Miryana Grigorova

Peter Imkeller

Youssef Ouknine

Marie-Claire Quenez

OPTIMAL STOPPING WITH f -EXPECTATIONS: THE IRREGULAR CASE

Keywords: backward stochastic dierential equation, optimal stopping, f -expectation, nonlinear expectation, aggregation, dynamic risk measure, American option, strong E f -supermartingale, Snell envelope, reected backward stochastic dierential equation, comparison theorem, Tanaka-type formula, general ltration 1

We consider the optimal stopping problem with nonlinear f -expectation (induced by a BSDE) without making any regularity assumptions on the payo process ξ and in the case of a general ltration. We show that the value family can be aggregated by an optional process Y . We characterize the process Y as the E f -Snell envelope of ξ. We also establish an innitesimal characterization of the value process Y in terms of a Reected BSDE with ξ as the obstacle. To do this, we rst establish some useful properties of irregular RBSDEs, in particular an existence and uniqueness result and a comparison theorem.

where T S,T denotes the set of stopping times valued a.s. in [S, T ] and E f S,τ (•) denotes the conditional f -expectation/evaluation at time S when the terminal time is τ .

The above non-linear problem has been introduced in [14] in the case of a Brownian ltration and a continuous nancial position/pay-o process ξ and applied to the (nonlinear) pricing of American options. It has then attracted considerable interest, in particular,

1. Introduction. The classical optimal stopping probem with linear expectations has been largely studied. General results on the topic can be found in El [START_REF] Karoui | Les aspects probabilistes du contrôle stochastique. École d'été de Probabilités de Saint-Flour IX[END_REF] ([12]) where no regularity assumptions on the reward process ξ are made.

In this paper, we are interested in a generalization of the classical optimal stopping problem where the linear expectation is replaced by a possibly non-linear functional, the so-called f -expectation (f -evaluation), induced by a BSDE with Lipschitz driver f . For a stopping time S such that 0 ≤ S ≤ T a.s. (where T > 0 is a xed terminal horizon), we dene (1.1) V (S) := ess sup τ ∈T S,T E f S,τ (ξ τ ), due to its links with dynamic risk measurement (cf., e.g., [START_REF] Bayraktar | Optimal Stopping for Dynamic Convex Risk Measures[END_REF]). In the case of a nancial position/payo process ξ, only supposed to be right-continuous, this non-linear optimal stopping problem has been studied in [START_REF] Quenez | Reected BSDEs and robust optimal stopping for dynamic risk measures with jumps[END_REF] (the case of Brownian-Poisson ltration), and in [START_REF] Bayraktar | Optimal stopping for Non-linear Expectations[END_REF] where the non-linear expectation is supposed to be convex. To the best of our knowledge, [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF] is the rst paper addressing the stopping problem (1.1) in the case of a non-rightcontinuous process ξ (with a Brownian-Poisson ltration); in [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF] the assumption of rightcontinuity of ξ from the previous literature is replaced by the weaker assumption of rightuppersemicontinuity (r.u.s.c.).

In the present paper, we study problem (1.1) in the case of a general ltration and without making any regularity assumptions on ξ, which allows for more exibility in the modelling (compared to the cases of more regular payos and/or of particular ltrations).

The usual approach to address the classical optimal stopping problem (i.e., the case f ≡ 0 in (1.1)) is a a direct approach, based on a direct study of the value family (V (S)) S∈T 0,T .

An important step in this approach is the aggregation of the value family by an optional process. The approach used in the literature to address the non-linear case (where f is not necessarily equal to 0) is an RBSDE-approach, based on the study of a related Reected BSDE and on linking directly the solution of the Reected BSDE with the value family (V (S), S ∈ T 0,T ) (and thus avoiding, in particular, more technical aggregation questions).

This approach (cf., e.g., [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF], [START_REF] Quenez | Reected BSDEs and robust optimal stopping for dynamic risk measures with jumps[END_REF]) requires at least the uppersemicontinuity of the reward process ξ which we do not have here (cf. also Remark 10.1).

Neither of the two approaches is applicable in the general framework of the present paper and we adopt a new approach which combines some aspects of both the approaches. Our combined approach is the following: First, with the help of some results from the general theory of processes, we show that the value family (V (S), S ∈ T 0,T ) can be aggregated by a unique right-uppersemicontinuous optional process (V t ) t∈[0,T ] . We characterize the value process (V t ) t∈[0,T ] as the E f -Snell envelope of ξ, that is, the smallest strong E fsupermartingale greater than or equal to ξ. Then, we turn to establishing an innitesimal characterization of the value process (V t ) t∈[0,T ] in terms of a Reected BSDE where the pay-o process ξ from (1.1) plays the role of a lower obstacle. We emphasize that this RBSDE-part of our approach is far from mimicking the one from the r.u.s.c. case; we have to rely on very dierent arguments here due to the complete irregularity of the process ξ.

Let us recall that Reected BSDEs have been introduced by El Karoui et al. in the seminal paper [START_REF] Karoui | Reected solutions of Backward SDE's and related obstacle problems for PDE's[END_REF] in the case of a Brownian ltration and a continuous obstacle, and then generalized to the case of a right-continuous obstacle and/or a larger stochastic basis than the Brownian one in [START_REF] Hamadène | Reected BSDE's with discontinuous barrier and application[END_REF], [START_REF] Crépey | Reected and doubly reected BSDEs with jumps: a priori estimates and comparison[END_REF], [START_REF] Hamadène | Backward stochastic dierential equations with jumps and random obstacle[END_REF], [START_REF] Essaky | Reected backward stochastic dierential equation with jumps and RCLL obstacle[END_REF], [START_REF] Hamadène | Reected backward SDEs with general jumps[END_REF], [START_REF] Quenez | Reected BSDEs and robust optimal stopping for dynamic risk measures with jumps[END_REF]. In [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF], we have formulated a notion of Reected BSDE in the case where the obstacle is only right-uppersemicontinuous (but possibly not right-continuous) and the ltration is the Brownian-Poisson ltration have shown existence and uniqueness of the solution. In the present paper, we show that the existence and uniqueness result from [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF] still holds in the case of a completely irregular obstacle and a general ltration. In the recent preprint [START_REF] Klimsiak | Reected BSDEs with regulated trajectories[END_REF], existence and uniqueness of the solution (in the Brownian framework) is shown by using a dierent approach, namely a penalization method.

We also establish a comparison result for RBSDEs with irregular obstacles and general ltration. Due to the complete irregularity of the obstacles and the presence of jumps, we are led to using an approach which diers from those existing in the literature on comparison of RBSDEs (cf. also Remark 9.2); in particular, we rst prove a generalization of Gal'chouk-Lenglart's formula (cf. [START_REF] Gal'chouk | Optional martingales[END_REF] and [START_REF] Lenglart | Tribus de Meyer et théorie des processus[END_REF]) to the case of convex functions, which we then astutely apply in our framework in order to establish the comparison theorem. We also show an E f -Mertens decomposition for strong E f -supermartingales, which generalizes to our framework the ones provided in the literature (cf. [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF] or [START_REF] Bouchard | A general Doob-Meyer-Mertens decomposition for g-supermartingale system[END_REF]). This result, together with our comparison theorem, helps in the study of the non-linear operator Ref f which maps a given (completely irregular) obstacle to the solution of the RBSDE with driver f . By using the properties of the operator Ref f , we show that Ref f [ξ], that is, the (rst component of the) solution to the Reected BSDE with irregular obstacle ξ and driver f , is equal to the E f -Snell envelope of ξ, from which we derive that it coincides with the value process (V t ) t∈[0,T ] of problem (1.1).

Finally, we give a nancial application to the problem of pricing of American options with irregular pay-o in an imperfect market model. In particular, we show that the superhedging price of the American option with irregular pay-o ξ is characterized as the solution of an associated RBSDE (where ξ is the lower obstacle). Some examples of digital American options are given as particular cases.

The rest of the paper is organized as follows: In Section 2 we give some preliminary definitions and some notation. In Section 3 we revisit the classical optimal stopping problem with irregular pay-o process ξ and a general ltration. We rst give some general results such as aggregation, Mertens decomposition of the value process, Skorokhod conditions satised by the associated non decreasing processes; then, we characterize the value process of the classical problem in terms of the solution of a Reected BSDE associated with a general ltration, with completely irregular obstacle and with a driver f which does not depend on the solution. In Section 4, we prove existence and uniqueness of the solution for general Lipschitz driver f , an irregular obstacle ξ and a general ltration. In Section 5, we present the formulation or our non-linear optimal stopping problem (1.1). In Section 6, we provide some results on the particular case where the payo ξ is right-uppersemicontinuous (r.u.s.c). , from which we derive an E f -Mertens decomposition of E f -strong supermartingales in the (general) framework of a general ltration (cf. Section 7). We then turn to the study of the case where ξ is completely irregular. Section 8 is devoted to the direct part of our approach to this problem; in particular, we present the aggregation result and the Snell characterization. Section 9 is devoted to establishing some properties of Reected BSDEs with completely irregular obstacles, which will be used to establish an innitesimal characterization of the value process of our problem (1.1) in the completely irregular case; more precisely, we rst provide a comparison theorem (Subsection 9.2); then, using this result together with the E f -Mertens decomposition, we establish useful properties of the non-linear operator Ref f (Subsection 9.3). In Section 10, using the results shown in the previous sections, we derive the innitesimal characterization of the value of the non-linear optimal stopping problem (1.1) with a completely irregular payo ξ in terms of the solution of our general RBSDE from Section 4. In Section 11 we give a nancial application to the pricing of American options with irregular pay-o in an imperfect market model with jumps; we also give a useful corollary of the innitesimal characterization, namely, a priori estimates with universal constants for RBSDEs with irregular obstacles and a general ltration.

2. Preliminaries. Let T > 0 be a xed positive real number. Let E = R n \ {0}, E = B(R n \ {0}), which we equip with a σ-nite positive measure ν. Let (Ω, F, P ) be a probability space equipped with a right-continuous complete ltration IF = {F t : t ∈ [0, T ]}. Let W be a one-dimensional IF -Brownian motion W , and let N (dt, de) an IF -Poisson random measure with compensator dt ⊗ ν(de), supposed to be independent from W . We denote by Ñ (dt, de) the compensated process, i.e. Ñ (dt, de) := N (dt, de) -dt ⊗ ν(de). We denote by

P (resp. O) the predictable (resp. optional) σ-algebra on Ω × [0, T ]. The notation L 2 (F T )
stands for the space of random variables which are F T -measurable and square-integrable. For t ∈ [0, T ], we denote by T t,T the set of stopping times τ such that P (t ≤ τ ≤ T ) = 1.

More generally, for a given stopping time S ∈ T 0,T , we denote by T S,T the set of stopping times τ such that P (S ≤ τ ≤ T ) = 1.

We use also the following notation:

• L 2 ν is the set of (E , B(R))-measurable functions : E → R such that 2 ν := E | (e)| 2 ν(de) < ∞. For ∈ L 2 ν , k ∈ L 2 ν , we dene
, k ν := E (e)k (e)ν(de).

• IH 2 is the set of R-valued predictable processes φ with φ 2 IH 2 := E T 0 |φ t | 2 dt < ∞. • IH 2 ν is the set of R-valued processes l : (ω, t, e) ∈ (Ω × [0, T ] × E) → l t (ω, e) which are
predictable, that is (P ⊗E , B(R))-measurable, and such that l 2 [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF], we denote by S 2 the vector space of R-valued optional (not necessarily cadlag) [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF], the mapping |||•||| S 2 is a norm on S 2 , and S 2 endowed with this norm is a Banach space.

IH 2 ν := E T 0 l t 2 ν dt < ∞. • As in
processes φ such that |||φ||| 2 S 2 := E[ess sup τ ∈T 0 |φ τ | 2 ] < ∞. By Proposition 2.1 in
• Let M 2 be the set of square integrable martingales M = (M t ) t∈[0,T ] with M 0 = 0. This is a Hilbert space equipped with the scalar product (M, M )

M 2 := E[M T M T ] (= E[ M, M T ] = E( [M, M ] T )), for M, M ∈ M 2 (cf., e.g., [37] IV.3). For each M ∈ M 2 , we set M 2 M 2 := E(M 2 T ). • Let M 2,
⊥ be the subspace of martingales h ∈ M 2 satisfying h, W • = 0, and such that, for all predictable processes l ∈ IH 2 ν , (2.1)

h,

• 0 E l s (e) Ñ (dsde) t = 0, 0 ≤ t ≤ T a.s.

Remark 2.1 Note that condition (2.1) is equivalent to the fact that the square bracket process [ h , • 0 E l s (e) Ñ (dsde) ] t is a martingale (cf. the Appendix for additional comments on condition (2.1)).

Recall also that the condition h, W • = 0 is equivalent to the orthogonality of h (in the sense of the scalar product (•, •) M 2 ) with respect to all stochastic integrals of the form • 0 z s dW s , where z ∈ IH 2 (cf. e.g. , [START_REF] Protter | Stochastic Integration and Dierential Equations (Stochastic Modelling and Applied Probability[END_REF] IV. 3 Lemma 2). Similarly, the condition (2.1) is equivalent to the orthogonality of h with respect to all stochastic integrals of the form

• 0 E l s (e) Ñ (dsde), where l ∈ IH 2 ν (cf., e.g. , Lemma 12.1 in the Appendix).

We recall the following orthogonal decomposition property of martingales in M 2 (cf. Lemma III.4.24 in [START_REF] Jacod | Grundlehren der Mathematischen Wissenschaften[END_REF]).

Lemma 2.1 For each M ∈ M 2 , there exists a unique triplet

(Z, l, h) ∈ IH 2 × IH 2 ν × M 2,⊥ such that (2.2) M t = t 0 Z s dW s + t 0 E l t (e) Ñ (dt, de) + h t , ∀ t ∈ [0, T ] a.s.
Denition 2.1 (Driver, Lipschitz driver) A function f is said to be a driver if

• f : Ω × [0, T ] × R 2 × L 2 ν → R (ω, t, y, z, k ) → f (ω, t, y, z, k ) is P ⊗ B(R 2 ) ⊗ B(L 2 ν )-measurable, • E[ T 0 f (t, 0, 0, 0) 2 dt] < +∞.
A driver f is called a Lipschitz driver if moreover there exists a constant K ≥ 0 such that dP ⊗ dt-a.e. , for each

(y 1 , z 1 , k 1 ) ∈ R 2 × L 2 ν , (y 2 , z 2 , k 2 ) ∈ R 2 × L 2 ν , |f (ω, t, y 1 , z 1 , k 1 ) -f (ω, t, y 2 , z 2 , k 2 )| ≤ K(|y 1 -y 2 | + |z 1 -z 2 | + k 1 -k 2 ν ).
Denition 2.2 (BSDE, conditional f -expectation) We have (cf., e.g., Remark 12.1 in the Appendix) that if f is a Lipschitz driver and if ξ is in L 2 (F T ), then there exists a unique solution (X, π, l, h) ∈ S 2 × IH 2 × IH 2 ν × M 2,⊥ to the following BSDE:

-dX t = f (t, X t , π t , l t )dt -π t dW t -E l t (e) Ñ (dt, de) -dh t ; X T = ξ.
For t ∈ [0, T ], the (non-linear) operator

E f t,T (•) : L 2 (F T ) → L 2 (F t ) which maps a given terminal condition ξ ∈ L 2 (F T )
to the position X t (at time t) of the rst component of the solution of the above BSDE is called conditional f -expectation at time t. As usual, this notion can be extended to the case where the (deterministic) terminal time T is replaced by a (more general) stopping time τ ∈ T 0,T , t is replaced by a stopping time S such that S ≤ τ a.s. and the domain L 2 (F T ) of the operator is replaced by L 2 (F τ ).

We now pass to the notion of Reected BSDE. Let T > 0 be a xed terminal time. Let f be a driver. Let ξ = (ξ t ) t∈[0,T ] be a process in S 2 . We dene the process (ξ t ) t∈]0,T ] by ξ t := lim sup s↑t,s<t ξ s , for all t ∈]0, T ]. We recall that ξ is a predictable process (cf. [START_REF] Dellacherie | Probabilité et Potentiel[END_REF]Thm. 90,page 225]). The process ξ is left uppersemicontinuous and is called the left upper-semicontinuous envelope of ξ.

Denition 2.3 (Reected BSDE) A process (Y, Z, k, h, A, C) is said to be a solution to the reected BSDE with parameters (f, ξ), where f is a driver and ξ is a process in

S 2 , if, 1 (Y, Z, k, h, A, C) ∈ S 2 × IH 2 × IH 2 ν × M 2,⊥ × S 2 × S 2 , -dY t = f (t, Y t , Z t , k t )dt + dA t + dC t--Z t dW t - E k t (e) Ñ (dt, de) -dh t , 0 ≤ t ≤ T, (2.3)
Y T = ξ T a.s., and Y t ≥ ξ t for all t ∈ [0, T ], a.s., A is a nondecreasing right-continuous predictable process with A 0 = 0 and such that

T 0 1 {Y t->ξ t } dA c t = 0 a.s. and (Y τ --ξ τ )(A d τ -A d τ -) = 0 a.s. for all predictable τ ∈ T 0,T , (2.4)
C is a nondecreasing right-continuous adapted purely discontinuous process with C 0-= 0

and such that (Y τ -ξ τ )(C τ -C τ -) = 0 a.s. for all τ ∈ T 0,T .

(2.5)

Here A c denotes the continuous part of the process A and A d its discontinuous part.

Equations (2.4) and (2.5) are referred to as minimality conditions or Skorokhod conditions.

For real-valued random variables X and X n , n ∈ IN , the notation "X n ↑ X" stands for "the sequence (X n ) is nondecreasing and converges to X a.s.". For a ladlag process φ, we denote by φ t+ and φ t-the right-hand and left-hand limit of φ at 1 As usual, equation (2.3) means that a.s. , for all t ∈ [0, T ], we have:

Yt = YT + T t f (s, Ys, Zs, ks)ds - T t ZsdWs - T t E
ks(e) Ñ (ds, de) -hT + ht + AT -At + CT --Ct-.

3.

The classical optimal stopping problem. In this section, we revisit the classical (linear) optimal stopping problem with irregular pay-o process and a general ltration.

3.1. The classical linear optimal stopping problem revisited. Let (ξ t ) t∈[0,T ] be a process belonging to S 2 , called the reward process or the pay-o process. For each S ∈ T 0,T , we (3.1) Lemma 3.1 (i) There exists a ladlag optional process (v t ) t∈[0,T ] which aggregates the family (v(S)) S∈T 0,T (i.e. v S = v(S) a.s. for all S ∈ T 0,T ). Moreover, the process (v t ) t∈[0,T ] is the smallest strong supermartingale greater than or equal to 2 For each S ∈ T 0,T and for each λ ∈]0, 1[, the process

(ξ t ) t∈[0,T ] . (ii) We have v S = ξ S ∨ v S+ a.s. for all S ∈ T 0,T . (iii)
(v t ) t∈[0,T ] is a martingale on [S, τ λ S ], where τ λ S := inf{t ≥ S , λv t (ω) ≤ ξ t }.
Proof.

These results are due to classical results of optimal stopping theory. For a sketch of the proof of the rst two assertions, the reader is referred to the proof of Proposition A.5 in the Appendix of [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF] (which still holds for a general process ξ ∈ S 2 ). The last 2 Note that in the case of a not necessarily non-negative pay-o process ξ this result holds up to a translation by the martingale XS := E[ess sup τ ∈T 0,T ξ - τ |FS] (cf. e.g. Appendix A in [START_REF] Kobylanski | [END_REF]). More precisely, the property holds for ṽ := v + X and ξ = ξ + X.

assertion corresponds to a result of optimal stopping theory (cf. [START_REF] Maingueneau | Temps d'arrêt optimaux et théorie générale[END_REF], [START_REF] Karoui | Les aspects probabilistes du contrôle stochastique. École d'été de Probabilités de Saint-Flour IX[END_REF] or Lemma 2.7 in [START_REF] Kobylanski | Optimal stopping time problem in a general framework[END_REF]). Its proof is based on a penalization method (used in convex analysis), introduced by Maingueneau (1978) (cf. the proof of Theorem 2 in [START_REF] Maingueneau | Temps d'arrêt optimaux et théorie générale[END_REF]), which does not require any regularity assumption on the reward process ξ.

Remark 3.1 It follows from (ii) in the above lemma that ∆ + v S = 1 {v S =ξ S } ∆ + v S a.s. Remark 3.2 Let us note for further reference that Maingueneau's penalization approach for showing the martingale property on [S, τ λ S ] (property (iii) in the above lemma) relies heavily on the convexity of the problem. Lemma 3.2 (i) The value process V of Lemma 3.1 belongs to S 2 and admits the following (Mertens) decomposition:

(3.2)

v t = v 0 + M t -A t -C t-, for all t ∈ [0, T ] a.s.,
where M ∈ M 2 , A is a nondecreasing right-continuous predictable process such that

A 0 = 0, E(A 2
T ) < ∞, and C is a nondecreasing right-continuous adapted purely discontinuous process such that

C 0-= 0, E(C 2 T ) < ∞.
(ii) For each τ ∈ T 0,T , we have

∆C τ = 1 {vτ =ξτ } ∆C τ a.s. (iii) For each predictable τ ∈ T 0,T , we have ∆A τ = 1 {v τ -= ξ τ } ∆A τ a.s. Proof.
By Lemma 3.1 (i), the process (v t ) t∈[0,T ] is a strong supermartingale. Moreover, by using martingale inequalities, it can be shown that E[ess sup S∈T 0,T |V S | 2 ] ≤ c|||ξ||| 2 S 2 . Hence, the process (v t ) t∈[0,T ] is in S 2 (a fortiori, of class (D)). Applying Mertens decomposition for strong supermartingales of class (D) (cf., e.g., [START_REF] Dellacherie | Probabilités et Potentiel, Théorie des Martingales[END_REF]Appendix 1,Thm.20,equalities (20.2)])

gives the decomposition (3.2), where M is a cadlag uniformly integrable martingale, A is a nondecreasing right-continuous predictable process such that A 0 = 0, E(A T ) < ∞, and C is a nondecreasing right-continuous adapted purely discontinuous process such that C 0-= 0, E(C T ) < ∞. Based on some results of Dellacherie-Meyer [START_REF] Dellacherie | Probabilités et Potentiel, Théorie des Martingales[END_REF] (cf., e.g., Theorem A.2 and Corollary A.1 in [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF]), we derive that A ∈ S 2 and C ∈ S 2 , which gives the assertion (i).

Let τ ∈ T 0,T . By Remark 3.1 together with Mertens decomposition (3.2), we get ∆C τ = -∆ + v τ a.s. It follows that ∆C τ = 1 {vτ =ξτ } ∆C τ a.s. , which corresponds to (ii). Assertion (iii) (concerning the jumps of A) is due to El Karoui 3 ([12, Proposition 2.34]) 3 Note that the proof in El Karoui [START_REF] Karoui | Les aspects probabilistes du contrôle stochastique. École d'été de Probabilités de Saint-Flour IX[END_REF] is given for nonnegative pay-o ξ. To pass from this to the more general case where ξ might take also negative values, we apply the result by El Karoui [START_REF] Karoui | Les aspects probabilistes du contrôle stochastique. École d'été de Probabilités de Saint-Flour IX[END_REF] with ξ := ξ + X (which is non-negative) and ṽ := v + X, where the process X = (Xt) is dened by

Xt := E[ess sup τ ∈T 0,T ξ - τ |Ft].
We then notice that the Mertens process (A, C) from the Mertens decomposition of v is the same as the Mertens process ( Ã, C) from the Mertens decomposition of ṽ (indeed, only the Its proof is based on the equality A S = A τ λ S a.s. , for each S ∈ T 0,T and for each λ ∈]0, 1[ (which follows from Lemma 3.1 (iii) together with Mertens decomposition (3.2)).

The following minimality property for the continuous part A c is well-known from the literature in the "more regular" cases (cf., e.g., [START_REF] Kobylanski | Erratum: Optimal stopping time problem in a general framework[END_REF] for the right-uppersemicontinuous case).

In the case of completely irregular ξ, this minimality property was not explicitly available.

Only recently, it was proved by [START_REF] Klimsiak | Reected BSDEs with regulated trajectories[END_REF] (cf. Proposition 3.7) in the Brownian framework. Here, we generalize the result of [START_REF] Klimsiak | Reected BSDEs with regulated trajectories[END_REF] to the case of a general ltration by using dierent analytic arguments. Lemma 3.3 The continuous part A c of A satises the equality T 0 1 {v t->ξ t } dA c t = 0 a.s.

Proof.

As for the discontinuous part of A, the proof is based on Lemma 3.1 (iii) , and also on some analytic arguments similar to those used in the proof of Theorem D13 in Karatzas and Shreve (1998) ( [START_REF] Karatzas | Methods of mathematical nance[END_REF]).

We have to show that T 0 (v t--ξ t )dA c t = 0 a.s. Lemma 3.1 (iii) yields that for each S ∈ T 0,T and for each λ ∈]0, 1[, we have A S = A τ λ S a.s.

Without loss of generality, we can assume that for each ω, the map t → A c t (ω) is continuous, that the map t → v t (ω) is left-limited, and that, for all λ ∈]0,

1[∩Q and t ∈ [0, T [∩Q, we have A t (ω) = A τ λ t (ω).
Let us denote by J (ω) the set on which the nondecreasing function t → A c t (ω) is at:

J (ω) := {t ∈]0, T [ , ∃δ > 0 with A c t-δ (ω) = A c t+δ (ω)}
The set J (ω) is clearly open and hence can be written as a countable union of disjoint intervals:

J (ω) = ∪ i ]α i (ω), β i (ω)[. We consider (3.3) Ĵ (ω) := ∪ i ]α i (ω), β i (ω)] = {t ∈]0, T ] , ∃δ > 0 with A c t-δ (ω) = A c t (ω)}.
We have

T 0 1 Ĵ (ω) dA c t (ω) = i (A c β i (ω) (ω) -A c α i (ω) (ω)) = 0. Hence, the nondecreasing function t → A c t (ω) is at on Ĵ (ω). We now introduce K(ω) := {t ∈]0, T ] s.t. v t-(ω) > ξ t (ω)}
We next show that for almost every ω, K(ω) ⊂ Ĵ (ω), which clearly provides the desired result. Let t ∈ K(ω). Let us prove that t ∈ Ĵ (ω). By (3.3), we thus have to show that martingale parts of the two decompositions dier by X). Moreover, we see that the set {vτ-= ξ τ } is the same as the set where v is replaced by ṽ and ξ is replaced by ξ (this is due to the fact that X is a martingale and thus has left limits; so Xt = Xt-).

there exists δ > 0 such that A c t-δ (ω) = A c t (ω). Since t ∈ K(ω), we have v t-(ω) > ξ t (ω). Hence, there exists δ > 0 and λ ∈]0, 1[∩Q such that t -δ ∈ [0, T [∩Q and for each r ∈ [t -δ, t[, λv r (ω) > ξ r (ω). By denition of τ λ t-δ (ω), it follows that τ λ t-δ (ω) ≥ t. Now, we have

A c τ λ t-δ (ω) = A c t-δ (ω). Since the map s → A c s (ω) is nondecreasing, we get A c t (ω) = A c t-δ (ω),
which implies that t ∈ Ĵ (ω). We thus have K(ω) ⊂ Ĵ (ω), which completes the proof. Remark 3.3 We note that the martingale property from assertion (iii) of Lemma 3.1 is crucial for the proof of the minimality conditions for the process A (namely, for the proofs of Lemma 3.2 assertion (iii), and for Lemma 3.3).

3.2. The classical linear optimal stopping problem with an additional instantaneous reward. In this subsection, we extend the previous results to the case where, besides the reward process ξ, there is an additional running (or instantaneous) reward process f ∈ IH 2 . More precisely, let (ξ t ) t∈[0,T ] be a process belonging to S 2 , called the reward process or the pay-o process. Let f = (f t ) t∈[0,T ] be a predictable process with E[ T 0 f 2 t dt] < +∞, called the instantaneous reward process. For each S ∈ T 0,T , we dene the value V (S) at time S by V (S) := ess sup

τ ∈T S,T E[ξ τ + τ S f u du | F S ]. (3.4)
This is equivalent to

V (S) + S 0 f u du := ess sup τ ∈T S,T E[ξ τ + τ 0 f u du | F S ]. (3.5)
Hence, the results of the previous subsection can be applied with ξ • replaced by ξ • +

• 0 f u du and v(S) replaced by V (S) + S 0 f u du. Here is a brief summary. Lemma 3.4 (i) There exists a ladlag optional process (V t ) t∈[0,T ] which aggregates the family (V (S)) S∈T 0,T (i.e. V S = V (S) a.s. for all S ∈ T 0,T ). Moreover, the process (V t + t 0 f u du) t∈[0,T ] is the smallest strong supermartingale greater than or equal to

(ξ t + t 0 f u du) t∈[0,T ] . (ii) We have V S = ξ S ∨ V S+ a.
s. for all S ∈ T 0,T . Remark 3.4 It follows from (ii) in the above lemma that ∆ + V S = 1 {V S =ξ S } ∆ + V S a.s. Lemma 3.5 (i) The value process V of Lemma 3.4 belongs to S 2 and admits the following (Mertens) decomposition:

(3.6)

V t = V 0 - t 0 f u du + M t -A t -C t-, for all t ∈ [0, T ] a.s.,
where M ∈ M 2 , A is a nondecreasing right-continuous predictable process such that

A 0 = 0, E(A 2
T ) < ∞, and C is a nondecreasing right-continuous adapted purely discontinuous process such that C 0-= 0, E(C 2 T ) < ∞.

(ii) For each τ ∈ T 0,T , we have ∆C τ = 1 {Vτ =ξτ } ∆C τ a.s. (iii) For each predictable τ ∈ T 0,T , we have ∆A τ = 1 {V τ -= ξ τ } ∆A τ a.s. Lemma 3.6 The continuous part A c of A satises the equality T 0 1 {V t->ξ t } dA c t = 0 a.s.

3.3. Characterization of the value function as the solution of an RBSDE. In this subsection, we show, using the above lemmas, that the value process V of the classical optimal stopping problem (3.4) solves the RBSDE from Denition 2.3 with parameters the driver process (f t ) and the obstacle (ξ t ). We also prove the uniqueness of the solution of this RB-SDE. To this aim, we rst provide a priori estimates for RBSDEs in our general framework. Lemma 3.7

(A priori estimates) Let (Y 1 , Z 1 , k 1 , h 1 , A 1 , C 1 ) (resp. (Y 2 , Z 2 , k 2 , h 2 , A 2 , C 2 )) ∈ S 2 ×IH 2 ×IH 2 ν ×M 2,⊥ ×S 2 ×S 2 be a solution to the RBSDE associated with driver f 1 (ω, t) (resp. f 2 (ω, t)) and with obstacle ξ. We set Ỹ := Y 1 -Y 2 , Z := Z 1 -Z 2 , Ã := A 1 -A 2 , C := C 1 -C 2 , k := k 1 -k 2 , h := h 1 -h 2 , and f (ω, t) := f 1 (ω, t) -f 2 (ω, t). There exists c > 0 such that for all ε > 0, for all β ≥ 1 ε 2 we have Z 2 β ≤ ε 2 f 2 β , k 2 ν,β ≤ ε 2 f 2 β and h 2 β,M 2 ≤ ε 2 f 2 β . (3.7) ||| Ỹ ||| 2 β ≤ 4ε 2 (1 + 12c 2 ) f 2 β . (3.8)
Proof.

The proof is given in the Appendix.

Using these a priori estimates, the lemmas from the previous subsection, and the orthogonal martingale decomposition (Lemma 2.1), we derive the following "innitesimal characterization" of the value process V .

Theorem 3.1 Let V be the value process of the optimal stopping problem (3.4). Let A and C be the non decreasing processes associated with the Mertens decomposition (3.6) of V . There exists a unique triplet Z,k,h,A,C) is a solution of the RBSDE from Denition 2.3 associated with the driver process f (ω, t, y, z, k ) = f t (ω) and the obstacle (ξ t ). Moreover, the solution of this RBSDE is unique.

(Z, k, h) ∈ IH 2 × IH 2 ν × M 2,⊥ such that the process (V,

Proof.

By Lemma 3.4 (ii), the value process V corresponding to the optimal stopping problem (3.4) satises V T = V (T ) = ξ T a.s. and V t ≥ ξ t , 0 ≤ t ≤ T , a.s. By Lemma 3.5 (ii), the process C of the Mertens decomposition of V (3.6) satises the minimality condition (2.5). Moreover, by Lemma 3.5 (iii) and Lemma 3.6, the process A satises the minimality condition (2.4). By Lemma 2.1, there exists a unique triplet

(Z, k, h) ∈ IH 2 × IH 2 ν × M 2,⊥ such that dM t = Z t dW t + E k t (e) Ñ (dt, de) + dh t . The process (V, Z, k, h, A, C) is thus a solution of the RBSDE (2.
3) associated with the driver process (f t ) and the obstacle ξ.

It remains to show the uniqueness of the solution. Using the a priori estimates from Lemma 3.7, together with classical arguments (cf. step 5 of the proof of Lemma 3.3 in [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF]), we obtain the desired result.

We are interested in generalizing this result to the case of the optimal stopping problem (1.1) with non-linear f -expectation (associated with a non-linear driver f (ω, t, y, z, k )). To this purpose, we rst establish an existence and uniqueness result for the RBSDE from Denition 2.3 in the case of a general (non-linear) Lipschitz driver f (ω, t, y, z, k ).

4. Existence and uniqueness of the solution of the RBSDE with an irregular obstacle and a general ltration in the case of a general driver. In Theorem 3.1, we have shown that, in the case where the driver does not depend on y, z, and k , the RBSDE from Denition 2.3 admits a unique solution. Using this result together with the above a priori estimates from Lemma 3.7, we derive the following existence and uniqueness result in the case of a general Lipschitz driver f (t, y, z, k).

Theorem 4.1 (Existence and uniqueness) Let ξ be a process in S 2 and let f be a Lipschitz driver. The RBSDE with parameters (f, ξ) from Denition 2.3 admits a unique solution

(Y, Z, k, h, A, C) ∈ S 2 × IH 2 × IH 2 ν × M 2,⊥ × S 2 × S 2 .
Proof.

For each β > 0, we denote by B 2 β the Banach space S 2 × IH 2 × IH 2 ν which we equip with the norm (•,

•, •) B 2 β dened by (Y, Z, k) 2 B 2 β := |||Y ||| 2 β + Z 2 β + k 2 ν,β , for (Y, Z, k) ∈ S 2 × IH 2 × IH 2
ν . We dene a mapping Φ from B 2 β into itself as follows: for a given (y, z, l) ∈ B 2 β , we set Φ(y, z, l) := (Y, Z, k), where Y, Z, k are the rst three components of the solution (Y, Z, k, h, A, C) to the RBSDE associated with driver process f (s) := f (s, y s , z s , l s ) and with obstacle ξ. The mapping Φ is well-dened by Theorem 3.1. Using the a priori estimates from Lemma 3.7 and similar computations as those from the proof of Theorem 3.4 in [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF], we derive that Φ is a contraction for the norm • B 2 β . By the xed point theorem in the Banach space B 2 β , the mapping Φ thus admits a unique xed point, which corresponds to the unique solution of the RBSDE with parameters (f, ξ).

Remark 4.1 In [START_REF] Klimsiak | Reected BSDEs with regulated trajectories[END_REF], the above existence and uniqueness result is shown in a Brownian framework by using a penalization method. Our approach provides an alternative proof of this result.

We now provide a useful property of the solution of an RBSDE, which will be used in the sequel. Lemma 4.1 (E f -martingale property of Y ) Let ξ be a process in S 2 and let f be a Lipschitz driver. Let (Y, Z, k, h, A, C) be the solution to the reected BSDE with parameters (f, ξ) as in Denition 2.3. For each S ∈ T 0,T and for each ε > 0, we set

τ ε S := inf{t ≥ S , Y t ≤ ξ t + ε}.
The process

(Y t ) is an E f -martingale on [S, τ ε S ].
Proof: The proof in our case of a general ltration is identical to that of Lemma 4.1 (statement (ii)) in [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF] and is given here for the convenience of the reader 4 . By denition of τ ε S , we have: for a.e. ω ∈ Ω, for all t ∈ [S(ω),

τ ε S (ω)[, Y t (ω) > ξ t (ω) + ε.
Hence, by the Skorokhod condition for A, we have that for a.e. ω ∈ Ω, the function t

→ A c t (ω) is constant on [S(ω), τ ε S (ω)[; by continuity of almost every trajectory of the process A c , A c • (ω) is constant on the closed interval [S(ω), τ ε S (ω)],
for a.e. ω. Furthermore, (again by the Skotokhod condition for A), for a.e. ω ∈ Ω, the function t

→ A d t (ω) is constant on [S(ω), τ ε S (ω)[. Moreover, Y (τ ε S ) -≥ ξ (τ ε S ) -+ ε a.s. , which implies that ∆A d τ ε S = 0 a.s. Finally, for a.e. ω ∈ Ω, for all t ∈ [S(ω), τ ε S (ω)[, ∆C t (ω) = C t (ω) -C t-(ω) = 0; therefore, for a.e. ω ∈ Ω, for all t ∈ [S(ω), τ ε S (ω)[, ∆ + C t-(ω) = C t (ω) -C t-(ω) = 0, which implies that, for a.e. ω ∈ Ω, the function t → C t-(ω) is constant on [S(ω), τ ε S (ω)[
. By left-continuity of almost every trajectory of the process (C t-), we get that for a.e. ω ∈ Ω, the function Remark 4.2 Note that in the case where ξ is nonnegative, the above result holds true also on the stochastic interval [S, τ λ S ], where λ ∈ (0, 1) and τ λ S := inf{t ≥ S : λY t ≤ ξ t }. Note that in the case of non-negative obstacle, we have also andτ λ S is nite a.s. 5. Optimal stopping with non-linear f -expectation: formulation of the problem. Let (ξ t ) t∈[0,T ] be a process in S 2 . Let f be a Lipschitz driver. For each S ∈ T 0,T , we dene the value at time S by (5.1)

t → C t-(ω) is constant on the closed interval [S(ω), τ ε S (ω)]. Thus, for a.e. ω ∈ Ω, the map t → A t (ω) + C t-(ω) is constant on [S(ω), τ ε S (ω)].
Y ≥ 0 (as Y ≥ ξ ≥ 0); hence, λY T ≤ Y T = ξ T a.s.
V (S) := ess sup τ ∈T S,T E f S,τ (ξ τ ).
We make the following assumption on the driver (cf., e.g., Theorem 4.2 in [START_REF] Quenez | BSDEs with jumps, optimization and applications to dynamic risk measures[END_REF]).

Assumption 5.1 Assume that dP ⊗ dt-a.e. for each (y, z,

k 1 , k 2 ) ∈ R 2 × (L 2 ν ) 2 , f (t, y, z, k 1 ) -f (t, y, z, k 2 ) ≥ θ y,z,k 1 ,k 2 t , k 1 -k 2 ν , where θ : [0, T ] × Ω × R 2 × (L 2 ν ) 2 → L 2 ν ; (ω, t, y, z, k 1 , k 2 ) → θ y,z,k 1 ,k 2 t (ω, •) is a P ⊗ B(R 2 ) ⊗ B((L 2 ν ) 2 )-measurable mapping, satisfying θ y,z,k 1 ,k 2 t (•) ν ≤ C for all (y, z, k 1 , k 2 ) ∈ R 2 × (L 2 ν ) 2 , dP ⊗ dt-a.e.
, where C is a positive constant, and such that θ y,z,k 1 ,k 2 t (e) ≥ -1, for all

(y, z, k 1 , k 2 ) ∈ R 2 × (L 2 ν ) 2 , dP ⊗ dt ⊗ dν(e) -a.e.
The above assumption is satised if, for example, [START_REF] Dumitrescu | Generalized Dynkin Games and Doubly reected BSDEs with jumps[END_REF]). We recall that under Assumption 5.1 on the driver f , the functional E f S,τ (•) is nondecreasing (cf. [START_REF] Quenez | BSDEs with jumps, optimization and applications to dynamic risk measures[END_REF]Thm. 4.2] and Remark 12.1).

f is of class C 1 with respect to k such that ∇ k f is bounded (in L 2 ν ) and ∇ k f ≥ -1 (cf. Proposition A.2. in
As mentioned in the introduction, the above optimal stopping problem has been largely studied: in [START_REF] Karoui | Non-linear Pricing Theory and Backward Stochastic Dierential Equations[END_REF], and in [START_REF] Bayraktar | Optimal Stopping for Dynamic Convex Risk Measures[END_REF], in the case of a continuous pay-o process ξ; in [START_REF] Quenez | Reected BSDEs and robust optimal stopping for dynamic risk measures with jumps[END_REF] and [START_REF] Bayraktar | Optimal stopping for Non-linear Expectations[END_REF] in the case of a right-continuous pay-o; and recently in [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF] in the case of a rightuppersemicontinuous pay-o process ξ. In this section, we do not make any regularity assumptions on ξ (cf. also Remark 2.2). If we interpret ξ as a nancial position process and -E f (•) as a dynamic risk measure (cf.,e.g., [START_REF] Peng | Nonlinear expectations, nonlinear evaluations and risk measures[END_REF], [START_REF] Rosazza-Gianin | Risk measures via g-expectations[END_REF]), then (up to a minus sign) V (S) can be seen as the minimal risk at time S. As also mentioned in the introduction, the absence of regularity allows for more exibility in the modelling. If, for instance, we consider a situation where the jump times of the Poisson random measure model times of default (which, being totally inaccessible, cannot be foreseen), then, the complete lack of regularity allows to take into account an immediate non-smooth, positive or negative, impact on ξ after the default occurs.

If we interpret ξ as a payo process, and E f (•) as a non linear pricing rule, then the optimal stopping problem (5.1) is related to the (non linear) pricing problem of the American option with payo ξ. The absence of regularity allows us to deal with the case of American options with irregular payos, such as American digital options (cf. Section 11.1 for details). On the other hand, the fact that the ltration is not necessarily the natural ltration associated with W and N allows to incorporate some additional information in the modelling (such as, for example, default risks or other economic factors).

We begin by addressing the simpler case where the payo is assumed to be right u.s.c. This preliminary study of the right u.s.c. case will allow us to establish an E f -Mertens decomposition for strong E f -supermartingales with respect to a general ltration (extending the existing results from the literature; cf. [START_REF] Bouchard | A general Doob-Meyer-Mertens decomposition for g-supermartingale system[END_REF] and [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF]). This will be an important result for the treatment of the non-linear optimal stopping problem in the case of a completely irregular pay-o. We have (6.1)

Y τ ε S ≤ ξ τ ε S + ε a.s.
Proof: The proof of this result in our case of a general ltration is identical to that from [17, Lemma 4.1(i)] in the case of a Brownian-Poisson ltration. We give again the arguments here in order to emphasize the important role of the right-uppersemicontinuity assumption on ξ. By way of contradiction, we suppose P (Y

τ ε S > ξ τ ε S +ε) > 0. By the Skorokhod condition for C, we have ∆C τ ε S = C τ ε S -C (τ ε S )-= 0 on the set {Y τ ε S > ξ τ ε S + ε}. On the other hand, due to Remark 2.3, ∆C τ ε S = Y τ ε S -Y (τ ε S )+ . Thus, Y τ ε S = Y (τ ε S )+ on the set {Y τ ε S > ξ τ ε S + ε}.
Hence, (6.2)

λY (τ ε S )+ > ξ τ ε S on the set {Y τ ε S > ξ τ ε S + ε}.
We will obtain a contradiction with this statement. Let us x ω ∈ Ω. By denition of τ ε S (ω), there exists a non-increasing sequence

(t n ) = (t n (ω)) ↓ τ ε S (ω) such that Y tn (ω) ≤ ξ tn (ω) + ε, for all n ∈ IN . Hence, lim sup n→∞ Y tn (ω) ≤ lim sup n→∞ ξ tn (ω) + ε.
As the process ξ is right-uppersemicontinuous , we have lim sup n→∞ ξ tn (ω) ≤ ξ τ ε S (ω). On the other hand, as

(t n (ω)) ↓ τ ε S (ω), we have lim sup n→∞ Y tn (ω) = Y (τ ε S )+ (ω). Thus, Y (τ ε S )+ (ω) ≤ ξ τ ε S (ω) + ε, which is in contradiction with (6.2). We conclude that Y τ ε S ≤ ξ τ ε S + ε a.s.
With the help of the previous lemma together with Lemma 4.1, we derive the following result.

Theorem 6.1 (Characterization theorem in the r.u.s.c. case) Let (ξ t ) t∈[0,T ] be a process in S 2 , supposed to be right u.s.c. Let (Y, Z, k, h, A, C) be the solution to the reected BSDE with parameters (f, ξ) as in Denition 2.3.

• For each stopping time S ∈ T 0 , we have5 (6.3)

Y S = ess sup τ ∈T S,T E f S,τ (ξ τ ) a.s.
• Moreover, the stopping time τ ε S dened by (4.1), that is,

τ ε S = inf{t ≥ S, Y t ≤ ξ t + ε}, satises (6.4) Y S ≤ E f S,τ ε S (ξ τ ε S ) + Lε a.s. ,
where L is a constant which only depends on T and the Lipschitz constant K of f . In other words, τ ε S is an Lε-optimal stopping time for problem (6.3).

Proof: The arguments are classical. Let us show the inequality (6.4). Since by Lemma 4.1, the process

(Y t ) is an E f -martingale on [S, τ ε S ], we get Y S = E f S,τ ε S (Y τ ε S ) a.
s. Since ξ is right u.s.c. , we can apply Lemma 6.1. Using this, the monotonicity property of the conditional f -expectation and the a priori estimates for BSDEs (cf. [START_REF] Quenez | BSDEs with jumps, optimization and applications to dynamic risk measures[END_REF] which still hold in our case of a general ltration), we derive that

Y S = E f S,τ ε S (Y τ ε S ) ≤ E f S,τ ε S (ξ τ ε S + ε) ≤ E f S,τ ε S (ξ τ ε S ) + Lε a.s.,
where L is a positive constant depending only on T and the Lipschitz constant K of the driver f ; this gives the desired inequality (6.4). Moreover, as ε is an arbitrary nonnegative number, we get Y S ≤ ess sup τ ∈T S,T E f S,τ (ξ τ ) a.s. It remains to show the converse inequality. Let τ ∈ T S,T . By Lemma 12.2 in the Appendix, the process (Y t ) is a strong E f -supermartingale. Hence, for each τ ∈ T S,T , we have

Y S ≥ E f S,τ (Y τ ) ≥ E f S,τ (ξ τ ) a.s.
, where the second inequality follows from the inequality Y ≥ ξ and the monotonicity property of E f (•) (with respect to terminal condition). By taking the supremum over τ ∈ T S,T , we get Y S ≥ ess sup τ ∈T S,T E f S,τ (ξ τ ) a.s. We thus derive the desired equality (6.3), which completes the proof.

We now investigate the question of the existence of optimal stopping times for the optimal stopping problem (6.3). We rst provide an optimality criterion. Lemma 6.2 (Optimality criterion) Let (ξ t , 0 ≤ t ≤ T ) be a process6 in S 2 and let f be a predictable Lipschitz driver satisfying Assumption 5.1. Let S ∈ T 0,T and τ * ∈ T S,T .

If Y is a strong E f -martingale on [S, τ * ] with Y τ * = ξ τ * a.s., then the stopping time τ * is optimal at time S (i.e. Y S = E f S,τ * (ξ τ * ) a.s.). The converse statement also holds true, if, in addition, the inequality from Assumption 5.1 is strict (that is, θ y,z,k 1 ,k 2 t > -1).

Proof: The proof of this result in the case of a Brownian-Poisson ltration can be found in [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF]Proposition 4.1 ]. The proof in our case of a general ltration is identical and is therefore omitted.

We now show that if ξ is assumed to be r.u.s.c. and also l.u.s.c. along stopping times, then there exists an optimal stopping time.

Let S ∈ T 0 . Let us recall the denition of τ ε S from before:

τ ε S := inf{t ≥ S , Y t ≤ ξ t + ε}.
We notice that τ ε S is non-increasing in ε. Let (ε n ) be a non-increasing positive sequence converging to 0. We set τS := lim n→∞ ↑ τ εn S .

The random time τS is a stopping time in T S .

We also set τ 0 S := inf{t ≥ S , Y t = ξ t }. We notice that τ εn S ≤ τ 0 S a.s. for all n. Hence, by passing to the limit, we get τS ≤ τ 0 S a.s.

In the following theorem we show that, under the additional assumption that ξ is l.u.s.c. along stopping times, the stopping time τS is an optimal stopping time at time S. We also show that the stopping times τS and τ 0 S coincide. Theorem 6.2 (Existence of optimal stopping time) Let (ξ t , 0 ≤ t ≤ T ) be an r.u.s.c. process in S 2 and let f be a predictable Lipschitz driver satisfying Assumption 5.1. We assume, in addition, that (ξ t ) is l.u.s.c. along stopping times. Then, the stopping time τS is S-optimal, in the sense that it attains the supremum in (6.3)

lim sup n→∞ E f S,τ εn S ξ τ εn S ≤ E f S,τ S lim sup n→∞ ξ τ εn S ≤ E f S,τ S ξ τS a.s.,
where the last inequality follows from (6.5) and from the monotonicity of E f S,τ S (•). On the other hand, from Eq. (6.4) in Theorem 6.1, we have Y S ≤ lim sup n→∞ E f S,τ εn S ξ τ εn S a.s. From this, together with (6.6), we get Y S ≤ E f S,τ S ξ τS a.s., which shows that τS is an optimal stopping time.

Let us now prove the equality τS = τ 1 S a.s. We have already noticed that τS ≤ τ 1 S a.s. It remains to show the converse inequality. Note that for each S ∈ T 0,T , Y S is equal a.s. to the value at time S of the linear optimal stopping problem associated with the pay-o process (ξ t ) and the instantaneous reward process ( ft ) dened by ft (ω, t)

:= f (ω, t, Y t-(ω), Z t (ω), k t (ω)),
that is

Y S = ess sup τ ∈T S,T E[ξ τ + τ S fu du | F S ] a.s.. (6.7)
It is not dicult to see that τS is also optimal for this linear optimal stopping problem. Now, from classical results on linear optimal stopping, τ 0 S is the minimal optimal stopping time for problem (6.7); hence, we have τS ≥ τ 0 S a.s., which completes the proof. Proposition 6.1 Let (ξ t , 0 ≤ t ≤ T ) be an r.u.s.c. process in S 2 and let f be a predictable Lipschitz driver. We assume, in addition, that (ξ t ) is l.u.s.c. along stopping times. Let (Y, Z, k, h, A, C) be the solution to the reected BSDE with parameters (f, ξ) as in Denition 2.3. Then, the process A is continuous.

Proof: Given the solution (Y, Z, k, h, A, C) to the reected BSDE with parameters (f, ξ),

we dene the process f by

f (ω, t) := f (ω, t, Y t-(ω), Z t (ω), k t (ω)).
The process f is a predictable process in IH 2 . From the denition of f and from Denition 2.3, we see that (Y, Z, k, h, A, C) is the solution of the RBSDE with driver process f and obstacle ξ. By Theorem 3.1 (on RBSDEs with given driver process and linear optimal stopping), we have that, for all S ∈ T 0 ,

Y S = ess sup τ ∈T S,T E[ξ τ + τ S fu du | F S ] a.s., (6.8)
which is equivalent to Y S + S 0 fu du = ess sup τ ∈T S,T E[ξ τ + τ 0 fu du | F S ] a.s. From results on classical optimal stopping with linear expectations, we deduce that A is continuous, as (ξ t ) is r.u.s.c. and l.u.s.c. along stopping times (cf., e.g., Proposition B.10 in [START_REF] Kobylanski | Optimal stopping time problem in a general framework[END_REF] 8 ).

7. E f -Mertens decomposition of strong E f -supermartingales with respect to a general ltration. By using the above characterization of the solution of the RB-SDE with an r.u.s.c. obstacle as the value function of the non-linear optimal stopping problem (5.1) (cf. Theorem 6.1), we derive an E f -Mertens decomposition of strong E fsupermartingales, which generalizes the one provided in [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF] (cf. Theorem 5.2 in [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF]) to the case of a general ltration. 9

As mentioned before, this is an important property in the present work which will allow us to address the non-linear optimal stopping problem in the completely irregular case (cf. Section 9.3, more precisely the proof of Proposition 9.1, and also Theorem 10.1).

Theorem 7.1 (E f -Mertens decomposition) Let (Y t ) be a process in S 2 . Let f be a Lipschitz driver satisfying Assumption 5.1. The process (Y t ) is a strong E f -supermartingale if and only if there exists a nondecreasing right-continuous predictable process A in S 2 with A 0 = 0 and a nondecreasing right-continuous adapted purely discontinuous process C in S 2 with C 0-= 0, as well as three processes Z ∈ IH 2 , k ∈ H 2 ν and h ∈ M 2,⊥ , such that a.s. for all t ∈ [0, T ],

(7.1) -dY t = f (t, Y t , Z t , k t )dt+dA t +dC t--Z t dW t - E k t (e) Ñ (dt, de)-dh t , 0 ≤ t ≤ T.
This decomposition is unique. Moreover, a strong E f -supermartingale is necessarily r.u.s.c.

Proof: Assume that (Y t ) is a strong E f -supermartingale. By the same arguments as in [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF] (cf. Lemma 5.1 in [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF]), it can be shown that the process (Y t ) is r.u.s.c. Let S ∈ T 0 . Since (Y t ) is a strong E f -supermartingale, we derive that for all τ ∈ T S , we have Y S ≥ E f S,τ (Y τ ) a.s. We get Y S ≥ ess sup τ ∈T S E f S,τ (Y τ ) a.s. Now, by denition of the essential supremum, Y S ≤ ess sup τ ∈T S E f S,τ (Y τ ) a.s. because S ∈ T S . Hence, Y S = ess sup τ ∈T S E f S,τ (Y τ ) a.s. By Theorem 6.1, the process (Y t ) coincides with the solution of the reected BSDE associated with the (r.u.s.c.) obstacle (Y t ), and thus admits the decomposition (7.1).

The converse follows from Lemma 12.2 in the Appendix. [START_REF] Dellacherie | Probabilités et Potentiel, Théorie des Martingales[END_REF]. Optimal stopping with non-linear f -expectation in the completely irregular case: the direct part of the approach. We now turn to the study of the non-linear optimal stopping problem (5.1) in the more dicult case where (ξ t ) is completely irregular.

Since the process (ξ t ) is not r.u.s.c. , the inequality Y τ ε S ≤ ξ τ ε S + ε (i.e. inequality (6.1))

does not necessarily hold (not even in the simplest case of linear expectations; cf., e.g., [START_REF] Karoui | Les aspects probabilistes du contrôle stochastique. École d'été de Probabilités de Saint-Flour IX[END_REF]). This prevents us from adopting here the approach used in the r.u.s.c. case to prove an innitesimal characterization of the value of the non-linear optimal stopping problem in terms of the solution of an RBSDE. Thus, when ξ is completely irregular, we have to proceed dierently. We use a combined approach which consists in a direct part and an RBSDE-part. This section is devoted to the direct part of our approach to the non-linear optimal stopping problem (5.1).

8.1. Preliminary results on the value family. Let us rst introduce the denition of an admissible family of random variables indexed by stopping times in T 0,T (or T 0,T -system in the vocabulary of Dellacherie and Lenglart [START_REF] Dellacherie | Sur des problèmes de régularisation, de recollement et d'interpolation en théorie des processus[END_REF]).

Denition 8.1 We say that a family U = (U (τ ), τ ∈ T 0,T ) is admissible if it satises the following conditions 1. for all τ ∈ T 0,T , U (τ ) is a real-valued F τ -measurable random variable.

2. for all τ, τ ∈ T 0,T , U (τ ) = U (τ ) a.s. on {τ = τ }. Moreover, we say that an admissible family U is square-integrable if for all τ ∈ T 0,T , U (τ ) is square-integrable. Lemma 8.1 (Admissibility of the family V ) The family V = (V (S), S ∈ T 0,T ) dened in (5.1) is a square-integrable admissible family.

Proof: The proof uses arguments similar to those used in the "classical" case of linear expectations (cf., e.g., [START_REF] Kobylanski | Optimal multiple stopping time problem[END_REF]), combined with some properties of f -expectations. For each S ∈ T 0,T , V(S) is an F S -measurable square-integrable random variable, due to the denitions of the conditional f -expectation and of the essential supremum (cf. [START_REF] Neveu | Martingales à Temps Discret[END_REF]). Let us prove Property 2 of the denition of admissibility. Let S and S be two stopping times in T 0,T . We set A := {S = S } and we show that V (S) = V (S ), P -a.s. on A. For each τ ∈ T S,T , we set τ A := τ 1 A + T 1 A c . We have τ A ≥ S a.s. By using the fact that S = S a.s. on A, the fact that τ A = τ a.s. on A, and a standard property of conditional f -expectations (cf., e.g., Proposition A.3 in [START_REF] Grigorova | Optimal stopping and a non-zero-sum Dynkin game in discrete time with risk measures induced by BSDEs, to appear in Stochastics[END_REF] which can be extended without diculty to the framework of general ltration), we obtain

1 A E f S,τ [ξ τ ] = 1 A E f S ,τ [ξ τ ] = E f τ 1 A S ,T [ξ τ 1 A ] = E f τ A 1 A S ,T [ξ τ A 1 A ] = 1 A E f S ,τ A [ξ τ A ] ≤ 1 A V (S ),
where f τ (t, y, z, k ) := f (t, y, z, k )1 {t≤τ } . By taking the ess sup over T S,T on both sides, we get 1 A V (S) ≤ 1 A V (S ). We obtain the converse inequality by interchanging the roles of S and S . Lemma 8.2 (Optimizing sequence) For each S ∈ T 0,T , there exists a sequence (τ n ) n∈N of stopping times in T S,T such that the sequence (E f S,τn (ξ τn )) n∈N is nondecreasing and

V (S) = lim n→∞ ↑ E f
S,τn (ξ τn ) a.s.

Proof: Due to a classical result on essential suprema (cf. [START_REF] Neveu | Martingales à Temps Discret[END_REF]), it is sucient to show that, for each S ∈ T 0,T , the family (E S,τ (ξ τ ), τ ∈ T S,T ) is stable under pairwise maximization.

Let us x S ∈ T 0,T . Let τ ∈ T S,T and τ ∈ T S,T . We dene T 0,T ) is said to be a right-uppersemicontinuous (along stopping times) family if, for any (τ n ) nonincreasing sequence in T 0,T and any τ in T 0,T such that τ = lim ↓ τ n , we have U (τ ) ≥ lim sup n→∞ U (τ n ) a.s. Lemma 8.3 Let U := (U (S), S ∈ T 0,T ) be an E f -supermartingale family. Then, (U (S), S ∈ T 0,T ) is a right-uppersemicontinuous (along stopping times) family.

A := { E f S,τ (ξ τ ) ≤ E f S,τ (ξ τ ) } and ν := τ 1 A + τ 1 A c . We have A ∈ F S and ν ∈ T S,T . We compute 1 A E f S,ν (ξ ν ) = E f ν 1 A S,T (ξ ν 1 A ) = E f τ 1 A S,T (ξ τ 1 A ) = 1 A E f S,τ (ξ τ ) a.s. Similarly, we show 1 A c E f S,ν (ξ ν ) = 1 A c E f S,τ (ξ τ ). It follows that E f S,ν (ξ ν ) = E f S,τ (ξ τ )1 A + E f S,τ (ξ τ )1 A c = E f S,τ (ξ τ )∨ E f S,τ (ξ τ ),
Proof: Let τ ∈ T 0,T and let (τ n ) ∈ T IN 0,T be a nonincreasing sequence of stopping times such that lim n→+∞ τ n = τ a.s. and for all n ∈ IN , τ n > τ a.s. on {τ < T }, and such that lim n→+∞ U (τ n ) exists a.s. As U is an E f -supermartingale family and as the sequence

(τ n ) is nonincreasing, we have E f τ,τn (U (τ n )) ≤ E f τ,τ n+1 (U (τ n+1 )) ≤ U (τ ) a.s. Hence, the sequence (E f τ,τn (U (τ n ))) n is nondecreasing and U (τ ) ≥ lim ↑ E f τ,τn (U (τ n )).
U (τ ) ≥ lim n→+∞ E f τ,τn (U (τ n )) = E f τ,τ ( lim n→+∞ U (τ n )) = lim n→+∞ U (τ n ) a.s.
By Lemma 5 of Dellacherie and Lenglart [START_REF] Dellacherie | Sur des problèmes de régularisation, de recollement et d'interpolation en théorie des processus[END_REF] 10 , the family (U (S)) is thus right-uppersemicontinuous (along stopping times).

Theorem 8.1 The value family V = (V (S), S ∈ T 0,T ) dened in (5.1) is an E f -supermartingale family. In particular, V = (V (S), S ∈ T 0,T ) is a right-uppersemicontinuous (along stopping times) family in the sense of Denition 8.3.

Proof: We know from Lemma 8.1 that V = (V (S), S ∈ T 0,T ) is a square-integrable admissible family. Let S ∈ T 0,T and S ∈ T S,T . We will show that E f S,S (V (S )) ≤ V (S) a.s., which will prove that V is an E f -supermartingale family. By Lemma 8.2, there exists a sequence (τ n ) n∈N of stopping times such that τ n ≥ S a.s. and V (S ) = lim n→∞ ↑ E f S ,τn (ξ τn ) a.s. By using this equality, the property of continuity of BSDEs, and the consistency of conditional f -expectation, we get

E f S,S (V (S )) = E f S,S ( lim n→∞ ↑ E f S ,τn (ξ τn )) = lim n→∞ E f S,S (E f S ,τn (ξ τn )) = lim n→∞ E f S,τn (ξ τn ) ≤ V (S).
Hence, V is an E f -supermartingale family. This property, together with Lemma 8.3, yields that V is a right-uppersemicontinuous (along stopping times) family.

8.2. Aggregation and Snell characterization. Using the above results on the value family V = (V (S), S ∈ T 0,T ), we show the following theorem, which generalizes some results of classical optimal stopping theory (more precisely, the assertion (i) from Lemma 3.4) to the case of an optimal stopping problem with f -expectation.

Theorem 8.2 (Aggregation and Snell characterization) There exists a unique rightuppersemicontinuous optional process, denoted by (V t ) t∈[0,T ] , which aggregates the value family V = (V (S), S ∈ T 0,T ). Moreover, (V t ) t∈[0,T ] is the E f -Snell envelope of the pay-o process ξ, that is, the smallest strong E f -supermartingale greater than or equal to ξ.

Proof: By Theorem 8.1, the value family V = (V (S), S ∈ T 0,T ) is a right-uppersemicontinuous family (or a right-uppersemicontinuous T 0,T -system in the vocabulary of Dellacherie-Lenglart [START_REF] Dellacherie | Sur des problèmes de régularisation, de recollement et d'interpolation en théorie des processus[END_REF]). Applying Theorem 4 of Dellacherie-Lenglart ([6]), gives the existence of a unique (up to indistinguishability) right-uppersemicontinuous optional process (V t ) t∈[0,T ] which aggregates the value family (V (S), S ∈ T 0,T ). From this aggregation property, namely the property V S = V (S) a.s. for each S ∈ T 0,T , and from Theorem 8.1, we deduce that the process 10 The chronology Θ (in the vocabulary and notation of [START_REF] Dellacherie | Sur des problèmes de régularisation, de recollement et d'interpolation en théorie des processus[END_REF]) which we work with here is the chronology of all stopping times, that is, Θ = T0,T ; hence [Θ] = Θ = T0,T .

(V t ) t∈[0,T ] is a strong E f -supermartingale. Moreover, we have V S = V (S) ≥ ξ S a.s. for each S ∈ T 0,T , which implies that V t ≥ ξ t , for all t ∈ [0, T ], a.s.

Let us now prove that the process (V t ) t∈[0,T ] is the smallest strong E f -supermartingale greater than or equal to ξ. Let (V t ) t∈[0,T ] be a strong E f -supermartingale such that V t ≥ ξ t , for all t ∈ [0, T ], a.s. Let S ∈ T 0,T . We have V τ ≥ ξ τ a.s. for all τ ∈ T S,T . Hence, E f S,τ (V τ ) ≥ E f S,τ (ξ τ ) a.s., where we have used the monotonicity of the conditional f -expectation. On the other hand, by using the strong E f -supermartingale property of the process (V t ) t∈[0,T ] , we have V S ≥ E f S,τ (V τ ) a.s. for all τ ∈ T S,T . Hence, V S ≥ E f S,τ (ξ τ ) a.s. for all τ ∈ T S,T . By taking the essential supremum over τ ∈ T S,T in the inequality, we get V S ≥ ess sup τ ∈T S,T E f S,τ (ξ τ ) = V (S) = V S a.s. Hence, for all S ∈ T 0,T , we have V S ≥ V S a.s., which yields that V t ≥ V t , for all t ∈ [0, T ], a.s. The proof is thus complete. 9. Non-linear Reected BSDE with completely irregular obstacle and general ltration: useful properties. Our aim now is to establish an innitesimal characterization for the non-linear problem (5.1) in terms of the solution of a non-linear RBSDE (thus generalizing Theorem 3.1 from the classical linear case to the non-linear case). In order to do so, we need to establish rst some results on non-linear RBSDEs with completely irregular obstacles, in particular, a comparison result for such RBSDEs. This section is devoted to these results (this is the RBSDE-part of our approach to problem (5.1)). The results from this section extend and complete our work from [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF], where an assumption of rightuppersemicontinuity on the obstacle is made. Let us note that the proof of the comparison theorem from [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF] cannot be adapted to the completely irregular framework considered here; instead, we rely on a Tanaka-type formula for strong (irregular) semimartingales which we also establish. Remark 9.1 (A "bottle-neck" of the direct approach) One might wonder whether the innitesimal characterization for the non-linear optimal stopping problem (5.1) can be obtained by pursuing the direct study of the value process (V t ) of problem (5.1), similarly to what was done in the classical linear case in Sub-section 3.1. In the classical case, we applied Mertens decomposition for (V t ); then, we showed directly the minimality properties for the processes A d and A c (cf. Lemmas 3.2 and 3.3) by using the martingale property on the interval [S, τ λ S ] from Lemma 3.1(iii), which itself relies on Maingueneau's penalization approach (cf. also Remarks 3.3 and 3.2). In the non-linear case, Mertens decomposition is generalized by the E f -Mertens decomposition (cf. Theorem 7.1). However, the analogue in the non-linear case of the martingale property of Lemma 3.4[(iii)] (namely, the E f -martingale property) cannot be obtained via Maingueneau's approach (not even in the case of nonnegative ξ and under the additional assumption f (t, 0, 0, 0) = 0 which ensures the non-negativity of E f ) due to the lack of convexity of the functional E f . 9.1. Tanaka-type formula. The following lemma will be used in the proof of the comparison theorem for RBSDEs with irregular obstacles. The lemma can be seen as an extension of Theorem 66 of [37, Chapter IV] from the case of right-continuous semimartingales to the more general case of strong optional semimartingales. Lemma 9.1 (Tanaka-type formula) Let X be a (real-valued) strong optional semimartingale with decomposition X = X 0 +M +A+B, where M is a local (cadlag) martingale, A is a right-continuous adapted process of nite variation such that A 0 = 0, B is a left-continuous adapted purely discontinuous process of nite variation such that B 0 = 0. Let f : R -→ R be a convex function. Then, f (X) is a strong optional semimartingale. Moreover, denoting by f the left-hand derivative of the convex function f , we have

f (X t ) = f (X 0 ) + ]0,t] f (X s-)d(A s + M s ) + [0,t[ f (X s )dB s+ + K t ,
where K is a nondecreasing adapted process (which is in general neither left-continuous nor right-continuous) such that

∆K t = f (X t ) -f (X t-) -f (X t-)∆X t and ∆ + K t = f (X t+ ) -f (X t ) -f (X t )∆ + X t .
Proof: Our proof follows the proof of Theorem 66 of [37, Chapter IV] with suitable changes.

Step 1. We assume that X is bounded; more precisely, we assume that there exists N ∈ IN such that |X| ≤ N . We know (cf. [START_REF] Protter | Stochastic Integration and Dierential Equations (Stochastic Modelling and Applied Probability[END_REF]) that there exists a sequence (f n ) of twice continuously dierentiable convex functions such that (f n ) converges to f , and (f n ) converges to f from below. By applying Gal'chouk-Lenglart's formula (cf., e.g., Theorem A.3 in [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF]) to f n (X t ), we obtain for all τ ∈ T 0,T

(9.1) f n (X τ ) = f n (X 0 ) + ]0,τ ] f n (X s-)d(A s + M s ) + [0,τ [ f n (X s )dB s+ + K n τ , a.s.
, where (9.2)

K n τ := 0<s≤τ f n (X s ) -f n (X s-) -f n (X s-)∆X s + 0≤s<τ f n (X s+ ) -f n (X s ) -f n (X s )∆ + X s + 1 2 ]0,τ ] f n (X s-)d M c , M c s a.s.
We show that (K n τ ) is a convergent sequence by showing that the other terms in Equation (9.1) converge. The convergence ]0

,τ ] f n (X s-)d(A s + M s ) -→ n→∞ ]0,τ ] f (X s-)d(A s + M s )
is shown by using the same arguments as in the proof of [37, Thorem 66, Ch. IV]. The convergence of the term [0,τ [ f n (X s )dB s+ , which is specic to the non-right-continuous case, is shown by using dominated convergence. We conclude that (K n τ ) converges and we set K τ := lim n→∞ K n τ . The process (K t ) is adapted as the limit of adapted processes. Moreover, we have from Eq. ( 9.2) and from the convexity of f n that, for each n, K n t is nondecreasing in t. Hence, the limit K t is nondecreasing.

Step 2. We treat the general case where X is not necessarily bounded by using a localization argument similar to that used in [START_REF] Protter | Stochastic Integration and Dierential Equations (Stochastic Modelling and Applied Probability[END_REF]Th. 66 (Y,Z,k,h,A,C) (resp. (Y ,Z ,k ,h ,A ,C )) be the solution of the RBSDE associated with obstacle ξ (resp. ξ ) and with driver f (resp.

f ). If ξ t ≤ ξ t , 0 ≤ t ≤ T a.s. and f (t, Y t , Z t , k t ) ≤ f (t, Y t , Z t , k t ), 0 ≤ t ≤ T dP ⊗ dt-a.s., then, Y t ≤ Y t , 0 ≤ t ≤ T a.s. Proof: We set Ȳt = Y t -Y t , Zt = Z t -Z t , kt = k t -k t , Āt = A t -A t , Ct = C t -C t , ht = h t -h t , and ft = f (t, Y t-, Z t , k t ) -f (t, Y t-, Z t , k t ). Then, -d Ȳt = ft dt + d Āt + d Ct--Zt dW t - E kt (e) Ñ (dt, de) -d ht , with ȲT = 0.
Applying Lemma 9.1 to the positive part of Ȳt , we obtain (9.3)

Ȳ + t = - ]t,T ] 1 { Ȳs->0} Zs dW s - ]t,T ] E 1 { Ȳs->0} ks (e) Ñ (ds, de) - ]t,T ] 1 { Ȳs->0} d hs + ]t,T ] 1 { Ȳs->0} fs ds + ]t,T ] 1 { Ȳs->0} d Ās + [t,T [ 1 { Ȳs>0} d Cs + (K t -K T ).
We set δ t :=

f (t,Y t-,Zt,kt)-f (t,Y t-,Zt,kt) Y t--Y t- 1 { Ȳt-=0} and β t := f (t,Y t-,Zt,kt)-f (t,Y t-,Z t ,kt) Zt-Z t 1 { Zt =0} .
Due to the Lipschitz-continuity of f , the processes δ and β are bounded. We note that ft =

δ t Ȳt + β t Zt + f (Y t-, Z t , k t ) -f (Y t-, Z t , k t ) + ϕ t , where ϕ t := f (Y t-, Z t , k t ) -f (Y t-, Z t , k t ).
Using this, together with Assumption 5.1, we obtain

(9.4) ft ≤ δ t Ȳt + β t Zt + γ t , kt ν , +ϕ t 0 ≤ t ≤ T, dP ⊗ dt -a.e.,
where we have set γ t := θ Y t-,Z t ,k t ,kt t

. For τ ∈ T 0,T , let Γ τ,• be the unique solution of the following forward SDE dΓ τ,s = Γ τ,s-[δ s ds+β s dW s + E γ s (e) Ñ (ds, de)] with initial condition (at the initial time τ ) Γ τ,τ = 1. To simplify the notation, we denote Γ τ,s by Γ s for s ≥ τ .

By applying Gal'chouk-Lenglart's formula to the product (Γ t Ȳ + t ), and by using that h c , W = 0, we get (9.5)

Γ τ Ȳ + τ = -(M θ -M τ ) -
For the second term, we have -

θ τ Γ s-1 { Ȳs->0} dA c s ≤ 0. We also have - θ τ Γ s-dK c s ≤ 0 and - θ τ Γ s dK d,+ s ≤ 0. Hence, (9.6) Ȳ + τ ≤ -(M θ -M τ ) - θ τ Γ s ( Ȳ + s-δ s + Zs 1 { Ȳs->0} β s -fs 1 { Ȳs->0} )ds + τ <s≤θ Γ s-1 { Ȳs->0} ∆ Ās - θ τ Γ s-dK d,- s - τ <s≤θ ∆Γ s ∆ Ȳ + s .
We compute the last term

τ
∆Γ s ∆ Ȳ + s = = τ <s≤θ Γ s-1 { Ȳs->0} γ s (p s ) ks (p s ) - τ <s≤θ Γ s-γ s (p s )(1 { Ȳs->0} ∆ Ās -∆K d,- s -1 { Ȳs->0} ∆ hs ) = θ τ E Γ s-1 { Ȳs->0} γ s (e)
ks (e)N (ds, de) -

τ <s≤θ Γ s-γ s (p s )(1 { Ȳs->0} ∆ Ās -∆K d,- s -1 { Ȳs->0} ∆ hs ) = θ τ E Γ s-1 { Ȳs->0} γ s (e) ks (e) Ñ (ds, de) + θ τ Γ s-1 { Ȳs->0} γ s , ks ν ds - τ <s≤θ Γ s-1 { Ȳs->0} γ s (p s )∆ Ās + τ <s≤θ Γ s-γ s (p s )∆K d,- s + τ <s≤θ Γ s-1 { Ȳs->0} γ s (p s )∆ hs .
By plugging this expression in equation ( 9.6) and by putting together the terms in "ds", the terms in "dK d,- s ", and the terms in "∆ Ās ", we get (9.8)

Ȳ + τ ≤ -(M θ -M τ ) - θ τ Γ s-( Ȳ + s-δ s + Zs 1 { Ȳs->0} β s + 1 { Ȳs->0} γ s , ks ν -fs 1 { Ȳs->0} )ds + τ <s≤θ Γ s-1 { Ȳs->0} (1 + γ s (p s ))∆ Ās - τ <s≤θ Γ s-(1 + γ s (p s ))∆K d,- s -( Mθ -Mτ ) - θ τ d[ h , • 0 E Γ s-1 { Ȳs->0} γ s (e) Ñ (ds, de) ] s ,
where Mt := t 0 E Γ s-1 { Ȳs->0} γ s (e) ks (e) Ñ (ds, de). Note that by classical arguments (as for M above), the stochastic integral M is a martingale, equal to zero in expectation.

We have -

θ τ Γ s-( Ȳ + s-1 { Ȳs->0} δ s + Zs 1 { Ȳs->0} β s + 1 { Ȳs->0} γ s , ks ν -fs 1 { Ȳs->0} )ds ≤ θ τ Γ s-
θ τ Γ s-1 { Ȳs->0} ϕ s ds is nonpositive, as ϕ s = f (Y s , Z s , k s ) -f (Y s , Z s , k s ) ≤ 0 dP ⊗ ds-a.s
. by the assumptions of the theorem. We conclude that E[ Ȳ + τ ] ≤ 0, which implies Ȳ + τ = 0 a.s. The proof is thus complete.

Remark 9.2 Note that due to the irregularity of the obstacles, together with the presence of jumps, we cannot adopt the approaches used up to now in the literature (see e.g. [START_REF] Karoui | Reected solutions of Backward SDE's and related obstacle problems for PDE's[END_REF], [START_REF] Crépey | Reected and doubly reected BSDEs with jumps: a priori estimates and comparison[END_REF], [START_REF] Quenez | Reected BSDEs and robust optimal stopping for dynamic risk measures with jumps[END_REF] and [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF]) to show the comparison theorem for our RBSDE. between processes are to be understood in the "up to indistinguishability"-sense.

We recall the notion of a strong E f -supermartingale.

Denition 9.2 Let φ be a process in S 2 . Let f be a Lipschitz driver. The process φ is said to be a strong

E f -supermartingale (resp. a strong E f -martingale) , if E f σ,τ (φ τ ) ≤ φ σ a.s. (resp. E f σ,τ (φ τ ) = φ σ a.s.) on σ ≤ τ , for all σ, τ ∈ T 0,T .
Using the above comparison theorem and the E f -Mertens decomposition for strong (r.u.s.c.) E f -supermartingales in the case of a general ltration (cf. Theorem 7.1), we show that the operator Ref f satises the following properties. 1. The operator Ref f is nondecreasing, that is, for ξ, ξ ∈ S 2 such that ξ ≤ ξ we have

Ref f [ξ] ≤ Ref f [ξ ]. 2. If ξ ∈ S 2 is a (r.u.s.c.) strong E f -supermartingale, then Ref f [ξ] = ξ. 3. For each ξ ∈ S 2 , Ref f [ξ] is a strong E f -supermartingale and satises Ref f [ξ] ≥ ξ.
Proof: The rst assertion follows from our comparison theorem for reected BSDEs with irregular obstacles (Theorem 9.1).

Let us prove the second assertion. Let ξ be a (r.u.s.c.) strong E f -supermartingale in S 2 . By denition of Ref f , we have to show that ξ is the solution of the reected BSDE associated with driver f and obstacle ξ. By the E f -Mertens decomposition for strong (r.u.s.c.) E f -supermartingales in the case of a general ltration (Theorem 7.1), together with Lemma

2.1, there exists (Z, k, h, A, C) ∈ IH 2 × IH 2 ν × M 2,⊥ × S 2 × S 2 such that -dξ t = f (t, ξ t , Z t , k t )dt -Z t dW t - E k t (e) Ñ (dt, de) -dh t + dA t + dC t-, 0 ≤ t ≤ T,
where A is predictable right-continuous nondecreasing with A 0 = 0, and C is adapted right-continuous nondecreasing and purely discontinuous, with C 0-= 0. Moreover, the Skorokhod conditions (for RBSDEs) are here trivially satised. Hence, ξ = Ref With the help of the above proposition, we show that the process Ref f [ξ], that is, the rst component of the solution of the RBSDE with (irregular) obstacle ξ, is characterized in terms of the smallest strong E f -supermartingale greater than or equal to ξ. In the case of a right-continuous left-limited obstacle ξ the above characterization has been established in [START_REF] Quenez | Reected BSDEs and robust optimal stopping for dynamic risk measures with jumps[END_REF]; it has been generalized to the case of a right-upper-semicontinuous obstacle in [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF]Prop. 4.4]. Let us note however that the arguments of the proofs given in [START_REF] Quenez | Reected BSDEs and robust optimal stopping for dynamic risk measures with jumps[END_REF] and in [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF] cannot be adapted to our general framework. [START_REF] Dumitrescu | Game options in an imperfect market with default[END_REF]. Innitesimal characterization of the value process in terms of an RBSDE in the completely irregular case. The following theorem is a direct consequence of Theorem 9.2 and Theorem 8.2. It gives "an innitesimal characterization" of the value process (V t ) t∈[0,T ] of the non-linear problem (5.1). Theorem 10.1 (Characterization in terms of an RBSDE) Let (ξ t ) t∈[0,T ] be a process in S 2 and let f be a Lipschitz driver satisfying Assumption 5.1. The value process (V t ) t∈[0,T ] aggregating the family V = (V (S), S ∈ T 0,T ) dened by (5.1) coincides (up to indistinguishability) with the rst component (Y t ) t∈[0,T ] of the solution of our RBSDE with driver f and obstacle ξ. In other words, we have, for all S ∈ T 0,T , (

Y S = V S = ess sup τ ∈T S,T E f S,τ (ξ τ ) a.s. 10.1)
By using this theorem, we derive the following corollary, which generalizes some results of classical optimal stopping theory (more precisely, the assertions (ii) and (iii) from Lemma 3.4) to the case of an optimal stopping problem with (non-linear) f -expectation.

Remark 10.1 Let us summarize our two-part approach to the non-linear optimal stopping problem (5.1) in the case where ξ is completely irregular: First, we have applied a direct approach to the problem (5.1), which consists in showing that the value family (V (S)) S∈T 0,T can be aggregated by an optional process (V t ) t∈[0,T ] and, then, in characterizing (V t ) as the E f -Snell envelope of the (completely irregular) pay-o process (ξ t ). On the other hand, we have applied an RBSDE-approach which consists in establishing some results on RBSDEs with completely irregular obstacles (in particular, existence, uniqueness, and a comparison result) and some useful properties of the operator Ref f , 11 and then in using these properties to show that the unique solution (Y t ) of the RBSDE is equal to the E f -Snell envelope of the completely irregular obstacle. We have then deduced from those two parts (the direct part and the RBSDE-part) that (Y t ) and (V t ) coincide, which gives an innitesimal characterization for the value process (V t ).

Finally, let us put together some of the results for the non-linear optimal stopping problem (5.1): i) • For any reward process ξ ∈ S 2 , we have the innitesimal characterization

V t = Y t = Ref f t [ξ]
, for all t, a.s. (Theorem 10.1).

• Also, (V t ) t∈[0,T ] is the E f -Snell envelope of the pay-o process ξ (Theorem 8.2).

ii) If, moreover, ξ is right-u.s.c. , then, for any S ∈ T 0,T , for any ε > 0, there exists an Lεoptimal stopping time for the problem at time S. (Theorem 6.1). iii) If, moreover, ξ is also left-u.s.c. along stopping times, then, for any S ∈ T 0,T , there exists an optimal stopping time for the problem at time S (Theorem 6.2). 11 We emphasize that the proof of these properties (cf. Proposition 9.1) relies heavily on the E f -Mertens decomposition for strong E f -supermartingales (cf. Theorem 7.1), which is obtained as a direct consequence of the preliminary result (Theorem 6.1) established in the r.u.s.c. case.

11. Applications of Theorem 10.1.

11.1. Application to American options with a completely irregular payo. In the following example, we set E := R, ν(de) := λδ 1 (de), where λ is a positive constant, and where δ 1 denotes the Dirac measure at 1. The process N t := N ([0, t] × {1}) is then a Poisson process with parameter λ, and we have Ñt := Ñ ([0, t] × {1}) = N t -λt.

We assume that the ltration is the natural ltration associated with W and N .

We consider a nancial market which consists of one risk-free asset, whose price process S 0 satises dS 0 t = S 0 t r t dt, and two risky assets with price processes S 1 , S 2 satisfying:

dS 1 t = S 1 t -[µ 1 t dt + σ 1 t dW t + β 1 t d Ñt ]; dS 2 t = S 2 t -[µ 2 t dt + σ 2 t dW t + β 2 t d Ñt ].
We suppose that the processes σ 1 , σ 2 , β 1 , β 2 , r, µ 1 , µ 2 are predictable and bounded, with

β i t > -1 for i = 1, 2. Let µ t := (µ 1 , µ 2
) and let Σ t := (σ t , β t ) be the 2 × 2-matrix with rst column σ t := (σ 1 t , σ 2 t ) and second column β t := (β 1 t , β 2 t ) . We suppose that Σ t is invertible and that the coecients of Σ -1 t are bounded.

We consider an agent who can invest his/her initial wealth x ∈ R in the three assets. For i = 1, 2, we denote by ϕ i t the amount invested in the i th risky asset. A process ϕ = (ϕ 1 , ϕ 2 ) belonging to H 2 × H 2 ν will be called a portfolio strategy. The value of the associated portfolio (or wealth) at time t is denoted by X x,ϕ t (or simply by X t ). In the case of a perfect market, we have

dX t = (r t X t + ϕ 1 t (µ 1 t -r t ) + ϕ 2 t (µ 2 t -r t ))dt + (ϕ 1 t σ 1 t + ϕ 2 t σ 2 t )dW t + (ϕ 1 t β 1 t + ϕ 2 t β 2 t )d Ñt = (r t X t + ϕ t (µ t -r t 1))dt + ϕ t σ t dW t + ϕ t β t d Ñt ,
where 1 = (1, 1) . More generally, we will suppose that there may be some imperfections in the market, taken into account via the nonlinearity of the dynamics of the wealth and encoded in a Lipschitz driver f satisfying Assumption 5.1 (cf. [START_REF] Karoui | Non-linear Pricing Theory and Backward Stochastic Dierential Equations[END_REF] or [START_REF] Dumitrescu | Game options in an imperfect market with default[END_REF] for some examples).

More precisely, we suppose that the wealth process X x,ϕ t (also X t ) satises the forward dierential equation:

(11.1) -dX t = f (t, X t , ϕ t σ t , ϕ t β t )dt -ϕ t σ t dW t -ϕ t β t d Ñt , ; X 0 = x, or, equivalently, setting Z t = ϕ t σ t and k t = ϕ t β t , (11.2) -dX t = f (t, X t , Z t , k t )dt -Z t dW t -k t d Ñt ; X 0 = x. Note that (Z t , k t ) = ϕ t Σ t , which is equivalent to ϕ t = (Z t , k t ) Σ -1 t .
Remark 11.1 Note that the wealth process X x,ϕ is an E f -martingale, since X x,ϕ is the solution of the BSDE with driver f , terminal time T and terminal condition X x,ϕ T .

Let us consider an American option associated with terminal time T and payo given by a process (ξ t ) ∈ S 2 . As is usual in the literature, the option's superhedging price at time 0, denoted by u 0 , is dened as the minimal initial wealth enabling the seller to invest in a portfolio whose value is greater than or equal to the payo of the option at all times. More precisely, for each initial wealth x, we denote by A(x) the set of all portfolio strategies ϕ ∈ H 2 × H 2 ν such that X x,ϕ t ≥ ξ t , for all t ∈ [0, T ] a.s. The superhedging price of the American option is thus dened by (11.3) u 0 := inf{x ∈ R, ∃ϕ ∈ A(x)}. 12Using the innitesimal characterization of the value function (5.1) (cf. Theorem 10.1), we

show the following characterizations of the superhedging price u 0 , as well as the existence of a superhedging strategy.

Proposition 11.1 Let (ξ t ) be an optional process such that E[ess sup τ ∈T 0 |ξ τ | 2 ] < ∞.

(i) The superhedging price u 0 of the American option with payo (ξ t ) is equal to the value function V (0) of our optimal stopping problem (1.1) at time 0, that is (11.4)

u 0 = sup τ ∈T 0,T E f 0,τ (ξ τ ).
(ii) We have u 0 = Y 0 , where (Y, Z, k, h, A, C) is the solution of the reected BSDE (2.3) (with h = 0).

(iii) The portfolio strategy φ, dened by φt = (Z t , k t ) Σ -1 t , is a superhedging strategy, that is, belongs to A(u 0 ).

In the case of a perfect market (for which f is linear) and a regular pay-o, the above result reduces to a well-known result from the literature (cf. [START_REF] Hamadène | Mixed zero-sum stochastic dierential game and American game options[END_REF]). Even in the case of a perfect market, our result for a completely irregular pay-o is new.

Proof: The proof relies on Theorem 10.1 and similar arguments to those in [START_REF] Dumitrescu | Game options in an imperfect market with default[END_REF] (in the case of game options with RCLL payos and default). Note rst that, by Theorem 10.1, we have sup τ ∈T 0,T E f 0,τ (ξ τ ) = Y 0 . In order to prove the three rst assertions of the above theorem, it is thus sucient to show that u 0 = Y 0 and φ ∈ A(Y 0 ).

We rst show that φ ∈ A(Y 0 ). By (11.2), the value X Y 0 , φ of the portfolio associated with initial wealth Y 0 and strategy φ satises: We now show that Y 0 = u 0 . Since φ ∈ A(Y 0 ), by denition of u 0 (cf. (11.3)), we derive that Y 0 ≥ u 0 . Let us now show that u 0 ≥ Y 0 . Let x ∈ R be such that there exists a strategy ϕ ∈ A(x). We show that x ≥ Y 0 . Since ϕ ∈ A(x), we have X x,ϕ t ≥ ξ t , for all t ∈ [0, T ] a.s. For each τ ∈ T we thus get the inequality X x,ϕ τ ≥ ξ τ a.s. By the non decreasing property of E f together with the E f -martingale property of X x,ϕ (cf. Remark 11.1), we thus get x = E f 0,τ (X x,ϕ τ ) ≥ E f 0,τ (ξ τ ). By taking the supremum over τ ∈ T 0,T , we derive that x ≥ sup τ ∈T 0,T E f 0,τ (ξ τ ) = Y 0 , where the equality holds by Theorem 10.1. By denition of u 0 as an inmum (cf (11.3)), we get u 0 ≥ Y 0 , which, since Y 0 ≥ u 0 , yields that u 0 = Y 0 .

dX Y 0 , φ t = -f (t, X Y 0 , φ t , Z t , k t )dt + dM t , with initial condition X Y 0 , φ 0 = Y 0 ,
We now give some examples of American options with completely irregular pay-o.

Example 11.1 We consider a pay-o process (ξ t ) of the form ξ t := h(S 1 t ), for t ∈ [0, T ], where h : R → R is a (possibly irregular) Borel function such that the process (h(S t )) is optional and (h(S 1 t )) ∈ S 2 . In general, the pay-o (ξ t ) is a completely irregular process. By the rst two statements of Proposition 11.1, the superhedging price of the American option is equal to the value function of the optimal stopping problem (11.4), and is also characterized as the solution of the reected BSDE (2.3) with obstacle ξ t = h(S 1 t ).

If h is an uppersemicontinuous function on R, then the process (h(S 1 t )) is optional, since an u.s.c. function can be written as the limit of a (non increasing) sequence of continuous functions. Moreover, the process (h(S 1 t )) is right-u.s.c. and also left-u.s.c. along stopping times. The right-uppersemicontinuity of (ξ t ) follows from the fact that the process S 1 is rightcontinuous; the left-uppersemicontinuity along stopping times of (ξ t ) follows from the fact that S 1 jumps only at totally inaccessible stopping times. In virtue of Proposition 11.1, last statement, there exists in this case an optimal exercise time for the American option with payo ξ t = h(S 1 t ). A particular example is given by the American digital call option (with strike K > 0), where h(x) := 1 [K,+∞[ (x). The function h is u.s.c. on R. The corresponding payo process ξ t := 1 S 1 t ≥K is thus r.u.s.c and left-u.s.c. along stopping times in this case, which implies the existence of an optimal exercise time.

In the case of the American digital put option (with strike K > 0), the corresponding payo ξ t := 1 S 1 t <K is not r.u.s.c. We note that the pay-o of the American digital call and put options is in general neither left-limited nor right-limited. 11.2. An application to RBSDEs. The characterization (Theorem 10.1) is also useful in the theory of RBSDEs in itself: it allows us to obtain a priori estimates with universal constants for RBSDEs with completely irregular obstacles. Proposition 11.2 (A priori estimates with universal constants) Let ξ and ξ be two processes in S 2 . Let f and f be two Lipschitz drivers satisfying Assumption 5.1 with common Lipschitz constant K > 0. Let (Y, Z, k) (resp. (Y , Z , k )) be the three rst components of the solution of the reected BSDE associated with driver f (resp. f ) and obstacle ξ (resp. ξ ). Let Y := Y -Y , ξ := ξ -ξ , and

δf s := f (s, Y s , Z s , k s ) -f (s, Y s , Z s , k s ). Let η, β > 0 with β ≥ 3 η + 2K and η ≤ 1 K 2 .
For each S ∈ T 0,T , we have Proof: The proof is divided into two steps.

Step 1: For each τ ∈ T 0,T , let (X τ , π τ , l τ ) (resp. (X τ , π τ , l τ )) be the solution of the BSDE associated with driver f (resp. f ), terminal time τ and terminal condition ξ τ (resp. ξ τ ). Set X τ := X τ -X τ . By an estimate on BSDEs (cf. Proposition A.4 in [START_REF] Quenez | BSDEs with jumps, optimization and applications to dynamic risk measures[END_REF]), we have Step 2: Note that (Y , Z , k ) is the solution the RBSDE associated with obstacle ξ and driver f (t, y, z, k) + δf t . By applying the result of Step 1 to the driver f (t, y, z, k) and the driver f (t, y, z, k) + δf t (instead of f ), we get the desired result.

12. Appendix. Let M, M ∈ M 2 . Recall that M M -[M, M ] is a martingale, and that M, M is dened as the compensator of the integrable nite variation process [M, M ].

Using these properties we derive the following equivalent statements (cf., e.g., [START_REF] Protter | Stochastic Integration and Dierential Equations (Stochastic Modelling and Applied Probability[END_REF] IV.3 for details): 13 For the convenience of the reader, we state the following equivalences, which, to our knowledge, are not explicitly specied in the literature. Lemma 12.1 For each h ∈ M 2 , the following properties are equivalent:

M, M t = 0, 0 ≤ t ≤ T a.s. ⇔ [M, M ] • is a martingale ⇔ M M is a martingale.
(i) For all predictable process l ∈ IH 2 ν , we have h , • 0 l s (e) Ñ (dsde) t = 0, 0 ≤ t ≤ T a.s. (ii) For all predictable process l ∈ IH 2 ν , we have ( h , • 0 E l s (e) Ñ (dsde) ) M 2 = 0.

(iii) M P N (∆h • | P) = 0, where M P N ( . | P) is the conditional expectation given P := P ⊗ E under the Doleans' measure M P N associated to probability P and random measure N . 14 Proof: Let us show that (i) ⇔ (ii). By denition of the scalar product (•, •) M 2 , we have

( h , • 0 E l s (e) Ñ (dsde) ) M 2 = E[ h ,
• 0 E l s (e) Ñ (dsde) T ]. Hence, (i) ⇒ (ii). Let us show that (ii) ⇒ (i). If for all l ∈ IH 2 ν , E[ h , • 0 l s (e) Ñ (dsde) T ] = 0, then, for each bounded predictable process ϕ ∈ IH 2 , we have

E[ T 0 ϕ t d h, • 0 E l s (e) Ñ (dsde) t ] = E[ h , • 0 E ϕ s l s (e) Ñ (dsde) T ] = 0.
since, for each M ∈ M 2 , ϕ • h, M = h, ϕ.M (using the notation of [START_REF] Dellacherie | Probabilités et Potentiel, Théorie des Martingales[END_REF] or [START_REF] Jacod | Calcul Stochastique et Problèmes de martingales[END_REF]). By [START_REF] Dellacherie | Probabilités et Potentiel, Théorie des Martingales[END_REF] (Chap 6 II Th. 64 p141), this implies that the integrable-variation predictable process A • := h , • 0 l s (e) Ñ (dsde) • is equal to 0, which gives that (ii) ⇒ (i). Hence (i) ⇔ (ii). It remains to show that (ii) ⇔ (iii). Note rst that ( h , • Proof of Lemma 3.7: Let β > 0 and ε > 0 be such that β ≥ 1 ε 2 . We note that ỸT = ξ T -ξ T = 0; moreover, we have -d Ỹt = f (t)dt+d Ãt +d Ct--Zt dW t -E kt (e) Ñ (dt, de)-d ht .

Thus we see that Ỹ is an optional strong semimartingale in the vocabulary of [START_REF] Gal'chouk | Optional martingales[END_REF] with decomposition Ỹ = Ỹ0 +M +A+B, where M t := t 0 Zs dW s + t 0 E ks (e) Ñ (ds, de)+ ht , A t := -t 0 f (s)ds -Ãt and B t := -Ct-. Applying Gal'chouk-Lenglart's formula (more precisely Corollary A.2 in [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF]) to e βt Ỹ 2 t , and using that ỸT = 0, and the property h c , W = 0, we 13 In this case, using he terminology of [START_REF] Protter | Stochastic Integration and Dierential Equations (Stochastic Modelling and Applied Probability[END_REF] IV.3, the martingales M and M are said to be strongly orthogonal.

Note also that, if M, M ∈ M 2 , using the terminology of [START_REF] Protter | Stochastic Integration and Dierential Equations (Stochastic Modelling and Applied Probability[END_REF] IV.3, the martingales M and M are said to be weakly orthogonal if ( M , M ) M 2 = 0, that is E[MT M T ] = 0. e βs Ỹsd hs .

By the same arguments as in [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF] (cf. the proof of Lemma 3.2 in [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF] for details), since β ≥ 1 ε 2 , we obtain the following estimate for the sum of the rst and the second term on the r.h.s. of equality (12.1): -]t,T ] β e βs ( Ỹs ) e βs Ỹs-

dA 1 s = ]t,T ] e βs (Y 1 s--Y 2 s-)dA 1 s = ]t,T ] e βs (Y 1 s--ξ s )dA 1 s + ]t,T ] e βs (ξ s -Y 2 s-)dA
E[∆ Ãτ ∆ hτ /F τ -] = ∆ Ãτ E[∆ hτ /F τ -] = 0. We thus get E[ 0<s≤T e βs ∆ Ãs ∆ hs ] = 0.
By applying (12.4) with t = 0, and by taking expectations on both sides of the resulting inequality, we obtain Ỹ

2 0 + Z 2 β + k 2 ν,β + h 2 β,M 2 ≤ ε 2 f 2 β . We deduce that Z 2 β ≤ ε 2 f 2 β , k 2 ν,β ≤ ε 2 f 2 β and h 2 β,M 2 ≤ ε 2 f 2 β ,
] ≤ 1 4 ||| Ỹ ||| 2 β +4c 2 Z 2 β ,
and a similar estimate for the last term in (12.5). By (12.5), we thus have

1 4 ||| Ỹ ||| 2 β ≤ ε 2 f 2 β +4c 2 ( Z 2 β + k 2 ν,β + h 2 β,M 2 ).
= 0. Let (Y, Z, k, h) ∈ S 2 × H 2 × H 2 ν × M 2,⊥ satisfy -dY t = f (t, Y t , Z t , k t )dt + dA t + dC t--Z t dW t - E k t (e) Ñ (dt, de) -dh t , 0 ≤ t ≤ T.
Then the process (Y t ) is a strong E f -supermartingale.

The proof is omitted since it relies on the same arguments as those used in the proof of the same result shown in [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF] in the particular case when the ltration is associated with W and N (cf. Proposition A.5 in [START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF]), as well as on some specic arguments, due to the general ltration, which are similar to those used in the proof of the previous lemma.

13. Complements: The strict value. In this section we give some complements on a closely related (non-linear) optimal stopping problem.

Let S be a stopping time in T 0,T . We denote by T S + the set of stopping times τ ∈ T 0,T with τ > S a.s. on {S < T } and τ = T a.s. on {S = T }. The strict value V + (S) (at time S) of the non-linear optimal stopping problem is dened by

V + (S) := ess sup τ ∈T S + E f S,τ (ξ τ ).

(13.1)

We note that V + (S) = ξ T a.s. on {S = T }.

Using the same arguments as for the value family (V (S)) S∈T 0,T , we show that Proposition 13.1 The strict value family (V + (S)) S∈T 0,T is a strong E f -supermartingale family. There exists a unique right-uppersemicontinuous optional process, denoted by (V + t ) t∈[0,T ] , which aggregates the family (V + (S)) S∈T 0,T . The process (V + t ) t∈[0,T ] is a strong E f -supermartingale.

The following theorem connects the above strict value process (V + t ) t∈[0,T ] with the process of right-limits (V t+ ) t∈[0,T ] , where (V t ) denotes as before the value process of our non-linear problem (5.1). Theorem 13.1 (i) The strict value process (V + t ) is right-continuous. (ii) For all S ∈ T 0,T , V + S = V S+ a.s. (iii) For all S ∈ T 0,T , V S = V + S ∨ ξ S a.s.

The proof of the theorem uses the following preliminary result which states that the strict value process (V + t ) is right-continuous along stopping times in E f -conditional expectation.

Lemma 13.1 (Right-continuity along stopping times in E f -conditional expectation)

The strict value process (V + t ) is right-continuous along stopping times in E f -expectation, in the sense that for each θ ∈ T 0,T , and for each sequence of stopping times (θ n ) n∈N belonging to T 0,T such that θ n ↓ θ, we have For the proof, we recall the following classical statement:

Remark 13.1 Let (Ω, F, P ) be a probability space. Let A ∈ F. Let (X n ) be a sequence of real valued random variables. Suppose that (X n ) converges a.s. on A to a random variable X. Then, for each ε > 0, lim n→+∞ P ({|X -X n | < ε} ∩ A) = P (A).

From this property, it follows that for each ε > 0, there exists n 0 ∈ N such that for all n ≥ n 0 , P ({|X -X n | < ε} ∩ A) ≥ P (A) 2 .

Proof of Lemma 13.1: Let n ∈ N. Let us rst consider the simpler case where θ < T a.s. In this case, since θ ∈ T θ + , we have θ > θ a.s. Hence, we have Ω = ∪ n∈N ↑ {θ > θ n } a.s. Dene the stopping time θ n := θ 1 {θ >θn} + T 1 {θ ≤θn} . We note that θ n ∈ T θ + n for each n ∈ N. Moreover, lim n→∞ θ n = θ a.s. and lim n→∞ ξ θn = ξ θ a.s. By the continuity property of E f with respect to terminal condition and terminal time, we get lim n→∞ E f θ,θn (ξ θn ) = E f θ,θ (ξ θ ) a.s. Let us now consider a general θ ∈ T 0,T . On the set {θ = T }, we have θ n = θ a.s. for all n. Hence, on {θ = T }, we have lim n→∞ E f θ,θn (V + θn ) = V + θ a.s. On the set {θ < T }, using the same arguments as above with θ n = θ 1 {θ >θn}∩{T >θ} + T 1 {θ ≤θn}∪{T =θ} , we show the inequality (13.4). The proof is thus complete.

We are now ready to prove the theorem.

Proof of Theorem 13.1: The proof of (i) is based on the previous Lemma 13.1 and on a result from the general theory of processes. Let S ∈ T 0,T and let (S n ) be a non-increasing sequence of stopping times in T S+ with lim ↓ S n = S a.s. By applying Lemma 13.1 and the continuity property of E f -expectations with respect to the terminal condition and to the terminal time, we get

V + S = lim n→∞ E f S,Sn (V + Sn ) = E f S,S ( lim n→∞ V + Sn ) = lim n→∞ V + Sn ,
where we have used that lim n→∞ V + Sn exists, as (V + t ) is a strong E f -supermartingale, and hence has right limits. The above equality shows that the process (V + t ) is right-continuous along stopping times. By Proposition 2 in [START_REF] Dellacherie | Sur des problèmes de régularisation, de recollement et d'interpolation en théorie des processus[END_REF], we conclude that (V + t ) is right-continuous. We now show (ii). Let S ∈ T 0,T . Let (S n ) be a non-increasing sequence of stopping times in T S + with lim ↓ S n = S a.s. We know that V τ ≥ V + τ a.s., for all τ ∈ T 0,T . Hence, V Sn ≥ V + Sn a.s., for all n. We derive that lim n→∞ V Sn ≥ lim n→∞ V + Sn a.s. Using this and the right-continuity of V + established in (i), gives V S+ ≥ V + S a.s. In order to show the converse inequality, we rst show (13.8)

E f S,S n (V Sn ) ≤ V + S a.s. for all n.

We x n and we take (τ p ) ∈ T Sn an optimizing sequence for the problem with value V Sn , i.e. V Sn = lim p→∞ E f Sn,τp (ξ τp ). We have (13.9)

E f S,Sn (V Sn ) = E f S,Sn ( lim p→∞ E f Sn,τp (ξ τp )) = lim p→∞ E f S,Sn (E f Sn,τp (ξ τp )) a.s., where we have used the continuity property of E f S,S n (•) with respect to the terminal condition (recall that here n is xed). Using the consistency property of E f -expectations, we get E f S,Sn (E f Sn,τp (ξ τp )) = E f S,τp (ξ τp ) ≤ V + S a.s. (where for the inequality we have used that τ p ∈ T S + ). From this, together with equation (13.9), we derive the desired inequality (13.8). From inequality (13.8), together with the continuity of E f -expectations with respect to the terminal time and the terminal condition, we derive V + S ≥ lim n→∞ E f S,S n (V Sn ) = E f S,S (V S+ ) = V S+ a.s. Hence, V + S ≥ V S+ a.s., which, together with the previously shown converse inequality, proves the equality V S+ = V + S a.s. Statement (iii) is a direct consequence of part (ii) (which we have just shown), together with Remark 2.3 and Theorem 10.1.

Remark 13.2 By the same arguments as those of the proof of statement (i) in the above Theorem 13.1, the following general statement can be shown: A strong E f -supermartingale is right-continuous if it is right-continuous along stopping times in E f -conditional expectation.

14. Aclnowledgements. The authors are very grateful to Klébert Kentia for his helpful remarks. The authors are also indebted to Sigurd Assing for his helpful comments, and to Marek Rutkowski and Tianyang Nie for useful discussions.

  dene the value v(S) at time S by v(S) := ess sup τ ∈T S,T E[ξ τ | F S ].

  which shows the stability under pairwise maximization and concludes the proof. Denition 8.2 (E f -supermartingale family) An admissible square-integrable family U := (U (S), S ∈ T 0,T ) is said to be an E f -supermartingale family if for all S, S ∈ T 0,T such that S ≤ S a.s., E f S,S (U (S )) ≤ U (S) a.s. Denition 8.3 (Right-uppersemicontinuous family) An admissible family U := (U (S), S ∈

Proposition 9 . 1 (

 91 Properties of the operator Ref f ) Let f be a Lipschitz driver satisfying Assumption 5.1. The operator Ref f : S 2 → S 2,rusc , dened in Denition 9.1, has the following properties:

  f [ξ], which is the desired conclusion. It remains to show the third assertion. By denition, the process Ref f [ξ] is equal to Y , where (Y, Z, k, h, A, C) is the solution our reected BSDE. Hence, Ref f [ξ] = Y admits the decomposition (7.1), which, by Theorem 7.1, implies that Ref f [ξ] = Y is a strong E f -supermartingale. Moreover, by denition, Ref f [ξ] = Y is greater than or equal to the obstacle ξ.

Theorem 9 . 2 (

 92 The operator Ref f and the E f -Snell envelope operator) Let ξ be a process in S 2 and let f be a Lipschitz driver satisfying Assumption 5.1. The rst component Y = Ref f [ξ] of the solution to the reected BSDE with parameters (ξ, f ) coincides with the E f -Snell envelope of ξ, that is, the smallest strong E f -supermartingale greater than or equal to ξ. Proof: By the third assertion of Proposition 9.1, the process Y = Ref f [ξ] is a strong E f -supermartingale satisfying Y ≥ ξ. It remains to show the minimality property. Let Y be a strong E f -supermartingale such that Y ≥ ξ. We have Ref f [Y ] ≥ Ref f [ξ], due to the nondecreasingness of the operator Ref f (cf. Proposition 9.1, 1st assertion). On the other hand, Ref f [Y ] = Y (due to Proposition 9.1, 2nd assertion) and Ref f [ξ] = Y . Hence, Y ≥ Y , which is the desired conclusion.

2 ≤

 2 e β(T -S) E[ess sup τ ∈T S,T ξ τ 2 |F S ] + ηE[ T S e β(s-S) (δf s ) 2 ds|F S ] a.s.

S ) 2 ≤S ) 2 ≤

 22 e β(T -S) E[ξ 2 | F S ] + ηE[ T S e β(s-S) [(f -f )(s, X τ s , π τ s , l τ s )] 2 ds | F S ] a.s. e β(T -S) E[ess sup τ ∈T S,T ξ τ 2 |F S ] + ηE[ T S e β(s-S) (f s ) 2 ds|F S ] a.s., where f s := sup y,z,k |f (s, y, z, k) -f (s, y, z, k)|. Now, by Theorem 10.1, we have Y S = ess sup τ ∈T S,T X τ S and Y S = ess sup τ ∈T S,T X τ S . We thus get |Y S | ≤ ess sup τ ∈T S,T |X τ S |. By (11.6), we derive the inequality (11.5) with δf s replaced by f s .

0

  E l s (e) Ñ (dsde) ) M 2 = E([ h , • 0 E l s (e) Ñ (dsde) ] T ) = E( [0,T ]×E ∆h s l s (e)N (dsde)) = M P N (∆h • l • ). Property (ii) can thus be written as M P N (∆h • l • ) = 0 for all l • ∈ IH 2 ν , which means that M P N (∆h • | P) = 0. Hence, (ii) ⇔ (iii).

  θ,θn (V + θn ) = V + θ a.s.

  We denote by ∆ + φ t := φ t + -φ t the size of the right jump of φ at t, and by ∆φ t := φ t -φ t- the size of the left jump of φ at t. Remark 2.2 In the particular case where ξ has left limits, we can replace the process (ξ t ) by the process of left limits (ξ t-) in the Skorokhod condition(2.4).Remark 2.3 If(Y, Z, k, h, A, C) is a solution to the RBSDE dened above, by(2.3), we have∆C t = Y t -Y t+ , which implies that Y t ≥ Y t+ , for all t ∈ [0, T ). Hence, Y is r.u.s.c. Moreover, from C τ -C τ -= -(Y τ + -Y τ ), combined with the Skorokhod condition (2.5), we derive (Y τ -ξ τ )(Y τ + -Y τ ) = 0,a.s. for all τ ∈ T 0,T . This, together with Y τ ≥ ξ τ and Y τ ≥ Y τ + a.s., leads to Y τ = Y τ + ∨ ξ τ a.s. for all τ ∈ T 0,T . Denition 2.4 Let τ ∈ T 0 . An optional process (φ t ) is said to be right upper-semicontinuous

t.

(resp. left upper-semicontinuous) along stopping times if for all stopping time τ ∈ T 0 and for all non increasing (resp. non decreasing) sequence of stopping times (τ n ) such that τ n ↓ τ (resp. τ n ↑ τ ) a.s. , φ τ ≥ lim sup n→∞ φ τn a.s..

  6. Optimal stopping with non-linear f -expectation: the right u.s.c. case. Let f be a Lipschitz driver satisfying Assumption 5.1. The following result relies crucially on an assumption of right-uppersemicontinuity of ξ. Lemma 6.1 Let ξ be a process in S 2 , supposed to be right u.s.c. Let (Y, Z, k, h, A, C) be the solution to the reected BSDE with parameters (f, ξ) as in Denition 2.3. Let S ∈ T 0,T and let ε > 0. Let τ ε S be the stopping time dened by (4.1), that is, τ ε

S := inf{t ≥ S , Y t ≤ ξ t + ε}.

  . Moreover, τS = τ 0 S a.s.

	Proof: As (ξ t ) is l.u.s.c. along stopping times, we have
	(6.5)	lim sup n→∞	ξ τ εn S	≤ ξ τS a.s.
	By applying Fatou's lemma for (non-reected) BSDEs (cf. Lemma A.5 in [11]	7 ), we obtain
	(6.6)

  , Ch. IV]. 9.2. Comparison theorem. Theorem 9.1 (Comparison) Let ξ ∈ S 2 , ξ ∈ S 2 be two processes. Let f and f be Lipschitz drivers satisfying Assumption 5.1. Let

  <s≤θ ∆Γ s ∆ Ȳ + s . Let (p s ) be the point process associated with the Poisson random measure N (cf. [8, VIII Section 2. 67], or [24, Section III d]). We have ∆Γ s = Γ s-γ s (p s ) and ∆ Ȳ + s = 1 { Ȳs->0} ks (p s ) -1 { Ȳs->0} ∆ Ās + ∆K d,-

	Hence,
	(9.7)
	τ <s≤θ

s + 1 { Ȳs->0} ∆ hs .

  >0} (1 + γ s (p s ))∆ Ās is nonpositive, due to 1 + γ s ≥ 0, to the Skorokhod condition for ∆A s and to ∆A s ≥ 0 (the details are similar to those for d C in the reasoning above). Since h ∈ M 2,⊥ , by Remark 2.1, we derive that the expectation of the last term of the above inequality (9.8) is equal to 0. Moreover, the term

1 { Ȳs->0} ϕ s ds, due to the inequality (9.4). The termτ <s≤θ Γ s-(1+γ s (p s ))∆K d,- s is nonpositive, as 1 + γ s ≥ 0 by Assumption 5.1. The term τ <s≤θ Γ s-1 { Ȳs-

  9.3. Non-linear operator induced by an RBSDE. Snell characterization. We introduce the non-linear operator Ref f (associated with a given non-linear driver f ) and provide some useful properties. In particular, we show that this non-linear operator coincides with the E f -Snell envelope operator (cf. Theorem 9.2). Denition 9.1 (Non-linear operator Ref f ) Let f be a Lipschitz driver. For a process (ξ t ) ∈ S 2 , we denote by Ref f [ξ] the rst component of the solution to the Reected BSDE with (lower) barrier ξ and with Lipschitz driver f .The operator Ref f [•] is well-dened due to Theorem 4.1. Moreover, Ref f [•] is valued in S 2,rusc , where S 2,rusc := {φ ∈ S 2 : φ is r.u.s.c.} (cf. Remark 2.3). In the following proposition we give some properties of the operator Ref f . Note that equalities (resp. inequalities)

  where M t := t 0 Z s dW s + t 0 k s d Ñs . Moreover, since Y is the solution of the reected BSDE (2.3) (with h = 0), we have dY t = -f (t, Y t , Z t , k t )dt + dM t -dA t -dC t-. Applying the comparison result for forward dierential equations, we derive that X Y 0 , φ t ≥ Y t , for all t ∈ [0, T ] a.s. Since Y t ≥ ξ t , we thus get X Y 0 , φ t ≥ ξ t for all t ∈ [0, T ] a.s. It follows that φ ∈ A(Y 0 ).

  2 ds + 2 ]t,T ] e βs Ỹs f (s)ds ≤ ε 2 ]t,T ] e βs f 2 (s)ds. We also have that [t,T [ e βsỸs d Cs ≤ 0 and ]t,T ] e βs Ỹsd Ãs ≤ 0. We give the detailed arguments for the second inequality (the arguments for the rst are similar). We have

]t,T ] e βs

Ỹsd Ãs = ]t,T ] e βs Ỹs-dA 1 s -]t,T ] e βs Ỹs-dA 2 s . For the rst term, we write ]t,T ]

  The rst summand is equal to 0 due to the Skorokhod condition for A 1 . Hence, ]t,T ] e βs Ỹs-dA 1 s ≤ 0. By similar arguments, we see that -]t,T ] e βs Ỹs-dA 2 , we have used the fact that the processes A • and N (•, de) "do not have jumps in common", since A (resp. N (•, de)) jumps only at predictable (resp. totally inaccessible) stopping times.By adding the term ]t,T ] e βs || ks || 2 ν ds + t<s≤T e βs (∆ hs ) 2 on both sides of inequality (12.3), by using the above computation and the well-known equality [ h] t = hc t + (∆ h) 2 (where M is given by (12.2)).By classical arguments, which use Burkholder-Davis-Gundy inequalities, we can show that the local martingale M is a martingale. Moreover, since h ∈ M 2,⊥ , by Remark 2.1, we derive that the expectation of the last term of the above inequality (12.4) is equal to 0. Furthermore, since h is a martingale, for each predictable stopping time τ , we have E[∆ hτ /F τ -] = 0 (cf., e.g., Chapter I, Lemma (1.21) in[START_REF] Jacod | Calcul Stochastique et Problèmes de martingales[END_REF]). Moreover, since à is pre-

								s , we
	get
	(12.4)
	e βt Ỹ 2 t +	e βs Z2 s ds +		e βs || ks || 2 ν ds +	e βs d[ h] s ≤ ε 2	e βs f 2 (s)ds -(M T -M t )
	]t,T ]			]t,T ]		]t,T ]	]t,T ]
		-2	e βs ∆ Ãs ∆ hs -2	T	d[ h ,	•	e βs	ks (e) Ñ (ds, de) ] s ,
				t<s≤T			t	0 E
	with M t = Mt + ]t,T ] e βs	E	k2 s (e) Ñ (ds, de)
	(12.3)
	e βt Ỹ 2 t +	e βs Z2 s ds +		e βs d hc	s ≤ ε 2
	]t,T ]			]t,T ]		]t,T ]
	from which we derive estimates for	Z 2 β , k 2 ν,β , h 2 β,M 2 , and then an estimate for ||| Ỹ ||| 2 β .
	Estimate for Z 2 β , k 2 ν,β and h 2 β,M 2 . Note rst that we have:
	e βs (∆ hs ) 2 +	e βs || ks || 2 ν ds -		e βs (∆ Ãs ) 2
	t<s≤T	]t,T ]						t<s≤T
		-	e βs	k2 s (e) Ñ (ds, de) -2
		]t,T ]			E

1 s . The second summand is nonpositive as Y 2 s-≥ ξ s (which is due to Y 2 s ≥ ξ s , for all s). s ≤ 0. Hence, ]t,T ] e βs Ỹsd Ãs ≤ 0. The above observations, together with equation (12.1), yield that a.s., for all t ∈ [0, T ], e βs f 2 (s)ds -( MT -Mt ) -t<s≤T e βs (∆ Ỹs ) 2 , t<s≤T e βs (∆ Ỹs ) 2 = -t<s≤T e βs ∆ Ãs ∆ hs -2 t<s≤T e βs ks (p s )∆ hs , wheredictable, ∆ Ãτ is F τ --measurable (cf., e.g., Chap I (1.40)-(1.42) in [24]), which implies that

  which are the desired estimates(3.7).Let us consider the third term of the r.h.s. of the inequality(12.5). By Burkholder-Davis-Gundy inequalities, we have E[ess sup τ ∈T 0,T | By using similar arguments, we get 2E[ess sup τ ∈T 0,T

																τ 0 e βs	Ỹs-Zs dW s
	Estimate for ||| Ỹ |||
	Using rst Chasles' relation for stochastic integrals, then taking the essential supremum
	over τ ∈ T 0,T and the expectation on both sides of the above inequality, we obtain
	(12.5) E[ess sup τ ∈T 0,T	e βτ Ỹ 2 τ ] ≤ ε 2 f 2 β + 2E[ess sup τ ∈T 0,T	|	0	τ	e βs	τ ∈T 0,T Ỹs-Zs dW s |] + 2E[ess sup	|	0	τ	e βs	Ỹs-d hs |]
							+ 2E[ess sup	|			e βs
								τ ∈T 0,T		]0,τ ]
													τ 0 e βs	Ỹs-d hs |] ≤ cE[	T 0 e 2βs Ỹ 2 s-d[ h] s ]. This
	inequality and the trivial inequality ab ≤ 1 2 a 2 + 1 2 b 2 lead to
	2E[ess sup τ ∈T 0,T	|	0	τ	e βs	Ỹs-d hs |] ≤ E	  1 2	ess sup τ ∈T 0,T	e βτ Ỹ 2 τ	8c 2	0	T	e βs d[ h] s	  ≤	1 4	||| Ỹ ||| 2 β +4c 2 h 2 β,M 2 .

2

β . From inequality (12.3) we derive that, for all τ ∈ T 0,T , a.s., e βτ Ỹ 2 τ ≤ ε 2 ]τ,T ] e βs f 2 (s)ds -( MT -Mτ ), where M is given by (12.2).

E

Ỹs-ks (e) Ñ (ds, de)|].

  We note that this proof shows that the estimates(3.7) and(3.8) also hold in the simpler case of a non reected BSDE. From this result, together with Lemma 2.1, and using the same arguments as in the proof of Theorem 4.1, we easily derive the existence and the uniqueness of the solution of the non reected BSDE with general ltration from Denition 2.2. Similarly, we can show the comparison result for non reected BSDEs with general ltration under the Assumption 5.1. Lemma 12.2 Let f be a Lipschitz driver satisfying Assumption 5.1. Let A be a nondecreasing right-continuous predictable process in S 2 with A 0 = 0 and let C be a nondecreasing right-continuous adapted purely discontinuous process in S 2 with C 0-

	Using the estimates for Z 2 β , k 2 ν,β and h 2 β,M 2 (cf.
	(3.7)), we thus get ||| Ỹ ||| 2 β ≤ 4ε 2 (1 + 12c 2 ) f 2 β , which is the desired result.
	Remark 12.1

  By the consistency property of E f , we have Now, since the process (V+ t ) is a strong E f -supermartingale, we have E f θ n+1 ,θn (V + θn ) ≤ V + θ n+1a.s. Using this inequality, together with equality (13.3) and the monotonicity of E f Since this inequality holds for each n ∈ N, we derive that the sequence of random variablesE f θ,θn (V + θn ) n∈N is nondecreasing. Moreover, since the process (V + t ) is a strong E f -supermartingale, we have E f θ,θn (V + θn ) ≤ V + θ a.s. for each n ∈ N. By taking the limit as n tend to +∞, we thus get lim Suppose, by way of contradiction, that this inequality does not hold. Then, there exists a constant α > 0 such that the event A dened byAs for the value function, there exists an optimizing sequence (τ p ) p∈N for the strict value function V + θ , that is, such that, for each p ∈ N, τ p ∈ T θ + , and such that By Remark 13.1 (applied with ε = α 2 ), we derive that there exists p 0 ∈ N such that the event B dened byB := {V + θ ≤ E f

	It remains to show the converse inequality:
	(13.4)		lim n→∞	↑ E f θ,θn (V + θn ) ≥ V + θ	a.s.
		A := { lim n→∞	↑ E f θ,θn (V + θn ) ≤ V + θ -α}
	satises P (A) > 0. By denition of A, we have
	(13.5)	lim n→∞	↑ E f θ,θn (V + θn ) + α ≤ V + θ	a.s. on A.
			V + θ = lim
					θ,τp 0	(ξ τp 0 ) +	α 2	} ∩ A
		V + θ ≤ E f θ,θ (ξ θ ) +	α 2	a.s. on B.
	(13.3) By the inequality (13.5), we derive that E f θ,θn (V + θn ) = E f θ,θ n+1 E f θ n+1 ,θn (V + θn ) (13.6) lim n→∞ ↑ E f θ,θn (V + θn ) + α 2 ≤ E f θ,θ (ξ	a.s.
					θ,θ n+1	, we
	obtain
		E f θ,θn (V + θn ) ≤ E f θ,θ n+1 (V + θ n+1 ) a.s.
			n→∞	↑ E f θ,θn (V + θn ) ≤ V + θ	a.s.

p→∞ ↑ E f θ,τp (ξ τp ) a.s.

satises P (B) ≥ P (A) 2 . Denoting τ p 0 by θ , we have θ ) a.s. on B.

  By Remark 13.1, we derive that there exists n 0 ∈ N such that the event C dened by By the inequality (13.6), we derive that (13.7) the last inequality follows from the fact that θ n 0 ∈ T θ +

		lim n→∞	↑ E f θ,θn (V + θn ) +	α 4	≤ E f θ,θn 0	(ξ θn 0	) a.s. on C.
	Now, by the consistency of E f , we have
		E f θ,θn 0	(ξ θn 0	) = E f θ,θn 0	E f θn 0 ,θn 0	(ξ θn 0	) ≤ E f θ,θn 0	(V + θn 0	) a.s.,
	V + θn 0	. By (13.7), we thus derive that				n 0	and from the denition of
		lim n→∞	↑ E f θ,θn (V + θn ) +	α 4	≤ E f θ,θn 0	(V + θn 0	) a.s. on C,
			C := {|E f θ,θ (ξ θ ) -E f θ,θn 0	(ξ θn 0	)| ≤	α 4	} ∩ B

satises P (C) > 0.

where which gives a contradiction. Hence, the desired inequality (13.4) holds.

We note that the proof of Lemma 4.1 (statement (ii)) in[START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF] does not require the assumption of r.u.s.c. of ξ.

In other words, the process (Yt) aggregates the value family (V (S), S ∈ T0) dened by(5.1), that is YS = V (S) a.s. for all S ∈ T0,T .

Let us emphasize that this optimality criterion holds true without an assumption of rightupppersemicontinuity of the process ξ.

Note that Fatou's lemma for (non-reected) BSDEs, shown in[START_REF] Dumitrescu | Mixed generalized Dynkin game and stochastic control in a Markovian framework[END_REF] in the case of a Brownian-Poisson ltration, still holds true in our framework of a general ltration.

Note that Proposition B.10 in[START_REF] Kobylanski | Optimal stopping time problem in a general framework[END_REF] also holds true in the case where the reward process is not necessarily nonnegative.

An E f -Mertens decomposition was also shown in[START_REF] Bouchard | A general Doob-Meyer-Mertens decomposition for g-supermartingale system[END_REF] (at the same time as in[START_REF] Grigorova | Reected BSDEs when the obstacle is not right-continuous and optimal stopping[END_REF]) in the case of a driver f (t, y, z) which does not depend on k by using a dierent approach.

This model includes the case of a perfect market, for which f is a linear driver given by f (t, y, z, k) = -r t y -(z, k) Σ -1 t (µ t -r t 1).

As shown in assertion (iii) of Proposition 11.1, the inmum in (11.3) is always attained.