B Ä A such that F " F B ˝rB . Thus, for every a R B, F is G a -measurable and then D a F " 0. This implies that

}DF} 2 L 2 pAˆE A q " E « ÿ aPA |D a F| 2 " E « ÿ aPB |D a F| 2 † 8,
hence pD a F,a P Aq defines an element of L 2 pE A ˆAq.

Definition 2.1.5. The set of simple processes, denoted by S 0 p`2pAqq is the set of random variables defined on E A ˆA of the form

U " ÿ aPB U a 1 a ,
for B a finite subset of A and such that U a belongs to S for any a P B.

The key formula for the sequel is the so-called integration by parts. It amounts to compute the adjoint of D in L 2 pE A ˆAq. We denote the domain of D in L 2 pAq by D, the closure of the class of cylindrical functions with respect to the norm

}F} 1,2 " ´}F} 2 L 2 pAq `}DF} 2 L 2 pE A ˆAq ¯1 2 .
We could as well define p-norms corresponding to L p integrability. However, for the current applications, the case p " 2 is sufficient and the apparent lack of hypercontractivity of the Ornstein-Uhlenbeck semi-group (see below Section 2.1.2) lessens the probable usage of other integrability order. Since D is defined as a closure, it is often useful to have a general criterion to ensure that a functional F, which is not cylindrical, belongs to D. The following criterion exists as is in the settings of Wiener and Poisson spaces.

Lemma 2.1.8 (‹). If there exists a sequence pF n q nPN ˚of elements of D such that 1.

F n converges to F in L 2 pE A q, 2. sup n }DF n } D is finite,
then F belongs to D and DF " lim nÑ8 DF n in D.

Divergence

We can now introduce the adjoint of D, often called the divergence as for the Lebesgue measure on R n , the usual divergence is the adjoint of the usual gradient.

Definition 2.1.9 (‹ Divergence). Let Dom "

! U P L 2 pE A ˆAq : D c °0, @ F P D, |xDF, Uy L 2 pE A ˆAq | § c }F} L 2 pAq
) .

For any U belonging to Dom , U is the element of L 2 pAq characterized by the following identity xDF, Uy L 2 pE A ˆAq " E rF Us , for all F P D.

The integration by parts formula (2.1.1) entails that for every U P Dom ,

U " ÿ aPA D a U a .
The expression of ,a st h es u mo fthe partial derivatives,a si nR n , allows to legitimize the name of divergence we give it.

In the setting of Malliavin calculus for Brownian motion, the divergence of adapted processes coincides with the Itô integral and the square moment of U is then given by the Itô isometry formula. We now see how this extends to our situation.

Definition 2.1.10 (‹). The Hilbert space Dp`2pAqq is the closure of S 0 p`2pAqq with respect to the norm

}U} 2 Dp`2pAqq " E « ÿ aPA |U a | 2 `E « ÿ aPA ÿ bPA |D a U b | 2 .
In particular, this means that the map DU "p D a U b ,a , bP Aq is Hilbert-Schmidt as a map from L 2 pE A ˆAq into itself. As a consequence, for two such maps DU and DV,t h em a p DU ˝DV is trace-class (see Yosida [START_REF] Yosida | Functional analysis[END_REF]) with

tracepDU ˝DVq" ÿ a,bPA pD a U b qp D b V a q.
The next formula is the counterpart of the Itô isometry formula for the Brownian motion, sometimes called the Weitzenböck formula (see Privault [117, Remark 2.1.12. It must be noted that compared to the analogue identity for the Brownian and the Poisson settings, the present formula is slightly different. For both processes, with corresponding notations, we have

} U} 2 L 2 pAq "}U} 2 L 2 pE A ˆAq `tracepDU ˝DUq.
The absence of the term }U} L 2 pAq gives to our formula a much stronger resemblance to the analogue equation for the Lebesgue measure. As in this latter case, we do have here 1 " 0 whereas for the Brownian motion, it yields the Itô integral of the constant function equal to one. If A " N, let F n " tX k ,k § nu and assume that U is adapted, i.e. for all n • 1, U n P F n . Then, D n U k " 0 as soon as n °k, hence

E " p Uq 2 ‰ " E « 8 ÿ n"1 ´Un ´E rU n | F n´1 s ¯2 , i.e. E "
p Uq 2 ‰ is the L 2 pNˆE N q-norm of the innovation process associated to U, which appears in filtering theory.

Ornstein-Uhlenbeck semi-group and generator

Having defined a gradient and a divergence, one may consider the Laplacian-like operator defined by L "´ D, which is also called the number operator in the settings of Gaussian Malliavin calculus. Definition 2.1.13 (‹). The number operator, denoted by L, is defined on its domain

Dom L " # F P L 2 pAq : E « ÿ aPA |D a F| 2 † 8 + by LF "´ DF "´ÿ aPA D a F. (2.1.3)
The map L can be viewed as the generator of a symmetric Markov process X, which is ergodic, whose stationary probability is P A .A s s u m efi r s tt h a tA is finite. Consider pZ t q tPR `aP oisson process on the half-line of rate |A|, and the process X t "pX 1,t , ¨¨¨, X N,t ,t• 0q which evolves according to the following rule: At a jump time of Z,

• Choose randomly (with equiprobability) an index a P A,

• Replace X a by an independent random variable X 1 a distributed according to P a . For every x P E A , a P A, set x a "p x 1 , ¨¨¨,x a´1 ,x a`1 , ¨¨¨,x |A| q. The generator of the Markov process X is clearly given by

|A| ÿ aPA 1 |A| ª Ea ´Fpx a ,x 1 a q´Fpxq ¯dP a px 1 a q"´ÿ aPA D a Fpxq. (2.1.4)
The factor |A| is due to the intensity of the Poisson process Z which jumps at rate |A|,t h e factor |A| ´1 is due to the uniform random choice of an index a P A. Thus, for a finite set A, L coincides with the generator of X.

Remark 2.1.14. This result is in fact transposable in terms of a regular Markov process of parameters p⌫, Qq where ⌫ is a probability measure on a state space E and Q at r a n s i t i o n matrix.The Markov process is contructed in a trajectorial way (see for instance Decreusefond and Moyal [START_REF] Decreusefond | Modélisation et analyse stochastiques des réseaux de télécommunication Hermès[END_REF]). Let E " E A , X 0 be a random variable with law ⌫ and define the matrix

Q " ´q`x , px a ,zq, px, px a ,zqq P E A ˆE a A ˆEa ¯as follows : $ & % qpx, xq"| A| q `x, px a ,zq ˘" dP a pzq |A| ,zP E a
that is well defined since ∞ aPA ∞ zPEa q `x, px a ,zq ˘" 1. As a reminder, the generator of a regular Markov process X of parameters p⌫, Qq is given for f P L 8 pE A q by A Q Fpx, yq"qpx, xq ÿ Denote by P "pP t q tPR `the semi-group of X ; for any x P E A , for any bounded f :E A Ñ R,

P t Fpxq"E rFpX t q|X 0 " xs .
Then, pP t ,t • 0q is a strong Feller semi-group on L 8 pE A q. This result still holds when E A is countable.

Theorem 2.1.15 (‹). For any countable set A, L defined as in (2.1.3) generates a strong Feller continuous semi-group pP t q t•0 on L 8 pE A q.

As a consequence, there exists a Markov process X whose generator is L as defined in (2.1.3).

It admits as a core (a dense subset of its domain) the set of cylindrical functions.

From the sample-path construction of X, the next result is straightforward for A finite and can be obtained by a limit procedure for A countable. P xa a,t " e ´t xa `p1 ´e´t q P a .

For any x P E A ,a n yt °0,

P t Fpxq" ª E A Fpyqb aPA dP xa a,t py a q.
It follows easily that pP t q t•0 is ergodic and stationary :

lim tÑ8 P t Fpxq" ª E A
FdP and X 0 law " P ùñ X t law " P.

We then retrieve the classical formula (in the sense that it holds as is for Brownian motion and Poisson process) of commutation between D and the Ornstein-Uhlenbeck semi-group.

Theorem 2.1.17 (‹). Let F P L 2 pE A q.F o re v e r ya P A, x P E A , D a P t Fpxq"P t D a Fpxq.

(2.1.5)

Functional identities

This section is devoted to several functional identities which constitute the crux of the matter if we want to do some computations with our new tools. These can be associated to two areas of interest, which are also linked in our context: representation theorems and concentration inequalities.

The first part of this section is devoted to the statement of a representation identity, the analog to the Clark-Ocone formula and its corollaries (for original functional identities see Bakry Gentil and Ledoux [START_REF] Bakry | Analysis and Geometry of Markov Diffusion operators[END_REF], Ledoux [START_REF] Ledoux | Concentration of measure and logarithmic sobolev inequalities[END_REF], Talagrand [START_REF] Talagrand | Transportation cost for gaussian and other product measures[END_REF]): Poincaré's inequality and a covariance identity. The Clark formula is also the initial point to lead to a log-Sobolev inequality, strongly reminding that obtained for Poisson processes (see Wu [START_REF] Wu | A new modified logarithmic Sobolev inequality for Poisson point processes and several applications[END_REF]), together with a concentration inequality.

It is classical that the notion of adaptability is linked to the support of the gradient.

Lemma 2.2.1 (‹). Assume that A " N and let F n " tX k ,k § nu.F o ra n yF P D, F is F k -measurable if and only if D n F " 0 for any n °k.A sac o n s e q u e n c e ,DF " 0 if and only if F " E rFs.

It is also well known that, in the Brownian and Poisson settings, D and conditional expectation commute.

Lemma 2.2.2 (‹). For any

F P D,f o ra n yk • 1,w eh a v e D k E rF | F k s"E rD k F | F k s . (2.2.1)
The Brownian martingale representation theorem says that a martingale adapted to the filtration of a Brownian motion is in fact a stochastic integral. The Clark formula gives the expression of the integrand of this stochastic integral in terms of the Malliavin gradient of the terminal value of the martingale. We here have the analogue formula.

Theorem 2.2.3 (‹ Clark formula). For A " N and F P D,

F " E rFs`8 ÿ k"1 D k E rF | F k s .
Since Malliavin calculus is agnostic to any time reference, we do not even assume that we have an order on the product space. It is not a major feature since a countable set A is by definition one-to-one with the set of natural integers and thus inherits of at least one order structure. However, this added degree of freedom appears to be useful (see the Clark decomposition of the number of fixed points of a random permutations in Section 3.1) and bears strong resemblance with the different filtrations which can be put on an abstract Wiener space, via the notion of resolution of the identity (see Üstünel and Zakai [START_REF] Üstünel | The construction of filtrations on abstract Wiener space[END_REF]).

Corollary 2.2.4 (‹). Within the assumptions of theorem 2.2.3 ; if A is finite and if there is no privileged order on A,w ec a nw r i t e

F " E rFs`ÿ

BÄA ˆ|A| |B| ˙´1 1 |B| ÿ bPB D b E rF | X B s .
Remark 2.2.5. The chaos decomposition is usually deduced from Clark formula by iteration.

If we apply Clark formula to E rF | F k s, we get

D k E rF | F k s" 8 ÿ j"1 D k D j E rF | F j^k s"¨¨¨"D k E rF | F k s ,
and we get back to our starting point, since j °k implies D j E rF | F k s"0 in view of Lemma 2.2.1. Furthermore, the same holds when k °j since it is easily seen that

D j D k " D k D j .F o rj " k, simply remark that D k D k " D k .
Hence, it seems that we cannot go further this way to find a potential chaos decomposition.

As mentioned in the Introduction, it may be useful to reverse the time arrow. Choose an order on A so that A can be seen as N. Then, let q F n " tX k ,k°nu.

and for any n Pt0, ¨¨¨, N ´1u,

q F N n " q F n £ F N and q F N k " F 0 "tH, E A u for k • N. Note that q F N 0 " F N and as in Lemma 2.2.1, F is q F k -measurable if and only if D n F " 0 for any n § k. Theorem 2.2.6 (‹). For every F in D, F " E rFs`8 ÿ k"1 D k E " F | q F k´1 ‰ .
In the present context, the next result is a Poincaré type inequality as it gives a bound for the variance of F in terms of the oscillations of F. In other context, it turns out to be called the Efron-Stein inequality (see Boucheron, Lugosi and Massart [START_REF] Boucheron | Concentration inequalities[END_REF]). It can be noted that both the statement and the proof are similar in the Brownian and Poisson settings.

covpF, Gq"E « ÿ kPA D k E rF | F k s D k G . (2.2.2)
As for the other versions of the Malliavin calculus (Brownian, Poisson and Rademacher), from (2.1.5), can be deduced another covariance identity.

Theorem 2.2.9 (‹). For any F, G P D,

covpF, Gq"E « ÿ kPA D k F ª 8 0 e ´tP t E rD k G | F k s dt . (2.2.3)
Then, using the so-called Herbst principle, we can derive a concentration inequality, in its large deviation formulation, which, as usual, requires an L 8 bound on the derivative of the functional to be valid.

Theorem 2.2.10 (‹ Concentration inequality). Let F for which there exists an order on A with

M " sup XPE A 8 ÿ k"1 |D k FpXq| E r|D k FpXq| | F k s † 8.
Then, for any x • 0,w eh a v e

PpF ´E rFs • xq § exp ˆ´x 2
2M ˙Ïn the Gaussian case, the concentration inequality is deduced from the logarithmic Sobolev inequality. This does not seem to be feasible in the present context because D is not a derivation, i.e. does not satisfy DpFGq"FDG`GDF. There is so some interest to study an inequality such as

Ent µ rFs § C E µ " |DF| 2 F ⇢ , (2.2.4)
that can be stated if, for instance, µ is the Bernoulli measure on t0, 1u or the Poisson measure on N and D the associated usual discrete gradient in each case. We still have an LSI identity comparable to (2.2.4). For the proof of it, we follow closely the proofs of Nicolas Privault [START_REF] Privault | Chaotic and variational calculus in discrete and continuous time for the Poisson process[END_REF] and Liming Wu [START_REF] Wu | A new modified logarithmic Sobolev inequality for Poisson point processes and several applications[END_REF]. They are based on two ingredients: the Itô formula and the martingale representation theorem. We get an ersatz of the former but the latter seems inaccessible as we do not impose the random variables to live in the same probability space and to be real valued. Should it be the case, to the best of our knowledge, the martingale representation formula is known only for the Rademacher space (see Williams [139, Section 15.1]), which is exactly the framework of Nicolas Privault [START_REF] Privault | Chaotic and variational calculus in discrete and continuous time for the Poisson process[END_REF]. This lack of a predictable representation explains the conditioning in the denominator of (2.2.5).

Theorem 2.2.11 (‹ Logarithmic Sobolev inequality). Let a positive random variable G P L log LpE A q.T h e n ,

E rG log Gs´E rGs log E rGs § ÿ kPA E " |D k G| 2 E rG | G k s ⇢ .
(2.2.5)

In the usual vector calculus on R 3 , the Helhmoltz decomposition stands that a sufficiently smooth vector field can be resolved in the sum of a curl-free vector field and a divergence-free vector field. We have here the exact counterpart with our definition of gradient.

Theorem 2.2.12 (‹ Helhmoltz decomposition). Let U P Dp`2pAqq.T h e r ee x i s t sau n i q u e couple p', Vq where ' P L 2 pE A q and V P L 2 pE A ˆAq such that E r's"0, V " 0 and

U a " D a ' `Va ,
for any a P A.

We choose to highlight the similarities and differences of the classical functional inequalities obtained in the Gaussian and Poisson settings and their analogues in our discrete framework through the following summary table. The identities relating to "Gaussian space", held on any Gaussian fields (we keep the notations used on subsection 1.3.1) except from the Clark formula stated here with respect to the standard Brownian motion. Those relating to "Poisson space" are true in any Poisson space constructed on an underlying measurable space pX,µq where µ is a -finite measure, apart from the Clark formula, given here for pN t q tPR `aP oisson point process on R `with intensity 1. Most of the identites stated in Gaussian spaces can be retrieved in Bakry, Gentil and Ledoux [START_REF] Bakry | Analysis and Geometry of Markov Diffusion operators[END_REF],Ledoux [START_REF] Ledoux | Concentration of measure and logarithmic sobolev inequalities[END_REF], Nourdin and Peccati [START_REF] Nourdin | Normal Approximations with Malliavin Calculus: From Stein's Method to Universality[END_REF] and for Poisson spaces in Last and Penrose [START_REF] Last | Lectures on the Poisson Process[END_REF], Peccati and Reinert [START_REF] Peccati | Stochastic analysis for Poisson point processes: Malliavin calculus[END_REF], Wu [START_REF] Wu | A new modified logarithmic Sobolev inequality for Poisson point processes and several applications[END_REF]. The logarithmic Sobolev inequality for differentiable Wiener functionals F,

EntrF 2 s § 2E " }rF} 2 H ‰ , ( 2
1 n! n ´E" r pnq F ⇢ ¯F " E rFs`ÿ n•1 1 n! n ´E" D pnq F ‰ ¯No decomposition decomposition Clark F " E rFs`ª 8 0 E rD t F | F t s dB t F " E rFs`ª 8 0 E rD t F | F t sp dN t ´dtq F " E rFs`8 ÿ k"1 D k E rF | F k s . formula Isometry } U} L 2 pΩq "}U} 2 L 2 pΩq `E rtrprU ˝rUqs } U} 2 L 2 pΩq "}U} 2 L 2 pΩq `E rtrpDU ˝DUqs } U} 2 L 2 pΩq " E rtrpDU ˝DUqs formula Modified log-Sobolev EntpFq § E rxrF, r log Fy H s EntpFq § E " ª E minpF ´1|D x F| 2 , EntpFq § 8 ÿ k"1 E " |D k F| 2 E rF | G k s ⇢ inequality D x FD x log

Dirichlet structures

Despite apparent dissimilarities of structures, such as the lack of chaotic decomposition in our construction (see remark 2.2.5), we nevertheless succeeded in connecting our elaboration to the preexiting theories for Brownian motion and Poisson processes; this through Dirichlet structures. Inspirated by the binding identities between Malliavin calculus and Dirichlet forms, we provide to any countable product of probability spaces a Dirichlet structure naturally induced by our construction. In the two following subsections, we state that the usual Poisson and Brownian Dirichlet structures (see Bouleau and Hirsch [START_REF] Bouleau | Dirichlet forms and analysis on Wiener space[END_REF]), associated to their respective gradient, can be retrieved as limits of the structures induced by our formalism. We borrow for that the idea of convergence of Dirichlet structures to Bouleau [START_REF] Bouleau | Théorème de Donsker et formes de Dirichlet[END_REF].

On pE A , P A q, we have already implicitly built a Dirichlet structure, i.e. a Markov process X, a semi-group P and a generator L (see subsection 2.1.2). It remains to define the Dirichlet form E A such that E A pFq"E rF LFs for any sufficiently regular functional F.

Definition 2.3.1 (‹). For F P D, define

E A pFq"E « ÿ aPA |D a F| 2 "}DF} 2 L 2 pE A ˆAq .
The integration by parts formula means that this form is closed. Since we do not assume any property on E a for any a P A and since we do not seem to have a product rule formula for the gradient, we cannot assert more properties for E A . However, following [START_REF] Bouleau | Théorème de Donsker et formes de Dirichlet[END_REF], we now show that we can reconstruct the usual gradient structures on Poisson and Wiener spaces as well chosen limits of our construction. For these two situations, we have a Polish space W , equipped with B its Borelean -field and a probability measure P. There also exists a Dirichlet form E defined on a set of functionals D. Let pE N , A N q be a sequence of Polish spaces, all equipped with a probability measure P N and their own Dirichlet form E N , defined on D N . Consider maps U N from E N into W such that pU N q ˚PN , the pullback measure of P N by U N , converges in distribution to P. We assume that for any F P D,t h em a pF ˝UN belongs to D N . The image Dirichlet structure is defined as follows. For any F P D,

E U N pFq"E N pF ˝UN q.
We adapt the following definition from Bouleau [START_REF] Bouleau | Théorème de Donsker et formes de Dirichlet[END_REF].

Definition 2.3.2. With the previous notations, we say that ppU N q ˚PN , N P N ˚q converges as a Dirichlet distribution whenever for any F P Lip XD, lim NÑ8 E U N pFq"EpFq.

Brownian motion

We now consider P as the Wiener measure on W B " C 0 pr0, 1s; Rq. Denote T "r 0, 1s. Let ph k ,kP N ˚q be an orthonormal basis of the Cameron-Martin space H 1 (defined by (1.3.1)).

For any function F:W B Ñ R cylindrical of the form

F " f pWpg 1 q, ¨¨¨, Wpg n qq,
where Wpgq coincides with the Wiener integral of g and f belongs to the Schwartz space SpR n q. As a reminder, for t P R `, W `1r0,ts ˘" Bptq, where B stands for the standard Brownian motion on R and for h P H 1 ,

r h F " n ÿ k"1 Bf Bx k p B v 1 , ¨¨¨, B v n q h k . (2.3.1)
The map r is closable from L 2 pW B q to L 2 pW B ; H 1 q. Thus, it is meaningful to define D 1,2 B as the closure of cylindrical functions for the norm

}F} 1,2 " ´}F} 2 L 2 pW B q `}rF} 2 L 2 pW B ;H 1 q ¯1 2 . Definition 2.3.3. A function F:W Ñ R is said to be H-C 1 if • for almost all ! P W B , h fi ›Ñ Fp! `hq is a continuous function on H 1 ,
• for almost all ! P W B , h fi ›Ñ Fp! `hq is continuously Fréchet differentiable and this Fréchet derivative is continuous from

H 1 into R b H 1 .
We still denote by rF the element of H 1 such that

d d⌧ Fp! `⌧ hq ˇˇˇ⌧ "0 "xrFp!q,h y H 1 .
For N P N ˚,l e t e N k ptq" ? N 1 rpk´1q{N,k{Nq ptq and h N k ptq"

ª t 0 e N k psqds.
The family ph N k ,k " 1, ¨¨¨, Nq is then orthonormal in H 1 .F o r pM k ,k " 1, ¨¨¨, Nq a sequence of independent identically distributed random variables, centered with unit variance, the random walk

! N ptq" N ÿ k"1 M k h N k ptq, for all t Pr0, 1s,
is known to converge in distribution in W B to P. Let E N " R N equipped with the product measure P N "b N k"1 ⌫ where ⌫ is the standard Gaussian measure on R.W ed e fi n et h em a p U N as follows:

U N :E N ›Ñ W B m "pm 1 , ¨¨¨,m N q fi ›Ñ N ÿ k"1 m k h N k .
It follows from our definition that: Lemma 2.3.4 (‹). For any F P L 2 pW B ; Rq,

E U N pFq" N ÿ k"1 E " ´Fp! N q´E 1 " Fp! N pkq `M1 k h N k q ı¯2 ⇢ ,
where ! N pkq " ! N ´Mk h N k and M 1 k is an independent copy of M k .T h ee x p e c t a t i o ni st a k e no n the product space R N`1 equipped with the measure P N b ⌫.

The definition of Lipschitz function we use here is the following:

Definition 2.3.5. A function F:W B Ñ R is said to be Lipschitz if it is H-C 1 and for almost all ! P W B , |xrFp!q,h y| § } 9 h} L 1 .
In particular since e N k • 0, this implies that

|xrFp!q,h N k y| § h N k p1q´h N k p0q" 1 ? N ; k " 1, ¨¨¨, N. For F P D B X H-C 1 ,w eh a v e Fp! `hq´Fp!q"xrFp!q,h y H 1 `}9 h} L 1 "p!,hq, (2.3.2)
where "p!,hq is bounded and goes to 0 in L 2 , uniformly with as } 9 h} L 1 tends to 0.

Theorem 2.3.6 (‹). For any

F P D B X H-C 1 , E U N pFq NÑ8 ›››Ñ E " }rF} 2 H 1 ‰ " EpFq.
Remark 2.3.7. The error caculus theory is provided by the following principle: the variance of the error on a random variable X, represented by ΓpXq, is thus attached to it. Then if the sequence of pairs pX n , ΓpX n qq converges suitably, it converges necessarily to a pair pX, ΓpXqq.

We can interpret the result of Theorem 2.3.6 in this perspective: the approximation of the Brownian motion by a random walk is validated by the convergence of the associated error structures.

Poisson point process

Let Y be a compact Polish space and N Y be the set of weighted configurations, i.e. the set of integer valued, locally finite measures on Y. Such a measure is of the form

! " 8 ÿ n"1 p n " ⇣n ,
where p⇣ n q nPN ˚is a set of distinct points in Y with no accumulation point, pp n q nPN ˚any sequence of positive integers. The topology on N Y is defined by the semi-norms

p f p!q" ˇˇˇˇ8 ÿ n"1 p n f p⇣ n q ˇˇˇˇ,
2) the ⇣ N k 's are sufficiently well spread so that we have convergence of Riemann sums : For any continuous and M-integrable function f : Y Ñ R,w eh a v e

N ÿ k"1 f p⇣ N k q p N k NÑ8 ›››Ñ ª f pxqdMpxq. (2.3.6)
Take f " 1 implies that ∞ k p N k tends to 1 as N goes to infinity. For any N P N ˚and any k Pt1, ¨¨¨, Nu, let µ N k be the Poisson distribution on N, of parameter p N k . In this situation, let

E N " N N with µ N "b N k"1 µ N k .
That means we have independent random variables M N 1 , ¨¨¨, M N N , where M N k follows a Poisson distribution of parameter p N k for any k Pt1, ¨¨¨, Nu. We turn these independent random variables into a point process by the map U N defined as

U N : N N ›Ñ N Y pm 1 , ¨¨¨,m N q fi ›Ñ N ÿ k"1 m k " ⇣ N k . Lemma 2.3.8 (‹). For any F P D P , E U N pFq" N ÿ m"1 8 ÿ `"0 E » - ˜8 ÿ ⌧ "0 ´Fp! N pmq ``" ⇣ N m q´Fp! N pmq `⌧" ⇣ N m q ¯µN m p⌧ q ¸2fi fl µ N m p`q, (2.3.7)
where ! N pmq "

∞ k‰m M N k " ⇣ N k .
Proof. According its very definition,

E U N pFq" N ÿ m"1 E » -˜Fp! N pmq `MN m " ⇣ N m q´8 ÿ ⌧ "0 Fp! N pmq `⌧" ⇣ N m qµ N m p⌧ q ¸2fi fl .
The result follows by conditioning with respect to M N m , whose law is µ N m .

Since the vague topology on N Y is metrizable, one could define Lipschitz functions with respect to this distance. However, this turns out to be not sufficient for the convergence to hold. Proof of Theorem 2.1.6. The process tracepDUq"p D a U a q aPB belongs to L 2 pE A ˆAq:u s i n g the Jensen inequality, we have:

} tracepDUq} 2 L 2 pE A ˆAq " E « ÿ aPB |D a U a | 2 § 2 ÿ aPB E " U 2 a ‰ † 8. (2.4.1) Moreover, xDF, Uy L 2 pE A ˆAq " E « ÿ aPA pF ´E rF | G a sq U a " E « ÿ aPB pF ´E rF | G a sq U a " E « F ÿ aPB pU a ´E rU a | G a sq ,
since the conditional expectation is a projection in L 2 pE A q.

Proof of corollary 2.1.7. Let pF n q nPN ˚be a sequence of random variables defined on S such that F n converges to 0 in L 2 pE A q and the sequence DF n converges to ⌘ in L 2 pE A ˆAq. Let U be a simple process. From the integration by parts formula (2.1.1)

E « ÿ aPA D a F n U a " E « F n ÿ aPA D a U a
where ÿ aPA D a U a P L 2 pE A q in view of (2.4.1). Then,

x⌘, Uy L 2 pE A ˆAq " lim nÑ8 E « F n ÿ aPA D a U a " 0,
for any simple process U. It follows that ⌘ " 0 and then the operator D is closable from

L 2 pE A q to L 2 pE A ˆAq.
Proof of Lemma 2.1.8. Since sup n }DF n } D is finite, there exists a subsequence which we still denote by pDF n ,n • 1q weakly convergent in L 2 pE A ˆAq to some limit denoted by ⌘.F o r k °0, let n k be such that }F m ´F} L 2 † 1{k for m • n k . The Mazur's Theorem implies that there exists a convex combination of elements of pDF m ,m

• n k q such that › › › M k ÿ i"1 ↵ k i DF m i ´⌘ › › › L 2 pE A ˆAq † 1{k.
Moreover, since the ↵ k i are positive and sums to 1,

› › › M k ÿ i"1 ↵ k i F m i ´F › › › L 2 pE A q § 1{k.
We have thus constructed a sequence

F k " M k ÿ i"1 ↵ k i F m i
such that F k tends to F in L 2 and DF k converges in L 2 pE A ˆAq to a limit. By the construction of D, this means that F belongs to D and that DF " ⌘.

Proof of Lemma 2.1.11. For U and V in S 0 p`2pAqq, from the integration by parts formula,

E r U Vs"xD pUq, Vy L 2 pE A ˆAq " E « ÿ aPA D a p Uq V a " E » - ÿ pa,bqPA 2 V a D a pD b U b q fi fl " E » - ÿ pa,bqPA 2 V a D b pD a U b q fi fl " E » - ÿ pa,bqPA 2 pD b V a qpD a U b q fi fl " E rtracepDU ˝DVqs . It follows that E " U 2 ‰ § }U} 2 Dp`2pAqq
. Then, by density, Dp`2pAqq Ä Dom and Eqn. (2.1.2) holds for U and V in Dom .

Proof of Theorem 2.1.15. To prove the existence of pP t q tPR `for a countable set, we apply the Hille-Yosida theorem: Theorem 2.4.1 (Hille-Yosida). Al i n e a ro p e r a t o rL on L 2 pE A q is the generator of a strongly continuous contraction semigroup on L 2 pE A q if and only if

1. Dom L is dense in L 2 pE A q.
2. L is dissipative i.e. for any °0, F P Dom L,

} F ´LF} L 2 pE A q • }F} L 2 pE A q . 3. Imp Id ´Lq is dense in L 2 pE A q.
We know that S Ä Dom L and that S is dense in L 2 pE A q, then so does Dom L. Let pA n q nPN ˚an increasing sequence of subsets of A such that î n•1 A n " A.F o rF P L 2 pE A q, let F n " E rF | F An s. Since pF n q nPN ˚is a square integrable F-martingale, F n converges to F both almost-surely and in L 2 pE A q.F o ra n yn P N ˚, F n depends only on X An .A b u s i n gt h e notation, we still denote by F n its restriction to E An so that we can consider L n F n where L n is defined as above on E An . Moreover, according to Lemma 2.2.2, D a F n " E rD a F | F An s, hence

2 }F n } 2 L 2 pE A q § } F n ´Ln F n } 2 L 2 pE An q " E » - ˜ F n `ÿ aPA D a F n ¸2fi fl " E » -E « F `ÿ aPA D a F ˇˇF An 2 fi fl nÑ8 ›››Ñ} F ´LF} 2 L 2 pE A q .
Therefore, point (2) is satisfied. Since A n is finite, there exists G n P L 2 pE An q such that F n "p Id ´Ln qG n pX An q" G n pX An q`ÿ aPAn D a G n pX An q " Gn pX A q`ÿ aPAn D a Gn pX A q" Gn pX A q`ÿ aPA D a Gn pX A q, where Gn pX A q"G n pX An q depends only on the components whose index belongs to A n . This means that F n belongs to the range of Id ´L and we already know it converges in L 2 pE A q to F.

Proof of Theorem 2.1.17. For A finite, a P A, let G a,t :" pX x b b,t ,b‰ aq so that For A infinite, let pA n q nPN ˚an increasing sequence of finite subsets of A such that î n•1 A n " A.F o rF P L 2 pE A q, let F n " E rF | F An s. Since P is a contraction semi-group, for any t, P t F n tends to P t F in L 2 pE A q as n goes to infinity. From the Mehler formula, we known that P t F n " P n t F n where P n is the semi-group associated to P n , hence

D
D a P t F n " D a P n t F n " P n t D a F n . (2.4.2)
Moreover,

E « ÿ aPAn |D a P t F n | 2 " ÿ aPAn E " |P t D a F n | 2 ‰ § ÿ aPAn E " |D a F n | 2 ‰ " ÿ aPAn E " |E rD a F | F An s| 2 ‰ § ÿ aPAn E " |D a F| 2 ‰ § }DF} 2 D .
According to Lemma [2. 1.8], this means that P t F belongs to D. Let n go to infinity in (2.4.2) yields (2.1.5).

Remark 2.4.2. An alternative proof consists in using the identity P t F " e ´tL F and then the equivalent expression of P t as

P t F " ÿ nPN t n n! L n F. ( 2
LD a F " D a D a F ÿ bPA,b‰a D b D a F " D a ´Da F `ÿ bPA,b‰a D b F ¯" D a LF,
we get by induction that L n D a " L n D a on S. We conclude by plotting this in (2.4.3) and using a density argument.

Proofs of Section 2.2

Proof of Lemma 2.2.1. Let k P A.A s s u m et h a tF P F k . Then, for every n °k, F is G nmeasurable and D n F " 0.

Let F P D such that D n F " 0 for every n °k. Then F is G n -measurable for any n °k.F r o m the equality F k " ì n°k G n , it follows that F is F k -measurable. Proof of Lemma 2.2.2. For any k P N ˚, F k X G k " F k´1 , hence D k E rF | F k s"E rF|F k s´E rF | F k´1 s"E rD k F | F k s .
The proof is thus complete.

Proof of Theorem 2.2.3. Let F a F n -measurable random variable. It is clear that

F ´E rFs" n ÿ k"1 pE rF | F k s ´E rF | F k´1 sq " n ÿ k"1 D k E rF | F k s .
For F P D, apply this identity to F n " E rF | F n s to obtain

F n ´E rFs" n ÿ k"1 D k E rF | F k s .
Remark that for `°k, in view of Lemma 2.2.1,

E rD k E rF | F k s D `E rF | F `ss " E rD `Dk E rF | F k s E rF | F `ss " 0, (2.4.4) since D k E rF | F k s is F k -measurable. Hence, we get E " |F ´E rFs| 2 ‰ • E " |F n ´E rFs| 2 ‰ " n ÿ k"1 E " D k E rF | F k s 2 ı .
Thus, the sequence pD k E rF | F k s ,kP N ˚q belongs to `2pNq and the result follows by a limit procedure.

Proof of Corollary 2.2.4. We now analyze the non-ordered situation. If A is finite, each bijection between A and t1, ¨¨¨,nu defines an order on A. Hence, there are |A| ! possible filtrations. Each term of the form

D i k E rF | X i 1 , ¨¨¨, X i k s
appears pk ´1q! p|A|´kq! times since the order of X i 1 , ¨¨¨, X i k´1 is irrelevant to the conditioning. The result follows by summation then renormalization of the identities obtained for each filtration.

Proof of Theorem 2.2.6. Remark that

D k E " F ˇˇq F N k´1 ı " E " F ˇˇq F N k´1 ı ´E " F ˇˇq F N k´1 X G k ı " E " F ˇˇq F N k´1 ı ´E " F ˇˇq F N k ı .
For F P F N , since the successive terms collapse, we get

F ´E rFs"E " F ˇˇq F N 0 ı ´E " F ˇˇq F N N ı " N ÿ k"1 D k E " F | q F N k´1 ı " 8 ÿ k"1 D k E " F ˇˇq F N k´1 ı ,
by the very definition of the gradient map. As in (2.4.4), we can show that for any N,

E " D k E " F ˇˇq F N k´1 ı D `E " F ˇˇq F N `´1 ıı " 0, for k ‰ `.
Consider F N " E rF | F N s and proceed as in the proof of Lemma 2.2.3 to conclude.

Proof of Corollary 2.2.7. According to (2.4.4) and (2.2.1), we have

VarpFq"E » -ˇˇˇˇÿ kPA D k E rF | F k s ˇˇˇˇ2 fi fl " E « ÿ kPA ˇˇD k E rF | F k s ˇˇ2 " E « ÿ kPA ˇˇE rD k F | F k s ˇˇ2 § E « ÿ kPA E " |D k F| 2 | F k ‰ " E « ÿ kPA |D k F| 2 ,
where the inequality follows from then Jensen inequality.

Proof of Theorem 2.2.8. Let F, G P D, the Clark formula entails covpF, Gq"E rpF ´E rFsqpG ´E rGsqs

" E » - ÿ k,`PA D k E rF | F k s D `E rG | F `sfi fl " E « ÿ kPA D k E rF | F k s D k E rG | F k s " E « ÿ kPA D k FD k E rG | F k s
where we have used (2.4.4) in the third equality and the identity D k D k " D k in the last one.

Proof of Theorem 2.2.9. Let F, G P L 2 pE A q.

covpF, Gq"E « ÿ

kPA pD k E rF|F k sqpD k E rG|F k sq " E « ÿ kPA D k E rF|F k s ˆ´ª 8 0 LP t E rG|F k s dt ˙ " ª 8 0 E « ÿ kPA D k E rF|F k s ˜ÿ `PA D `Pt E rG|F k s dt ¸ " ª 8
✓ ˇˇE " Fe ✓F ıˇˇˇ" ✓ ˇˇˇˇE « ÿ kPA D k FD k E " e ✓F | F k ı ˇˇˇ § ✓ ÿ kPA E " |D k F| ˇˇD k E " e ✓F | F k ıˇˇˇı .
Recall that

D k E " e ✓F | F k ı " E 1 " E " e ✓F | F k ı ´E " e ✓FpX k ,X 1 k q | F k ıı " E " E 1 " e ✓F ´e✓FpX k ,X 1 k q ı ˇˇF k ı " E " e ✓F E 1 " 1 ´e´✓∆ k F ı ˇˇF k ı where ∆ k F " F ´FpX k , X 1 k q so that D k F " E 1 r∆ k Fs. Since px fi Ñ 1 ´e´x q is concave, we get D k E " e ✓F | F k ı § E " e ✓F p1 ´e´✓D k F q|F k ı § ✓ E " e ✓F |D k F||F k ı . Thus, ˇˇE " Fe ✓F ıˇˇˇ § ✓ E « e ✓F 8 ÿ k"1 |D k F| E r|D k F||F k s § M ✓ E " e ✓F ı .
By Gronwall lemma, this implies that

E " e ✓F ı § exp ˆ✓2 2 M ˙¨ Hence, PpF ´E rFs • xq"Ppe ✓pF´ErFsq q • e ✓x q § exp ´´✓x `✓2 2 M ¯.
Optimize with respect to ✓ gives ✓ opt " x{M, hence the result.

Proof of Theorem 2.2.11. We follow closely the proof of [START_REF] Wu | A new modified logarithmic Sobolev inequality for Poisson point processes and several applications[END_REF] for Poisson process. Let G P L 2 pE A q be a positive random variable such that DG P L 2 pE A ˆAq. For any non-zero integer n, define G n " minpmaxp 1 n , Gq,nq,f o ra n yk, L k " E rG n |F k s and L 0 " E rG n s. We have,

L n log L n ´L0 log L 0 " n´1 ÿ k"0 L k`1 log L k`1 ´Lk log L k " n´1 ÿ k"0 log L k pL k`1 ´Lk q`n ´1 ÿ k"0 L k`1 plog L k`1 ´log L k q.
Note that `log L n pL n`1 ´Ln q ˘nPN and `Ln`1 ´Ln ˘nPN are pF n q nPN -martingales, hence

E rL n log L n ´L0 log L 0 s " E « n´1 ÿ k"0 L k`1 log L k`1 ´Lk`1 log L k ´Lk`1 `Lk " E « n´1 ÿ k"0 L k`1 log L k`1 ´Lk log L k ´plog L k `1qpL k`1 ´Lk q " E « n´1 ÿ k"0 `pL k , L k`1 ´Lk q ,
where the function `is defined on Θ "tpx, yqPR 2 : x °0,x`y °0u by `px, yq"px `yq logpx `yq´x log x ´plog x `1qy.

Since `is convex on Θ, it comes from the Jensen inequality for conditional expectations that

n´1 ÿ k"0 E r`pL k , L k`1 ´Lk qs " n´1 ÿ k"0 E r`pE rG n | F k s , D k`1 E rG n | F k`1 sqs " n ÿ k"1 E r`pE rG n | F k´1 s , E rD k G n | F k sqs § n ÿ k"1 E rE r`pE rG n | G k s , D k G n q|F k ss " n ÿ k"1 E r`pE rG n | G k s , D k G n qs " 8 ÿ k"1 E r`pE rG n | G k s , D k G n qs .
We know from [START_REF] Wu | A new modified logarithmic Sobolev inequality for Poisson point processes and several applications[END_REF] that for any non-zero integer k, `pE rG n | G k s , D k G n q converges increasingly to `pE rG | G k s , D k Gq P-a.s., hence by Fatou Lemma,

E rG log Gs´E rGs log E rGs § 8 ÿ k"1 E r`pE rG | G k s , D k Gqs .
Furthermore, for any px, yqPΘ, `px, yq § |y| 2 {x, then,

E rG log Gs´E rGs log E rGs § 8 ÿ k"1 E " |D k G| 2 E rG | G k s ⇢ The proof is thus complete.
Proof of Theorem 2.2.12. We first prove the uniqueness. Let p', Vq and p' 1 , V 1 q two convenient couples. We have D a p' ´'1 q"V 1 a ´Va for any a P A and ∞ aPA D a pV 1 a ´Va q"0, hence

0 " E « p' ´'1 q ÿ aPA D a pV 1 a ´Va q " E « ÿ aPA D a p' ´'1 qpV 1 a ´Va q " E « ÿ aPA pV 1 a ´Va q 2 .
This implies that V " V 1 and Dp' ´'1 q"0. The Clark formula (Theorem 2.2.3) entails that 0 " E r' ´'1 s " ' ´'1 .

We now prove the existence. Since E rD a ' | G a s"0, we can choose

V a " E rU a | G a s ,
which implies D a ' " D a U a , and guarantees V " 0. Choose any ordering of the elements of A and remark that, in view of (2.4.4),

E » - ˜8 ÿ k"1 E rD k U k | F k s ¸2fi fl " E » - ˜8 ÿ k"1 D k E rU k | F k s ¸2fi fl " E « 8 ÿ k"1 ´Dk E rU k | F k s ¯2 § 8 ÿ k"1 E " |D k U k | 2 ‰ § }U} 2 Dp`2pAqq , hence ' " 8 ÿ k"1 E rD k U k | F k s ,
defines a square integrable random variable of null expectation, which satisfies the required property.

Proofs of Section 2.3

Proof of Theorem 2.3.6. For F P D B X H-C 1 , in view of (2.3.2), we have

Fp! N q´Fp! N pkq `M1 k h N k q"pM k ´M1 k qxrFp! N pkq q,h N k y H 1 `Mk ´M1 k ? N "p! N pkq ,h N k q.
Hence,

N ÿ k"1 E " ´Fp! N q´E 1 " Fp! N pkq `M1 k h N k q ı¯2 ⇢ " N ÿ k"1 E « ˆMk xrFp! N pkq q,h N k y H 1 `E1 " M k ´M1 k ? N "p! N pkq ,h N k q ⇢˙2 " N ÿ k"1 E " xrFp! N pkq q,h N k y 2 H 1 ı `Rem, and
Rem § c N N ÿ k"1 E " "p! N pkq ,h N k q 2 ı NÑ8 ›››Ñ 0,
by the Césaro theorem. It follows that E U N pFq has the same limit as

N ÿ k"1 E " xrFp! N pkq q,h N k y 2 H 1 ı .
As N goes to infinity, we add more and more terms to the random walk, so that the influence of one particular term becomes negligible. The following result is well known (see Bouleau, Proposition 3 in [START_REF] Bouleau | Théorème de Donsker et formes de Dirichlet[END_REF]): for any k Pt1,...,Nu, for any bounded and ',

E " pM k q'p! N q ‰ NÑ8 ›››Ñ E r pM k qs E r'p!qs .
Since }rF} H 1 belongs to L 8 and }h N k } 8 tends to 0, this entails that for any k,

lim NÑ8 E " xrFp! N pkq q,h N k y 2 H 1 ı " lim NÑ8 E " xrFp! N q,h N k y 2 H 1 ‰ " lim N Ñ8 E " }⇡ V N rFp! N q} 2 H 1 ‰ ,
where ⇡ V N is the orthogonal projection in H 1 onto spanth N k ,k" 1, ¨¨¨, Nu. We conclude by dominated convergence.

Proof of Theorem 2.3.10. Starting from (2.3.7), the terms with ⌧ " 0 can be decomposed as

e ´2p N m N ÿ m"1 E " ´Fp! N pmq `"⇣ N m q´Fp! N pmq q ¯2⇢ µ N m p1q`R N 0 .
Since F belongs to TV ´Lip,

R N 0 § N ÿ m"1 8 ÿ `"2 l 2 µ N m plq § c 1 Npp N q 2 E " pPoissonpp N q`2q 2 ‰ § c 2 Npp N q 2 ,
where the c 1 and c 2 are irrelevant constants. As Np N is bounded, R N 0 goes to 0 as N grows to infinity. For the very same reasons, the sum of the terms of (2.3.7) with ⌧ • 1 converge to 0, thus

lim NÑ8 E U N pFq" lim NÑ8 N ÿ m"1 e ´2p N m E " ´F p! N pmq `"⇣ N m q´Fp! N pmq q ¯2⇢ p N m .
Consider now the space N ⇣ Y " N Y ˆt ⇣ N k ,k " 1, ¨¨¨, Nu with the product topology and probability measure PN

" P N b ∞ k p N k " ⇣ N k . Let : N Y ˆt⇣ N k ,k" 1, ¨¨¨, Nu ›Ñ E p!, ⇣q fi ›Ñ ´Fp! ´p!p⇣q´1q" ⇣ q´Fp! ´!p⇣q" ⇣ q ¯2.
Then, we can write

N ÿ m"1 E " ´Fp! N pmq `"⇣ N m q´Fp! N pmq q ¯2⇢ p N m " ª N ⇣ Y p!, ⇣qd PN p!, ⇣q.
Under PN , the random variables ! and ⇣ are independent. Equation (2.3.6) means that the marginal distribution of ⇣ tends to M (assumed to be a probability measure at the very beginning of this construction). Moreover, we already know that P N converges in distribution to P. Hence, PN tends to PbM as N goes to infinity. Since F is in TV ´Lip, is continuous and bounded, hence the result.

Chapter 3

Applications : representation formulas, convergence theorems, finance

The last chapter is a patchwork which pieces are made of the applications inherited from our construction. The diversity of these comes from the variety of the fields explored. The first subsection deals with the declination of Clark formula for U -statistics, such that we show that the Hoeffding decomposition can be interpreted as an avatar of. The second one is motivated by the statement of a discrete analogue of the Stein-Malliavin criterion (1.4.1). In that perspective, we follow the footsteps of the path "from Stein's method to Stein-Dirichlet-Malliavin structures" described in the first chapter. In the last part we challenge our formalism with an application in finance : the computation of the greek Rho in the trinomial model.

Representations

We now show that our Clark formula yields an interesting decomposition of random variables. For U -statistics, it boils down to the Hoeffding decomposition.

Definition 3.1.1. For m, n P N ˚, let h : R m Ñ R be a symmetric function, and X 1 , ¨¨¨, X n , n random variables supposed to be independent and identically distributed. The U -statistics of degree m and kernel h is defined, for any n • m by

U n " UpX 1 , ¨¨¨, X n q" ˆn m ˙´1 ÿ APprns,mq hpX A q
where prns,mq denotes the set of ordered subsets A Ä rns"t 1, ¨¨¨,nu, of cardinality m. More generally, for a set A Ä rns, pA,mq denotes the set of subsets of A with m elements.

If E r|hpX 1 , ¨¨¨, X m q|s is finite, we define h m " h and for 1 § k § m ´1,

h k pX 1 , ¨¨¨, X k q"E rhpX 1 , ¨¨¨, X m q|X 1 , ¨¨¨, X k s .
Let ✓ " E rhpX 1 , ¨¨¨, X m qs, consider g 1 pX 1 q"h 1 pX 1 q´✓, and

g k pX 1 , ¨¨¨, X k q"h k pX 1 , ¨¨¨, X k q´✓ ´k´1 ÿ j"1 ÿ
BPprks,jq g j pX B q, 85 for any 1 § k § m. Since the variables X 1 , ¨¨¨, X n are independent and identically distributed, and the function h is symmetric, the equality

E rhpX AYB q|X B s"E rhpX CYB q|X B s ,
holds for any subsets A and C of rnszB, of cardinality n ´k.

Theorem 3.1.2 (‹ Hoeffding decomposition of U -statistics, [START_REF] Koroljuk | Theory of U -statistics[END_REF]). For any integer n,w eh a v e

U n " ✓ `m ÿ k"1 H pkq n (3.1.1)
where H pkq n is the U -statistics based on kernel g k , i.e. defined by

H pkq n " ˆn k ˙´1 ÿ
BÄprns,kq

g k pX B q.
As mentioned above, reversing the natural order of A, provided that it exists, can be very fruitful. We illustrate this idea by the decomposition of the number of fixed points of a random permutation under Ewens distribution. It could be applied to more complex functionals of permutations but to the price of increasingly complex computations. For every integer N, denote by S N the space of permutations on t1, ¨¨¨, Nu. We always identify S N as the subgroup of S N`1 stabilizing the element N `1. For every k Pt1, ¨¨¨, Nu, define J k "t1, ¨¨¨,ku and

J " J 1 ˆJ2 ˆ¨¨¨ˆJ N .
The coordinate map from J to J k is denoted by I k . Following the construction designed by Serguei Kerov [START_REF] Kerov | Harmonic analysis on the infinite symmetric group[END_REF], we have:

Theorem 3.1.

(‹).

There exists a natural bijection Γ between J and S N .

Proof. To a sequence pi 1 , ¨¨¨,i N q where i k P J k , we associate the permutation Γpi 1 , ¨¨¨,i N q"pN,i N q˝pN ´1,i N´1 q ...˝p2,i 2 q.

where pi, jq denotes the transposition between the two elements i and j. To an element N P S N , we associate i N " N pNq. Then, N is a fixed point of N´1 "p N,i N q˝ N , hence it can be identified as an element N´1 of S N´1 . Then, i N´1 " N´1 pN ´1q and so on for decreasing indices. It is then clear that Γ is one-to-one and onto.

In Kerov [START_REF] Kerov | Harmonic analysis on the infinite symmetric group[END_REF], Γ is described by the following rule: start with permutation 1 "p 1q, if at the N-th step of the algorithm, we have i N " N then the current permutation is extended by leaving N fixed, otherwise, N is inserted in N´1 just before i N in the cycle of this element. This construction is reminiscent of the Chinese restaurant process (see Arratia, Barbour and Tavaré [START_REF] Arratia | Poisson process approximations for the Ewens sampling formula[END_REF]) where i N is placed immediately after N. An alternative construction of permutations is known as the Feller coupling (see Arratia, Barbour and Tavaré [START_REF] Arratia | Poisson process approximations for the Ewens sampling formula[END_REF]). In our notations, it is given by Definition 3.1.4 (Ewens distribution). For some t P R `, for any k Pt 1, ¨¨¨, Nu, consider the measure P k defined on J k by

P k ptjuq " $ ' ' ' & ' ' ' % 1 t `k ´1 if j ‰ k, t t `k
´1 for j " k. Under the distribution P "b k P k , the random variables pI k ,k" 1, ¨¨¨, Nq are independent with law given by PpI k " jq"P k ptjuq, for any k. The Ewens distribution of parameter t on S N , denoted by P t , is the push-forward of P by the map Γ.

A moment of thought shows that a new cycle begins in the first construction for each index where i k " k. Moreover, it can be shown that: Theorem 3.1.5 (see Kerov [START_REF] Kerov | Harmonic analysis on the infinite symmetric group[END_REF]). For any P S N ,

P t pt uq " t cycp q pt `1qpt `2qˆ¨¨¨ˆpt `N ´1q
, where cycp q is the number of cycles of .

For any F, a measurable function on S N , we have the following diagram

pJ , b N k"1 P k q pS N , P t q R Γ r F " F ˝Γ F
We denote by i "p i 1 , ¨¨¨,i N q a generic element of J and by " Γpiq. Let C 1 p q denote the number of fixed points of the permutation and r C 1 " C 1 ˝Γ.F o r a n y k P J N ,t h e random variable U k p q is the indicator of the event (k is a fixed point of ) and let r U N k " U k ˝Γ. The Clark formula with reverse filtration shows that we can write r U N k as a sum of centered orthogonal random variables as in the Hoeffding decomposition of U -statistics (see Theorem 3.1.2). Theorem 3.1.6 (‹). For any k Pt1, ¨¨¨, Nu,

r U k " 1 pI k "kq 1 pIm‰k, mPtk`1,¨¨¨,Nuq . (3.1.2)
and under P t , r U N k is Bernoulli distributed with parameter tp k ↵ k ,w h e r ef o ra n yk Pt1, ¨¨¨, Nu,

p k " 1 t `k ´1 and ↵ k " N π j"k`1 j ´1 t `j ´1 Moreover, r U N k " tp k ↵ k `´1 pI k "kq ´tp k ¯N π m"k`1 1 pIm‰kq ´tp k N´k´1 ÿ j"1 t `k ´1 t `k `j ´2 ´1pI k`j "kq ´pk`j ¯N´k π `"j`1 1 pI k``‰ kq . Since r C 1 " N ÿ k"1 r U N k ,
we retrieve the result of [START_REF] Arratia | Logarithmic combinatorial structures: a probabilistic approach[END_REF]:

E " r C 1 ı " tN t `N ´1 ,
and the following decomposition of r C 1 can be easily deduced from the previous theorem.

Theorem 3.1.7 (‹). We can write

r C 1 " t ˆ1 ´t ´1 N `t ´1 ˙`N ÿ `"1 D `Ũ N ``N ÿ `"2 t t ``´2 D `˜`´1 ÿ k"1 N π m"`1 pIm‰kq " t ˆ1 ´t ´1 N `t ´1 ˙`N ÿ `"1 p1 pI `"`q ´t t ``´1 q N π m"``1 1 pIm‰`q ´N´1 ÿ `"2 t t ``´2 `´1 ÿ k"1 ˆ1pI `"kq ´1 t ``´1 ˙N π m"``1
1 pIm‰kq .

Remark 3.1.8. Note that such a decomposition with the natural order on N would be infeasible since the basic blocks of the definition of r C 1 , namely the Ũk , are anticipative (following the vocabulary of Gaussian Malliavin calculus), i.e. r U k P pI k``, `" 0, ¨¨¨, N ´kq.

This decomposition can be used to compute the variance of r C 1 . To the best of our knowledge, this is the first explicit, i.e. not asymptotic, expression of it. Theorem 3.1.9. For any t P R,w eg e t Varr r

C 1 s" Nt t `N ´1 ˜t t `N ´1 `1 ´2t 2 N N ÿ k"1 1 t `k ´1 ¸.
We retrieve Varr r

C 1 s ›››Ñ NÑ8 t,
as can be expected from the Poisson limit.

Stein-Malliavin criterion for Normal and Gamma approximations

In this section we confront our model to the Stein-Dirichlet-Malliavin method in order to provide in particular a discrete analogue of the Stein-Malliavin criterions for Normal and Gamma approximations. Such approximation results derived from this method for functionals of Rademacher (see for instance Nourdin, Peccati and Reinert [START_REF] Nourdin | Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos[END_REF], Zheng [START_REF] Zheng | Normal approximation and almost sure central limit theorem for nonsymmetric rademacher functionals[END_REF]), Poisson (see for instance Decreusefond, Schulte and Thäle [START_REF] Decreusefond | Functional Poisson approximation in Kantorovich-Rubinstein distance with applications to U-statistics and stochastic geometry[END_REF] Lachièze-Rey and Peccati [START_REF] Lachièze-Rey | Fine Gaussian fluctuations on the Poisson space II: rescaled kernels, marked processes and geometric U-statistics[END_REF], Peccati et al. [START_REF] Peccati | Stein's method and Normal approximation of Poisson functionals[END_REF]) or Gaussian random variables (see Nualart and Peccati [START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF]) or processes (see Coutin and Decreusefond [32,[START_REF] Coutin | Higher order expansions via Stein's method[END_REF]). Our results take place among them; however, to the best of our knowledge, when Q is the distribution of a family of independent random variables, the distance distpF ˚Q, Pq (where dist " d 1 , d 2 or dist Kol as the case may be) is evaluated through exchangeable pairs or couplings, which means to construct an ad-hoc structure for each situation at hand. We intend to give here an exact analogue of (1.4.1) in this situation using only our newly defined operator D, that makes our approach original.

Remark 3.2.1. In what follows, we deal with functions F defined on E A , that means that F is a function of X A and as such, we should use the notation FpX A q.F o rt h es a k eo fn o t a t i o n s , we identify F and FpX A q. The distribution of F is denoted P F " F ˚PA .

Theorem 3.2.2 (‹). Let P denote the standard Gaussian distribution on R.F o ra n yF: E A Ñ R such that E rFs"0 and F P Dom D.T h e n ,

d 1 pP, P F q § E «ˇˇˇˇˇ1 ´ÿ aPA D a F p´D a L ´1qF ˇˇˇˇ `ÿ aPA E "ª E A ´F ´FpX A a ; xq ¯2dP a pxq| D a L ´1F| ⇢ .
The proof of this version follows exactly the lines of the proof of Theorem 3.1 in Nourdin and Peccati [START_REF] Nourdin | Stein's method on wiener chaos[END_REF], Peccati, Solé, Taqqu and Utzet [START_REF] Peccati | Stein's method and Normal approximation of Poisson functionals[END_REF] but we can do slightly better by changing a detail in the Taylor expansion.

Theorem 3.2.3 (‹). Let P denote the standard Gaussian distribution on R.F o ra n yF: E A Ñ R such that E rFs"0 and F P Dom D.T h e n ,

d 1 pP, P F q § sup PLip 2 E « pFq´ÿ aPA pFpX 1 a qqD a Fp´D a L ´1qF `ÿ aPA E "ª E A ´F ´FpX A a ; xq ¯2dP a pxq| D a L ´1F| ⇢ , (3.2.1)
where X 1 a " X A a YtX 1 a u. Remark 3.2.4. This theorem provides a Berry-Essen type bound in the Wasserstein distance for the Normal approximation by any functional of random variables and generalises in the certain sense that got by Sourav Chatterjee ([24]). Raphaël Lachièze-Rey and Giovanni Peccati stated a remarkable bound in the Kolmogorov distance for the same approximation in terms of another difference operator ∆ ¨; this is defined for any ordered subset C of rr 1, Nss by

∆ C FpX A , X 1 A q"FpX A q´F C pX A , X 1 A q,
and contructed by iteration of the operator

∆ tau FpX A , X 1 A q"FpX A q´FpX 1 1 , ¨¨¨, X 1 a´1 , X a , X 1 a`1 , ¨¨¨, X 1 N q.
Remark 3.2.5. We can point out however that, for lack of chaos decomposition, we could not rule the equivalent of the fourth order moment theorem (see [START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF], [START_REF] Nourdin | Classical and free fourth moment theorems: Universality and thresholds[END_REF]), appearing in the normal approximations of number of nonlinear functionalities acting on random fields, such as Poisson ones (see [START_REF] Döbler | Fourth moment theorems on the Poisson space: analytic statements via product formulae[END_REF]) or relative to fractional Brownian motion (see [START_REF] Nourdin | Multivariate normal approximation using stein's method and malliavin calculus[END_REF]) to mention but two examples.

This formulation may seem cumbersome, but it easily gives a close to the usual bound in the Lyapounov central limit theorem, with a non optimal constant (see [START_REF] Goldstein | Bounds on the constant in the mean central limit theorem[END_REF]).

Corollary 3.2.6 (‹ Lyapounov). Let pX n q nPN ˚be a sequence of thrice integrable, independent random variables. Denote

2 n " VarpX n q,s 2 n " n ÿ j"1 2 j and Y n " 1 s n n ÿ j"1
pX j ´E rX j sq .

Then,

d 1 pP, P Yn q § 2p ? 2 `1q s 3 n n ÿ j"1 E " |X j ´E rX j s | 3 ‰ .
Remark 3.2.7. If we use Theorem 3.2.2, we get

d 1 pP, P Yn q § E «ˇˇˇˇˇ1 ´n ÿ j"1 X 2 j s 2 n ˇˇˇˇ `2 s 3 n n ÿ j"1 E " |X j ´E rX j s| 3 ‰ ,
and the quadratic term is easily bounded only if the X i 's are such that E " X 4 i ‰ is finite, which in view of Corollary 3.2.6 is a too stringent condition.

The functional which appears in the central limit theorem is the basic example of U -statistics or homogeneous sums. If we want to go further and address the problem of convergence of more general U -statistics (or homogeneous sums), we need to develop a similar apparatus for the Gamma distribution. Recall that the Gamma distribution of parameters r and has density f r,`p xq" r

Γprq

x r´1 e ´`x 1 R `pxq.

Let Y r,`" Γpr, `q, it has mean r{`and variance r{`2. Denote by Y r,`" Y r,`´r {`. As described in Graczyk et al. [START_REF] Graczyk | Higher order Riesz transforms, fractional derivatives, and Sobolev spaces for Laguerre expansions[END_REF], Z " Y r,`" Y r,`´r {`if and only if E rL r,`f pZqs " 0 for any f once differentiable, where

L r,`f pyq" 1 ´y `r `¯f 1 pyq´yfpyq. The Stein equation L r,`f pyq"gpyq´E " gpY r,`q ‰ (3.2.2)
has a solution f g which satisfies

}f g } 8 § }g 1 } 8 , }f 1 g } 8 § 2 max ˆ1, 1 r ˙}g 1 } 8 and }f 2 g } 8 § 2 ˆmax ˆ , r ˙}g 1 } 8 `}g 2 } 8 ˙, (3.2.3)
noting that f g is solution of (3.2.2) if and only if h g : x fi Ñ 1 f ´x ´r ¯solves xh 1 pxq`pr ´ xqhpxq"gpxq´E rgpY r, qs , studied in [START_REF] Arras | A stroll along the gamma[END_REF][START_REF] Döbler | The gamma stein equation and non-central de jong theorems[END_REF].

Theorem 3.2.8 (‹). Let F is the set of twice differentiable functions with first and second derivative bounded by 1.T h e r ee x i s t sc °0 such that for any F P Dom D with E rFs"0,

d 1 pP F , P Y r, q § c E «ˇˇˇˇˇ1 F `r 2 ´ÿ aPA D a Fp´D a L ´1qF ˇˇˇˇ `c ÿ aPA E "ª E A ´FpX A q´FpX A a ; xq ¯2dP a pxq| D a L ´1F| ⇢ . (3.2.4)
This theorem reads exactly as Theorem 1.5 in [START_REF] Döbler | The gamma stein equation and non-central de jong theorems[END_REF] for Poisson functionals and is proved in a similar fashion.

Remark 3.2.9. The generalization of this result to multivariate Gamma distribution could be considered in a forthcoming paper. The difficulty lies in the regularity estimates of the solution of the Stein equation associated to multivariate Gamma distribution, which require lengthy calculations.

An homogeneous sum of order d is a functional of independent identically distributed random variables pX 1 , ¨¨¨, X Nn q, of the form

F n pX 1 , ¨¨¨, X Nn q" ÿ 1 §i 1 ,¨¨¨,i d §Nn f n pi 1 , ¨¨¨,i d q X i 1 ...X i d
where pN n ,n • 1q is a sequence of integers which tends to infinity as n does and the functions f n are symmetric on t1, ¨¨¨, N n u d and vanish on the diagonal. The asymptotics of these sums have been widely investigated and depend on the properties of the function f n .F o rd " 2,s e e for instance Götze and Tikhomirov [START_REF] Götze | Asymptotic distribution of quadratic forms and applications[END_REF]. In Nourdin, Peccati and Reinert [START_REF] Nourdin | Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos[END_REF], the case of any value of d is investigated through the prism of universality: roughly speaking (see Theorem 4.1), if F n pG 1 , ¨¨¨, G Nn q converges in distribution when G 1 , ¨¨¨, G Nn are standard Gaussian random variables then F n pX 1 , ¨¨¨, X Nn q converges to the same limit whenever the X i 's are centered with unit variance and finite third order moment and such that

max i ÿ 1 §i 2 ,¨¨¨,i d §Nn f 2 n pi, i 2 , ¨¨¨,i d q nÑ8 ›››Ñ 0.
For Gaussian random variables, the functional F n belongs to the d-th Wiener chaos. Combining the algebraic rules of multiplication of iterated Gaussian integrals and the Stein-Malliavin method, it is proved in Nourdin and Peccati [START_REF] Nourdin | Noncentral convergence of multiple integrals[END_REF] that F n pG 1 , ¨¨¨, G Nn q converges in distribution to a chi-square distribution of parameter ⌫ if and only if

E " F 2 n ‰ nÑ8 ›››Ñ 2⌫ and E " F 4 n ‰ ´12 E " F 3 n ‰ ´12⌫ 2 `48⌫ nÑ8 ›››Ñ 0.
We obtain here a related result for d " 2 (for the sake of simplicity though the method is applicable for any value of d) and a general distribution without resorting to universality. Let A "t1, ¨¨¨,nu.F o rf, g :A 2 Ñ R, symmetric functions vanishing on the diagonal, define the two contractions by pf ‹ 1 1 gqpi, jq"

ÿ kPA f pi, kqgpj, kq, pf ‹ 1 2 gqpiq" ÿ jPA f pi, jqgpi, jq. (3.2.5)
Theorem 3.2.10 (‹). Let X A "t X i , 1 § i § nu be a col lection of centered independent random variables with unit variance and finite moment of order 4. Define

FpX A q" ÿ pi,jqPA ‰ f pi, jq X i X j
where pi, jqPA ‰ means that we enumerate all the couples pi, jq in A 2 with distinct components and f is a symmetric function which vanishes on the diagonal. Let ⌫ " ∞ pi,jq f 2 pi, jq.T h e n , there exists c ⌫ °0 such that

d 2 2 pP F , P Ȳ⌫{2,1{2 q § c ⌫ E " X 4 1 ‰ 2 ˆ» - ÿ pi,aqPA 2 f 4 pi, aq`}f ‹ 1 2 f } 2 L 2 pAq `}f ´f ‹ 1 1 f } 2 L 2 pA 2 q fi fl . (3.2.6)
We now introduce Inf a pf q, called the influence of the variable a,b y

Inf a pf q" ÿ iPA f 2 pi, aq. Remark that ÿ iPA f 4 pi, aq § ÿ aPA ÿ i f 2 pi, aq ÿ j f 2 pj, aq " ÿ aPA ÿ i f 2 pi, aq Inf a pf q § ⌫ max aPA Inf a pf q.
The same kind of computations can be made for }f ‹ 1 2 f } 2 L 2 pAq . As a consequence, we get the following corollary.

Corollary 3.2.11 (‹). With the same notations as above,

d 2 2 pP F , P Y ⌫{2,1{2 q § c ⌫ E " X 4 1 ‰ 2 " max aPA Inf a pf q`}f ´f ‹ 1 1 f } 2 L 2 pA 2 q ⇢ .
The supremum of the influence is the quantity which governs the distance between the distributions of F n pG 1 , ¨¨¨, G Nn q and F n pX 1 , ¨¨¨, X Nn q in Nourdin, Peccati and Reinert [START_REF] Nourdin | Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos[END_REF], thus it is not surprising that it still appears here.

Remark 3.2.12. It may be interesting to compare the result of Theorem 3.2.10 with Theorem 4.3 (by taking a dimension d " 2) of Nguyen Tien Dung [START_REF] Dung | Explicit rates of convergence in the multivariate clt for nonlinear statistics[END_REF], which led to a publication slightly subsequent to ours. Similar assumptions are done on the "contractions" (without being explicitely mentioned in [START_REF] Dung | Explicit rates of convergence in the multivariate clt for nonlinear statistics[END_REF]) of the components of matrix pf pi, jqq i,j but more regularity on test functions that must be three differentiable is needed in [START_REF] Dung | Explicit rates of convergence in the multivariate clt for nonlinear statistics[END_REF] whereas we request twice differentiability in our statement. This multivariate CLT for quadratic forms is in fact studied as an application of a general work on multivariate CLT for nonlinear statistics, which Nguyen Tien Dung gives explicit bounds in Theorems 3.1 and 3.2. These are obtained using Stein's method and by means of a difference operator which definition coincides with our discrete gradient (see Definition 2.1.2), reminiscent also of the works of G. Lugosi, P. Massart, C. Houdré, N. Privault, or S. Chatterjee. Without constructing a discrete Malliavin calculus as in our formalism, Nguyen Tien Dung stated a covariance formula and an approximate chain rule on which are based his results. The use of the Slepian's interpolation method cleverly allows to bypass the computational difficulties involved in upper dimension, and its combination with Stein's method is advantageous.

Theorem 3.2.13. The moments of F (up to the fourth one) are linked by the relation:

E " F 4 ‰ ´12E " F 3 ‰ ´12⌫ 2 `48⌫ " ÿ pi,jqPA ‰ f 4 pi, jq E " X 4 ‰ 2 `6 ÿ pi,j,kqPA ‰ f 2 pi, jqf 2 pi, kqE " X 4 ‰ `12 E " X 3 ‰ 2 $ & % ÿ pi,j,kqPA ‰ f 2 pi, jq f pi, kq f pk, jq´ÿ pi,jqPA ‰ f 3 pi, jq
, .

-

´48 $ & % ÿ pi,j,kqPA ‰ f pi, jqf pi, kqf pk, jq´f 2 pi, jq , . - ´12 ÿ pi,jqPA ‰ f 4 pi, jq.
(3.2.7) Remark 3.2.14. The Cauchy-Schwarz inequality entails that the properties

E " F 4 n ‰ ´12E " F 3 n ‰ ´12⌫ 2 `48⌫ nÑ8 ›››Ñ 0 and d 2 pP F , P Ȳ⌫{2,1{2 q nÑ8 ›››Ñ 0
share the same sufficient condition:

ÿ pi,aqPA ‰ f 4 pi, aq`}f ‹ 1 2 f } 2 L 2 pAq `}f ´f ‹ 1 1 f } 2 L 2 pA 2 q nÑ8 ›››Ñ 0.
However, we cannot go further and state a fourth moment theorem as we know, that for Benoulli random variables, F n may converge to Y ⌫{2,1{2 while the RHS of (3.2.6) does not converge to 0.

As another corollary of Theorem 3.2.10, we obtain the KR distance between a degenerate Ustatistics of order 2 and a Gamma distribution. Compared to the more general result stated by Christian Döbler and Giovanni Peccati (Theorem 1.1 in [START_REF] Döbler | The gamma stein equation and non-central de jong theorems[END_REF]), the computations are here greatly simplified by the absence of exchangeable pairs. Theorem 3.2.15 (‹). Let A "t 1, ¨¨¨,nu and pX i ,i P Aq af a m i l yo fi n d e p e n d e n ta n d identically distributed real-valued random variables such that

E rX 1 s"0, E " X 2 1 ‰ " 2 and E " X 4 1 ‰ † 8.

Consider the random variable

F " 2 n ´1 ÿ pi,jqPA ‰ X i X j .
Then, there exists c °0,i n d e p e n d e n to fn,s u c ht h a t

d 2 ´PF , P Y 1{2,1{2 2 ¯ § c 2 ? n E " X 4 1 ‰ . (3.2.8)
Proof. Take f n pi, jq"2{pn ´1q and apply Theorem 3.2.10.

Remark 3.2.16. The proof of Theorem 3.2.10 is rich of insights. In Gaussian, Poisson or Rademacher contexts, the computation of L ´1F is easily done when there exists a chaos decomposition since L operates as a dilation on each chaos (see [START_REF] Nourdin | Stein's method on wiener chaos[END_REF][START_REF] Nourdin | Normal Approximations with Malliavin Calculus: From Stein's Method to Universality[END_REF][START_REF] Peccati | Stein's method and Normal approximation of Poisson functionals[END_REF]). In Lemma 3.4 of [START_REF] Reitzner | Central limit theorems for U -statistics of Poisson point processes[END_REF], Matthias Schulte and Matthias Reitzner provide a formula for the operator L ´1 of Poisson driven U -statistics, not resorting to the chaos decomposition. It is based on the fact that L applied to a U -statistics F of order k yields kF plus a U -statistics of order pk ´1q.

Then, the construction of an inverse formula can be made by induction. In our framework, the action of L on a U -statistics yields kF plus a U -statistics of order k so that no induction seems possible. However, for an order kU-statistics which is degenerate of order pk ´1q,w e have LF " kF.F o rk " 2, this hypothesis of degeneracy is exactly the sufficient condition to have a convergence towards a Gamma distribution.

Finance and Malliavin calculus

The first one is the computation of one of the Greeks, so named with reference to the Greek letters by which they are denoted. In the framework of a financial market, these quantities simulate the sensitivity of the price of derivatives -such as options -to a change in underlying parameters of the model. Their computation is performed by the use of the Malliavin integration by parts formula; in the framework of Black-Scholes model, see for instance Nualart [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF], Montero and Kohatsu-Higa [START_REF] Kohatsu-Higa | Malliavin calculus in finance[END_REF]. The idea is to rewrite Rho (the one that interests us here)

⇢ " BΦpS T q Br " E Q " Bf pS T q Br ⇢ (3.3.1)
where Q denotes the risk-neutral probability measure and f : R `Ñ R is a function, as E Q rf pS T q ⇡s, where the weight ⇡ does not depend of the payoff. The transformation step can be advantageously made simple using integration by parts in the sense of Malliavin calculus as shown first by Éric Fournié, Jean-Michel Lasry, Jérôme Lebuchoux, Pierre-Louis Lions and Nizar Touzi in [START_REF] Fournié | Applications of malliavin calculus to monte carlo methods in finance[END_REF] and [START_REF] Fournié | Applications of Malliavin calculus to Monte Carlo methods in finance[END_REF].

The application of our formalism to the computation of Rho takes shape in the trinomial model described as follows ; given T P N ˚, denote N T " N Xr0,Ts and consider the state space E T " ± N T t"1 E t , where the E t are the 3-elements sets E t "t 1, 0, 1u (d " 1), equipped with the product b tPN T µ t where µ t pt1uq " p, µ t pt´1uq " q and µ t pt0uq " 1 ´p ´q.

The distribution of the canonical process pX t q tPN T is given by PpX t " 1q"p, PpX t "´1q"q and PpX t " 0q"1 ´p ´q.

The trinomial model, is, within this frame, a simplified discrete market model consisting of two assets: ' a riskless asset pA t q tPN T , ' a risky asset pS t q tPN T , where pA t q tPN T modelises the riskless asset with deterministic initial value A 0 " a 0 and is defined by

A n " a 0 p1 `rq n , (3.3.2)
whereas the stock price pS t q tPN T with (deterministic) initial value S 0 " 1 is given by

S t " $ & % p1 `bqS t´1 if X t " 1 S t´1 if X t " 0 p1 `aqS t´1 if X t "´1
pP Tri q with a and b real such that ´1 † a † r † b. Define the sequence pS t q tPN T of discounted stock prices by

S t " 1 p1 `rq n S t , p0 † r † bq.
Assume in this section that P verifies the risk-neutral condition i.e. bp `aq " r, (3.3.3) which ensures that the sequence pS t q tPN T is a pP, Fq-martingale.

The aim of this subsection is to compute, in the frame of trinomial model, the greek called ⇢, i.e. the sensitivity of the price with respect to the parameter r that is

d dr E rf pS N prqqs " lim hÑ0 1 h `E rf pS N pr `hqqs ´E rf pS N prqqs ˘,
by considering the price as a function of r. The other ones, with respect to which it could have been interesting to study the variations, such as the initial value s 0 (if S 0 " s 0 )o rt h e volatility are fixed here.

To this end, consider P the reference probability under which corresponds to the undisturbed evolution of the price and P h the probability corresponding to a h-pertubation. Denote by E and E h the expectations respectively taken with respect to P and P h .

This leads thus to compute, for a given functional F,

lim hÑ0 1 h `Eh rFs´E rFs ˘. (3.3.4)
Considering a small enough variation h, the existence of a random variable L h , nonnegative P-p.s. such that dP h " L h dP, and so that p3.3.4q is equal to

E " F dL h dh ˇˇh "0 ⇢ .
We can thus establish:

Theorem 3.3.1 (‹). Let the process pS t q 0 §t §N T i whose evolution is given by pP Tri q.F o ra n y bounded functional f P F N ,

d dr E rf pS N prqqs " E " f pS N prqq dL h dh ˇˇh "0 ⇢ ,
where the expression of dL h dh ˇˇh "0 is provided by

dL h dh ˇˇh "0 " N T ÿ t"1 # ˆkprq`gprq p 1 pUt"1q `kprq´gprq q 1 pUt"´1q ´gprq s 1 pU k "0q ˙t´1 π `"1 L h `pU `q+ . (3.3.5)
where k and g are the functions defined on R `by kprq" ? ∆t p✓qp p✓q´✓ 1 p✓qq 2 3 p✓q and gprq" ∆t✓ p✓q´ 1 p✓qp1 `∆t✓ 2 q 3 p✓q .

Proofs of chapter 3

3.4.1 Proofs of Section 3.1

Proof of Theorem 3.1.2. Take care that in the argument of h, all the sets are considered as ordered : when we write B Y C, we implicitly reorder its elements, for instance hpX t1,3uYt2u q"hpX 1 , X 2 , X 3 q.

Apply the Clark formula, for any integer n.F o rn " 1, it is straightforward that

U n ´✓ " ˆn m ˙´1 ÿ APprns,mq ÿ BÄA ˆm |B| ˙´1 1 |B| ÿ bPB D b E rhpX A q|X B s " ˆn m ˙´1 ÿ BÄrns ˆm |B| ˙´1 1 |B| ÿ bPB ÿ AÅB APprns,mq D b E rhpX A q|X B s " ˆn m ˙´1 ÿ BÄrns ˆm |B| ˙´1 1 |B| ÿ bPB ÿ CPprnszB,m´|B|q D b E rhpX BYC q|X B s .

It remains to prove that

g 1 pX 1 q"hpX 1 q´✓ " D 1 E rhpX 1 q|X 1 s .
Assume the existence of an integer n such that (3.4.1) holds for any set of cardinality n.I n particular, for any l Prn `1s

m ÿ k"1 ˆm k ˙Hpkq A `" ˆn m ˙´1 ÿ BÄrA `s,|B| §m ˆm |B| ˙´1 1 |B| ÿ bPB ÿ CPprA `szB,m´|B|q D b E rhpX BYC q|X B s ,
where A `"r n `1szt`u. Let m such that m § n. Then,

m ÿ k"1 ˆm k ˙Hpkq n`1 " m ÿ k"1 ˆm k ˙ˆn `1 k ˙´1 1 n `1 ´k n`1 ÿ l"1 ÿ BPprA `s,kq g k pX B q " 1 n `1 n`1 ÿ l"1 m ÿ k"1 ˆm k ˙ˆn k ˙´1 ÿ BPprA `s,kq g k pX B q " 1 n `1 n`1 ÿ l"1 ˆn m ˙´1 ˆÿ BÄrA `s,|A `| §m ˆm |B| ˙´1 1 |B| ÿ bPB ÿ CPprA `szB,m´|B|q D b E rhpX BYC q|X B s " n `1 ´m n `1 ˆn m ˙´1 ˆÿ BÄrn`1s,|B| §m ˆm |B| ˙´1 1 |B| ÿ bPB ÿ CPprn`1szB,m´|B|q D b E rhpX BYC q|X B s " ˆn `1 m ˙´1 ˆÿ BÄrn`1s,|B| §m ˆm |B| ˙´1 1 |B| ÿ bPB ÿ CPprn`1szB,m´|B|q D b E rhpX BYC q|X B s ,
where we have used in the first line that each subset B of rn `1s of cardinality k appears in n `1 ´k different subsets A `(for l Prn `1szB), and in the same way, in the penultimate line, that each subset B Y C of rn `1s of cardinality m appears in n `1 ´m different subsets A (for `Prn `1szB Y C). Eventually, the case m " n `1 follows from

n`1 ÿ k"1 ÿ BPprn`1s,kq g k pX B q"hpX rn`1s q´✓ " ÿ BÄrn`1s ˆn `1 |B| ˙´1 1 |B| ÿ bPB D b E " hpX rn`1s q|X B ‰ ,
by applying the Clark formula to h.

Proof of Theorem 3.1.6. By the previous construction, for

i "pi 1 , ¨¨¨,i N qPpI k " kqX N £ m"k`1 pI m ‰ kq,
the permutation " Γpiq admits k as a fixed point. Hence, #

pI k " kqX N £ m"k`1 pI m ‰ kq + Ä p ŨN k " 1q.
As both events have cardinality pN ´1q!, they do coincide. The values of p k and ↵ k are easily computed since the random variables pI m ,k § m § Nq are independent. According to Theorem 2.2.6,

r U N k " E " r U N k ı `N ÿ `"1 D `E " Ũk | q F `´1 ı " E " r U N k ı `N ÿ `"1 E " r U N k | q F `´1 ı ´E " r U N k | q F `ı . Since r U N k P q F k´1 , D `E " Ũk | q F `´1 ı " 0 for ` † k.F o r`" k, we get E « 1 pI k "kq N π m"k`1 1 pIm‰kq | I k ,I k`1 , ¨¨¨ ´E « 1 pI k "kq N π m"k`1 1 pIm‰kq | I k`1 ,I k`2 , ¨¨¨ " ´1pI k "kq ´Pk ptkuq ¯N π m"k`1 1 pIm‰kq . For `" k `1, E « 1 pI k "kq N π m"k`1 1 pIm‰kq | I k`1 ,I k`2 , ¨¨¨ ´E « 1 pI k "kq N π m"k`1 1 pIm‰kq | I k`2 ,I k`3 , ¨¨¨ " tp k ´1pI k`1 ‰kq ´Pk`1 ptku c q ¯N π m"k`2 1 pIm‰kq "´tp k ´1pI k`1 "kq ´Pk`1 ptkuq ¯N π m"k`2 1 pIm‰kq .
The subsequent terms are handled similarly and the result follows.

Proof of Theorem 3.1.7. By the very definition of r

C 1 ,w eh a v e r C 1 " E " r C 1 ı `N ÿ k"1 N ÿ `"k D `E " r U N k | q F `´1 ı . (3.4.2)
For k " `,

E " r U N k | q F `´1 ı " r U N k and for `°k, E " r U N k | q F `´1 ı " t t `k ´1 ˆ1 ´1 t `k ˙... ˆ1 ´1 t ``´2 ˙N π m"`1 pIm‰kq " t t ``´2 N π m"`1 pIm‰kq . It is straightforward that `°k, D `˜N π m"`1 pIm‰kq ¸" ˆ1pI `‰kq ´p1 ´1 t ``´1 q ˙N π m"``1 1 pIm‰kq "´ˆ1 pI `"kq ´1 t ``´1 ˙N π m"``1 1 pIm‰kq .
The result then follows by direct computations.

Proof of Theorem 3.1.9. Recall that for j ‰ l,

D `E " r U N k | q F `´1 ı and D j E " r U N m | q F j´1 ı are or- thogonal in L 2 .
In view of (3.4.2), according to the integration by parts formula, we have

Varr r C 1 s" N ÿ k"1 N ÿ m"1 N ÿ `"k N ÿ j"m E " D `E " ŨN k | q F `´1 ı D j E " ŨN m | q F j´1 ıı " N ÿ k"1 N ÿ m"1 N ÿ `"k_m E " D `E " r U N k | q F `´1 ı D `E " r U N m | q F `´1 ıı " 2 N ÿ k"1 N ÿ m"k`1 N ÿ `"m E " r U N k D `E " r U N m | q F `´1 ıı `E « N ÿ k"1 N ÿ `"k ŨN k D `E " ŨN k | q F `´1 ı .
Then, for `• m °k,

E " r U N k D `E " r U N m | q F `´1 ıı "´t t ``´2 E » -1 pI k "kq N π p"k`1 1 pIp‰kq ˆ1pI `"mq ´1 t ``´1 ˙N π j"``1 1 pI j ‰mq fi fl "´t P k ptkuq t ``´2 ˆP`p tmuq ´1 t ``´1 ˙E » - `´1 π p"k`1 1 pIp‰kq fi fl E » - N π p"``1 1 pIpRtk,muq fi fl " 0, since, for any l • m °k E " 1 pI `"mq 1 pI `‰kq ‰ " E " 1 pI `"mq ‰ " P `ptmuq " 1 t ``´1
.

Furthermore, for `°k,

E " ŨN k D `E " ŨN k | q F `´1 ıı "´t t ``´2 E » -1 pI k "kq N π p"k`1 1 pIp‰kq ˆ1pI `"kq ´1 t ``´1 ˙N π p"``1 1 pIp‰kq fi fl " t pt ``´1qpt ``´2q P k ptkuqE » - N π p"k`1 1 pIp‰kq fi fl " t 2 pt ``´1qpt ``´2qpt `N ´1q , as ± N p"k`1 1 pIp‰kq 1 pI `"kq " 0, for `°k. Finally, for `" k, we get E " ŨN k D `E " ŨN k | q F `´1 ıı " E » -1 pI k "kq N π p"k`1 1 pIp‰kq ˆ1pI k "kq ´t t `k ´1 ˙N π p"k`1 1 pIp‰kq fi fl " ˆt t `k ´1 ´t2 pt `k ´1q 2 ˙t `k ´1 t `N ´1 " tpk ´1q pt `k ´1qpt `N ´1q ¨ It follows that Varr r C 1 s" t 2 t `N ´1 N ÿ k"1 N ÿ `"k`1 1 pt ``´1qpt ``´2q `t t `N ´1 N ÿ k"1 k ´1 t `k ´1 " t t `N ´1 ˜Nt t `N ´1 `N ´2t N ÿ k"1 1 t `k ´1 ¸.
The proof is thus complete.

Proofs of Section 3.2

Proof of Theorem 3.2.3. We have to compute

sup 'PT E " ' 1 pFq´F'pFq ‰ ,
where T is the set of twice differentiable functions with second order derivative bounded by 2. Since F is centered,

E rF'pFqs " E " LL ´1F 'pFq ‰ " ÿ aPA E " p´D a L ´1qFD a 'pFq ‰ .
The trick is to use the Taylor expansion taking the reference point to be X 1 a instead of X A . This yields

D a 'pFq"E 1 " 'pFpX A qq ´'pFpX 1 a , X 1 a qq ‰ " ' 1 pFpX 1 a qqD a F `R, where R " 1 2 ª 1 0 E 1 " ' 2 ´✓FpX 1 a q`p1 ´✓qFpX A q ¯´FpX A q´FpX 1 a q ¯2⇢ d✓. Hence E " ' 1 pFq´F'pFq ‰ " E « ' 1 pFq´ÿ aPA ' 1 pFpX 1 a qq D a Fp´D a L ´1qF `ÿ aPA E " R p´D a L ´1qF ‰ .
The rightmost term of the the latter equation easily yields the rightmost of (3.2.1). Since }' 2 } 8 † 2, it is clear that ' 1 belongs to Lip 2 hence the formulation of the distance with a supremum.

Proof of Corollary 3.2.6. Without loss of generality, we can assume that X i is centered for any i • 1. Remark that

D j X k " # 0 if j ‰ k, X k if j " k.
Hence LY n " Y n and Y n " L ´1Y n . According to Theorem 3.2.3,

d 1 pP, P Yn q § sup PLip 2 E « pFq´1 s 2 n ÿ iPA ˆF´Y n ´Xi ´X1 i s n ¯˙X 2 i `1 s 3 n n ÿ j"1 E "ª E A `Xi ´x˘2 dP i pxq| X i | ⇢ . By independence, since is 2-Lipschitz continuous, ˇˇˇˇE « pFq´1 s 2 n ÿ iPA ˆF´Y n ´Xi ´X1 i s n ¯˙X 2 i ˇˇˇ" ˇˇˇˇ1 s 2 n ÿ iPA 2 i E " pFq´ ´FpY n ´Xi ´X1 i s n q ¯⇢ˇˇˇˇ § 2 s 3 n ÿ iPA 2 i E " |X i ´X1 i | ‰ § 2 ? 2 s 3 n ÿ iPA 3 i . Moreover, E "ª E A ´Xi ´x¯2 dP i pxq| X i | ⇢ " E " |X i | 3 ‰ ` 2 E r|X i |s § E " |X i | 3 ‰ ` 3 § 2 E " |X i | 3 ‰ ,
according to the Hölder inequality. Hence the result.

Proof of Theorem 3.2.8. According to the principle of Stein's method, we have to estimate

E " 1 ´'pFq`r `¯´F' 1 pFq ⇢ , (3.4.3)
where ' and its derivatives satisfy (3.2.3). For any a P A, thanks to the Taylor expansion,

´Da 'pFq"E 1 " 'pFpX a , X 1 a qq ´'pFpXqq ‰ "´' 1 pFqD a F `R, (3.4.4)
where

R " 1 2 ª 1 0 p1 ´✓qˆE 1 " ' 2 ´p1 ´✓qFpXq`✓FpX a ,X 1 a q ¯´FpXq´FpX a , X 1 a q ¯2⇢ d✓. (3.4.5)
According to (2.1.1) and to the definition of L,

E rF'pFqs " E " LL ´1F 'pFq ‰ " E " ´ pDL ´1Fq'pFq ‰ " E " xD'pFq, ´DL ´1Fy L 2 pAq ‰ . (3.4.6)
Plug (3.4.4) into (3.4.6):

E " xD'pFq, ´DL ´1Fy L 2 pAq ‰ "´ÿ aPA E " D a 'pFq D a pL ´1Fq ‰ "´ÿ aPA E " ' 1 pFq D a FD a pL ´1Fq ‰ `ÿ aPA E " RD a pL ´1Fq ‰ " E " ' 1 pFqxDF, ´DL ´1Fy L 2 pAq ‰ `E " xR, ´DL ´1Fy L 2 pAq ‰ . Then, ˇˇˇE " 1 `pF `r `q' 1 pFq´F'pFq ⇢ ˇˇ § ˇˇˇE " ' 1 pFq ´1 `pF `r `q´xDF, ´DL ´1Fy L 2 pAq ¯⇢ˇˇˇˇ`ˇˇE " xR, ´DL ´1Fy L 2 pAq q ‰ˇ" B 1 `B2 .
Since ' 1 is bounded, we get

B 1 § }' 1 } 8 E " ˇˇ1 `pF `r `q´xDF, ´DL ´1Fy L 2 pAq ˇˇ⇢
and from (3.4.5), we deduce that

B 2 § }' 2 } 8 ÿ aPA E " |D a F| 2 |D a L ´1F| ‰ .
The proof follows from (3.4.3) and (3.2.3).

Proof of Theorem 3.2.10. For any a P A,

D a pX i X j q" $ ' & ' % X a X j if a " i X i X a if a " j 0 otherwise.
Then,

D a F " ÿ pi,aqPA ‰ f pi, aq X i X a `ÿ pj,aqPA ‰ f pa, jq X a X j " 2 ÿ pi,aqPA ‰ f pi, aq X i X a , so that LF "´ÿ aPA D a F "´2F and L ´1F "´F 2 .
With our notations, the first term of the right-hand-side of (3.2.4) becomes

E » - ˇˇˇˇˇ2 F `2⌫ ´2 ÿ aPA ÿ pi,jqPA 2 f pi, aqf pj, aq X 2 a X i X j ˇˇˇˇˇfi fl § 2 ÿ i"1 A i , (3.4.7)
where

A 1 " 2 E » -ˇˇÿ pi,aqPA 2 f 2 pi, aqpX 2 a X 2 i ´1q ˇˇfi fl , A 2 " 2 E » - ˇˇˇˇˇF ´ÿ aPA ÿ pi,jqPA ‰ f pi, aqf pj, aq X 2 a X i X j ˇˇˇˇˇfi fl .
We first control A 1 . According to the Cauchy-Schwarz inequality,

A 2 1 § 4 E » - ÿ pi,aqPA 2 ÿ pj,cqPA 2 f 2 pi, aqf 2 pj, cqpX 2 a X 2 i ´1qpX 2 c X 2 j ´1q fi fl § 4pA 11 `A12 q,
where

A 11 " E » - ÿ pi,aqPA 2 f 4 pi, aqpX 2 a X 2 i ´1q 2 fi fl , A 12 " E » - ÿ aPA ÿ pi,jqPA ‰ f 2 pi, aqf 2 pj, aqpX 2 a X 2 i ´1qpX 2 a X 2 j ´1q
fi fl , by orthogonality of the X i 's. On the one hand,

A 11 § ÿ pi,aqPA 2 f 4 pi, aqE " ´X2 a X 2 i ´1¯2 ⇢ " `E " X 4 1 ‰ 2 ´1˘ÿ pi,aqPA 2 f 4 pi, aq. (3.4.8)
On the other hand,

A 12 " E » - ÿ pi,jq ‰ PA 2 ÿ aPA f 2 pi, aqf 2 pj, aqpX 2 a X 2 i ´1qpX 2 a X 2 j ´1q fi fl § ÿ pi,jq ‰ PA 2 ÿ aPA f 2 pi, aqf 2 pj, aqE " pX 2 a X 2 i ´1qpX 2 a X 2 j ´1q ‰ " `E " X 4 1 ‰ ´1˘ÿ pi,aqPA 2 f 2 pi, aq ÿ j‰i f 2 pj, aq § `E " X 4 1 ‰ ´1˘} f ‹ 1 2 f } 2 L 2 pAq . (3.4.9)
In a similar way, A 2 § A 21 `A22 , where

A 21 " 2 E » -ˇˇˇˇÿ pi,jqPA ‰ f pi, jqX i X j ´ÿ pi,jqPA ‰ ÿ aPA f pi, aqf pj, aqX i X j ˇˇˇˇfi fl , A 22 " 2 E » -ˇˇˇˇÿ pi,jqPA ‰ ÿ aPA f pi, aqf pj, aqX i X j ´X2 a ´E " X 2 a ‰ ¯ˇˇˇˇfi fl .
As above,

A 2 21 § 4 E » - ¨ÿ pi,jqPA ‰ ´f pi, jq´ÿ aPA f pi, aqf pj, aq ¯Xi X j '2 fi fl " 4 }f ´f ‹ 1 1 f } 2 2 . (3.4.10)
Furthermore, according to Cauchy-Schwarz inequality and by independence, we have

A 22 § 2 ÿ pi,jqPA ‰ E « |X i X j | ˇˇÿ aPA f pi, aqf pj, aqpX 2 a ´1q ˇˇ § 2 E » - ´ÿ pi,jqPA ‰ ÿ aPA f pi, aqf pj, aqpX 2 a ´1q ¯2fi fl 1{2 § 2 ¨ÿ pi,jqPA ‰ ÿ aPA f pi, aq 2 f pj, aq 2 E " X 4 a ´1‰ '1{2 § 2 `E " X 4 1 ‰ ´1˘1 {2 }f ‹ 1 2 f } L 2 pAq . (3.4.11)
The remainder term is given by

A 3 " ÿ aPA E "ª E A ´FpX A q´FpX A a ; xq ¯2dP a pxq| D a L ´1F| ⇢ .
Once again, using the orthogonality, we have

G a pX A q" ª E A ´FpX A q´FpX A a ; xq ¯2dP a pxq " 4 E 1 « ´ÿ iPA f pi, aqX i X a ´ÿ iPA f pi, aqX i X 1 a ¯2 " 4 E 1 « pX a ´X1 a q 2 ´ÿ iPA f pi, aqX i ¯2 " 4 ´ÿ iPA f pi, aqX i ¯2 E 1 " pX a ´X1 a q 2 ‰ " 4 ´ÿ iPA f pi, aqX i ¯2 `X2 a `1˘.
Thus,

E « ÿ aPA G a pX A q 2 " 16 E « ÿ aPA ´ÿ iPA f pi, aqX i ¯4 `X2 a `1˘2 " 16 `E " X 4 1 ‰ `3˘E " X 4 1 ‰ ÿ aPA ÿ iPA f 4 pi, aq `96 `E " X 4 1 ‰ `3˘ÿ aPA ÿ pi,jqPA ‰ f 2 pi, aqf 2 pj, aq § 16 `E " X 4 1 ‰ `3˘2 ÿ aPA ÿ iPA f 4 pi, aq`96 `E " X 4 1 ‰ `3˘} f ‹ 1 2 f } 2 L 2 pAq . (3.4.12) Moreover, ÿ aPA E " |D a L ´1F| 2 ‰ " 1 4 ÿ aPA E " |D a F| 2 ‰ " ÿ aPA E » - ¨ÿ pi,aqPA ‰ f pi, aq X i X a '2 fi fl " ÿ pi,aqPA ‰ f 2 pi, aq"⌫. ( 3

Proofs of Section 3.3

Proof of Theorem 3.3.1. As to constuct the mesure L h , define the measure of probability P h on E T by P h " Â N T t"1 P h t where P h t pU t " 1q"p ˚" p `∆h p, P h t pU t "´1q"q ˚" q `∆h q, and P h t pU t " 0q"s ˚" s `∆h s where s " 1 ´p ´q and with $ ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' % ∆ h p " 1 2 3 p✓q `?∆t p✓qr p✓q´✓ 1 p✓qs `2r∆t✓ p✓q´ 1 p✓qp1 `∆t✓ 2 qs ˘h "rkprq`gprqsh

∆ h s "´∆ t✓ p✓q´ 1 p✓qp1 `∆t✓ 2 q 3 p✓q h "´gprqh ∆ h q " 1 2 3 p✓q `´?
∆t p✓qr p✓q´✓ 1 p✓qs `2r∆t✓ p✓q´ 1 p✓qp1 `∆t✓ 2 qs ˘h "r´kprq`gprqsh, since ∆ h p `∆h s `∆h q " 0.

Define thus L h " ± N T t"1 L h t pU t q, where L h t is the random variable defined on E t by

L h t pU t q" p p 1 tUt"1u `qq 1 tUt"´1u `ss 1 tUt"0u , (3.4.14)
and such that the expectation under P h is defined by

E h rFs"E rf pU 1 ,...,U N T qL h s"E « f pU 1 ,...,U N T q N T π t"1 L h t pU t q ,
for any functional F.

Noting that E rL h s"1, it follows from the Clark formula (2.2.3)

L h ´1 " N T ÿ t"1 E rD k L h pU q|F t s " N T ÿ t"1 # `Lt pU t q´1 ˘t´1 π `"1
L `pU `q+ , since, for any r P N T ,

E rD k L h pUq|F r s"E » - N T π `"1 L h `pU `q´E h rL h pU t qs ˆNT π `"1,`‰t L h `pU `qˇF k fi fl " ´Lh t pU t q´1 ¯t´1 π `"1 L h `pU `q.
Note thus that for any t P N T ,

L h t pU t q´1 " ∆ h p p 1 pUt"1q `∆h q q 1 pUt"´1q `∆h s s 1 pUt"0q " rkprq`gprqsh p 1 pUt"1q `rkprq´gprqsh q 1 pUt"´1q ´gprqh s 1 pUt"0q ,
to state (3.3.5). Hence the result.

Part II

Malliavin calculus and enlargement of filtrations in discrete time: additional utility of an insider in a ternary model 107

The starting point for our second work arises in a discrete incomplete market, where we consider two agents with different information levels; the first one, called insider, possesses from the beginning extra information whereas the second one, the ordinary agent, bases his investment decisions on the public information flow. We ask the question, as the common thread of this second part:

How to quantify the additional utility of an insider in a incomplete discrete market model? (3.4.15) The framework in which lies the insider problem is that of a financial market embodied by a probability space pΩ, F, Pq. Its mathematical traduction boils down to the quantitative comparison of a regular agent whose information coincides with the natural filtration F and an insider who possesses some extra information hidden in a random variable G from the beginning ot the trading interval N T "r 0,TsXN. Two main approaches were developped to deal with this problem. Jorge A León, Reyla Navarro and David Nualart [START_REF] León | An anticipating calculus approach to the utility maximization of an insider[END_REF] stated that insider's portfolio satisfied an anticipative stochastic equation and used the tools of anticipating stochastic calculus to determine its maximal utility. The alternative approach, that we have chosen to follow is to address the problem from the point of view of martingale theory, as first envisaged by Karatzas and Pikovsky [START_REF] Pikovsky | Anticipative portfolio optimization[END_REF]. They made the insider evolve in a parallel probability space pΩ, G, Pq where G is an enrichment of the initial filtration F and encompasses the additional information which benefits him. They converted the initial problem into that of a grossissement de filtrations one (see Jacod [START_REF] Jacod | Grossissement initial, hypothèse h et théorème de girsanovr[END_REF]), Yor ([141]) in which they exploited the powerful techniques. In the 90's the question of enlargement of filtrations aroused great interest and generated many works. Plenty applications in finance resulted from the related theories; to name but a few, the questions of arbitrage or indifference pricing can be addressed in terms of initial enlargement, whereas credit risk, or insurance issues can be modelised by a progressive enlargement situation.

In a continuous setting, the Jacod's condition indicates that the absolutely continuity of the conditional laws of G with respect to its law is a sufficient criterion to the preservation of semimartingales. Transposing these works in the referred financial context, Imkeller highlighted the crucial role played by the information drift, i.e. the drift to eliminate in order to preserve the martingale property, and remarkably described it by the logarithmic Malliavin trace of the conditional laws of G with respect ot F. In discrete time, the question of elargement of filtration seems easier. Indeed, Jacod's hypothesis is de facto satisfied in a discrete setting and we are directly provided with a "bridge formula" thanks to Doob's decomposition (see Blanchet-Scalliet, Jeanblanc and Romero [START_REF] Blanchet-Scalliet | Enlargement of filtration in discrete time[END_REF]). This brings us to the question: can we provide an analogue of the information drift in a discrete context and express it in terms of Malliavin derivative?

In the very beginning we tried to fit our formalism to address the insider's problem in a the underlying trinomial model (see its definition in section 3.3). The difficulty came to our Clark-Ocone formula (2.2.3) from which we cannot derive a Karatzas-Ocone hedging formula. Indeed, the F k -measurability of the term D k E rF|F k s appearing in (2.2.3) prevented us from defining for the F-predictable drift process we searched for. This observation was prone to replace the trinomial model with what we called a ternary model, equally distributed to the first one, but enjoying the properties of the jump processes on which it lies proved to be more conducive to state a predictable representation formula.

The second part consists of three chapters; the first one is an overview of enlargement of filtrations techniques and insider relative topics in familiar contexts. The new results are included in the two following chapters. The second chapter is devoted to the construction of Malliavin calculus for compound geometric processes. In the last one, we address the problem of insider trading in the so-called ternary market and apply our new formalism to provide a response to the problem 3.4.15.

Ω is the set of continuous functions on T starting at 0, A the -algebra of Borel sets with respect to the uniform convergence of compact subsets of T, F "pF t q tPT the natural filtration generated by pB t q tPT (assume A " F)a n dP the Wiener measure. The financial market lying on pΩ, F, Pq is defined through the couple of progressively measurable processes pr, q, where r stands for the mean rate of return and the volatility and satisfy the integrability conditions:

ª T 0 |r t | dt † 8 and ª T 0 | t | 2 dt † 8 , P ´a.s.
The evolution of the stock process pS t q tPT is given by:

S t " 1 `ª t 0 r u S u du `ª t 0 u S u dB u ,tP T, (4.1.1)
When the processes r et are deterministic and constant, the dynamics (4.1.1) of the underlying is a geometric Brownian motion and defines the so-called Black-Scholes model. Two agents act on the trading interval T:t h eordinary agent whose information level corresponds to the filtration F i.e. whose knowledge at time t is given by F t and the insider who enjoys an information overload encoded by a F T or possibly F T `"-measurable (for some small " °0) random variable G with values in a Polish space pΓ, G q. Thus, the insider disposes at time t an information given by the -algebra G t defined by

G t " F t _ pGq.
We denote G "p G t q tPT (and G 0 "p G t q tPr0,T q ) the insider's filtration. This depicts the easier case of enlargement of filtration called initial enlargement. The problem of progressive enlargement where G t " F t _ p F t and p F "p p Fq tPT is an another filtration distinct of F sounds to be more difficult, and won't be treated here.

The ordinary agent manages a F-portfolio determined by the proportion of the wealth invested in stocks. It is encoded by a F-progressively measurable process called strategy such that

ª T 0 | t r t | dt † 8 and ª T 0 | t r t | 2 dt † 8, pP, Fq´a.s,
which value Vp q satisfies the differential equation:

dV t p q V t p q " t dS t S t ,tPr0,Ts. (4.1.2)
The maximal logarithmic utility of the ordinary agent whose initial wealth is x is defined via the maximization problem:

Φ F pxq" max PF´portfolio E rlogpV x t p qs , (4.1.3)
The insider's portfolio process, value and maximal logarithmic utility are analogously defined by systematically replacing F by G. Then, the additional expected logarithmic of the insider with initial wealth x is defined by:

U pxq"Φ G pxq´Φ F pxq (4.1.4)
where

Φ G " max PG´portfolio E " logpV x t p G q ‰ . (4.1.5)
Introduce now the notion of arbitrage. From an economic point of view, an arbitrage designates a portfolio (i.e. the strategy it refers to) such that even if its initial value is zero, its terminal value is strictly positive, in other words, an "always winning strategy" called free lunch. Thus there exists an arbitrage opportunity in a financial market when making a profit without risk and without net investment of capital is possible. Mathematically, that translates as follows:

Definition 4.1.1. Given a filtration F, consider a F-semimartingale S. The model pS, Fq has no arbitrage if there exists a positive F-martingale L with L 0 " 1 such that LS is a F-martingale.

A financial market with no arbitrage opportunities is said to be arbitrage free. Two questions naturally arise: how to estimate the additional maximal logarithmic utility of the insider? Does the additional benefiting the insider provides him with an arbitrage opportunity?

For a given of F-portfolio , the solution of (4.1.2) expressed as

V t " V 0 exp ˆª t 0 s s dB s ´1 2 ª t 0 2 s 2 s ds `ª t 0 s r s ds
ṡuch that the solution of (4.1.3) from the ordinary agent's point of view is provided by Merton's formula

Φ F pxq" 1 2 E "ª T 0 r 2 s 2 s ⇢ ds,
which proof lies on the local martingale property of the stochastic integral ≥ t 0 s s dB s . Since the process pB t q tPT is not a G-martingale, the stochastic integral ≥ t 0 s s dB s is not a Gmartingale any more, so that the previous computations no longer hold. This led to the conservation of martingale properties face to an enlargement of filtrations.

Main instigator of these questions, Jacod showed (Jacod Théorème 2.1 [START_REF] Jacod | Grossissement initial, hypothèse h et théorème de girsanovr[END_REF]) that every continuous local F-martingale is a G 0 -martingale under the so-called Jacod's hypothesis : Assumption. The regular conditional law of G given F t is absolutely continuous with respect to the the law of G, P almost everywhere, for all t Pr0,Tq. Its reinforcement, assuming of equivalence of the involved distributions (instead of simple absolute continuity) enables to define the Radon-Nikodym density process of the conditional laws of G with respect to its law by

p c t p!q" dPpG P¨|F t qp!q dPpG P¨q pcq ; c P Γ, ! P Ω, (4.1.6)
that will be of relevance, and prone to exploit the techniques of Malliavin calculus. Provided Jacod's condition (4.1.6) holds, let the p G t q tPr0,T s be the G-progressively mesurable process defined by

G t " d dt xp t p¨,cq, B t y t p t p¨,cq ˇˇˇc "G
that we suppose to satisfy the additional integrability condition

ª T 0 | G t | dt † 8, P ´a.s.
Then the process

B " r B `ª 0 | G s | ds is a G-semimartingale with a G-Brownian motion r B.
The maximal logarithmic utility of the insider whose initial wealth is x defined by (4.1.4) can be expressed as

U " 1 2 E "ª T 0 ` G s ˘2 ds ⇢ ,
so that the knowledge in advance of the information contained in the random variable G is quantifiable in terms of energy of the so-called information drift G . Extending the Clark-Ocone formula to measure valued martingales in order to apply it to the conditional laws of G,

p t " p ò `ª t 0 D s p s dB s P.
Imkeller made clear the relationship between the random variable G and the information drift that he identified -via the remarkable formula -

G t " D t p t p¨,cq p t p¨,cq ˇˇc"G " D t `ln p t p¨,cq ˘ˇc "G ,
with a logarithmic Malliavin derivative trace of the conditional density.

Until the end of the section, consider a discrete setting which consists of a probability space pΩ, F, Pq where F "pF t q tPN T . The evolution of the discounted stock process pS t q tPN T initially defined by S 0 " 1 is didacted by pr t q tPN T through the equation :

∆S t S t " f t p↵ t , X t , ∆X t q,t P N T .
where ∆S t " S t ´St´1 (t P N 0 T ), pX t q tPN T is a F-adapted process and f t a function defined on R `ˆX t pΩqˆ∆X t pΩq. For instance, in the so-called Cox-Ross-Rubinstein model, pX t q tPN T is a Rademacher process and

f t pr t , X t , ∆X t q" 1 1 `r ´b1 tXt"1u `a1 tXt"´1u ´1¯( 4.1.7)
In other words, the dynamics of the stock price pS bin t q tPN T satisfying the recurrent relation

S bin t " " p1 `bqS bin t if X t " 1 p1 `aqS bin t if X t "´1
The ordinary agent manages a F-portfolio determined by the proportion of the wealth invested in stocks encoded by a F-progressively measurable process value Vp q (and discounted value Vp q) satisfies the self-financed condition given here by an equation in differences:

∆V t p q V t p q " t ∆S t S t ,tP N 0 T . (4.1.8)

HEDGING AND PORTFOLIO'S OPTIMIZATION IN INCOMPLETE MARKETS115

The difficulties inherent with the preservation of semimartingales are directly lifted as a consequence of Doob's decomposition. Indeed it is clear (see Blanchet-Scalliet, Jeanblanc and Romero [START_REF] Blanchet-Scalliet | Enlargement of filtration in discrete time[END_REF]) that any integrable process is a special semimartingale in any filtration with respect to which it is adapted. The specific case where the F T -measurable G takes its values in a countable set Γ arouses interest since the enlargement of filtrations can be also interpreted in terms of drift information. Within this framework, define for any c P Γ,t h eF-martingale p c by

p c t " PpG " c | F t q
Then the process S G defined by

S G t " S t ´t ÿ s"1 E " ∆S s ∆p c s | F s´1 ‰ ˇˇc "G p G s´1 (4.1.9)
is a G-martingale. We identify thus a drift information which one of the challenges of Chapter 3 will be to provide an expression in terms of Malliavin calculus (for a discrete setting to be precised).

Theorem 4.1.2. If the random variable G is not F 0 -measurable, the model pS, Gq is not arbitrage free.

Hedging and portfolio's optimization in incomplete markets

Along with the computation of the maximal utilities of the two traders (the ordinary agent and the insider) as indicated in the previous section, the determination of the financial strategy to follow in order to realize the maximum is of interest; starting with that of the traders themselves. In other words, assuming that Φ F pxq is known (for the ordinary agent), we wonder how to determine a F-predictable process which value Vp q satisfying (4.1.2) and such that (if T " N T in a discrete setting)

V 0 p q"x, V T p q"Φ F pxq and V t p q • 0, for any t P N T . This turns out to estimate the quantities of the risk-free and risky assets with which the agent must compose his portfolio to maximize his utility. This leads us to tackle with the question of hedging.Ahedge is basically an investment that protects the finances of a trading agent. Mathematically, this is the response to the problem: given a claim F, i.e. a F T -measurable random variable F, is there a F-predictable process such that (if T " N T in a discrete setting)

V T p q"V 0 p q`T ÿ t"1 t ∆S t " F and V t p q • 0 for any t P N T ?

The section consists of two parts; in the first one, we indicate how a martingale approach (via martingale representation theorem) performs advantageously to solve the problem of hedging in complete markets. Focusing on incomplete markets in the following one, we drop this approach for technical reasons (as it will be explained) and tackle the problem from the standpoint of optimization issues.

The financial market is complete if all claims are reachable i.e. the answer to the question 4.2.1 is always positive. This is the case of the Black-Scholes model (see (4.1.1)) in a continuous paradigm or the Cox-Ross-Rubinstein (see (4.1.7)) model in a discrete setting. The Fundamental Theorem of Asset Pricing holds in both frameworks; we implicitely evokated when we assumed in section 4.1 that the sequence of discounted prices pS t q tPT was a pP, Fq-martingale.

In its most basic form it reads as follows:

Theorem 4.2.1 (Fundamental Theorem of Asset Pricing). The financial market lying on pΩ, F, Pq is arbitrage-free if and only if there exits a probability measure P ‹ ,e q u i v a l e n tt o P,a n ds u c ht h a tt h es e q u e n c epS t q tPN T is a pP ‹ , Fq-martingale. The measure P ‹ is a riskneutral measure. If the market is further assumed to be complete, there is a unique risk-neutral measure.

The set of measures with respect to which the sequence of discounted prices pS t q tPT is a p¨, Fqmartingale will be called the set of martingale measures and denoted by C F . The subset of C F composed of the martingale measures equivalent to P is denoted M F . The Theorem 4.2.1 ensures that in a complete market, the set M F is reduced to the singleton tP ˚u.

The result is of key importance to tackle with the problems of pricing or hedging. Indeed, by systematically reason under risk-neutral measure(s) we benefit from all tools the martingale theory is provided with.

In the Black-Scholes model, the value of the portfolio "p↵, 'q denoted by the process Vp q is given at time t P T by

V t p q"↵ t A t `'t S t , (4.2.2)
where ↵ t and ' t stand respectively for the amount of the riskless asset A t and the amount of risky asset S t composing the portfolio at time t. for any t P N T .T h es t r a t e g y "p ↵, 'q satisfies the self-financing (4.2.3) assumption and simulates F.

In the Cox-Ross-Rubinstein which underlying Rademacher space is also equipped with a Malliavin calculus (2.0.2). The value of the portfolio process V p q is defined by (4.2.2) with the discrete time corresponding notations, and the self-financing assumption (4.1.8) boils down to: for any t P N T ´1, A t p↵ t`1 ´↵t q`S t p' t`1 ´'t q"0. (4.2.4)

In discrete time models, a strategy is admissible if it the self-financing (4.2.4) assumption and satisifes for any t P N T , V t p q • 0.

Again, the hedging formula comes out by the application of the Clark fomula to the claim F.

Then we can show that for any t P N T , the set F-martingale measures is described by a polyhedron which the J t " 2 t vertices depict the extremal measure P j t (j P J t ) that be can be written as P j t " â sPNt pP 0,Tri q j s pP 1,Tri q 1´ j s , where p j s q sPNt Pt0, 1u t . As a reminder, the problem to solve is: given an economic agent disposing of x euros at date t " 0 (initial budget constraint), determine the optimal admissible strategy allowing to maximize the value of the portfolio at time t P N T ´1. That is mathematically traduced by:

Φpxq" max PF´portfolio E " upV x,t p q ‰ (4.2.6)
where, for any t P N T ´1, V x,t p q stands for the discounted value at time t of a F-portfolio corresponding to the strategy "p↵ t , ' t q tPN T ´1 and starting at x P R ˚(the initial wealth), and u to an utility real function defined on R ˚or R striclty increasing and strictly concave.

Here consider the utility function u : x P R ˚fi Ñ log x, denote I "pu 1 q ´1 its "inverse" function and u ˚its Legendre-Fenchel tranform , i.e. the convex function defined by

u ˚prq" sup sPR ˚ upsq´sr ( . (4.2.7)
Let S F (resp. S F t ) the set of F-predictable self-financing strategies on N 0 T (resp. on N 0 t ). For x P R ˚, let Λ x,t " V:D P S F , V " V t p q and V 0 " x ( .

As seen above, the trinomial market is not complete and there exists a convex set M F t of Fmartingales measures on N 0 t with a finite number J t of extremal values pP j t ,jP J t q. Denote C F t the convex subset of M F t formed by the F-martingale measures equivalent to P.N o t ei n particular that for any strategy "p↵, 'q with initial value x is written by

V x,t p q"x `t ÿ s"1 ' s ∆S s ,
and satisfies E P j t rV x,t p qs " 0; j P J t . (4.2.8)

As precisely developped by Freddy Delbaen and Walter Schachermayer in [START_REF] Delbaen | The Mathematics of Arbitrage[END_REF], this optimization problem can be solved by a duality approach decomposable in the following steps :

Malliavin calculus for jump processes

The impossibility to state a martingale representation theorem for any sequence of independent random variables as explained in the introduction of the part II, and thus to provide an hedging formula for the trinomial model within our formalism, enhanced us to investigate the option of an alternative financial model. This "model quest" led us to design a ternary model equivalent in distribution to the Cox-Ross-Rubinstein's (see Remark 6.1.1). This new model lying on what we have called a geometric compound process suggests to focus on jump models to understand its behavior. These models are additionally worthy of interest ; contrary to the Black-Scholes model, they integrate the possibility of the occurrence of rare events (sudden change in the international environment for instance, market crashes, gaps or opening jumps).

For more details on the subject, see for instance the Chapter 1 of the book Rama Cont and Peter Tankov [START_REF] Cont | Nonparametric calibration of jump-diffusion option pricing models[END_REF]. Up to know, consider a probability sapce pΩ, A, Pq where are defined a standard Brownian motion pB t q tPR `, a Poisson process pN t q tPR `of intensity and a sequence of i.i.d. p´1, `8qvalued random variables V "p V t q tPR `. The -algebras generated by these three processes are supposed to be independent of each other. The filtration F "pF t q tPR `is defined by

F t " `Bs , N s , V ¨1t¨ §Nsu q,s § tq ˘.
The riskless asset pA t q tPR `is defined as in the Black-Scholes model (see 0.0.11) whereas the price of risky asset pS t q tPR `is occasionally disjoint with significant jumps which values and occurrences are respectively given by pV t q tPR `and pT t q tPR `(namely the times process associated to pN t q tPR `) and follows the Black-Scholes model between two jump times. Its dynamics characterizing a jump-diffusion process is provided in its differential form by: dS t " S t´`r dt ` dB t `dZ t ˘; t P R `,

where Z "pZ t q tPR `defined by

Z t " Nt ÿ s"1 V s , (4.3.1)
is a compound Poisson process.

When V is a derministic constant process, Z coincides with the standard Poisson process on R `. Dean of the family of the jump processes, the latter is equipped with an extensive Malliavin theory, born shortly after that on Wiener space. Indeed, its associated compensated process is in fact a normal martingale so that most of fundamental properties such as martingale representation theorem, decomposition in chaos, Itô formula are immediatly provided and make the theory powerful. For more details on Malliavin calculus for Poisson processes, see for instance chaos expansion (Last and Penrose [START_REF] Last | Martingale representation for poisson processes with applications to minimal variance hedging[END_REF], Privault [START_REF] Privault | Chaotic and variational calculus in discrete and continuous time for the Poisson process[END_REF]) for anticipative calculus (Nualart and Vives [START_REF] Nualart | Anticipative calculus for the Poisson process based on the Fock space[END_REF]).

When V denotes in a more general case an independent sequence of square-integrable random variables identically distributed with a probability distribution ⌫ defined on R, Z belongs to the family of compound Poisson processes, for which a stochastic calculus has already been designed. Extending what has been done before for the standard Poisson process, it is less rich in applications since the compound process does not enjoy all the properties of its illustrious elder. If the independence of increments they share makes possible a characterization in terms of Laplace's transform and to define stochastic integrals satisfying an Itô isometry property, it is however impossible to state a predictable representation property for compound Poisson processes in general. As suggested in the in the remark below, the associated compensated compound process pY t q tPR `does not satisfy a structure equation when the process V is not deterministic constant.

Remark 4.3.1. As explained in the last item the remark p. 95 in Privault [START_REF] Privault | Stochastic Analysis in Discrete and Continuous Settings[END_REF], the quadratic variation of the compensated compound F-martingale pY t q tPR `defined by

Y t " Y t ´ tE rV 1 s ? VarV 1 satisfies rY, Ys t " 1 a VarrV 1 s Nt ÿ s"1 |V s | 2 " 1 a VarrV 1 s ª t 0 |V 1`N s´| 2 V Ns dY s `E rV 1 s a VarrV 1 s ª t 0 |V 1`N s´| 2 V Ns ds.
If this last expression does not allow find a square integrable F-adapted process p t q tPR satisfying the structure equation Despite of this intrinsic absence of predictable representation property, a Malliavin calculus can be elaborated for compound Poisson processes seen as a particular case a the larger class of Lévy processes to wich it belongs (see chapter 6 in [START_REF] Privault | Stochastic Analysis in Discrete and Continuous Settings[END_REF], [START_REF] Oksendal | White noise of poisson random measures[END_REF], [START_REF] Di Nunno | White noise analysis for lévy processes[END_REF]). The formalism thus conceived gives in particular birth to an analogue Clark-Ocone formula, inescapable tool to address hedging and portfolios optimization problems in jump-diffusion models. See for instance the works of Elisa Alòs, Jorge A. León, and Josep Vives [START_REF] Alòs | An anticipating Itô formula for Lévy processes[END_REF], Ioannis Karatzas and Daniel Ocone [START_REF] Ocone | A generalized clark representation formula, with application to optimal portfolios[END_REF].

Conclusion

In the same spirit as the conclusion of chapter 1, we close this one by listing what we need to answer the question: how can we quantify the additional utility of the insider trading on a a financial market bore by a ternary model? The challenge is threefold: we have to understand the mechanics of the underlying compound geometric process, tanspose one of the methods usually used for the optimization of dynamic portfolios to our framework, and transfer all these tools to an insider's point of view. In other words, we need to be equipped with:

• A Malliavin calculus for compound geometric processes and derived functional identities: martingale representation formula, Clark-Ocone type formula...

• A method for portfolio optimization in the incomplete market designed by a ternary model.

• A technique of enlargement of filtrations in this particular discrete setting.

If the chapter 5 will be charged the elaboration of a stochastic variational calculus for compoung geometric process, the questions of enlargement of filtrations and portefolio optimization will be tackled in the last one to lead to the answer to question 3.4.15 of the second part.

Chapter 5

Malliavin calculus for compound geometric processes

As one of the different ways to construct it, we choose to developp a Malliavin calculus for geometric compound process by deriving it from a modified chaotic decomposition of square integrable functionals of that process.

Compound geometric processes

The general framework is that of a probability space pΩ, F, Pq described in this first subsection. We borrow the notations and presentation from the framework introduced for filtered Poisson processes ( [START_REF] Decreusefond | Anticipative calculus with respect to filtered Poisson processes[END_REF]). Let p ,pqPp0, 1q 2 , E "t´1, 1u, T P N ˚. Denote N t " N Xr1,ts and N 0 t " N t Yt0u for any t P N T . We consider pX, T , ⌫q the measured space defined by X " N T ˆE, T " PpXq and ⌫ppt, kqq " ÿ

s•1 s pttuq b `p 1 ptkuq `p1 ´pq ´1ptkuq ˘,
for any pt, kqPN T ˆt´1, 1u, ⌫p0, ¨q " 0 and ⌘ its marginal distribution with respect to the time variable ⌘ptq" 0 pttuq `ÿ s•1 s pttuq ; t P N T .

For any n P N ˚the tensor measure ⌫ bn is defined by:

⌫ bn pt n , k n q"1 Γn ppt n , k n qq n π i"1 ⌫ppt i ,k i qq, (5.1.1)
where pt n , k n q" `pt 1 ,k 1 q, ¨¨¨, pt n ,k n q ˘PpN T ˆEq n and Γ n " ! `pt 1 ,k 1 q, ¨¨¨, pt n ,k n q ˘PpN T ˆEq n q : @i ‰ j, t i ‰ t j )

The space of simple, integer-valued, locally finite measures on X is denoted by Ω.

Remark 5.1.1. The definition of the probability space pΩ, F, Pq from the measured space pX, T , ⌫q as described above is of key importance; it is through this underlying jump structure that we can afford to state a Karatzas-Ocone hedging formula for compound geometric

We draw our inspiration from the construction of Malliavin calculus for Gaussian fields, by embedding them in isonormal Gaussian processes. Similarly, and in order to encode the properties of family t∆R t,k ,tP N T ,kP Eu, we introduce an Hilbert space H , whose inner product is reproduced by the covariance structure of the family of random variables. Consider thus H the space composed of R 2 -valued processes u "p u ¨,k ,k P Eq such that u is Fpredictable and

}u} 2 H " E « ÿ kPE T ÿ s"0  k |u s,k | 2 .
Endowed with the scalar product

xu, vy H " E « ÿ kPE T ÿ s"0  k u s,k v s,k ,
the space pH , x¨, ¨yH q is hilbertian. The second Hilbert space is the set (of equivalence classes) of square-integrable F-martingales over N 0 T endowed by the scalar product xM, Ly M " E rxM T , L T y E s so that the space pM , }¨} M q is hilbertian. The Hilbert subspace of M consisting of the 0-mean martingales of M is denoted M 0 .

The simple stochastic integral can be defined with respect to this latter F-martingale: Definition 5.2.1 (‹). For any function f P `2pN T ˆEq, the stochastic integral J1 pf q is defined by J1 pf q" ÿ sPN T ÿ kPE f ps, kq ∆R s,k .

In particular, J1 satisfies the identity: J1 p1 pt,kq q"∆R t,k ; pt, kqPN T ˆE.

The isometry property verified by J1 enables to extend its definition to H .

Theorem 5.2.2 (‹). The stochastic integral defined as the application

J1 : H ›Ñ M 0 u fi ›Ñ J1 puq" ∞ sPN T ∞ kPE u s,k ∆R s,k
is an isometry. In other words, E " J1 puq J1 pvq ı "xu, vy H ; u, v P H .

The stochastic integral Jt 1 pf q of u P H is defined as Jt 1 puq" J1 pu 1 rr 0,tss q, and satisifies the following lemma:

Lemma 5.2.3 (‹). For u P H , E " J1 puq|F t ı " Jt 1 puq.

In order to define multiple stochastic integrals of random variables of the form (5.1.2), we can work in a space of symmetrical functions; indeed the occurrence of the jumps does not affect the value of F. Our construction follows closely that depicted by Nicolas Privault (see chapter 6 in [START_REF] Privault | Stochastic Analysis in Discrete and Continuous Settings[END_REF]); in a certain sense we transpose it in our context. The space `2pN T ˆEq ˝0 is by convention identified to R; let thus for any f P `2pN T ˆEq ˝0, J0 pf 0 q"f 0 .

Definition 5.2.4 (‹). For n P N T , let `2pN T ˆEq ˝n denote the subspace of `2pN T ˆEq bn " `2pN T ˆEq n composed of the functions f n : pN T ˆEq n Ñ R symmetric in their n variables, i.e. such that for any permutation ⌧ of t1, ¨¨¨,nu,

f n `pt ⌧ p1q ,k ⌧ p1q q, ¨¨¨, pt ⌧ pnq ,k ⌧ pnq q ˘" f n `pt 1 ,k 1 q, ¨¨¨, pt n ,k n q ˘,
for any pt 1 ,k 1 q, ¨¨¨, pt n ,k n qPN T ˆE. The space `2pN T ˆEq ˝n is endowed by the scalar product

xf n ,g n y `2pN T ˆEq ˝n " n! ª pN T ˆEq n f n pt n , k n q g n pt n , k n q d⌫ bn pt n , k n q " n! ÿ ptn,knqPpN T ˆEq n, † f n pt n , k n q g n pt n , k n q n π i"1  k i
where we denote pt n , k n q"ppt 1 ,k 1 q, ¨¨¨, pt n ,k n qq.

Theorem 5.2.5 (‹ Multiple stochastic integral). The multiple stochastic integral Jn pf n q of f n P `2pN T ˆEq ˝n is defined as

Jn pf n q"n! ÿ ptn,knqPpN T ˆEq n, † f n `pt 1 ,k 1 q, ¨¨¨, pt n ,k n q ˘n π i"1 ∆R t i ,k i . (5.2.1)
It satisfies the recurrence relation

Jn pf n q"n ÿ pt,kqPN T ˆE Jn´1 p⇡ n t,k f n q ∆R t,k (5.2.2)
where the function ⇡ n t,`i s defined on `2pN T ˆEq ˝n by " ⇡ n t,k f n ‰`p t 1 ,k 1 q, ¨¨¨, pt n´1 ,k n´1 q ˘" f n `pt 1 ,k 1 q, ¨¨¨, pt n´1 ,k n´1 q, pt, kq ˘1rr 0,tss n´1 pt 1 , ¨¨¨,t n´1 q Remark 5.2.6. The description of N † T by (5.1.3) implies that the definition of the multiple stochastic integral Jn pf n q holds if n P N T .

Moreover, the application Jn satisfies an isometry property which enables to extend it to H bn , the Hilbert space of pR 2 q n -valued and F-predictable processes u bn where u b v denotes the element of L 2 pΩ ˆΩ , P b Pq defined by pu b vq t,k p!, ! 1 q"u t,k p!q v t,k p! 1 q, and endowed with the scalar product

Modified chaos representation

Let H 0 " R and for n P N ˚, H n be the subspace of L 2 pΩq made of integrals of order n • 1:

H n " ! Jn pf n q ; f n P `2pN T ˆEq ˝n)
, and called modified chaos of order n. Let S denote the linear space spanned by multiple stochastic integrals i.e.

S " Span

# T §

n"0

H n + .
The completion of S in L 2 pΩq is denoted by the sum

T à n"0 H n . Lemma 5.3.1 (‹). For t P N T , L 0 pΩ, F t q"pH 0 '¨¨¨'H t q £ L 0 pΩ, F t q (5.3.1)
As a direct consequence of lemma 5.3.1, any random variable F P L 0 pΩ, F t q can be expressed as

F " E rFs`t ÿ n"1
Jn `fn 1 rr 0,tss n Ȃs the sequence of jump times is bounded by T and the space L 2 pΩq is characterized by Lemma 5.1.4, the chaotic decomposition naturally extends to L 2 pΩq without any recourse to a limit procedure or a density argument. Theorem 5.3.2 (‹). The space L 2 pΩq is provided with the modified chaos decomposition property

L 2 pΩq" T à n"0 H n .
(5.3.2)

In other words, any random variable F P L 2 pΩq can be expanded as

F " E rFs`T ÿ n"1
Jn pf n q.

(5. 3.3) In this subsection we aim at providing a decomposition of any square integrable random variable in terms of iterated integrals with respect to the family Z "t∆Z t,k ; pt, kqPN T ˆEu defined by

∆Z t,k " sgnpkq `1tp1,kqu p∆N t , V Nt q´ p k q"sgnpkq `1tp1,kqu p∆N t , W t q´ p k ¯.
The definition of Z is quite natural since

Y t " ÿ s §t ÿ kPE ∆Z s,k (5.3.4)
can be interpreted as the compensated F-martingale associated to the compound geometric process Y. The family Z is not orthogonal. The finite dimension of the relative spanned space, being equal to

1 `T ÿ s"1 2 s ˆˆT s ˙" 3 T ,
we can derive from it an orthogonal family through the Gram-Schmidt process. The derived orthogonal family R "t∆R t,k ; pt, kqPN T ˆEu is defined by

R 0 " 1, ∆R 1,1 " ∆Z 1,1 and ∆R 1,´1 " ∆Z 1,´1 ´ q 1 ´ p ∆Z 1,1 .
By induction, we have also for any t P N T zt1u, and where the first identity comes from the independence of ∆N t and F s for s P N t´1 .

∆R
Remark 5.3.3. Consider here the worthy of interest situations when the parameters and p take the extremal values " 1 or p " 1. Note that the cases " 0 or q " 1 are sort of the negatives of the latter cases and thus left to the reader. When is equal to 1, we retrieve the definition of the structure equation solution sequence pY t q tPN defined by Nicolas Privault (chapter 1 -1.4.6 in [START_REF] Privault | Stochastic Analysis in Discrete and Continuous Settings[END_REF]). Up to a constant of normalization, the variable Y t can be expressed with respect to the variables ∆Z t,¨b y Y t " 2 ? pq p∆Z t,1 `∆Z t,´1 q, so that our underlying jump process coincides with a Rademacher sequence. When p is equal to 1, the process pY t q tPN T defined by (5.3.4) boils down to a single compensated geometric process which properties could be soundly compared to these of a Poisson process on R `of intensity .

In what follows, we frequently refer to the Rademacher sequence (resp. the Poisson process on R `of intensity ) as a "guarantor" of our construction, and which we hope to retrieve the properties when letting equal to 1 (resp. p equal to 1).

The (multiple) stochastic integral with respect to Z defined for any g n P `2pN T ˆEq ˝n by

J n pg n q"n! ÿ ptn,knqPpN T ˆEq n, † g n `pt 1 ,k 1 q, ¨¨¨, pt n ,k n q ˘n π i"1 ∆Z t i ,k i
inherites of the properties of Jn by isomorphism between Span R and Span Z.

Remark 5.3.4. Let the application

1 † : pN T ˆEq n, † ›Ñt 0, 1u pt n , k n q fi ›Ñ 1 ptn,knq
We retrieve the remarkable and usual identity

J n `1 † ptn,knq ˘" n π i"1 ∆Z t i ,k i ; n P N ˚,
where pt n , k n q" `pt 1 ,k 1 q, ¨¨¨, pt n ,k n q ˘. This is of key importance; it basically means that we can recontruct the signal Y (which connection with the family R will be precised in section 5.3) by the means of the stochastic integral of elementary functions defined on pN T ˆEq n .I n particular for n " 1, J 1 p1 pt,kq q"∆Z t,k appears as a reminiscence of (1.3.2).

We can thus state that:

Theorem 5.3.5 (‹). Any random variable F P L 2 pΩq can be expressed as

F " E rFs`T ÿ n"1 J n pg n q.
where

gpt n , k n q" n ÿ m"1 ÿ AÄrns m, † π iPA ⇢ m 1 tk i "1u f `pt A ,
´kA q, pt rnszA , k rnszA q where pt n , k n q"p p t 1 ,k 1 q, ¨¨¨, pt n ,k n qq, rns"t 1, ¨¨¨,nu and the functions f n are the ones defined through (5.3.3) and by letting ⇢ "´ q 1 ´ .

Remark 5.3.6. The latter representation formula can be put in perspective compared to remark 4.3.1. Our framework is close to that of a compound Poisson process where the process |V| is constant equal to one. It is thus possible to state a modified structure equation which interpretation (via Theorem 5.3.5) can be tranposed in our context by: if it remains impossible to supply our compound geometric process with an usual chaotic decomposition for functionals of the form (5.1.2) (as for Brownian motion or in Poisson space), we can establish a modified decomposition in chaos in terms of multiple stochastic integrals of functions g n where are the images of functions f n (appearing in (5.1.2)), via a Gram-Schmidt process. The existence of this shift seems to be directly attributed to the presence of t in place of t in (4.3.2).

We introduce the Malliavin derivative as the annihilation operator acting on the space L 2 pΩq seen in terms of its chaotic expansion (5.3.2).

Definition 5.3.7. Let the linear, unbounded, closable operator D:L 2 pΩq ›Ñ L 2 pΩ ˆNT , Pq defined for any element J n pf n q of H n by D t,k J n pf n q"nJ n´1 `fn p‹, pt, kqq1 rr 1,t´1ss n, † ˘.

(5. 3.6) where the notation ‹ will be used to indicate the first k´1 variables `pt 1 ,k 1 q, ¨¨¨, pt n´1 ,k n´1 q ȏf f n `pt 1 ,k 1 q, ¨¨¨, pt n´1 ,k n´1 q, pt, kq ˘.

Let the application where is the function defined on N T ˆE such that p∆N t , W t q"∆N t 'pV t q.

⇡ t : Ω ›Ñ Ω ! fi
Remark 5.4.7. Note that for any t P N T , p∆N t , W t q"∆N t ˜˜ p1 ´ q p1 ´˜ q p p 1 tp∆Nt,Wtq"p1,1qu `˜ p1 ´ q p1 ´˜ q q q 1 tp∆Nt,Wtq"p1,´1qu ´1"

˜ p1 ´ q p1 ´˜ q ˆp p ∆Z t,1 ´q q ∆Z t,´1 ` ˙.
Remark 5.4.8. As in the Poisson case, we retrieve that the shift space appearing in Gaussian analysis (Cameron-Martin space for the Brownian motion in particular) is replaced here by a perturbation on what characterizes the jumps: their occurrence and their height, respectively parameterized by and p.

Link with the Malliavin calculus in Rademacher and in Poisson spaces

In the same vein as the remark 5.3.3, we devote this subsection to the comparison of our formalism to the Malliavin calculus in the Rademacher space (resp. the Poisson and Rademacher spaces) which we retrieve the structure when letting be equal to 1 (resp. p equal to 1).

Let first " 1. Basically, that means that the underlying geometric process jumps every time step. A Rademacher process pX t q tPN 0 T can be defined by letting X t " V t and the variables Y t by Y t " ∆Z t,1 `∆Z t,´1 ? 2pq " X t ´p `q ?

2pq , so that we can verifiy that pY t q tPN 0 T defines a F-normal martingale. If we let D the derivative

D t FpX 1 , ¨¨¨, X T q"D t,1 G ´T ÿ s"1 ÿ kPE ∆Z s,k ¯´D t,´1 G ´T ÿ s"1 ÿ kPE ∆Z s,k
where F and G as defined to satisty FpX 1 , ¨¨¨,

X T q"G ´∞T s"1 ∞ kPE ∆Z s,k ¯, we get D t Y s " 2 ? 2pq 1 ttu psq" c 2 pq p D t Y s i.e.
-up to a constant -the expression of the gradient p D defined on the Rademacher space (2.0.2). All identities and formulas, such as the Clark formula and the predictable representation (see Privault chapter 1), are inherited by construction. Consider now the case p " 1. From a first point of view, that means that all variables tV t ,tP N T u are deterministic (w.l.o.g. supposed equal to 1), and that the process pY t q tPN T defined by (5.3.4) matches with a single compensated geometric process. The construction developped still holds and, as expected, enables to state stronger identities that are reminiscent of these holding for Poisson point processes on the real line. To take notice of it, consider the following framework: H P designates the Hilbert space of R-valued and F-predictable processes u such that

}u} 2 H P " E « T ÿ t"0 p1 ´ q|u t | 2 .
Consider thus the space of simple, locally finite on N T integer-valued measures still denoted by Ω P . In particular, the family Z P "t∆Z P t ; t P N T u defined by

∆Z P t " 1 t∆Nt"1u
´ , is orthogonal for the canonical product on Ω P so that the stochastic integral defined as the application I 1 :

H P ›Ñ M P 0 u fi ›Ñ I 1 puq" ÿ tPN T u t ∆Z P t
is an isometry. We can establish that for any F P L 2 pΩ P q,

F " E rFs`ÿ For any pt n , s m qPp N T q n, † ˆp N T q m, † , there exists i 0 P N n such that t i 0 P t n zs n . By independence of the random variable ∆R t i 0 ,k i 0 with respect to the -algebra F t i 0 ´1, for any pu, vqPH bn ˆH bm ,

tPN 0 T E " D P t F | F t´1
E " Jn pu bn q Jm pv bm q ı " n! m! ÿ ptn,knqPpN T ˆEq n, † ÿ psm,lmqPpN T ˆEq m, † u bn pt n , k n q v bm ps m , l m q ˆE « n π i"1 m π j"1 ∆R t i ,k i ∆R s j ,`j " n! m! ÿ ptn,knqPpN T ˆEq n, † ÿ psm,lmqPpN T ˆEq m, † u bn pt n , k n q v bm ps m , l m q ˆE " ∆R t i 0 ,k i 0 ı E » - - n π i"1 i‰i 0 m π j"1 ∆R t i ,k i ∆R s j ,`j ˇˇˇF t i 0 ´1fi fl i 0 ´1 π i"1 i 0 ´1 π j"1 ∆R t i ,k i ∆R s j ,`j " 0,
and for m " n,

E " Jn pu bn q Jn pv bn q ı "pn!q 2 ÿ ptn,knqPpN T ˆEq n, † ÿ psn,lnqPpN T ˆEq n, † u bn pt n , k n q v bn ps n , l n q ˆE « n π i,j"1 ∆R t i ,k i ∆R s j ,`j "pn!q 2 ÿ ptn,knqPpN T ˆEq n, † lnPE n, † f n pt n , k n q g n pt n , l n qE « n π i,j"1 ∆R t i ,k i ∆R t i ,`i " n! xu, vy H bn ,
since E r∆R t,k ∆R t,`s " k 1 tku p`q. Hence the result.

Proofs of Section 5.3

Proof of Lemma 5.3.1. It suffices to note that H s X L 0 pΩ, F t q is generated by the orthogonal basis

t1uY # r π i"1 ∆R t i ,k i , 1 § t 1 † ¨¨¨ † t r § s, pk 1 , ¨¨¨,k r qPE r + (5.6.1)
Indeed it appears that the family can be expressed in terms of multiple integrals as

s π i"1 ∆R t i ,k i " Jn ´1 † tpt 1 ,k 1 q,¨¨¨,pts,ksqu ¯,
and moreover, by equality of the dimensions of the two spaces at stake is equal to 2 ˆt s ˙.W e conclude by remarking that the dimensions of the two spaces appearing in (5.3.1) are both and g n ppt n´1 , k n´1 q pt j ,k j q , pt j , ´1qq " n´1 ÿ

`"1 ÿ APprns j q `, † π iPA ⇢ `1tk i "1u f `pt A , ´kA , pt j , ´1qq, pt rns j zA , k rns j zA q ˘.

Last write

g n pt n , k n q"g n ppt n´1 , k n´1 q pt j ,k j q , pt j , 1qq1 pk j "1q `gn ppt n´1 , k n´1 q pt j ,k j q , pt j , ´1qq1 pk j "´1q

" n ÿ `"1 ÿ APprnsq `, † π iPA ⇢ `1tk i "1u f `pt A , ´kA , pt rnszA , k rnszA q to complete the proof.
Proof of Theorem 5.3.9. For any pt, kqPN T ˆE, consider

r t,k F " sgnpkq " Fp⇡ t p!qYpt, kqq ´Fp⇡ t p!qq ‰ r t,k Jn pf n q" n! ÿ ptn,knqPpN T ˆEq n, † f n `pt 1 ,k 1 q, ¨¨¨, pt n ,k n q ˘n π i"1 r t,k ∆R t i ,k i " n! ÿ pt t n ,k k n qPpN T ˆEq n´1, † f n `pt 1 ,k 1 q, ¨¨¨, pt, kq, ¨¨¨, pt n ,k n q ˘n π i"1 t i ‰t ∆R t i ,k i " n! ÿ pt n´1 ,k n´1 qPpN t T ˆEq n, † f n `pt n´1 , k n´1 q, pt
∆R t i ,`i ¸" ˜n ÿ i"1 1 tpt,kqu pt i ,k i q ¸n π i"1 t i ‰t ∆R t i ,`i .
Thus, for any F P S, D t,k F " sgnpkq " Fp⇡ t p!qYpt, kqq ´Fp⇡ t p!qq ‰ . By a limit procedure we extend this result to L 2 pΩq.

applied to

t " s ´1 and F " X s that X s " E rX s |F s´1 s`ÿ kPE E rD s,k X t |F s´1 s " X s´1 `ÿ kPE E rD s,k X s |F s´1 s
Then by letting u s,k " E rD s,k X s |F s´1 s , we get

X t " X 0 `t ÿ s"1 X s ´Xs´1 " X 0 `t ÿ s"1 ÿ kPE u s,k ∆R s,k .
Hence the result.

Proof of Theorem 5.4.6. The proof follows closely the proof of Theorem 19.16. in Privault [START_REF] Privault | Chapter 19 stochastic calculus for jump processes[END_REF] for compound Poisson processes. Let ' and P˜ ,p as defined in the theorem. For any

s P R ˚, E ˜ ,p rs Yt s" ˜1 ´˜ 1 ´ ¸t t ÿ n"0 E « s Yt n π k"1 p1 `'pV k qq ˇˇN t " n PpN t " nq " ˜1 ´˜ 1 ´ ¸t t ÿ n"0 ˆt n ˙ n p1 ´ q t´n E « n π k"1 p1 `'pV k qqs V k ˇˇN t " n "p1 ´˜ q t t ÿ n"0 ˆt n ˙ˆ 1 ´ ˙n ˜˜ p1 ´ q p1 ´˜ q ¸n n π k"1 ˆp p ¨ps `1 ´p 1 ´p ¨1 ´p s " t ÿ n"0 ˆt n ˙˜ n p1 ´˜ q t´n ˆps `1 ´p s ˙n " ˆ1 ´˜ `˜ ´ps `1 ´p s ¯˙t .
Hence the result.

whereas the stock price pS t q tPN T with (deterministic) initial value S 0 " 1 satisfies the equation:

∆S t " ⌘ t S t´1 ∆N t , (6.1.3)
where ⌘ t " b1 tWt"1u `a1 tWt"´1u , a and b are real numbers such that ´1 † a † r † b. The two assets evolute on a given finite time period N 0 T . The sequence pS t q tPN T of discounted stock prices is given by S t " 1 p1 `rq t S t , ; t P N T . Remark 6.1.1. The price process defined in our ternary model is identically distributed with the one of a well-chosen trinomial model. As a reminder, stock price pT t q tPN T is defined in this latter model by T 0 " 1 and verifies the recurrent relation:

T t " $ & % p1 `bqT t´1 if X t " 1 T t´1 if X t " 0 p1 `aqT t´1 if X t "´1
, where the process pX t q tPN T is distributed according to the measure P such that: PpX t " 1q"p, PpX t "´1q"q and PpX k " 0q"1 ´p ´q.

with pp, qqPp0, 1q 2 . Let p " p and q " p1 ´pq such that 1 ´p ´q " 1 ´ .

E

" s S t S t´1 ⇢ " E " s ⌘t∆Nt`1 ‰ " s b`1 p `sa`1 p1 ´pq`sp1 ´ q " s 1`b p `s1`a q `sp1 ´ q " E " s T t T t´1 ⇢ ,
and S 0 " T 0 . Thus the trinomial and the ternary models are equivalent in law. The second one, based on a jump process, lends itself more easily to the statement of an hedging formula, directly derived from Clark-Ocone formula (5.4.2). This motivated us to substitute it to the trinomial model for our further investigations.

Remark 6.1.2. It could be interesting to extend this ternary model to a "stochastic volatilitytype" ternary model by allowing the up and down factors p1 `bq and p1 `aq to be stochastic. We could thus imagine to define the processes pb kt{n ,k " 1, ¨¨¨,nq and pa kt{n ,k " 1, ¨¨¨,nq as well-defined functions of the (stochastic volatility) Cox-Ingersoll-Ross process p⌫ t q tPN T defined by

⌫ t " ⌫ 0 `ª t 0 p✓ ´⌫s q ds `ª t 0 ? ⌫ s dB ⌫ s ,tP T,
and so that, by letting n go to infinity, the "stochastic volatility ternary" model tends (at time t) to the classical Heston model lying on the stochastic process

H H t " 1 `ª t 0 rH s ds `ª t 0 ? ⌫ s H s dB H s ,tP T,
where r is the rate of return asset,  denotes the speed of reversion, is the volatility of the volatility, and B ⌫ , B H are Wiener processes with covariance ⇢dt.

Martingales measures in the ternary market model

Before exhibiting an hedging strategy for any for any simulating claim, we wonder about the completeness of the market. As explained by Runggaldier in Portfolio optimization in discrete time, like the trinomial model, our ternary model is an incomplete market. Indeed, the measure with respect to which the sequence of discounted prices is a F-martingale, is not unique. Given the process pS t q tPN T defined by ( 6 it appears that the discounted prices sequence is a F-martingale if the condition t pbp t `aq t q´r " 0, holds for any t P N T . As expected, the system " pbp t `aq t q"r p t `qt " 1 admits infinitely many solutions p t ,p t ,q t qPp 0, 1q 3 such that any triplet p t ,p t ,q t q forms a convex M F set (here a segment) characterized by its extremal points (independent of t), i.e. the measures

P 0 t " P 0 " ´1, r ´a b ´a , b ´r b ´a ¯and P 1 t " P 1 " ´r b , 1, 0 ¯, (6.1.4)
which are not equivalent to P but such that any convex combination

P " P 0 `p1 ´ qP 1 , (6.1.5)
is. Any measure defined on Ω and with respect to which the sequence S is a F-martingale is called a F-martingale measure.

Note that the extremal measures P 0 t and P 1 t are independent of t and F t ,s ot h a t ,i fT " 2 and ! "p! 1 , ! 2 qPΩ,

P p! i 1 , ! j 2 q"P p! i 1 qP p! j 2 |! i 1 q"P p! i 1 qP p! j 2
q and by induction

P p! i 1 1 , ¨¨¨, ! in n q" n π k"1 P p! i k k q.
If J t " 2 t denotes the number of extremal points P j t of the polyhedron of F-martingale measures, for any j P J t there exists p j k q kPNt Pt0, 1u t such that P j t " â sPNt pP 0 q j s pP 1 q 1´ j s (6.1.6)

For any t P N T , M F t denotes the set of F-martingales measures on N 0 t , whereas the set C F t composed of the convex combinations of extremal measures (6.1.6) matches with the convex subset of M F t formed by the F-martingale measures equivalent to P.

Remark 6.1.3. We can give interpetation to the definitions of the measures P 0 "p 0 ,p 0 ,q 0 q and P 1 "p 1 ,p 1 ,q 1 q introduced above. These stand for the "limit models". Actually when the probability parameters tend towards extreme values, we fall back on well-known models and complete markets. Under P 0 , the process p∆N t q tPN T is deterministic, constant equal to one. In other words, the process S jumps at each time so that it coicindes with the price process defined in the Cox-Ross-Rubinstein model (4.1.7). To make sure, keep the parameters a, b, r and define the underlying Rademacher process X bin by X bin ¨" 1 tX¨‰0u X ¨. We can check this is well-defined since:

PpX t " 1q"p 0 , PpX t "´1q"q 0 and PpX t " 0q"1 ´p0 ´q0 , so that the equalities PpX bin t " 1q`PpX bin t "1q"1 and PpX t " 0q"0 hold for any t P N T . The definition of pp 0 ,q 0 q" `pr ´aq{pb ´aq, pb ´rq{pb ´a˘q corresponds exactly to the risk-neutral measure in the binomial model. Under P 1 , the process pV t q tPN T is deterministic, constant equal to one, so that the compound process pZ t q tPN T coincides with a single geometric process of intensity . Once again, the resulting market is complete and the risk-neutral measure is obtained by letting p " r{b.

Enlargement of filtration

As mentioned in chapter 4, trying to quantify the additional information provided by the known in advance of a F T -measurable random variable G leads to tackle with an enlargement of the initial filtration. In a discrete setting, it seems to boil down to the exploitation of the Doob decomposition (see Section 4.1 and [START_REF] Blanchet-Scalliet | Enlargement of filtration in discrete time[END_REF])

In what follows, the random variable G is supposed to take its values in a measurable space pΓ, G q where Γ is a countable set. We define a particular process p G which is of key importance in many ways. First, it naturally appears in the Doob decomposition and enables to connect the eponymous information drift (as reminiscence with the continuous case) with the variable G. Besides, to study the martingales with respect to the enlarged filtration, since, as it will be proved, the t-th term of the process 1{p G preserves the martingale property of any pF, Pq-martingale on N 0 t . As such, optimizing the insider's portfolio (see subsection 6.4.1) requires to determine the crucial set of G-martingale measures on N 0 t equivalent to P.

Up to now, let F 0 (resp. G 0 )bethefiltrationsF 0 "pF t q tPN 0 T (resp. G 0 "pG t q tPN 0 T

), where the -algebras pF t q tPN 0 T are defined in (5.1.4) (resp. G t " F t _ G) and which distribution given F t for some t P N T is given by a family of F t -measurable random variables PpG " c | F t q,cP Γ ( .

The information drift process

As the set Γ is countable, the absolute continuity of law type condition (inescapable in the continuous case) is de facto verified in our discrete context. Indeed, any set C P G is of the form C " î cPC tG " cu and, for any t P N 0 T ´1,

PpG P C | F t q" ÿ cPC PpG " c | F t q" ÿ cPC PpG " c | F t q PpG " cq PpG " cq"E " p G t 1 A ‰ ,
where the random variable p G t is defined by p G t p!q" PpG P¨|F t qp!q PpG P¨q such that p c t p!q" PpG " c |F t qp!q PpG " cq , (6.2.1)

for any ! P Ω, c P Γ. Define also the process p⌫ G t q tPN T by

⌫ G t " PpG P¨|F t q"PpG P¨qp G t (6.2.2)
The combination of (6.2.2) and (4.1.9) (see section 2.2 in [START_REF] Blanchet-Scalliet | Enlargement of filtration in discrete time[END_REF]) applied to the pP, Fq-martingale pYq tPN T ensures that the process pY G t q tPN T defined by

Y G t " Y t ´t ÿ s"1 xY,p c t y s ˇˇc"G p G s´1 " Y t ´µG t (6.2.3)
is a pP, Gq-martingale. The process µ G thus defined is called the information drift, namely the drift to absorb in passing to the insider paradigm in order to preserve the martingale property of the initial compound geometric process. As designed by Peter Imkeller in [START_REF] Imkeller | Malliavin's calculus in insider models: Additional utility and free lunches[END_REF], we can traduct its connection to the random variable G thanks to the Malliavin derivative.

Theorem 6.2.1 (‹). The information drift µ G defined in (6.2.3) can be written as

µ G t " ÿ kPE ÿ `PE a k,`E rD t,`p c t s| c"G p G t´1
for any t P N T ,w h e r et h ef a m i l yta k,`, pk, `qPE 2 u is defined by a k,`" E r∆Z t,k ∆R t,`s ,i . e . a 1,1 " pp1 ´ pq,a 1,´1 " 0,a 1,´1 " 2 pq and a ´1,´1 " q p1 ´ q 1 ´ p .

The following result is incomplete. We chose to present it anyway for the interest it could arouse in possible future investigations. Actually, it would seem possible to interpret µ G in terms of drift on the initial process X at the cost of an additional assumption: Assumption. For any c P Γ, assume the existence of a couple p ˜ c , pc qPp0, 1q 2 such that

p c t " d P˜ ,p dP ˇˇˇF t that is, satisfying: p c t p c t´1 " 1 `1 ´˜ c 1 ´ ´ p∆N t , W t q´ ´˜ c 1 ´˜ c ¯. (6.2.4)
Then by Girsanov's theorem 5.4.6, the process X defined by

X t " Nt ÿ s"1 V s
is geometric compound process of parameters p ˜ G , pG q under the probability measure P˜ ,p .

Recall moreover (see [START_REF] Blanchet-Scalliet | Enlargement of filtration in discrete time[END_REF]): Theorem 6.2.2 (Blanchet-Scalliet, Jeanblanc, Roméro). Let pX t q tPN T be a pP, Fq-martingale and Q ap r o b a b i l i t ym e a s u r es u p p o s e dt ob ee q u i v a l e n tt oP on F t for any t P N T .

Let is a geometric compound process of parameters p G , pG q where pc is defined by (6.2.3) for any c P Γ. Remark 6.2.4. This result sounds interesting because it does the counterpart with the Brownian case. However, it remains incomplete in its current form; indeed we still don't know what meaning to give to a compound geometric process of random parameters p G ,p G q. We could imagine associating to the process X a sequence of parameters p G t ,p G t q tPN T defined in terms of conditional probabilities PpG P¨|F t q and in a Markov chain-like way... To be continued !

The martingale preserving measure

In this section, we focus on the F-adapted process 1{p G , of key importance afterwards. This is well defined; indeed, resulting from the countability of Γ, for any pt, cqPN 0 T ´1 ˆΓ,t h e random variable p c t is not null P-a.s. Note that for any t P N T ,t h e -algebra G t is generated by the set tA X B; AP F t , B P E u. Theorem 6.2.5 (‹).

1. The process 1{p G is a pP, G 0 q martingale.

2. For t P N 0 T ,t h e -algebras F t and pGq are independent under the probability measure defined for any A P G t by

Q t pAq"E " 1 p G t 1 A ⇢ .
3. For any t P N 0 T ´1,t h ep r o b a b i l i t ym e a s u r eQ t coincides with P on F t . Remark 6.2.6. The process pQ t q tPN 0 T thus defined is called the martingale preserving measure.

It is thus possible to establish the following result. Theorem 6.2.7 (‹). For fixed t P N 0 T ´1,a nypP, Fq-martingale on N 0 t is a pQ t , Gq-martingale and also a pQ t , Fq-martingale on N 0 t . Remark 6.2.8. The process 1{p G defines a probability measure, which density is absolutely continuous with respect to P.

Hedging formula in a ternary model

The value of the portfolio at time t P N 0 T is given by the random variable

V t " ↵ t A t `'t S t ,
where p↵ t , ' t q tPN T is a couple of predictable processes modeling respectively the amounts of riskless and risky assets held in the portfolio, and its discounted value by V 0 " V 0 and

V t " V 0 t π s"1 1 1 `rs V s ; t P N T .
The aim of this subsection is to exhibit an hedging formula, i.e. given a nonnegative F Tmeasurable random variable F (called claim), to determine a portfolio modelised by p↵ t , ' t q tPN 0 T such that V 0 °0, V t • 0 pt P N T ´1q, and V T " F.

In an incomplete market, all claims are not reachable; they have an intrisic risk. Face to the impossibility to perform a perfect hedge in the general case, we can only hope to reduce the a priori risk to this minimal component. The question of hedging in an incomplete market has been widely investigated for years (see for instance [START_REF] Dalang | Equivalent martingale measures and noarbitrage in stochastic securities market models[END_REF], [START_REF] Follmer | Hedging of contingent claims[END_REF] in continuous time, [START_REF] Schweizer | Variance-optimal hedging in discrete time[END_REF] in discrete time). The ternary model, as the trinomial one, is not complete; we choose to deal with the optimization problem

min px,'qPR ˚ˆΘ E " pF ´x ´VT p'qq 2 ‰ , (6.3.1)
where x is the initial capital and S F is the set of F-predictable admissible strategies. The mean-variance tradeoff process pK t q tPN T is defined by

K t " t ÿ s"1 `E rp∆S s q|F s´1 s ˘2 varr∆S s | F s´1 s ; t P N T .
Define also the discrete analogue of the minimal martingale measure (see Föllmer and Schweizer [START_REF] Follmer | Hedging of contingent claims[END_REF]), i.e. the signed measure p P defined on pΩ, Fq such that

d p P dP " N T π t"1 1 ´✓t ∆S t 1 ´✓t E r∆S t |F t´1 s , (6.3.2)
where, for any t P N T ,

✓ t " E r∆S t | F t´1 s E rp∆S t q 2 | F t´1 s .
Last, consider the Kunita-Watanabe decomposition of F (see Metivier [START_REF] Métivier | Semimartingales, volume 2 of de gruyter studies in mathematics[END_REF] or Schweizer [START_REF] Schweizer | Variance-optimal hedging in discrete time[END_REF]) i.e. the unique couple of processes p⇠ F , L F q where ⇠ F is a square-integrable admissible strategy and L F a F-martingale , strongly orthogonal to S, with null intial value such that

F " F 0 `ÿ tPN T ⇠ F t ∆S t `LF T P-a.s.
Within previous notations, Martin Schweizer gives an expression of the quadratic-loss minimizing strategy.

Theorem 6.3.1 [START_REF] Schweizer | Mean-variance hedging for general claims[END_REF]. Provided pK t q tPN T is deterministic, the solution of (6.3.1) is given by

x ˚" p E rFs and ' t " ⇠ F t `E r∆S t | F t´1 s E rp∆S t q 2 | F t´1 s p p E rF|F t s´x ˚´V t´1 p' ˚qq (6.3.3)
and p P is the minimal martingale measure defined by (6.3.2).M o r e o v e r , t h e q u o t a o f t h e riskless asset pA t q tPN T is given by ↵ 0 "p1 `rq ´T p E rFs{S 0 and for any t P N T ,

↵ t " ↵ t´1 ´p' t ´'t´1 qS t´1 A t´1 . Remark 6.3.2. If the contingent claim F is reachable, then ' ˚" ⇠ F . The term ⇠ F in (6.3.
3) can be interpreted as a pure hedging demand, whereas the second one can be viewed as a demand for mean-variance purposes (see Schweizer [START_REF] Schweizer | Mean-variance hedging for general claims[END_REF]).

We slot these results to our formalism to solve (6. ´ 2 p2p ´1q 2 is a deterministic constant. Lemma 6.3.4 ((‹) Kunita-Watanabe decomposition in ternary model). For any claim F P L 2 pΩq there exist a square-integrable admissible strategy ⇠ F and a F-martingale L F ,s t r o n g l y orthogonal to S,w i t hn u l li n t i a lv a l u es u c ht h a t

F " F 0 `ÿ tPN T ⇠ F t ∆S t `LF T P-a.s.
Moreover, for any t P N T ,

⇠ F t " ∞ kPE c k p E rD t,k F | F t´1 s }c} 2 2 , and
L F t " ÿ kPE c k `p E rD t,k F | F t´1 s´⇠ t ˘,
where c "pc 1 ,c ´1qPR 2 with c 1 "pb `a⇢q 1 and c 1 "´a ´1.

Define the minimal martingale measure p P

d p P dP " N T π t"1 1 ´✓∆S t 1 ´✓E r∆S t |F t´1 s , (6.3.4)
where ✓ "p1 ` pbp ´aqqq{}c} 2 2 .

Theorem 6.3.5 (‹ Loss quadratic minimizing strategy in ternary model). Let p P be the minimal martingale measure defined by (6.3.4) and let a claim F P L 2 pΩ, Fq.T h eq u a d r a t i c loss minimizing hedge px ˚, ' ˚q is given by x ˚" p E rFs and ' t " ⇠ F t `✓p p E rF|F t s´x ˚´V t´1 p' ˚qq where ⇠ F P Θ is given by the Kunita-Watanabe decomposition

Proof. Since the mean-variance process is deterministic by Lemma 6.3.3, it suffices to incorporate the result of Lemma 6.3.4 to Theorem 6.3.1. The process p↵ t q tPN T is defined by the self-financing condition (6.5.1).

6.4 Additional utility of the insider in a ternary model

Theorical results

Given an economic agent (resp. an insider) disposing of x (x °0) euros at date t " 0 (initial budget constraint), we want to determine the optimal admissible strategy allowing to maximize his portfolio value at time t P N T . This leads to consider the optimization problem from the agent's point of view:

Φ ag t pxq" max PF´portfolio E rupV x,t p qqs , (6.4.1)
and respectively from the insider's one:

Φ ins t pxq" max PG´portfolio E rupV x,t p qqs (6.4.2)
where u " log.

Utility of the economic agent and the insider before the deadline T

In this subsection we tackle problems (6.4.1) and (6.4.2) when t P N T ´1 (T P N ˚). Based on the equivalence in law of the trinomial model and ours (see remark 6.1.1), there is a bijection between M F , the convex set of F-martingale measures equivalent to P in our model, and the one existing in the trinomial. Indeed, for T " 1, the measures P "p ,p ,q q (defined by 6.1.5) and P ,Tri "p p Tri ,q Tri ,s Tri q ( Pr 0, 1s) are equivalent by letting " p Tri `qTri and p " ´1 p Tri . For a given T P N ˚, the set M F is the polyhedron characterized by its extremal points; these are constructed by tensorization of the measures (independent of t)

P 0 t " P 0 " ´1, r ´a b ´a , b ´r b ´a ¯and P 1 t " P 1 " ´r b , 1, 0 ¯.
By induction, the convex set of F-martingale measures equivalent to P in our model, and the one existing in the trinomial are equivalent. The following portfolio optimization result is directly deduced. Its proof which turns out to be the rewrite of that of [START_REF] Delbaen | The Mathematics of Arbitrage[END_REF] for the trinomial model (see more details in section 4.2) and won't be detailed. The optimization problem from the insider point of view (6.4.2) seems to be solved by transposing that of the agent to the enriched filtration G. That lies on the possibility to rewrite (6.4.2) as a primal problem as in (4.2.9). With this edict in mind, we have to identify the set of measures which respect to which the sequence pS s q sPNt is a p¨, Gq-martingale, namely the G-martingale measures. Denote thus by M G t and C G t (t P N T ), respectively the set of G-martingale measures on N 0 t , and the convex subset of M G t formed by the G-martingale measures equivalent to P on N 0 t . In the first subsection we established that for any t P N 0 T ´1,a n ypP, F 0 q-martingale on N 0 t is a pQ t , G 0 q-martingale where Q t is the measure defined by

Q t pAq" ª A 1 p G t dP ;AP G t .
For the sake of simplicity let Q " Q T . The aim of this subsection is to describe M G t , the set of G-martingale measures on N 0 t . As seen in subsection 4.2, the set of F-martingale measures on N 0 t consists of the convex combinations of tP j t ,jP J t u.F o ra n yj P J t , define the sequence of measures pQ j t q tPN T ´1 by Q j t pAq"

ª A 1 p G t dP j ;AP G t ,
where we let P j " P j T . Furthermore, follows from theorem 6.2.7 that a pP j , Fq-martingale on N 0 t is a pQ j t , Gq-martingale on N 0 t . In particular, the discounted prices process S is a pQ j t , Gq-martingale on N 0 t ;s od o e s( o nN 0 t ) V G x,¨p q viewed as the G-martingale transform V G x,t p q"x `t ÿ s"1

' G s ∆S s ,
where "p↵, ' G q belongs to S G , the set of G-predictable admissible strategies. Note besides that E Q j t rV G x,t p qs " x ; j P J t , (6.4.4) so that we can address the optimization problem as a constrainted one. Define for px, tqP R ˚ˆN 0 T ´1, Λ G x,t " ! V P S G : E Q j t rVs § x,@j P J t

) .

As a consequence once again of the surreplication theorem, (6.4. Given two probablities measures defined on the same space pΩ, Fq, D F pP||Qq designates the relative entropy of P with respect to Q on F and is defined by

D F pP||Qq" $ & % E " log ´dP dQ ¯ˇˇF ⇢ if P ! Q on F, ` 8 otherwise.
Theorem 6.4.4 (‹). The insider's additional expected logarithmic utility up to time t P N T ´1 is given by U t " D Gt pP||Q t q.

Remark 6.4.5. From a philosophical point of view, it is not very surprising to see the entropy appear when quantifying an additional... information.

Remark 6.4.6. Thus we recover the result already established by Jürgen Amendinger, Peter Imkeller and Martin Schweizer ( [START_REF] Amendinger | Additional logarithmic utility of an insider[END_REF]) in the continuous case; the additional expected logarithmic utility of the insider can be expressed in terms of relative entropy. They get more: by letting t go to T , they underscore that the result still holds at the deadline T . Obviously we cannot transpose this in a discrete setting. Since there is absolutely no reason why pP, Fq-martingale N 0 t should be a pQ, G 0 q-martingale, we need to address the problem at the deadline T from another perspective. We chose to dedicate the next section to this case.

Utility of the insider at expiry

In order to remove the difficulty previously evokated (see remark 6.4.6), we link the optimal strategy utility final value with this at the instant just before. Lemma 6.4.7 (‹). For any x P R ˚define Φ ins T pxq by considering (6.4.2) at the deadline T . Then, Φ ins T pxq" max PG´portfolio E rupV x,1 p qqs " Φ ins 1 pxq, where x " Φ ins T ´1pxq.

Comparison with the binomial and the Black-Scholes models

As mentioned in Remark 6.4.6, the formula giving the additional utility of the insider before the deadline is exactly the same in the continuous case (Black-Scholes model) and in the discrete ones (binomial, trinomial or our ternary model). As a matter of fact, this lies on the enlargement of filtrations tools that are comparable in continuous and discrete time. The study of the insider problem from the additional expected utility point of view is similar in all frameworks.

The substantial difference we have met between our model and the classical ones (Black-Scholes and binomial one) concern the computation of the argmax admissible strategy appearing in (6.4.2). This is not strictly speaking linked with the insider matter, but rather with the impossibility to perform perfect hedges, because of the incompleteness of our ternary model. In this last subsection we chose to investigate a specific case: G " 1 tS T Pr0,S 0 su .M o r a l l y ,t h i s means the insider knows wether the stock price will have increased or not at the end of the trading period. Let x P R ˚be the initial budget. As described in subsection 6.4.1 to cover all cases we need to compute:

• Φ ag 1 pxq and Φ ins 1 pxq,

• Φ ag T pxq (for T • 2),

• U t pxq for any t P N T ´1.

Remark 6.4.8. In fact the computation of Φ ag T pxq does not require to invoke Lemma 6.4.7 and can be done directly.

Computations of Φ ag

1 pxq and Φ ins 1 pxq We provide a method to compute explicitely " Φ ins 1 pxq, namely to solve the optimization problem when T " 1. As evokated in subsection 6.1.1, the set M F 1 of F-martingale measures equivalent to P consists of the convex combination of the form P " P 0 `p1 ´ qP 1 , where the two extremal measures are defined by (6.1.4). In other terms,

P "

´ `p1 ´ qr b , 1 ´ ` pr ´aq b ´a , pb ´rq b ´a ¯"p1 ´ , p , p1 ´p qq

In that specific case, it seems easier to directly solve the primal problem, i.e. to compute:

max VPΛ x,1
E rlogpVqs under the constraints E P j rVs § x ; @j Pt0, 1u, where V " x `' ∆S 1 " x `'`b 1 tp∆N 1 ,W 1 q"p1,1qu `a 1 tp∆N 1 ,W 1 q"p1,´1qu ˘, and ' is a F 0 -measurable random variable (since T " 1). The ternary model described in subsection 6.1 can be thus treated as a stochastic volatility model: the closer is to 1 the more volatile is the model. In the extreme case (when is equal to one), the model coincides with the Cox-Ross-Rubinstein one for which there exists an unique optimizer (see proposition 3.3.2. in [START_REF] Delbaen | The Mathematics of Arbitrage[END_REF]). On the other hand, when is near to zero, the choice of the strategy ' has little influence on ' ∆S 1 and thus on the computation of E rlogpVqs. This leads us to speculate that the optimal strategy in the ternary model coincides with that of the binomial model. To be convinced that the maxima are realized for the same strategy, note that E " logpVp'qq ı " E 0 " logpV bin 1 p'qq ı `p1 ´ q logpxq, (6.4.7)

where E 0 designates the expectation under P 0 , namely the unique risk-neutral probability measure in the Cox-Ross-Rubinstein model, and V bin 1 p'q designates the value of the strategy ' at time 1 i.e.

V bin 1 p'q"x `' ∆S bin 1 . The sequence of prices pS bin t q tPN T is defined in the "associated" binomial model (by letting " 1)b y : The corresponding optimal investment is given by V ag x,1 " x1 t∆N 1 "0u `xp p ˚1tp∆N 1 ,W 1 q"p1,1qu `xq q ˚1tp∆N 1 ,W 1 q"p1,´1qu . Considering the specific case p " q " 1 2 and " 1, we exactly retrieve the result stated by Freddy Delbaen et Walter Schachermayer (see example 3.3.2. in [START_REF] Delbaen | The Mathematics of Arbitrage[END_REF]): ⇡ t "´p1 `rq ppb ´rq`qpa ´rq pb ´rqpa ´rq " 1 2 p1 `rqpb `a ´2rq pb ´rqpr ´aq .

pS bin q # S bin
We can get the result using a variational argument: Theorem 6.4.12 (‹). The maximal expected logarithmic utility of the agent is given at maturity by Φ ag 1 pxq"logpV 0 q`E rlog prr `⇡˚p b∆Z t,1 ´a∆Z t,´1 qsqs , where, for any t P N T ,

⇡ ˚" 1 `r b ´a ˆpq ˚´qp p˚q˚.
The maximal logarithmic of the agent is performed when following the strategy ag "p↵ ag 1 , ' ag 1 q with ' ag 1 " x⇡ ˚" xp1 `rq b ´a ˆpq ˚´qp p˚q˚a nd ↵ ag 1 "

x ´S0 ' ag 1 .

Remark 6.4.13. As expected, the strategy which components are F 0 -measurable, does not depend of the expression of G.

The situation is different from the point of view of the insider who knows wether the stock price will have increased or not at the end of the trading period T " 1. In other words, he knows from the start wether it is worth investing in the "risky" asset; this appears in fact as a riskless one for him since he knows the outcome. One of two things must be true: either G " 1 namely the stock price does not increase and it may be better to invest the entire capital in the asset A,o rG " 0 and investing in S is more profitable. As a reminder the budget constraint can be written as

x " V 0 p q"↵ ins 0 `'ins 0 S 0 , and be transposed at time T " 1 to ↵ ins 1 `'ins 1 S 0 " x, by readjusting the portfolio under the self-financing condition. We get clearly:

↵ ins 1 " x1 tG"1u and ' ins 1 "

x S 0 1 tG"0u .

Computation of Φ ag T pxq, T • 2 As suggested in the Example 3.3.5. depicted by Freddy Delbaen and Walter Schachermayer in [START_REF] Delbaen | The Mathematics of Arbitrage[END_REF], the case T P N ˚can be treated by extending the previous results via the principle of dynamic programming. In a certain sense, we make of "a multiplicative concatenation" of what has been done in the case T " 1. The choice of this procedure can be justified by the independence of increments of the underlying jump process pN t q tPN T . Conserving the previous notations, define with and so on by downward induction. The iteration of (6.5.2) provides :

Θ
Φ bin T pxq"logpxq`Tk bin where k bin "´p logpp ˚q´q logpq ˚q`p logppq`q logpqq and then,

Φ ag

T pxq" plogpxq`Tk bin q`p1 ´ q logpxq. We have also Φ ag t pxq" plogpxq`tk bin q`p1 ´ q logpxq for any t P N T ´1. Computation of U t pxq, t P N T ´1 (T • 2)

Consider first the general case where G is a discrete random variable in values in Γ. In what follows, EntpGq designates the entropy of G namely the quantity EntpGq"´ÿ cPΓ log `PpG " cq ˘PpG " cq, and EntpG | F t q the conditional entropy EntpG | F t q"log `PpG " c | F t q ˘PpG " c | F t q.

Recalling the definition of the process 1{p G ,

U t pxq"E " logpp G t q ‰ " E « ÿ cPΓ logpp c t q PpG " c | F t q " E « ÿ cPΓ log `PpG " c | F t q ˘PpG " c | F t q ´ÿ cPΓ log `PpG " cq ˘E " E " 1 pG"cq | F t ‰‰ " E « ÿ
cPΓ log `PpG " c | F t q ˘PpG " c|F t q ´ÿ cPΓ log `PpG " cq ˘PpG " cq " EntpGq´EntpG | F t q.

where we get the second line by conditioning on F t . In particular, for G " 1 tS T Pr0,S 0 su

PpG " 1 | F T q"1 tG"1u " 1 tS T Pr0,S 0 su and for any t P N 0 T ´1,

PpG " 1 | F t q"PpS T Pr0, S 0 s|F t q " P ´ST ´ pq,a 1,´1 " 0,a ´1,1 " 2 pq and a ´1,´1 " q p1 ´ q 1 ´ p .

S
Hence the result. so that the process 1{p G is a pP, G 0 q-martingale.

Proof of Theorem 6.2.7. Let L "p L s q 1 §s §t a pP, Fq-martingale on N t .F o rr † t and s P rr r `1,tss let A " A r XtG P Bu an element of G r . Let E Qt denote the expectation taken with respect to Q t .

E Qt " 1 A L s ‰ " E Qt " 1 Ar L s ‰ E Qt " 1 tGPBu ‰ " E r1 Ar L s s E Qt " 1 tGPBu ‰ " E r1 Ar E rL s |F r ss E Qt " 1 tGPBu ‰ " E r1 Ar L r s E Qt " 1 tGPBu ‰ " E Qt " 1 Ar L r ‰ E Qt " 1 tGPBu ‰ " E Qt " 1 A L r ‰ ,
where we have used that the -algebras F t and pGq are independent under Q t in the first line, that P coincide with Q t on F t in the second one, and that pL s q 1 §s §t is a pP, Fq-martingale on N t in the third one. , where we applied the Clark decomposition to F in the second line, and by letting c "pc 1 ,c ´1q, c 1 "pb `a⇢q 1 and c ´1 "´a ´1.

Hence the result.

Proof of Theorem 6.3.5. As a reminder, the strategy "p ↵, 'q is self-financed and only if for any t P N T , A t p↵ t`1 ´↵t q"S t p' t`1 ´'t q, (6.5.1) so that V t´1 p q"↵ t A t´1 `'t S t´1 . Let ' 0 " 0. Assume the existence of an admissible strategy p↵, 'q which final value satisfies V T p q"↵ T A T `'T S T .

Let ⇡ be the F-predictable process such that ⇡ t " ' t S t´1 V t´1 p q for any t P N T . By definition, ∆V t p q"↵ t ∆A t `'t ∆S t " V t´1 p q " ↵ t ∆A t V t´1 p q `⇡t ´b1 tp∆Nt,Wtq"p1,1qu `a1 tp∆Nt,Wtq"p1,´1qu

¯⇢ " V t´1 p q ´↵t r A t´1

V t´1 p q `⇡t `b ∆Z t,1 ´a ∆Z t,´1 `r˘"

V t´1 p q `rp1 ´⇡t q`⇡ t `b ∆Z t,1 ´a ∆Z t,´1 `r˘" rV t´1 p q`V t´1 p q⇡ t `b ∆Z t,1 ´a ∆Z t,´1 ˘,

where we used in the third line that ↵ t A t´1 V t´1 p q `⇡t " 1. Then, ∆V t p q" V t´1 p q ⇡ t 1 `r ˆ´b ´a q 1 ´ p ¯∆R t,1 `a∆R t,´1 ṡo that V T p q"1 `T ÿ t"1

V t´1 p q ⇡ t 1 `r ˆ´b ´a q 1 ´ p ¯∆R t,1 `a∆R t,´1 ˙ by assuming (even if it means dividing by V 0 ), V 0 " 1. We get the equal expression

V T p q" T π t"1 " 1 `⇡t 1 `r ´´b ´a q 1 ´ p ¯∆R t,1 `a∆R t,´1 ¯⇢
The application of the derivative in both directions gives D t,1 V T p q"V t T p q ´b `a q 1 ´ p ¯⇡t and D t,´1 V T p q"´a ⇡ t V t T p q, by using D t,k ∆Z t,k " sgnpkq. This entails

p E " D t,1 V T | F t´1 ‰ " ´b `a q 1
´ p ¯⇡t V t´1 p q and p E " D t,´1 V T | F t´1 ‰ "´a ⇡ t V t´1 p q so that p E " D t,1 V T p q`D t,´1 V T p q|F t´1 ‰ " ´b ´a `a q 1 ´ p ¯⇡t V t´1 p q Then, by letting on the one hand ' 0 " 0 and ' t " V t´1 p q ⇡ t "p1 `rq t´1 , and, on the other one, ↵ 0 "p1 `rq ´T p E rFs{S 0 and for any t P N T , ↵ t " ↵ t´1 ´p' t ´'t´1 qS t´1 A t´1

we get a couple of F-predictable processes "p ↵, 'q such that satisfies the self-financing condition and of terminal value F.

Proofs of Section 6.4

Proof of Theorem 6.4.3. Transposing the result of Theorem 6.4.1 to solve the problem (6.4.5) we get

Φ G pxq"E " up p V G x,t q ı " L p p V G x,t , p y G x , x M G p y G x q,
where L is the Lagrangian associated to the primal problem 6.4. It is thus possible to compare the logarithmic utilities up to time t of the two agents and to quantify the insider's additional one. As for the economic agent, the sequence p⌘ j q jPJt is determined thanks to the initial budget constraints, as to say

x " E « dQ j t ∞ jPJt ⌘ j dQ j t " E « L t dQ j ∞ jPJt ⌘ j L t dQ j t " E « dP j t ∞ jPJt ⌘ j dP j t .
The sequence p⌘ j q jPJt is solution of the system satisfied by the sequence p j q jPJt ; by uniqueness of the solutions, for any j P J t , ⌘ j " j . Thus, where we have used that L " 1{p G is a pP, G 0 q-martingale.

U
Proof of Lemma 6.4.7. Consider a G-predictable admissible strategy ✓. By definition of Φ, Φ ins T ´1pxq °E rupV T ´1p✓qqs. Transposing once again the result of Theorem 3.2.1. of [START_REF] Delbaen | The Mathematics of Arbitrage[END_REF] to our model, we get `Φins T ˘1pxq °0. In particular for T " 1 and with x ✓ " E rupV T ´1p✓qqs,w e get Φ ins 1 pxq °Φins 1 px ✓ q.

This holds for any G-predictable admissible strategy ✓. Assume moreover the existence of a G-predictable admissible strategy ✓ such that Then, Φ ins 1 pxq § Φ ins T pxq. The result follows.

Proof of 6.4.9. By transposing the results established by (example 3.3.2 in [START_REF] Delbaen | The Mathematics of Arbitrage[END_REF]) for the binomial model to our "associated" one pS bin q, we get the optimal strategy ' ag " xp1 `rq b ´a ˆpq ˚´qp p˚qẘ here p ˚" r ´a b ´a and q ˚" b ´r b ´a define the well-known risk-neutral probability measure in the Cox-Ross-Rubinstein model. The corresponding optimal investment from the agent's point of view (in the ternary model) is thus

V ag

x,1 " x1 t∆N 1 "0u `xpb ´aq ppr ´aq 1 tp∆N 1 ,W 1 q"p1,1qu `xpb ´aq p1 ´pqpb ´rq 1 tp∆N 1 ,W 1 q"p1,´1qu

" x1 t∆N 1 "0u `xp p ˚1tp∆N 1 ,W 1 q"p1,1qu `xq q ˚1tp∆N 1 ,W 1 q"p1,´1qu

We get thus Φ bin 1 pxq"logpxq`k bin (6.5.2)

where k bin "´p logpp ˚q´q logpq ˚q`p logppq`q logpqq. Follows from (6.4.7) that, logpxq`k ag " plogpxq`k bin q`p1 ´ q logpxq that is k ag " k bi . Thus, from Φ ag 1 pxq" Φ bin 1 pxq`p1 ´ q logpxq so that Then, we make use of this new formalism to compute the additional utility of an insider in the new ternary model. The insider benefits from an extra information hidden in a random variable from the start of trading period. Using the toolbox of C. Blanchet-Scalliet, M. Jeanblanc and R. Romero as for the enlargement of filtrations in discrete settings, we define an analogue of the information drift and express it in terms of our geometric gradient. We show that the insider's additional expected logarithmic utility at time t can be expressed as the relative entropy of the initial measure with respect to the martingale preserving measure on rr 0,tss .W er e t r i e v ee x a c t l yt h er e s u l to fJ . Amendinger, P. Imkeller and M. Schweizer stated in the continuous case.

Φ

Definition 2 . 1 . 1 .

 211 Ar a n d o mv a r i a b l eF is said to be cylindrical if there exist a finite subset B Ä A and a function F B :E B ›Ñ L 2 pAq such that F " F B ˝rB , where r B is the restriction operator : r B :E A ›Ñ E B px a ,a P Aq fi ›Ñpx a ,a P Bq.

Theorem 2 . 1 . 6 (›››Ñ L 2 pAq 0 and DF n nÑ8 ››››››ÑL 2

 216nÑ82 ‹ Integration by parts). Let F P S.F o re v e r ys i m p l ep r o c e s sU,xDF, Uy L 2 pE A ˆAq " E « F ÿ aPA D a U a .(2.1.1)Thanks to the latter formula, we are now in position to prove the closability of D: for pF n q nPN å sequence of cylindrical functionals, ˆFnnÑ8 pE A ˆAq ⌘ ˙ùñ ⌘ " 0.Corollary 2.1.7 (‹). The operator D is closable from L 2 pAq into L 2 pE A ˆAq.

  zq´Fpxqqqpx, px a ,zqq so that we retrieve (2.1.4).

Corollary 2 . 2 . 7 (

 227 ‹ Poincaré or Efron-Stein inequality). For any F P D, VarpFq § }DF} 2 L 2 pE A ˆAq . Another corollary of the Clark formula is the following covariance identity. Theorem 2.2.8 (‹ Covariance identity). For any F, G P D,

.4. 3 )

 3 Moreover, as follows from D a D a " D a and D b D a " D a D b for any a, b P A that

  D b E rhpX BYC q|X B s . (3.4.1)

.4. 13 )

 13 Combine (3.4.8)-(3.4.13) to obtain (3.2.6).

6. 4 . 3

 43 Computations in the case G " 1 tS T Pr0,S 0 su

Corollary 6 . 4 .

 64 10 (‹). The optimal couple pp y x , x M p yx qPR ˆC F 1 defined in Theorem 6.4

  t,k F | F t´1 s b ´a `a q 1´ p

  soirées chez Paul, Sofia à la Belle Époque ou au Circus, signatures, théâtres ou (inclusif voire à l'intersection de) cinémas. . . Sur une frise presque-chronologique prennent place : Yvonne et Jacques Desvergez, Théo, Mathilde et Wallerand, la famille Laurent, Damien, Laure et Xavier, Florence, Lee, Camille, Camiche et Ming, Ingrid, Jean, Carole, la famille Larebière-Ducasse, Marie-Pascale, Julie et Fred, Adrien, la famille Bourgeois, Aurore, Jennifer, François, Bénédicte. Les derniers remerciements sont destinés à ma famille pour son irremplaçable présence. Merci à mes oncle et tantes Marie-Pascale, Nicolas et Isabelle pour leurs encouragements et soutien toujours affectueux. Pensée toute particulière pour mon oncle Bernard, le premier à m'avoir parlé de Paul Malliavin lors d'une de nos discussions hebdomadaires ! Un grand merci enfin. . . À Antonin pour la (presque familiale) entité encadrant notre amitié, dont on trouve au dénominateur : Rachmaninoff et Dream Koala, matrices de Pauli et paire duale (et une topologie discutable), joutes verbales au bureau, au Zango ou sur Teams/Whatsapp/Outlook,

à Tignes et au Revest, conférences du samedi et cinés du dimanche. . . et dont il m'est donné de mesurer quotidiennement (mais pas quantiquement) l'inestimable richesse ! À ma Doc-Soeur Stéphanie, dont j'admire les compétences professionnelles, suis une inconditionnelle des inégalables réparties, et apprécie le tempérament complémentaire, partition de nos jeux d'hier, de nos discussions d'aujourd'hui et de notre complicité de toujours. À mes parents pour leur soutien indéfectible, leur affection, tous les délicieux moments en famille dont ils ont été les instigateurs, et -si je ne pourrai jamais leur rendre -pour ce qu'ils m'ont donné : presque sûrement tout.

  Au is the family of quadratic errors on tX k ,kP Au. Actually, the quadratic error can be expressed as 2 F " ΓpFq"ΓpF, Fq, where Γ is the carré du champ operator defined by

						CONTENTS
	and t 2 k ,kP ΓpF, Gq"	1 2	´LpFGq´G LF ´F LG ¯.	(0.0.7)
	2 F " LpF 2 q´2F LpFq,	(0.0.6)
	where L is the differential operator
		L "	1 2	ÿ kPA	2 k	B 2 BX 2 k

  En fait, Wphq coïncide avec l'intégrale de Wiener de la fonction h P H relativement à B. L'espace de Wiener est donc entièrement caractérisé par le processus isonormal Gaussien tWphq,h P Hu qui peut lui être associé. Même si l'intégrale de Wiener lie naturellement l'espace de Wiener à une notion d'intégration dans le sens qui vient d'être défini, la possibilité de définir un opérateur de différentiation pour une large classe de fonctionnelles de Wiener est restée longtemps obscure. En effet, la définition en tant que dérivée de Fréchet, aussi naturelle qu'elle puisse paraître au premier abord, n'est pas pertinente : les fonctionnelles

de Wiener usuelles comme les intégrales de Wiener ou les solutions d'équations différentielles stochastiques avec des coefficients réguliers ne sont même pas continues pour la norme de l'espace de Wiener. La définition appropriée d'une dérivée directionnelle dans L p pPq provient du théorème de Cameron-Martin qui indique -implicitement -qu'elle n'a de sens que si les accroissements générant la dérivée sont réalisés dans les directions de l'espace vectoriel du même nom. L'opérateur D est alors défini sur l'espace S variables aléatoires cylindriques de la forme F " f pWph 1 q, ¨¨¨, Wph n qq ; h 1 , ¨¨¨,h n P H, (0.0.9) où f appartient à l'espace de Schwartz S pR n q. Par densité de l'espace S dans l'espace des fonctionnelles de Wiener intégrables, cette définition peut être étendue au domaine de D,noté D 1,2 . L'adjoint de l'opérateur D, appelé divergence et noté , coïncide avec l'intégrale de Wiener (resp. de Skorohod) pour des processus adaptés (resp. non-adaptés). Les opérateurs D et sont liés par l'identité fondamentale : E rxDF, Uy H s " E rF Us ;FP D 1,2 , U P Dom . Comme cela est détaillé notamment dans les ouvrages de Paul-André Meyer [89] ou Nobuaki Obata [107], il existe une approche alternative menant au calcul de Malliavin et reposant sur la représentation chaotique de Wiener-Itô. Elle est basée sur le développement de ce qui serait l'analogue (en dimension infinie) de la théorie de Schwartz où la mesure Gaussienne µ sur le dual E ˚d'un espace nucléaire E " S pRq se substituerait à la mesure de Lebesgue sur R n . L'espace E (resp. E ˚) des fonctionnelles de bruit blanc test (resp. généralisées) est construit par un plongement dense et continu dans L 2 pE ˚,µq (resp. par dualité) :

  La multiplicité des approches et la variété des espaces canoniques sur lesquels il opère donc, semblent dissuader l'éventuel projet d'un travail unificateur. Si la formalisation d'une construction universelle parait donc compromise, il demeure envisageable d'identifier un dénominateur commun à tous ces formalismes. Celui-ci serait plutôt à chercher du côté de la ter-

	CONTENTS CONTENTS
	aux dérivées partielles et d'équations différentielles stochastiques. Cela donna lieu à de nom-breuses applications passées en revue notamment par Martha Sanz-Solé dans [128] (resp. par de simuler les Grecques. Le travail pionner de Fournier et al. [54] a été transposé pour les suite pr t q tPT de taux d'intérêt et que S satisfait une équation différentielle stochastique (dans le cadre du modèle de Black-Scholes par exemple) ou une équation aux différences (dans le modèles à sauts (en finance) dirigés par des processus de Poisson par Nicolas Privault et al.
	Hu, Huang, Lê, Nualart and Tindel [65]) pour les EDP dirigées par un bruit coloré (resp. par cadre de modèles discrets) sur l'intervalle de trading T. Nous nous intéressons à certains pro-(voir [50],[121]) et par des processus de Lévy (notamment) par Marie-Pierre Bavouzet-Morel
	duits dérivés particuliers, dont la valeur dérive donc des performances de S, appelés options et Marouen Messaoud dans [15]. un bruit blanc en temps /bruit rugueux en espace avec un indice de Hurst H Pp 1{4, 1{2q). On peut également citer les travaux de Vlad Bally et Denis Talay [13], Yaozhong Hu, David (vanille). Une option est un contrat qui donne à l'acheteur (le propriétaire ou détenteur) le Fortement connectée à la théorie des semi-martingales, l'évaluation des options et en partic-
	Nualart et Jian Song [66], ou Shigeo Kusuoka et Daniel W. Stroock [78] pour des équations droit mais non l'obligation, d'acheter ou de vendre un actif risqué à un prix fixé d'avance K et ulier les problèmes de couverture ont également tiré bénéfice des contributions du calcul de
	différentielles stochastiques dirigées par le mouvement Brownien et ceux de David Nualart et à une date spécifique T . La valeur d'une option à échéance, correspondant à ce qui est finale-Malliavin. La réponse à la question de hedging ("couverture") est cruciale dans la mesure où
	Bruno Sausserau [101] pour les EDS dirigées par un mouvement Brownien fractionnaire avec ment perçu par l'acheteur, est appelée payoff ; pour une option, cette valeur est modélisée elle donne des éléments de compréhension aux acheteurs ou vendeurs pour composer et ajuster
	un indice de Hurst H Pp1{2, 1q. Les résultats du calcul de Malliavin furent également exploités pour étudier des équations de façon dynamique leur portefeuille dans le but de répliquer cette option, i.e. atteindre sa par une variable aléatoire ΦpS T q. Enfin, on considère que sur ce marché agissent des vendeurs et acheteurs considérés comme des investisseurs ordinaires (resp. des initiés) si les décisions valeur à échéance. Mathématiquement, cela revient à déterminer un processus à valeurs
	différentielles stochastiques non-causales formulées en termes d'intégrales stochastiques an-relatives à la composition de leur portefeuille (avec les deux types d'actifs à leur disposition) dans R 2 satisfaisant une condition d'auto-financement (voir le chapitre 4 du manuscrit pour
	ticipatives. En effet l'opérateur divergence permet de définir une extension de l'intégrale sont prises sur la base d'informations publiques (resp. en bénéficiant d'informations supplé-son expression dans les modèles de Black-Scholes et de Cox-Ross-Rubinstein)
	stochastique d'Itô à des intégrandes anticipatifs dans le cas du mouvement Brownien (voir mentaires).
	Skorohod [131]). Une formule d'anticipation est donnée dans Nualart et Pardoux [99] ou dans Les Grecques, ainsi dénommées par allusion aux lettres grecques par lesquelles elles sont
	une version antérieure (et méconnue) par Masayuki Hitsuda [64] qu'Elisa Alòs et David Nu-désignées, sont des instruments de calcul de sensibilité du prix d'un actif relativement à la
	alart améliorèrent dans [4]. Un calcul anticipatif d'abord développé selon cette approche pour modification de paramètres sous-jacents dont sa valeur dépend. Elles sont d'une utilité capi-
	le mouvement Brownien (voir Berger [17] ou dans le chapitre 3 de Nualart [98]) a depuis été tale pour l'agent car elle lui confèrent des éléments de contrôle de son exposition aux risques.
	étendu aux processus de Poisson (voir Decreusefond and Savy [40], Nualart et Vives [103]) Par exemple, Delta et Rho sont les Grecques mesurant respectivement la sensibilité au prix de
	et aux processus de Lévy (voir [3],[105]). Elisa Alòs, Olivier Mazet et David Nualart util-l'actif sous-jacent et au taux d'intérêt alors que Gamma désigne un indicateur du second ordre
	isèrent également le calcul de Malliavin (en particulier la divergence) pour donner un sens à quantifiant la variation du Delta relativement à une variation du prix sous-jacent. Pendant
	l'intégrale stochastique par rapport au mouvement Brownien fractionnaire avec un indice de longtemps, le calcul de ces indicateurs a été la seule prérogative des analystes qui s'y employ-
	aient par des méthodes de différences finies dans le cadre de la méthode de Monte-Carlo. Sans Hurst H Pp0, 1{2q et construire un calcul stochastique dans ce contexte. introduire précisément le formalisme (ce qui sera fait au chapitre 4) considérons une option de
	Statistique payoff K " ΦpS T q et dont la dynamique de la suite de prix de l'actif risqué sous-jacent pS t q tPT est donnée par le modèle de Black-Scholes :
	Un large éventail d'outils issus du calcul de Malliavin trouvèrent leur place dans des travaux en statistique ; nous en donnons un bref aperçu dans ce paragraphe. Par exemple, Ciprian A. Tudor et Frederi G. Viens [135] utilisèrent la décomposition chaotique pour proposer un adap-S t " 1 `ª t 0 rS u du 0 `ª t S u dB u ,tP T, (0.0.11)
	tateur pour le paramètre d'auto-similarité H du processus de Rosenblatt. Fabienne Comte et où les paramètres r et représentent respectivement le taux d'intérêt sans risque et la volatilité Nicolas Marie [29] montrèrent que l'intégrale de Skorohod (définie à partir de la divergence) est l'unique extension de l'intégrale d'Itô valable pour l'estimation non-paramétrique dans des des profits générés par l'actif risqué. La valeur de l'option à un instant donné t P T est notée V t . Le Delta s'exprime (dans le modèle Black-Scholes) par EDS dirigées par un mouvement Brownien fractionnaire d'indice de Hurst H Pp1{2, 1q.J o s é M. Corcuera et Arturo Kohatsu-Higa [31] tirèrent profit de la formule d'intégration par par-ties pour étudier l'inférence asymptotique d'un processus (à sauts) stochastique, tandis que ∆ " BV 0 BS 0 " e ´rT E Q " Φ 1 pS T q ⇢ BS T , (0.0.12) BS 0 Nicolas Privault et Anthony Réveillac l'utilisèrent à une autre fin : celle de d'estimer le drift où Q correspond à la mesure de probabilité risque neutre. En général, la fonction Φ n'est pas de processus Gaussiens à l'aide d'estimateurs de type Stein [118]. dérivable ; par exemple, le payoff d'un cal l européen (option d'achat à une date déterminée) calcul de Malliavin a trouvé sa place ; à titre d'exemple on peut citer l'étude des opérateurs de Hörmander (qui est l'application « historique »), plus généralement l'étude de la régularité des lois de probabilité de solutions d'EDS/EDPs, la finance, la méthode de Stein, le calcul d'erreur, le calcul anticipatif et plus récemment l'inférence statistique. Finance est ΦpS T q"p S T ´Kq `. Il serait donc accommodant de pouvoir réécrire ∆ sous la forme ∆ " e ´rT E Q rΦpS T q ⇡s,o ùl epoids ⇡ ne dépendrait pas du payoff. L'idée de base fut d'abord introduite par Éric Fournié, Jean-Michel Lasry, Jérôme Lebuchoux, Pierre-Louis Lions et Nizar Une grande variété des fruits nés de la rencontre du calcul de Malliavin et de la finance Touzi dans [54] et [55] qui, tirant avantage de la formule d'intégration par parties d'une part, sont décrits la monographie de Paul Malliavin et Anton Thalmaier (voir [86]). Trois de ces applications ont particulièrement éveillé notre intérêt : le calcul de Grecques, la gestion de de la chain rule ("règle de dérivation en chaîne") d'autre part, ont prouvé que
	Équations differentilles sotchatiques, équations aux dérivées partielles et calcul portefeuille, et le délit d'initié. Afin de placer ces points d'intérêt dans le vaste univers des mathématiques financières, re-∆ " e ´rT E Q rΦpS T q puqs , anticipatif streignons d'abord le cadre d'étude à celui d'un marché composé simplement de deux actifs : de telle sorte que le poids s'exprime comme l'intégrale de Skorohod de générateur u, ici déter-
	Gardant à l'esprit les résultats du papier pionner de Paul Malliavin, de nombreux auteurs ré-utilisèrent la condition de non-dégénérescence pour étudier la régularité des solutions d'équations un actif sans risque modélisé par une suite déterministe A "p A t q tPT et un actif risqué dont la suite des cours est donnée par S "p S t q tPT .S u p p o s o n sd ' a b o r dq u eA est défini par une ministe et constant égal à p T q ´1. Éric Benamou ajouta dans [16] des conditions nécessaires et suffisantes sur u permettant d'automatiser la génération d'un poids dans la perspective

minologie du calcul de Malliavin qui peut être introduite par analogie avec celle du calcul différentiel classique dans les espaces de Banach : aux vecteurs se substituent les trajectoires (indexées par le temps ou un espace de Hilbert) du processus canonique, aux fonctions les fonctionnelles opérant sur l'espace des trajectoires, tandis qu'on désigne volontiers par les termes gradient et divergence, les deux opérateurs phares de cette théorie : la dérivée au sens de Malliavin et son adjoint. Le semi-groupe d'Ornstein-Uhlenbeck et son générateur complètent la famille des opérateurs de Malliavin. L'existence d'une relation fondamentale liant gradient et divergence parachève la description du champ lexical du calcul Malliavin et apparaît comme sa pierre angulaire : la formule d'intégration par parties au sens de Malliavin, E rxDF, Uy R s"E rF Us , (0.0.10) où pΩ, A, Pq est l'espace de probabilité canonique et R est un espace de Hilbert générique à voir comme l'espace des perturbations lorsque DF est interprété comme une dérivée directionnelle (par exemple R est l'espace de Cameron-Martin pour le mouvement Brownien). Au cours de la préparation de cette thèse, deux espaces munis d'une structure de Malliavin suscitèrent spécialement notre intérêt : l'espace de Poisson et l'espace de Rademacher. Le formalisme de Malliavin développé dans ces deux cadres sera décrit puis comparé à nos contributions à la fin de cette introduction. Branches et fruits : les applications du calcul de Malliavin La rencontre détonante du calcul de Malliavin avec d'autres branches des probabilités ou de l'analyse a fait entrevoir la puissance de cet outil et l'étendue des applications dont il est garant des bases théoriques. Nombreux sont les champs dans lesquels le

  initié dans le modèle trinomial. Ce problème présente un intérêt particulier à plusieurs égards ; tout d'abord, et comme mentionné plus haut, les mathématiques financières sont le champ d'application phare du calcul de Malliavin. Si la couverture d'options avait déjà été étudiée dans le modèle Cox-Ross-Rubinstein (voir le chapitre 1 dans Privault[START_REF] Privault | Stochastic Analysis in Discrete and Continuous Settings[END_REF]) en utilisant la structure de Malliavin dont est équipé l'espace Rademacher, aucun projet similaire n'avait été réalisé jusqu'alors dans le cadre des arbres multinomiaux (avec au moins trois branches pour l'élargissement de filtration en temps discret par Catherine Blanchet-Scalliett, Monique Jeanblanc et Roméo Romo Romero, nous définissons l'analogue du drift d'information dont nous donnons une expression en termes de gradient géométrique. Nous montrons alors que l'utilité logarithmique espérée additionnelle de l'initié au temps t peut être exprimée comme l'entropie relative de la mesure initiale par rapport à la mesure préservant les martingales sur r0,ts. Nous retrouvons ainsi exactement le résultat établi par Jürgen Amendinger, Peter Imkeller et Martin Schweizer dans le cas continu. Nous illustrons ce résultat et achevons ce travail en effectuant des calculs explicites dans le cas précis où l'initié obtient une information supplémentaire sur la valeur finale de l'actif risqué.ManuscritLe manuscrit est organisé comme suit ; il se compose de deux parties, chacune d'entre elles se référant à l'un des travaux mentionnés ci-dessus. La première, intitulée Structures de Malliavin et Dirichlet pour les variables aléatoires indépendantes renvoit à un travail mené en collaboration avec Laurent Decreusefond et publié dans Stochastic Processes and their Applications en 2019 (voir

	où Γ est l' opérateur carré du champ défini par gradient comme opérateur d'annihilation opérant sur un espace décomposable en chaos, nous Calcul de Malliavin pour des processus géométriques composés et délit d'initié
	ΓpF, Gq" définissions le gradient discret comme un opérateur différence dont la k-ème coordonnée traduit 1 2 dans un modèle ternaire ´LpFGq´G LF ´F LG ¯. (0.0.15) L'utilisation de ces outils est plus pertinente lorsqu'il s'agit d'estimer la vitesse de convergence d'une suite de variables aléatoires pX n q nPN ˚vers une variable aléatoire X, toutes supposées être définies sur un espace de probabilité pΩ, A, Pq.E n e ffet, si la forme Γ est fermée, la théorie du calcul d'erreur se déploie en s'appuyant sur le principe suivant : si la suite de couples pX n , erreur sur X n q converge convenablement, elle converge nécessairement vers un l'opérateur L apparaissant dans (0.0.15) coïncide avec l'opérateur Laplacien/nombre L "´ D. muni d'un calcul de Malliavin formalisant ainsi le concept de structure de Dirichlet-Malliavin. moment d'ordre 4 dans [11]. structures de Dirichlet. De telles structures sont de facto définies sur tout espace canonique et Guillaume Poly ont exploité le lien entre les trois théories pour revisiter le théorème du D'autre part, notre formalisme se raccorde aux théories préexistantes en temps continu par les Dirichlet-Malliavin (voir Decreusefond [36]). En outre, Ehsan Azmoodeh, Simon Campese par cette correspondance que nous pouvons en revisiter certaines. implicitement dans la combinaison Stein-Malliavin pour aboutir à une méthode de Stein-par ailleurs au rôle prépondérant dans l'établissement d'inégalités fonctionnelles. C'est donc travers l'exemple 1.3.6. Cette relation suggère en fait que les formes de Dirichlet s'insèrent que les opérateurs nombre définis dans les deux contextes coïncident. L'opérateur nombre joue Ce point clef sera expliqué dans la section 1.3.2 et illustré dans le cas de l'espace Wiener à ité. Tout d'abord, en la connectant à son aînée sur l'espace de Rademacher. Bien que les définitions de gradient soient distinctes (pour les raisons précédemment évoquées), il apparaît couple pX, erreur sur Xq. Lorsque l'espace sous-jacent est muni d'une structure de Malliavin, l'idée de mesure de l'influence de la k-ème composante du processus initial sur la fonction qui L'objectif initial du second travail n'était pas de concevoir un calcul stochastique pour les s'y applique. S'en suit la définition de son adjoint, appelé divergence (s'exprime comme processus composés géométriques mais plutôt d'utiliser notre calcul discret de Malliavin pour somme des dérivées partielles) et qui satisfait une formule d'intégration par parties discrète. traiter des problèmes de délit d'initié. C'est également la raison pour laquelle le titre de la thèse La définition de l'opérateur nombre (appelé Laplacien dans le cas Gaussien), qui est égale-n'en fait pas mention. En effet, nous avons d'abord essayé de déployer notre artillerie pour at-ment générateur d'un semi-groupe de Markov, complète alors la définition des opérateurs de teindre une cible identifiée : le calcul de la valeur logarithmique additionnelle attendue l'utilité Malliavin. Nous justifions notre construction à plusieurs égards et lui conférons ainsi une forme de légitim-d'un
	En effet, les structures de Dirichlet Poissonienne et Brownienne usuelles associées à leur gra-La version dessinée de cet état de l'ar-bre se trouve page 17 et conclut cette partie. dient respectif s'écrivent comme limites des structures induites par notre formalisme. En reprenant cette métaphore, nous pouvons résumer les contributions de cette thèse à Enfin, nous dérivons de notre construction nombre d'identités fonctionnelles. Nous trans-l'addition de deux racines à cet arbre : la construction d'un calcul de Malliavin pour les posons ainsi l'identité de covariance, l'inégalité de log-Sobolev et l'inégalité de déviation clas-suites de variables aléatoires indépendantes d'une part, et pour les processus géométriques siques dans notre contexte. Nous revisitons d'autres auxquelles nous donnons une nouvelle composés d'autre part. Au delà de l'intérêt propre que chaque construction a pu susciter, interprétation. Par exemple, nous montrons que l'inégalité d'Efron-Stein peut être interprétée notre choix de développer un formalisme dans ces deux cas a été tout ou partie motivé par comme une inégalité de Poincaré ou que la décomposition de Hoeffding des U -statistiques peut les perspectives d'applications qu'il laissait entrevoir. Aussi, les fruits résultant de ces deux être vue comme un avatar de la formule de représentation de Clark. Ainsi, notre formalisme formalismes sont représentés en couleur dans la figure page 17 : couleur pourpre pour ceux trouve sa place dans le paysage du calcul de Malliavin ; il généralise ce qui avait été théorisé relatifs au cadre des variables aléatoires indépendantes, et en bleu pour ceux relatifs au cadre dans le cas Rademacher et se connecte avec les contextes Brownien et Poissonien par approx-des processus géométriques composés. imation des structures de Dirichlet induites d'une part, et par la ressemblances des identités
	fonctionnelles (comme la formule de Clark, inégalité de Poincaré. . . ) obtenues dans les trois
	fonction de quantités erronées tX k ,k P Au (supposées Calcul de Malliavin et structures de Dirichlet pour des variables aléatoires in-cas d'autre part. La seconde motivation pour élaborer cette construction provenait de la méthode de Stein, dépendentes avec qui, comme évoqué précédemment, branche des un tel formalisme serait susceptible de La motivation d'élaborer un calcul de Malliavin pour des variables aléatoires indépendantes se combiner avantageusement. Nous donnons une borne générale de type "Berry-Esseen" était double ; d'une part, pour lui-même et pour généraliser ce qui avait été exploré jusqu'alors pour la distance de Wasserstein entre la loi de fonctionnelles de variables aléatoires indépen-dans un cadre discret, et d'autre part pour les applications potentielles au contrôle des suites dantes et la loi Normale ; et par la même d'établir un critère de "Stein-Malliavin" analogue de variables aléatoires indépendantes auxquelles son élaboration pourrait conduire. à celui rencontré dans les cas Gaussien et Poisson. Nous formulons un critère similaire pour connues, petites et indépendantes) l'erreur quadratique potentielle à espérer sur F notée 2 F peut être écrite comme suit : 2 F " LpF 2 q´2F LpFq, l'approximation de la loi Gamma que nous appliquons au cas particulier de U -statistiques Jusqu'ici l'incursion du calcul de Malliavin dans un contexte discret était circonscrite à l'espace de Rademacher. L'approche chaotique par laquelle est développé un calcul de Malliavin dans d'ordre 2 et en obtenons une estimation de type de Jong. (0.0.14) où L est l'opérateur différentiel L " 1 2 ÿ kPA 2 k B 2 BX 2 k cet espace s'appuie sur l'existence de martingales normales ou novations (pour reprendre la Pour toutes ces raisons, il semble que notre structure de Dirichlet-Malliavin donne un cadre terminologie de Émery) associées au processus canonique. L'existence de novations requiert unificateur à de nombreux résultats disséminés dans la littérature et suggère que des résultats que les lois des variables aléatoires ne soient supportées que par deux points. L'expression sans lien apparent (Efron-Stein, paires échangeables etc.) sont en réalité les faces d'une même du gradient (0.0.8) de Rademacher n'a d'ailleurs de sens que si les variables aléatoires sont réelles ou que leur espace d'état est réduit à deux points. Dans le cas général que nous avons pièce... à "l'effigie" de la formule d'intégration par parties.
	et t 2 choisi d'investiguer, celui de suites de variables aléatoires sur lesquelles une simple hypothèse k ,k P Au est la famille des erreurs quadratiques sur tX k ,k P Au. En fait, l'erreur quadratique peut s'exprimer d'indépendance est émise, il convient de préciser que les espaces d'états sous-jacents peuvent
	2 F " ΓpFq"ΓpF, Fq, être différents (simplement supposés Polonais). Devant l'impossibilité de définir l'opérateur

à chaque étape). En outre, si de nombreux travaux traitent de l'élargissement de filtrations et des questions de délit d'initié en temps continu (voir les travaux d'Amendinger, Imkeller et Schweizer (

[START_REF] Amendinger | Additional logarithmic utility of an insider[END_REF]

), très peu ont été réalisés dans des modèles discrets. Certains résultats existent déjà pour l'élargissement de filtrations en temps discret. Catherine Blanchet-Scalliet, Monique Jeanblanc et Roméo Romo Romero ont montré dans

[START_REF] Blanchet-Scalliet | Enlargement of filtration in discrete time[END_REF]

que les résultats connus en temps continu s'étendent immédiatement dans un contexte discret. La plupart d'entre eux sont facilement obtenus grâce à la décomposition de Doob. À cet égard, nous souhaitions discuter de la possibilité d'interpréter le processus prévisible croissant apparaissant dans la décomposition de Doob comme le drift d'information associé à l'initié et l'exprimer à l'aide de la dérivée de Malliavin, à l'instar du travail de Peter Imkeller en temps continu

[START_REF] Imkeller | Malliavin's calculus and applications in stochastic control and finance[END_REF]

.

Si notre formalisme se prête bien au calcul de Grecques dans le modèle trinomial, il demeure en effet impossible d'en dériver une formule de Karatzas-Ocone pour la stratégie couverture et donc d'en tirer avantage pour l'évaluation d'options. Celle-ci est une conséquence directe du théorème de représentation de martingale qui ne peut être statué quand les lois des variables aléatoires (discrètes) sont portées par au moins trois points. Ce constat a motivé le développement d'un calcul de Malliavin pour les processus géométriques composés (dont les variables i.i.d. correspondantes sont à valeurs dans t´1, 0, 1u), sur lesquels baser un modèle ternaire équivalent en loi au modèle trinomial.

De par la structure de saut qui les sous-tend, les processus géométriques héritent de nombreuses propriétés des processus de Poisson, source d'inspiration pour élaborer un calcul de Malliavin. S'il demeure, comme attendu, impossible d'établir une décomposition en chaos à partir de martingales normales, (le carré des variables aléatoires i.i.d. étant constant), il existe une décomposition en pseudo-chaos au moyen d'intégrales multiples relatives à une famille non-orthogonale (mais orthogonalisable) de variables aléatoires. Le gradient géométrique est alors défini comme l'opérateur d'annihilation agissant sur cette pseudo-décomposition chaotique et tel que l'intégration stochastique (définie par rapport à la famille orthogonalisable) apparaît -à une transformation linéaire près -comme l'opération "inverse". En outre, cet opérateur coïncide avec un opérateur différence dont l'expression est plus propice à statuer des identités fonctionnelles comme la formule de représentation martingale, la formule d'Itô, et la formule de Ocone-Karatzas espérée.

Nous appliquons alors ce formalisme au calcul de l'utilité additionnelle d'un initié dans un modèle ternaire construit sur la dynamique d'un processus géométrique composé. Reprenant les outils développés

  Definition 1.2.8. Under previous assumptions, the energy function EpFq is defined for any random variable F P L 2 pµq such that the limit Remark 1.2.9. The possibility to define E in the fashion (1.2.3) is consequent of the integration by parts formula satisfied for any pF, GqPD Γ ˆDΓ by ª

	tÑ0 lim	t 1	ª

E

FpF ´Pt Fq dµ exists; that shapes the domain D E of E. The Dirichlet form E (similarly denoted for the purposes of notation) is the bilinear symmetric operator on D E defined by EpF, Fq"EpFq and EpF, Gq" ª E ΓpF, Gq dµ ; @pF, GqPD E ˆDE .

(1.2.3) E ΓpF, Gq dµ "

  Consider the classical Wiener space supported by the Banach space W B " C 0 pr0, 1s ; Rq and equipped with the Wiener measure P. Let B "p B t q tPr0,1s the standard Brownian motion defined on the probability space pW B , A

			ª t 0	9 f psqds
	Define the mapping
		E ݄ W B
	W :	h fi ›Ñ	ª 1	9 hpsq dB
			0

B , Pq. Consider the family E "tt ^¨,tPr0, 1su , such that the closure with respect to the norm }¨} H 1 induced by the scalar product xf, gy H 1 "x 9 f, 9 gy L 2 pTq , is a Hilbert space, called the Cameron-Martin space defined by H 1 " " f : r0, 1sÑR, D 9 f P L 2 pTq with f ptq" * and }f } H 1 "} 9 f } L 2 pTq . (1.3.1) s Within W,t h em a p p i n g1 r0,ts fi Ñ B t naturally extend to an isometry between H 1 and the space L 2 pW B , Pq. In particular we can retrieve the canonical process as the Wiener integral Bptq"Wpt ^¨q; t P T. (1.3.2) The process pWphqq hPH 1 defined by Wphq"Wphq ; h P H 1 , is an isonormal Gaussian process associated with the Hilbert space H 1 . The aim is to define the derivative rF of a square integrable random variable F:W B Ñ R. A natural idea is to explore the possibility to define it as a Fréchet derivative lim "Ñ0

  1,2 B .F o ra n yu P Dom , puq is the element of L 2 pW B q characterized by Remark 1.3.4. The divergence operator coincides with the Wiener integral on H 1 as highlighted in(1.3.6). Moreover, the special feature of Malliavin integration by parts is to include the adjoint of the derivative operator.

	E rF puqs " E rxrF,uy H 1 s ; @F P D 1,2 B .	(1.3.7)

The Malliavin calculus on the Wiener space can be elaborated in a different but equivalent way; as a reminder, the family of Hermite polynomials tH n ,n P Nu is such that the associated polynomial functions satisfy H 0 " 1 and H 1 n " nH n ,nP N ˚.

  1.3.2 Malliavin calculus and Dirichlet structures: an intrisic connectionDue to the existence of an underlying Markovian frame, on which they both rest, Malliavin calculus and Dirichlet structures appear to be the two sides of the same coin. By construction, a Dirichlet structure is naturally induced when we are provided by a Malliavin frame. It becomes possible to express the integration by parts formula by means of the carré du champ operator associated in the coupled Dirichlet structure.Let a probability space pΩ, A, Pq supposed to be endowed by a Malliavin structure on a Hilbert space pH, x¨, ¨yq that is, a gradient operator D, a divergence operator , an Ornstein-Uhlenbeck operator satisfying L "´ D and a integration by parts formula including (1.3.1). Let D 1,2 "tF P L 2 pΩq : }F} 1,2 "}F} L 2 pΩq `}DF} L 2 pΩ;Hq be the closure of cylindrical functions for the norm }¨} 1,2 . Then the quintuplet pΩ, A, P, D E , Eq where D

E " D 1,2 and EpF, Gq"E rxDF, DGy H s ;F , G P D E (1.3.9) is a Dirichlet structure. The associated carré du champ operator is defined on D E by ΓpF, Gq"xDF, DGy H

  ΓpF, Gq"rF.rG and EpF, Gq"E rrF.rGsIn order to manifest the intimate link between Malliavin calculus and Dirichlet forms, we introduce the definition of Dirichlet-Malliavin structure.Definition 1.3.7 (‹ Dirichlet-Malliavin structure). Consider a probability space pΩ, A, Pq endowed by a Malliavin framework, i.e. the modicum of operators derivative D, diver-

	,1s;Rq
	and for any pF, GqPD 1,2 B by a limit procedure,

gence , Laplacian L, Markov semi-group pP t q tPR `defined and connected to each other by (1.3.1), (1.3.2). Define a Dirichlet form satisfying on pΩ, A, Pq (1.3.9). The quintuplet pD, , L, pP t q tPR `, Eq thus defined is a Dirichlet-Malliavin structure.

Example 1.3.8 (Dirichlet-Malliavin structure on the Wiener space). On the one hand, consider the Malliavin frame on the Wiener space described by the operators Malliavin derivative r (2.3.1), divergence (see Definition 1.3.3), Laplacian L and the relationships that bind them to each other, as to say the integration by parts formula (1.3.6) and the identity L "´ D.O n the other hand, consider the functional operator L as the infinitesimal generator of a Markovian semi-group pP t q tPR `(the Ornstein-Uhlenbeck one) and define the Dirichlet form (from a potentialist point of view) pD E , Eq by

  .2.6) initially stated by Leonard Gross in[START_REF] Gross | Logarithmic sobolev inequalities[END_REF] can be retrieved via the application of the Clark formula. It does not exist in its original form (2.2.6) due to the absence of chain rule formula for discrete gradients. This also explains the lack of hypercontractivity property in Poisson and discrete settings, actually equivalent to (2.2.6). Ivan Nourdin, Giovanni Peccati and Xiaochuan Yang nevertheless ruled on an restricted hypercontractivity in the Poisson space in a recent work[START_REF] Nourdin | Restricted hypercontractivity on the poisson space[END_REF].In the table below we review the identities established above, and compare them to their analogues in the Gaussian/Brownian and Poisson cases.

		Table 2.1: Functional identities -Similarities and differences
		Gaussian space	Poisson space	IR variables
	Chaos	F " E rFs`ÿ n•1

  Definition 2.3.9. A function F: N Y Ñ R is said to be TV ´Lip if F is continuous for the vague topology and if for any!, ⌘ P N Y , |Fp!q´Fp⌘q| § dist TV p!, ⌘q,where dist TV represents the distance in total variation between two point measures, i.e. the number of distinct points counted with multiplicity.

	2.4 Proofs of chapter 2
	2.4.1 Proofs of Section 2.1
	Theorem 2.3.10 (‹). For any F P TV ´Lip X D P ,w i t ht h en o t a t i o n so fL e m m a[ 2 . 3 . 8 ] and (2.3.4),
	E N pFq	NÑ8 ›››Ñ EpFq.

  Theorem 4.2.2 (Hedging formula in the Black-Scholes model). Let a claim depicted by a random variable F P L 2 pΩq.L e tt h eF-adapted processes ↵ and ' defined by ' t " pe ´rpT ´tq E ‹ rD t F | F t s S t and ↵ t " e ´rpT ´tq E ‹ rD t F | F t s´' t S t A t

	The self-financing condition(4.1.8) can be
	rewritten as
	(4.2.3) In the Black-Scholes model lying a the Wiener space, the hedging formula is thus directly ↵ t dA t `'t dS t " 0; t P T.
	provided by the Clark formula under the unique neutral-risk probability measure P ‹ (see
	prop 1.14.4 of chapter 1 in Privault [117]).

  restrictive conditions |V t |"1 and E rV t s"0 for any t P R `).

	in the general case, it seems to remain possible to state a kind of modified structure equation
	rY, Ys t "	1 ?	Nt ÿ s"1	1 " t	0 `ª t	s dY s ,	(4.3.3)
	in the particular case where p|V| t q tPR `is deterministic constant equal to 1
	∆Y t "	t 2	˘dˆ t 2	˙2 ` t ∆t
	(or under
	rY, Ys t " t	0 `ª t		s dY s ,	(4.3.2)

  xu, vy H bn " n! E " xu bn ,v bn y `2pN T ˆEq ˝n ‰ . Theorem 5.2.7 (‹). The stochastic integral defined as the application Jn : H bn ›Ñ H n u bn fi ›Ñ Jn pu bn q"n

				T ÿ t"1	ÿ kPE	Jn´1 pu bn´1 t,k	q ∆R t,k
	satisfies the isometry formula
	E	"	ı Jn puq Jn pvq	"xu, vy H

bn ; @u, v P H bn .

  E r∆R s,k ∆Z t,1 s"E r∆R s,k E r∆Z t,1 |F s ss " 0 and E r∆R t,1 ∆Z t,´1 s" 2 pq,

	t,1 " ∆Z t,1	and	∆R t,´1 " ∆Z t,´1 ´	q 1 ´ p	∆Z t,1	(5.3.5)
	by noting that for any s P N t´1 ,

  • W s p⇡ t p!qYpt, kqq " k1 tsu ptq`W s p!q1 tsu c ptq and W s p⇡ t p!qq " W s p!q1 ttu c psq,• ∆N s p⇡ t p!qYpt, kqq " 1 ttu psq`∆N t p!q1 ttu c psq and ∆N s p⇡ t p!qq " ∆N t p!q1 ttu c psq.We deduce that, for any pt, kqPN T ˆE that on the one hand, ∆Z t,k p⇡ t p!qYpt, kqq ´∆Z t,k p⇡ t p!qq " sgnpkq and, on the other hand ∆Z t,k p⇡ t p!qYpt, ´kqq ´∆Z t,k p⇡ t p!qq " 1 tp∆Nt,Wtq"p1,kqu p⇡ t p!qYpt, ´kqq ´ p k ´r1 tp∆Nt,Wtq"p1,kqu p⇡ t p!qq ´ p k s"0, This entails in particular the remarkable identity: ∆R t,k p⇡ t p!qYpt, kqq ´∆R t,k p⇡ t p!qq " sgnpkq.

	The probability measure P ˜ ,p can be equivalently expressed by
	d P˜ ,p "	˜1 ´˜ 1 ´	¸t t π s"1	p1 ` p∆N s , W s qq dP,
			›Ñ	ÿ	ÿ	!ps, kq
				s‰t	kPE
	i.e. the restriction of ! to G t . The applications defined on Ω ˆ`N t `!, pt, kq ˘P Ω ˆ`N t ˆE˘b y	ˆE˘, and expressed for
	! fi Ñ ⇡ t p!qYpt, kq and ! fi Ñ ⇡ t p!q,	(5.3.7)
	can be interpreted as the applications acting on ! respectively by forcing a jump of height k
	at time t or forbiddening any jump at time t.
	Remark 5.3.8. In particular, the applications defined by (??) satisfy the remarkable identi-
	ties:

  PptX t " 1uq " Ppt∆N t " 1uq. Define the random walk pS t q tPN 0 T by S 0 " 0 and Proof of Proposition 5.2.7. Assume with no loss of generality that n °m.

	‰	∆Z P t
	where
	D P t F " Fp! p `ttuq ´Fp! p ´ttuq which is -up to a constant -a reminiscent of the gradient used by Laurent Decreusefond and (5.5.1)
	Ian Flint [37] on the Poisson space
	r such that
	X t " 2∆Z P t `2 ´1 " 2∆N t so that " S t " t ÿ s"1 1 `Xs 2 T gives " N The application of D to N "pN t q tPN 0 D t FpN ¨q" " F `⇡t N ¨Yt tu ˘´F `⇡t N ¨˘ı 1 rr t,T ss p¨q ´1 " F `S¨`1tX t"´1u 1 rr t,T ss p¨q ˘´F `S¨´1tX t"1u 1 rr t,T ss p¨q ˘,
	by noting that t∆N

t F " Fp! Yttuq ´Fp!zttuq.

From another point of view, consider the family of t´1, 1u-random variables tX t ,t P N T u t ; t P N T . t " 1u"t X t " 1u and t∆N t " 0u"t X t "1u. We retrieve -up to a constant -the expression of Malliavin derivative p D defined on the Rademacher space (2.0.2)

  .1.3) is identically distributed to the one of the trinomial model, we expect to reach the same incompletness result. By writing for any t P N T ´1,

	∆S t " " "	S t ´p1 `rqS t´1 p1 `rq t ⌘ t S t´1 ∆N t ´rS t´1 p1 `rq t rb1 tWt"1u `a1 tWt"´1u s ∆N t p1 `rq t	´r	ˆSt´1 ,

  The application of Theorem 6.2.2 to L " p G provides: Theorem 6.2.3 (‹). Assume Assumption 6.2.4 holds. Let X be a geometric compound process of parameters p ,pq.T h e nt h ep r o c e s sX G

	L t "	dQ dP	ˇˇF	t	.T h e nt h ep r o c e s sX Q defined by
					X Q t " X t	´t ÿ s"1	∆xX, Ly s s´1 L G	,
	is a pQ, Fq-martingale.
					X G t " X t	´t ÿ s"1	∆xX,p c y s ˇˇc"G p G s´1

  3.3) in the ternary model. Lemma 6.3.3. The mean-variance tradeoff process of the ternary model is deterministic.

Proof. For any

t P N T , `E r∆S t | F t´1 s ˘2 varr∆S t | F t´1 s " pE r1 `⌘t ∆N t |F t´1 sq 2 varr1 `⌘t ∆N t |F t´1 s " p1 ` p2p ´1qq 2

  Some of the computations rely on the following Theorem 3.2.1 in[START_REF] Delbaen | The Mathematics of Arbitrage[END_REF]: within the previous notations, the optimizers p V x,t and x M p yx,t exist and are unique. The measure x M p

	Remark 6.4.2. yx
	belongs to C F t and satisfies the remarkable relation:
	d x M p yx,t dP	"	1 p y x	u 1 p p V x,t q where p y x " `Φag t ˘1pxq.
	M p yx,t i.e. Further computations will rely on. Denote by t j j P J t u the coordinates family of x such that x M p jPJt yx,t " ÿ j P j t . (6.4.3)
	Theorem 6.4.1 (‹). There exists unique optimizers p V F x,t and x M p yx,t of the problems (4.2.6)
	and (4.2.11) such that x M p yx P C F t and p V F x,t " I ˜p y x	dP M p yx,t d x	¸.

  Theorem 6.4.3 (‹). There exists unique optimizers p ,t of the problems (4.2.6) and (4.2.11) such that x M G and t j ,jP J t u is exactly the family defined by (6.4.3). The insider's additional expected logarithmic utility up to time t P N T ´1 is defined by U t pxq" max

					V G x,t and x M G p y G
	p y G x ,t P C G t and
		p V G x,t " I ˆp y G x	dP d x p y G x ,t M G	˙.
	Moreover,	x M G p y G x ,t "	jPJt ÿ	j Q j t ,	(6.4.6)
	Additional utility before the deadline
						2) can be rewritten as the
	constrained optimization problem
	max VPΛ G x,t

E rupVqs under the constraints E Q j t rVs § x ; @j P J t .

(6.4.5)

x PG´portfolio E rupV x,t p qs ´max PF´portfolio E rupV x,t p qs .

  In the latter one, picking up the results established by Freddy Delbaen et Walter Schachermayer ([42] example 3.3.2.) to solve the optimization problem from the agent's point of view, we can state: Theorem 6.4.9 (‹). Consider the ternary model with T " 1 and let the numbers p ˚,q

						˚P
	p0, 1q be equal to: The maximal expected utility of the ordinary agent is given by p ˚" r ´a b ´a and q ˚" b ´r b ´a .
	Φ ag 1 pxq"logpxq`	" p log	ˆp p ˚˙`q log	ˆq q ˚˙ı ,
	and realized by the strategy	' ag "	xp1 `rq b ´a	ˆpq ˚´qp	p˚q˚.

t`1 "p 1 `bq S bin t with probability p S bin t`1 "p 1 `aq S bin t with probability 1 ´p

  "pF s q t §s §T and "p↵, 'q. From the i.i.d. property of the variables t∆S t ,tP N T u we infer that Θ ag t pxq"Φ ag t pxq, so that we can state its expression by the following induction system : ∆S 1 qs ˇˇpx"x`' T ´1 ∆S T ´1q

	$ & %	Φ ag T pxq"logpxq Φ ag t´1 pxq" sup PH 1 ´portfolio	E rΘ ag t px `'∆S t qs
	Indeed, for t " T ´1, Φ ag T ´1pxq"Θ	ag T ´1pxq"	sup PH T ´1´portfolio	E	" u px `'T ∆S T q ˇˇF T	´1ı
			"	sup PH 1 ´portfolio	E ru px `'1
	ag t pxq"	sup PHt´portfolio	E	«	u ˜x	s"t`1 `T ÿ	' s ∆S s	¸ˇˇˇF

t where H t

  t ˆSt Pr0, S 0 s ˇˇF t St where we have used in the last line the variable S T {S t has the same law as S T ´t and T T ´t by equivalence of our model and the trinomial model (see remark 6.1.1). The distribution of T t is given by `! p pq n p qq m p1 ´ q t´pn`mq for any c P S t pΩq.Proof of Theorem 6.2.1. The corollary 5.4.3 applied to p G at time s " t ´1 gives where we have got the second line by conditioning with respect to F t´1 and by defining the family ta k,`, pk, `qPE 2 u by a k,`" E r∆Z t,k ∆R t,`s , i.e.

									"
			P " P ´TT ´t P ´ST S t P " 0, " 0, S 0 c S 0 ı ˇˇF t c ı ˇˇF t ¯ˇˇc ¯ˇˇc "St
	PpT t " cq"	ÿ n`m``"t, n´m"c t pn,m,`qPN 3	t! n! m!
	6.5 Proofs of chapter 6
	6.5.1 Proofs of Section 6.2
		∆p G t "	ÿ kPE	E	"	D t,`p	G t | F t´1	‰	∆R t,`,
	Then we get, for any c P Γ, ∆xY,p c y t " E " ÿ « ÿ kPE kPE ÿ `PE " ÿ kPE ÿ `PE	∆Z t,k E rE rD t,`p ÿ `PE E rD t,`p c t |F t´1 s E r∆Z t,k ∆R t,`s s c t |F t´1 s ∆R t,` a k,`E rD t,`p c t s ,

"a 1,1 " pp1

  Proof of Theorem 6.2.5. The proof follows closely the one ofProposition 2.3 ([7]). For any A t P F t , B P E , cq"PpG P Bq. This yields Q t pA t XtG P Buq " PpG P BqPpA t q, for any t P N 0 T ´1.T a k i n gA t " Ω, then B " Γ providesQ t pA t XtG P Buq " Q t pG P BqQ t pA t q,and enables to establish 2. and 3. Let A " A s XtG P Bu an element of G s and t °s. By noting that A s P F t and using what preceedes,

			E	"	1 AtXtGPBu	1 p G T	⇢	" E	"	1 At E	"	1 tGPBu	1 t p G	ˇˇˇF t	⇢⇢
									" E r1 At s PpG P Bq " PpA t qPpG P Bq,
	where we used: E	"	1 tGPBu	1 p G t	ˇˇˇF t t p!q¨PpG " E ⇢ " ÿ c P ΓXB 1 p c t p!q ¨pc " 1 1 A p G
									" E	"	1 A	1 s p G	⇢	,

t

⇢

" PpA s qPpG P Bq " Q s pA s XtG P Buq

  6.5.2 Proofs of Section 6.3Proof of Lemma 6.3.4. The Kunita-Watanabe decomposition (see[START_REF] Dellacherie | Probabilities and potential b[END_REF], theorem VIII.51.) yields F|F s´1 s ∆R s,k `pb `a⇢q∆R t,1 ´a∆R t,´1 ˘| F t´1

	⇠ F t " " "	E t ÿ " E rF | F t s ∆S t | F t´1 p E rp∆S t q 2 | F t´1 s s"1 ÿ kPE E " p E rD s,k ı ı }c} 2 2 ÿ kPE c k p E rD t,k F|F t´1 s }c} 2 2

  ∞jPJt p⌘ j {p y G x qQ j t where p⌘ j q jPJt Pr0, 1s Jt is a J t -uplet summoned to one. Thus, for any j P J t ,

	We get p y G x " 1{x and x M G p y G x ,t "
	E Q j t	r p V G x,t s"E Q j t	» -I ˆp y G x	d x M G p y G x dP	˙fi fl
			»		fi
		" E Q j t	-dP d x M G p y G x »	fl	fi
		" E Q j t « " E ∞ -dP dQ j t dQ j d x dQ j t M G p y G x ,t t jPJt ⌘ j dQ j	fl
	unique minimiser of the value function (for y "	∞ jPJt ⌘ j fixed)	5, x M G y,t is the element of C G t
						¯,
		VPΛx,t	tE rupVqs ´yE M rV ´xsu

Ψpyq : M P M G t fi Ñ ´sup t .

  Résumé en françaisLe calcul de Malliavin est un calcul des variations stochastiques en dimension infinie initialement élaboré sur l'espace de Wiener et étendu par la suite à d'autres familles de processus tels que les processus de Poisson, de Rademacher, de Lévy. La multiplicité des approches et la variété des espaces canoniques sur lesquels il opère semblent dissuader l'éventuel projet d'un travail unificateur. Il demeure néanmoins possible d'identifier une terminologie commune à tous ces formalismes autour des notions d'opérateurs de Malliavin (gradient, divergence, semi-groupe d'Ornstein-Uhlenbeck) et de la relation fondamentale liant l'opérateur gradient et la divergence (défini comme adjoint du gradient) : la formule d'intégration par parties. On élabore dans cette thèse un calcul de Malliavin pour deux classes de processus discrets : les suites de variables aléatoires indépendantes (non nécessairement de même loi) et les processus géométriques composés. Dans le premier cadre, on équipe tout produit dénombrable d'espaces de probabilité d'une structure de Dirichlet-Malliavin discrète échafaudée sur une famille d'opérateurs de Malliavin (gradient discret, divergence, opérateur nombre), une formule d'intégration par parties, et les formes de Dirichlet naturellement induites dans ce contexte. On obtient les analogues discrets des principales identités fonctionnelles (formule de Clark-Ocone, identité de covariance, inégalité de log-Sobolev, de déviation. . . ) établies pour les processus Brownien et de Poisson et dont on retrouve par ailleurs les structures de Dirichlet usuelles comme limites de celles induites par notre formalisme. En exploitant la combinaison de la formule d'intégration par parties et de la méthode de Stein, comme l'ont conceptualisée I. Nourdin et G. Peccati, on établit des critères de Stein-Malliavin pour les approximations Normale et Gamma par toute fonctionnelle de variables aléatoires indépendantes. Ce calcul de Malliavin discret généralise ce qui avait été théorisé dans le cas Rademacher et donne un cadre unificateur à de nombreux résultats disséminés dans la littérature suggérant également que des résultats sans lien apparent (Efron-Stein, paires échangeables etc.) sont en réalité les faces d'une même pièce. La motivation du second travail provient de l'impossibilité constatée d'établir (via le précédent formalisme) une formule de Ocone-Karatzas pour le modèle trinomial. On lui substitue un modèle ternaire, sous-tendu par un processus géométrique composé (défini par une famille de variables i.i.d. à valeurs dans t´1, 0, 1u)e tq u il u ie s té q u i v a l e n t en loi. Pour des raisons techniques énoncées par M. Émery, il n'existe pas de décomposition chaotique. On en donne une version modifiée au moyen d'intégrales multiples relatives à une famille orthogonalisable de variables aléatoires. Les opérateurs gradient et divergence sont alors définis comme les opérateurs d'annihilation et de création agissant sur cette décomposition et vérifient une formule d'intégration par parties. Des analogues géométriques à la formule d'Itô, au théorème de Girsanov, et à la formule de représentation de martingale, et la formule de Ocone-Karatzas sont établis. Ce formalisme est appliqué au calcul d'utilité dans un modèle ternaire où agissent un agent ordinaire et un initié disposant d'un surplus d'information dès le début de la période de trading. En incorporant les techniques d'élargissement de filtration en temps discret développées par C. Blanchet-Scalliett, M. Jeanblanc et R. Romero à notre construction, on définit l'équivalent du drift d'information dans ce cadre et on montre the analogues of Itô's formula, Girsanov's theorem, martingale representation formula, and Ocone-Karatzas formula for compound geometric processes.

	Indeed, we have
		d x M p yx dP	p0q"pp y x q ´1 ˆu1 ´Vag x,1 p0q ¯" 1,
	so that	x M p yx p0q"	d x M p yx dP	p0qˆPp0q"1 ´ .
	Since x M p yx is an equivalent measure to P such that E x M p yx pS M q $ ' & ' % b x M p yx p!p1, 1qq `ax M p yx p!p1, ´1qq x M p yx p!p1, 1qq `x M p yx p!p1, ´1qq `x M p rS 1 ´S0 s"0 we have x M p yx p0q"1 ´ " r yx p0q"1
	and thus In other words,	x M p yx p!p1, 1qq "	r ´a b ´a	and x M p yx p!p1, ´1qq "	b	b	´r ´a .
	Hence the result.	x M p yx " 0 P 0 ` 1 P 1 "	b	b	´r ´r P 0	`rp1 ´ q b ´r P 1 .	(6.5.3)
		ag 1 pxq"logpxq`	" p log	ˆp p ˚˙`q log	ˆq q ˚˙ı .
	Hence the result.
	Proof. The optimal couple pp y x , x M p yx qPR ˆC F 1 defined in Theorem 6.4.7 is such that p y x " 1 x and x M p yx " ˆ1 ´ , r ´a b ´a , b ´r b ´a ˙

The development of tools to derive an upper-bound from the control of the new expression.

"p1q; N " N´1 ˝p ´1 N´1 pi N q, Nq.

Remerciements

Chapter 4

From a discrete insider model to Malliavin calculus for compound geometric processes

The chapter is organized as follows; in the first section we set up the question of insider's trading in a continuous setting and give some elements about enlargement of filtrations techniques. The second section is devoted to the problems of hedging and optimization of portfolios. In the following one, as to install our theory among what has be done, we introduce Malliavin calculus for jump processes. Last, we gather in the conclusion all the ingredients we need to solve the problem 3.4.15 we have chosen to investigate.

Troughout this chapter, we consider a simple financial market embodied by the couple of R `-valued processes, called assets, pA t , S t q tPT defined on the same probability space pΩ, F, Pq where F "pF t q tPT is a filtration (generally that generated by the canonical process) and such that :

• the process pA t q tPT is deterministic and modelises the risk-free asset,

• the process pS t q tPT is F-adapted and modelises the risky asset, associated to its price S t at time t P T.

The sequence of discounted prices pS t q tPT is defined by S t " A ´1 t S t (t P T). On this market, operate different agents so that T is called the trading interval ; in a continuous setting it is equal to r0,Ts (T P R)a n dt oN T " N Xr0,Ts (T P N ˚) in a discrete one.

Insider's trading in a continuous setting and initial enlargement

Consider the market model described below, and assume in this section only that the sequence of discounted prices pS t q tPT is a pF, Pq-martingale. The underlying question of risk-neutral measure will be addressed in the forthcoming section.

Until we transpose the problem into a discrete setting and that we mention it, let T "r0,Ts and consider the Wiener space pΩ, A, F, Pq equipped with the canonical process pB t q tPT , where A ´1 s´1 `Ss´1 p' s ´'s´1 q for any t P N T .T h es t r a t e g y "p↵, 'q is admissible and simulates F.

Up to now, let T " N T and consider the simple discrete financial market described in section 4.1. When the market is incomplete, additional difficulties arise. All claims are no longer simulable. The hedging approach using Clark formula in the complete Cox-Ross-Rubinstein model for instance (Theorem 4.2.3), does not stand anymore.

The initial problem (4.1.5) can be addressed in terms of portfolio's optimization, and leads our steps towards the duality approach. A very substantiated expression and comprehensive look at arbitrage-related issues can be found in the book of Freddy Delbaen and Walter Schachermayer [START_REF] Delbaen | The Mathematics of Arbitrage[END_REF]; our work is greatly inspired by it. We yield thereafter some of the key points of the proof they used to deal with the optimization problem (4.1.3) in the trinomial model (see the definition (6.1.1)). This is a textbook case of incomplete models.

As explained by Wolfang Runggaldier in [START_REF] Runggaldier | Portfolio optimization in discrete time[END_REF] the trinomial model modelises an incomplete market, where the risk-neutral probability measure, with respect to which the sequence of discounted prices is a p¨, Fq-martingale, is not unique. Let be a probability measure P 1 , equivalent to P and, for any t P N T ´1 define p t`1 " P 1 pX t`1 " 1 `b | F t q,q t`1 " P 1 pX t`1 " 1 `a | F t q and s t`1 " P 1 pX t`1 " 1 | F t q.

The measure P 1 is a F-martingale measure if and only if the triplet pp t`1 ,q t`1 ,s t`1 q satisfies pPq : # ´p1 `bqp t`1 `p1 `aqq t`1 `ps t`1 q ¯St "p1 `rqS t p n`1 `qn`1 `sn`1 " 1 ; @t P N T ´2.

The system pPq admits infinitely many solutions `pt ,q t ,s t ˘T PN T ´1 such that any triplet `pt ,q t ,s t ˘forms a convex M F set (here a segment) characterized by its extremal points (independent of t), i.e. the measures

which are not equivalent to P but such that any convex combination

is. Note that the extremal measures P 0 n and P 1 n are independent of t and F t ,s ot h a t ,i fN " 2 and ! "p! 1 , ! 2 qPΩ,

q and by induction

processes (6.3.5), and thus to circumvent the impossibility to do that for general independent random variables.

Let V 0 " 0, T 0 " 0, and for any t P N T , let T t denote the time of the t-th jump, V t the t-th mark, ⇠ t the t-th inter-arrival and pN t q tPN 0 T the geometric process defined by N 0 " 0 and

where pV t q tPN T and p⇠ t q tPN T are sequences of independent and identically distributed variables, and such that V t is a t´1, 1u-Bernoulli variable of parameter p, whereas ⇠ t is a geometric random variable of parameter . The canonical process ! can be thus expressed by !p0,kq"0 and !pt, kq" ÿ sPN T pTs,Vsq pt, kq"

where, for any t P N T ,t h er a n d o mv a r i a b l eW t is defined by

and ˝stands for a point at infinity. Note that any variable V t pt P N T q is independent of the underlying jump process pN t q tPN 0 T . Define also the sequence of increments p∆N t q tPN T by

The variables ∆N t and W t thus defined play a major part and we will often refer to them. Indeed, ∆N t indicates wether there is a jump at time t,a n d ,i fs o ,t h ev a r i a b l eW t gives its value (`1 or ´1) i.e indicates if the process goes up or down.

Remark 5.1.2. To slighten the notations, we omit some parenthesis; !pt, kq and ¨pt, kq stand respectively for !ppt, kqq and ¨ppt, kqq.

From now on we consider geometric functionals F of the form

with f n P L 1 pX n , ⌫ bn q and pt n , k n q" `pt 1 ,k 1 q, ¨¨¨, pt n ,k n q ˘Pp N T ˆEq n, † , where for any B P PpN T q, we denote B n, † "tpt 1 , ¨¨¨,t n q : t i P B,i P N T and 1 § t 1 † ¨¨¨ † t n u.

(5.1.3) and for A " B ˆE P T , pB ˆEq n, † "t `pt 1 ,k 1 q, ¨¨¨, pt n ,k n q ˘: pt i ,k i qPB ˆE,i P N T and t 1 † ¨¨¨ † t n u.

For instance, if we consider the functional defined as Fp!q"!pA ˆEq, for a given A "pt n , k n qPpN T ˆEq n, † we have

The canonical filtration F "p F t q tPN 0 T is the -field generated by the functionals of the form (5.1.2) and can be rewritten as

-.

(5.1.4)

The compound geometric process Y of parameters p ,pq is defined by

V s 1 tTs §tu .

(5.1.5)

Remark 5.1.3. The compound geometric process is the discrete analogue of the compound Poisson process. The underlying jump strucutre modelised by a Poisson process in this latter is replaced in our framework by a binomial/geometric process.

It is possible to define the space of square integrable functionals of the compound geometric process.

Lemma 5.1.4 (‹). The space L 2 pΩq is made of the functions

The set of simple processes, denoted by S 0 is the set of random variables of the form

Stochastic integrals

Throughout this subsection, assume the existence of a family t∆R t,k ,t P N T ,k P Eu of random variables satisfying the following hypotheses:

Functional identities

We provide our construction with the analogues of Itô formula, Clark formula, martingale representation theorem and Girsanov theorem. The compound process pY t q tPN T defined by (5.1.5) is equipped with the pathwise Itô formula provided by the following theorem:

Theorem 5.4.1 (‹ Itô formula). Let pY t q tPN T be the compound Poisson process defined by (5.1.5) and f be a R-valued function defined on N.T h e n ,f o ra n yt P N T ,

The face to face of theorems 5.4.2 and 5.4.5 highlights clearly their connection to each other (as for Brownian motion): while the latter one means that a martingale adapted to the compound geometric process filtration is in fact a stochastic integral, the first one gives the expression of the integrand of this stochastic integral in terms of the Malliavin derivative of the terminal value of the martingale.

Theorem 5.4.2 (‹ Clark formula). For any F P L 2 pΩq,

As a direct consequence of Lemma 5.2.3 and Clark formula (5.4.2) we get:

Corollary 5.4.3 (‹). For any t P N T and F element of L 2 pΩq,

Remark 5.4.4. The transposition of the Clark formula with respect to he family Z is written: for any F P L 2 pΩq,

where the family tb k,`, pk, `qPE 2 u defined by

as provided by (5.3.5).

From Corollary 5.4.3 we derive a predictable representation formula for geometric discrete-time martingales.

Theorem 5.4.5 (‹ Martingale representation theorem). Let pX t q tPN T be a F-martingale.

There exists a F-predictable process u "pu ¨,k ,kP Eq such that

(5.4.4)

In particular, for any pt, kqPN T ˆE,

Theorem 5.4.6 (‹ Girsanov theorem). Let p ˜ , pqPp0, 1q 2 and let the function ' defined on t´1, 1u such that

Then, under the probability measure

V s is a compound geometric process of parameters p ˜ , pq.

(by taking p k " in Corollary 1.6.3. in Privault [START_REF] Privault | Stochastic Analysis in Discrete and Continuous Settings[END_REF]) applied to a functional F of the form F " f pX 1 , ¨¨¨, X T q"gpS T q, f : t´1, 1u T Ñ R and g : R Ñ R smooth functions. Besides, the Clark formula (5.5.1) is exactly the same of the eponymous one on the Rademacher space (se Proposition 1.7.1. in Privault [START_REF] Privault | Stochastic Analysis in Discrete and Continuous Settings[END_REF]) noting that the definition of the process p∆Z P t q tPN T coincides with this of normal martingales through structure equations. All results arise from these definitions.

5.6 Proofs of chapter 5

" Jn `u1 rr 0,tss since the independence of the centered variables

Proof of Proposition 5.2.5. For any f n P `2pN T ˆEq ˝n,

Hence the result.

equal to

The proof is thus complete.

Proof of Theorem 5.3.5. By induction on n ; in what follows rns denotes the set t1, ¨¨¨,nu.

For n " 1 there exists f : t1uˆE Ñ R such that

gpp1, 1qq " f pp1, 1qq `⇢f pp1, ´1qq and gpp1, ´1qq " f pp1, ´1qq

Assume there is n P N T such that

Jm pf m q, and where for m

By definition, and applying induction hypothesis

Then p⇡ n t j ,k j g n ppt n , k n qq " gppt n , k n q pt j ,k j q q" n´1 ÿ

`"1 ÿ APprns j q `, † π iPA ⇢ `1tk i "1u f `pt A , ´kA q, pt rns j zA , k rns j zA q Noting that g n ppt n´1 , k n´1 q, pt j , 1qq " f n ppt n´1 , k n´1 q, pt j , 1qq `⇢ f n ppt n´1 , k n´1 q, pt j , ´1qq, where rns j "rnsztju, we get

´kA q, pt rns j zA , k rns j zA , pt j , ´1qq ˘,

Proofs of Section 5.4

Proof of Theorem 5.4.1. The proof follows closely that of (see Proposition 19.12 in Privault [START_REF] Privault | Chapter 19 stochastic calculus for jump processes[END_REF]) For any t P N T

Ns ∆N s " ∆N s . Hence the result.

Proof of Theorem 5.4.2. Let F P S ; follows from the chaotic decomposition of F together with the definition of the operator gradient that

where we have used lemma 5.2.3 to get the third line. Hence the result. The result to any random variable F P L 2 pΩq is extended by noting that

and using the continuity of the operator E rD t,k p¨q|F t´1 s.

Proof of Theorem 5.4.4. Let pX t q tPN T be a F-martingale. Follows from Corollaries 5.3.9 and

Chapter 6

The insider problem in the ternary model

Framework: the ternary model

In the perspective of exploiting our theorical results to focus on the question of the additional utility of an insider in what we call the ternary model, we adopt now the following notations; our model is driven by an underlying compound geometric pocess Z "p Z t q tPN T (N T " N Xr0,Ts) defined by

As a reminder (see Section 5.1 for more details), pN t q tPN 0 T is the geometric process defined by N 0 " 0 and

where T 0 " 0, p⇠ t q tPN T is a sequence of independent geometric random variables of parameter and pV t q tPN T is a sequence of independent t´1, 1u-Bernoulli variables of parameter p Pp0, 1q. The canonical process ! can be thus expressed by !p0,kq"0 and !pt, kq" ÿ sPN T pTs,Vsq pt, kq"

where, for any t P N T , the random variables ∆N t and W t are defined by

The financial market lying on the probability space denoted pΩ, F, Pq is the one defined in 4.1. The riskless asset pA t q tPN T with initial value A 0 " a 0 is defined by A t " a 0 p1 `rq t , (6.1.2)

alors que l'utilité logarithmique espérée additionnelle de l'initié au temps t peut être exprimée comme l'entropie relative de la mesure initiale par rapport à la mesure préservant les martingales sur rr 0,tss .O nr e t r o u v ea i n s il ' e x a c ta n a l o g u ed ur é s u l t a té t a b l i par J. Amendinger, P. Imkeller et M. Schweizer dans le cas continu.

Summary in english

Malliavin calculus is an infinite-dimensional variational calculus initially developed on the Wiener space. Further extended to other classes of processes, such as Poisson, Rademacher, or Lévy processes, it has reached a certain maturity. Cause of the multiplicity of approaches and the variety of canonical spaces, it seems to be difficult to propose a unifying work. We can nevertheless identify a terminology common to all these formalisms; this lies around the notions of Malliavin operators (gradient, divergence, Ornstein-Uhlenbeck semi-group) and the fundamental relationship between the operator gradient and the divergence (defined as the adjoint of the gradient): the integration by parts formula.

In this thesis work, we propose to develop a Malliavin calculus for two classes of discrete processes: sequences of independent random variables (not necessarily identically distributed) and compound geometric processes. Both are wholly or partly motivated from an application perspective.

In the first mentioned framework, we equip any countable product of probability spaces with a discrete Dirichlet-Malliavin structure. This lies on a family of Malliavin operators (discrete gradient, divergence, number operator), an integration by parts formula, and the induced Dirichlet forms. We obtain the discrete analogues of the classical functional identities (Clark-Ocone formula, covariance identity, deviation inequality. . . ) established for Brownian and Poisson processes. Besides, we retrieve the usual Poisson and Brownian Dirichlet structures associated to their respective gradient, as limits of the structures induced by our formalism. Re-using the combination of Malliavin calculus and Stein's method designed by I. Nourdin and G. Peccati, we provide Stein-Malliavin criterions for the Normal and the Gamma approximations by functionals of independent random variables. This discrete Dirichlet-Malliavin structure seems to give a unified framework for many results scattered in the literature so that we hope to bring new insights on why these apparently disjoint results (Efron-Stein, exchangeable pairs, etc.) are in fact multiple sides of the same coin. Faced with the impossibility to state (via the discrete Malliavin calculation) an Ocone-Karatzas formula for the trinomial model, we replace it by a ternary model ;e q u i v alent in law to the initial one, it is supported by a compound geometric process (with t´1, 0, 1u valued i.i.d. random variables). For technical reasons outlined by M. Émery, there is no chaotic decomposition. We state a modified decomposition in terms of multiple integrals with respect to an orthogonalisable family of random variables.The gradient and divergence operators are then defined as the annihilation and creation operators acting on this decomposition and verify an integration formula by parts. Our approach is close to the one developed for (compound) Poisson processes, with which the (compound) geometric process shares many properties (as a jump process). We give