Doubly Reflected BSDEs and ${\cal E}^{f}$-Dynkin games: beyond the right-continuous case

We formulate a notion of doubly reflected BSDE in the case where the barriers $\xi$ and $\zeta$ do not satisfy any regularity assumption and with a general filtration. Under a technical assumption (a Mokobodzki-type condition), we show existence and uniqueness of the solution. In the case where $\xi$ is right upper-semicontinuous and $\zeta$ is right lower-semicontinuous, the solution is characterized in terms of the value of a corresponding $\mathcal{E}^f$-Dynkin game, i.e. a game problem over stopping times with (non-linear) $f$-expectation, where $f$ is the driver of the doubly reflected BSDE. In the general case where the barriers do not satisfy any regularity assumptions, the solution of the doubly reflected BSDE is related to the value of ''an extension'' of the previous non-linear game problem over a larger set of ''stopping strategies'' than the set of stopping times. This characterization is then used to establish a comparison result and \textit{a priori} estimates with universal constants.


Introduction
Backward stochastic differential equations (BSDEs) have been introduced in the case of a linear driver in [3], and then generalized to the non-linear case by Pardoux and Peng [33]. The theory of BSDEs provides a useful tool for the study of financial problems such as the pricing of European options among others (cf., e.g., [12] and [13]). When the driver f is non-linear, a BSDE induces a useful family of non-linear operators, first introduced in [13] under the name of non linear pricing system, and later called f -evaluation (also, f -expectation) and denoted by E f (cf. [34]). Reflected BSDEs (RBSDEs) are a variant of BSDEs in which the solution is constrained to be greater than or equal to a given process called obstacle. RBSDEs have been introduced in [11] in the case of a Brownian filtration and a continuous obstacle, and links with (non-linear) optimal stopping problems with f -expectations have been given in [13]. RBSDEs have been generalized to the case of a not necessarily continuous obstacle and/or a larger filtration than the Brownian one by several authors [21], [5], [27], [15], [28], [37]. In all these works, the obstacle has been assumed to be right-continuous. The paper [18] is the first to study RBSDEs beyond the right-continuous case: there, we work under the assumption that the obstacle is only right-uppersemicontinuous. In [19], we address the case where the obstacle does not satisfy any regularity assumption. Existence and uniqueness of the solution in the irregular case is also shown in [30] (in the Brownian framework) by using a different approach. In [18] and [19], links with optimal stopping problems with f -expectations are also provided.
Doubly reflected BSDEs (DRBSDEs) have been introduced by Cvitanic and Karatzas in [6] in the case of continuous barriers and a Brownian filtration. The solutions of such equations are constrained to stay between two adapted processes ξ and ζ, called barriers, with ξ ≤ ζ and ξ T = ζ T . In the case of non-continuous barriers and/or a larger filtration, DRBSDEs have been studied by several authors, cf. [2], [23], [25], [26], [24], [5], [16], [28], [8]. In all of the above-mentioned works on DRBSDEs, the barriers are assumed to be at least right-continuous.
In the first part of the present paper, we formulate a notion of doubly RBSDEs in the case where the barriers do not satisfy any regularity assumption. We show existence and uniqueness of the solution of these equations. To this purpose, we first consider the case where the driver does not depend on the solution, and is thus given by an adapted process (f t ). We show that in this particular case, the solution of the DRBSDE can be written in terms of the difference of the solutions of a coupled system of two reflected BSDEs. We show that this system (and hence the Doubly Reflected BSDE) admits a solution if and only if the so-called Mokobodzki's condition holds (assuming the existence of two strong supermartingales whose difference is between ξ and ζ). We then provide a priori estimates for our doubly RBSDEs, by using Gal'chouk-Lenglart's formula (cf. Corollary A.2 in [18]). From these estimates, we derive the uniqueness of the solution of the doubly RBSDE associated with driver process (f t ). We then solve the case of a general Lipschitz driver f by using the a priori estimates and Banach fixed point theorem.
In the second part of the paper, we focus on links between the solution of the doubly reflected BSDE with irregular barriers from the first part and some related two-stoppergame problems. Let us first recall the "classical" Dynkin game problem which has been largely studied (cf., e.g., [1] for general results).
Let T 0 denote the set of all stopping times valued in [0, T ], where T > 0. For each pair (τ, σ) ∈ T 0 × T 0 , the terminal time of the game is given by τ ∧ σ and the terminal payoff, or reward, of the game (at time τ ∧ σ) is given by The criterion is defined as the (linear) expectation of the pay-off, that is, E [I(τ, σ)]. It is well-known that, if ξ is right upper-semicontinuous (right u.s.c) and ζ is right lowersemicontinuous (right l.s.c) and satisfy Mokobodzki's condition, this classical Dynkin game has a (common) value, that is, the following equality holds:  Moreover, under the additional assumptions that ξ and −ζ are left-uppersemicontinuous along stopping times and ξ t < ζ t , t < T , there exists a saddle point (cf. [1], [31]) 1 .
DRBSDEs and E f -Dynkin games: beyond right-continuity Furthermore, when the processes ξ and ζ are right-continuous, the (common) value of the classical Dynkin game is equal to the solution at time 0 of the doubly reflected BSDE with driver equal to 0 and barriers (ξ, ζ) (cf. [6], [26], [32]).
In the second part of the present paper, we consider the following generalization of the classical Dynkin game problem: For each pair (τ, σ) ∈ T 0 × T 0 , the criterion is defined by E f 0,τ ∧σ [I(τ, σ)], where E f 0,τ ∧σ (·) denotes the f -expectation at time 0 when the terminal time is τ ∧ σ. We refer to this generalized game problem as E f -Dynkin game 2 . This non-linear game problem has been introduced in [8] in the case where ξ and ζ are right-continuous under the name of generalized Dynkin game, the term generalized referring to the presence of a (non-linear) f -expectation in place of the "classical" linear expectation.
In the second part of the paper, we generalize the results of [8] beyond the rightcontinuity assumption on ξ and ζ. By using results from the first part of the present paper, combined with some arguments from [8], we show that if ξ is right-u.s.c. and ζ is right-l.s.c. , and if they satisfy Mokobodzki's condition, there exists a (common) value and this common value is equal to the solution at time 0 of the doubly reflected BSDE with driver f and barriers (ξ, ζ) from the first part of the paper. Moreover, under the additional assumption that ξ is left u.s.c. along stopping times and ζ is left l.s.c. along stopping times, we prove that there exists a saddle point for the E f -Dynkin game. Let us note that in the particular case when f = 0, our results on existence of a common value and on existence of saddle points correspond to the results from the literature on classical Dynkin games recalled above.
In the final part of the paper, we turn to the interpretation of our Doubly RBSDE in terms of a two-stopper-game in the general case where ξ and ζ do not satisfy any regularity assumption. This is technically a more difficult problem. Indeed, even in the simplest case where f = 0, we know from the litterature on classical Dynkin games (cf. e.g. [1]) that the game on stopping times with criterion E [I(τ, σ)] does not even (a priori) admit a common value, that is, the equality (1.2) does not necessarily hold; this is true, a fortiori, for the E f -Dynkin game (with non-linear f ). In order to interpret the solution of the doubly reflected BSDE with irregular barriers (ξ, ζ) we formulate "an extension" of the previous E f -Dynkin game problem over a larger set of "stopping strategies" than the set of stopping times T 0 . We show that this extended game has a common value which coincides with the solution of our general DRBSDE with irregular barriers. Using this result, we prove a comparison theorem and a priori estimates with universal constants for DRBSDEs with irregular barriers. The remainder of the paper is organized as follows: In Section 2, we introduce the notation and some definitions. In Section 3, we provide first results on doubly reflected BSDEs associated with a Lipschitz driver and barriers (ξ, ζ) which do not satisfy any regularity assumption; in particular, we show existence and uniqueness of the solution of this equation. Section 4 is dedicated to the interpretation of the solution in terms of a two-stopper game problem, first in the case when ξ is right u.s.c. and ζ is right l.s.c., then in the case where they do not satisfy any regularity assumption. In Section 5, point (cf. Remark 3.8 in [8]). Note also that when ξ and ζ do not satisfy any regularity assumption, there does not necessarily exist a value for the Dynkin game, that is, the equality (1.2) does not necessarily hold. 2 Note that this game problem is related to the pricing of game options in imperfect market models (cf. the end of Section 3 for more explanations).
. we provide a comparison theorem and a priori estimates with universal constants for our doubly reflected BSDEs with irregular barriers. The Appendix contains some useful results on reflected BSDEs with an irregular obstacle and also some of the proofs.

Preliminaries
Let T > 0 be a fixed positive real number. Let ν be a σ-finite positive measure on the measurable space (E, E ) = (R * , B(R * )). Let (Ω, F, P ) be a probability space equipped with a one-dimensional Brownian motion W and with an independent Poisson random measure N (dt, de) with compensator dt ⊗ ν(de). We denote byÑ (dt, de) the compensated process, i.e.Ñ (dt, de) := N (dt, de) − dt ⊗ ν(de). Let IF = {F t : t ∈ [0, T ]} be the (complete) natural filtration associated with W and N . The space L 2 (F T ) is the space of random variables which are F T -measurable and square-integrable. For t ∈ [0, T ], we denote by T t the set of stopping times τ such that P (t ≤ τ ≤ T ) = 1. More generally, for a given stopping time ν ∈ T 0 , we denote by T ν the set of stopping times τ such that DRBSDEs and E f -Dynkin games: beyond right-continuity Definition 2.2. Let A = (A t ) 0≤t≤T and A = (A t ) 0≤t≤T be two real-valued optional non-decreasing cadlag processes with A 0 = 0, A 0 = 0 and E[A T ] < ∞ and E[A T ] < ∞. We say that the random measures dA t and dA t are mutually singular, and we write dA t ⊥ dA t , if there exists D ∈ O such that: which can also be written as 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 t. 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. Definition 2.3. An optional process (φ t ) is said to be left upper-semicontinuous (resp. left lower-semicontinuous) along stopping times if for each τ ∈ T 0 , for each nondecreasing sequence of stopping times (τ n ) such that τ n ↑ τ , a.s. , we have φ τ ≥ lim sup n→∞ φ τn (resp. φ τ ≤ lim inf n→∞ φ τn ) a.s.
For the easing of the presentation, we define the relation ≥ for processes in S 2 as follows: for φ, φ ∈ S 2 , we write φ ≤ φ , if φ t ≤ φ t for all t ∈ [0, T ] a.s. Similarly, we define the relations ≤ and = on S 2 .
3 Doubly Reflected BSDE whose obstacles are irregular

Definition and first properties
Let T > 0 be a fixed terminal time (as before). Let f be a driver. Let ξ = (ξ t ) t∈[0,T ] and ζ = (ζ t ) t∈[0,T ] be two left-limited processes in S 2 such that ξ t ≤ ζ t , 0 ≤ t ≤ T, a.s. and ξ T = ζ T a.s. A pair of processes (ξ, ζ) satisfying the previous properties will be called a pair of admissible barriers, or a pair of admissible obstacles.

Remark 3.2.
Let us note that in the following definitions and results we can relax the assumption of existence of left limits for the processes ξ and ζ. All the results still hold true provided we replace the process (ξ t− ) t∈]0,T ] by the process (lim sup s↑t,s<t ξ s ) t∈]0,T ] and the process (ζ t− ) t∈]0,T ] by the process (lim inf s↑t,s<t ζ s ) t∈]0,T ] .

2)
A and A are nondecreasing right-continuous predictable processes with A 0 = A 0 = 0,  C and C are nondecreasing right-continuous adapted purely discontinuous processes with dA t ⊥ dA t and dC t ⊥ dC t . Let us note that if (Y, Z, k, A, C, A , C ) satisfies the above definition, then the process Y has left and right limits.

Remark 3.3. When
A and A (resp. C and C ) are not required to be mutually singular, they can simultaneously increase . The constraints dA t ⊥ dA t and dC t ⊥ dC t will allow us to obtain the uniqueness of the nondecreasing processes A, A , C and C without the strict separability condition ξ < ζ. We note also that, due to Eq.
This, together with the condition dC t ⊥ dC t gives ∆C t = (Y t+ − Y t ) − for all t a.s., and ∆C t = (Y t+ − Y t ) + for all t a.s. On the other hand, since in our framework the filtration is quasi-left-continuous, martingales have only totally inaccessible jumps. Hence, for ). This, together with the condition dA t ⊥ dA t , ensures that for each predictable τ ∈ T 0 , ∆A d τ = (∆Y τ ) − and ∆A d τ = (∆Y τ ) + a.s. We note also that Y can jump (on the left) at totally inaccessible stopping times; these jumps of Y come from the jumps of the stochastic integral with respect toÑ in (3.1). Proposition 3.1. Let f be a driver and (ξ, ζ) be a pair of admissible obstacles. Let (Y, Z, k, A, C, A , C ) be a solution to the doubly reflected BSDE with parameters (f, ξ, ζ).
is left upper-semicontinuous (resp. left lower-semicontinuous) along stopping times, then the process A (resp. A ) is continuous.
Proof. Let us show the first assertion. Let τ ∈ T 0 . By the previous Remark 3. 3 s. Since C and C satisfy the Skorokhod condition (3.4), we have s. The first assertion thus holds.
Let us show the second assertion. Suppose that ξ is right-continuous. Let τ ∈ T 0 . We show ∆C τ = 0 a.s. As seen above, we have where the last equality follows from the right-continuity of ξ. Since Y ≥ ξ, we derive that ∆C τ = 0 a.s. This equality being true for all τ ∈ T 0 , it follows that C = 0. Similarly, it can be shown that if ζ is right-continuous, then C = 0. Hence, the second assertion holds.
It remains to show the third assertion. Suppose that ξ is left u.s.c.along stopping times. Let τ ∈ T 0 be a predictable stopping time. We show ∆A τ = 0 a.s. By the previous Remark 3. 3 The (last) inequality in the above computation follows from the inequality ξ τ − ≤ ξ τ a.s., which is due to the assumption of left u.s.c.of ξ (cf. Remark 2.1). Since ξ ≤ Y , we derive ∆A τ ≤ 0 a.s. , which implies that ∆A τ = 0 a.s. This equality being true for every predictable stopping time τ ∈ T 0 , it follows that A is continuous. Similarly, it can be shown that if ζ is left lower-semicontinuous along stopping times, then A is continuous, which ends the proof.

Remark 3.4 (Right-continuous case).
It follows from the second assertion in the above proposition that if ξ and ζ are right-continuous, then C = C = 0. In this case, our Definition 3.1 corresponds to the one given in the literature on DRBSDEs (cf. e.g. [8]).
be a solution to the DRBSDE associated with driver f and with a pair of admissible barriers (ξ, ζ). By taking the conditional expectation with respect to F t in the equality (3.1), we derive that Y = H −H , where H and H are the two nonnegative strong supermartingales given by Remark 3.5. The above reasoning gives us that Mokobodzki's condition is a necessary condition for the existence of a solution to the DRBSDE.

The case when f does not depend on the solution
Let us now investigate the question of existence and uniqueness of the solution to the DRBSDE defined above in the case where the driver f does not depend on y, z, and k , that is, f = (f t ), where (f t ) is a process belonging to IH 2 .

Equivalent formulation
We first show that the existence of a solution to the DRBSDE associated with driver process f = (f t ) is equivalent to the existence of a solution to a coupled system of reflected BSDEs.
be a solution to the DRBSDE associated with driver f (ω, t) and with a pair of admissible barriers (ξ, ζ).
Hence, the Skorokhod condition (3.3) satisfied by A can be written: s. It follows that (X f , π, l, A, C) is the solution of the reflected BSDE associated with driver 0 and obstacle (X f +ξ f )I [0,T ) (cf. Prop. 6.3 in the Appendix) 4 .
DRBSDEs and E f -Dynkin games: beyond right-continuity By similar arguments we get that (X f , π , l , A , C ) is the solution of the reflected BSDE associated with driver 0 and obstacle (X f −ζ f )I [0,T ) .
We have thus shown that (3.11) where Ref is the operator induced by the RBSDE with driver 0 (cf. Definition 6.1 in the Appendix). We conclude that the existence of a solution to the DRBSDE with parameters (f, ξ, ζ) (where f is a driver process) implies the existence of a solution to the coupled system of RBSDEs (3.11). We will see in the following proposition that the converse statement also holds true. Proposition 3.3. The DRBSDE associated with driver process f = (f t ) ∈ IH 2 and with a pair of admissible barriers (ξ, ζ) has a solution if and only if there exist two processes X · ∈ S 2 and X · ∈ S 2 satisfying the coupled system of RBSDEs: In this case, the optional process Y defined by gives the first component of a solution to the DRBSDE.

Proof.
The "only if part" of the first assertion has been proved above. Let us prove the "if part" of the first statement, together with the second statement. Let X · ∈ S 2 and X · ∈ S 2 be two processes satisfying the coupled system (3.12). Let (π, l, A, C) (resp. (π , l , A , C )) be the vector of the remaining components of the solution to the RBSDE whose first component is X (resp. whose first component is X ). We note that equations (3.8) and (3.9) hold for X and X (in place of X f and X f ). We define the optional process Y as in (3.13). Since by assumption X and X belong to S 2 , it follows that X and X are realvalued, which implies that the process Y is well-defined. From (3.13) and the property X T = X T = 0 a.s., we get Y T = ξ T a.s. From the system (3.12) we get X t ≥ X t +ξ f t and Moreover, the processes A, C (resp. A , C ) satisfy the Skorokhod conditions for RBS-DEs. More precisely, for A and C we have: for all τ ∈ T 0 , ∆ By applying the same arguments to A and C , we get ∆C τ = 1 {Yτ =ζτ } ∆C τ a.s. for all τ ∈ T 0 and  (ds, de). From this, together with the definition of Y and equations (3.8) and (3.9) for X and X , we obtain where Z := π − π + π, k := l − l + l, α := A − A and γ := C − C . If dA t ⊥ dA t and dC t ⊥ dC t , then (Y, Z, k, A, C, A , C ) is a solution to the doubly reflected BSDE with parameters (f, ξ, ζ), which gives the desired result.
Otherwise, by the canonical decomposition of RCLL processes with integrable variation (cf. Proposition A.7 in [8]), there exist two nondecreasing right-continuous predictable (resp. optional) processes B and B (resp. D and D ) belonging to S 2 such that α = B−B (resp. γ = D − D ) with dB t ⊥ dB t (resp. dD t ⊥ dD t ). Moreover, dB t < < dA t , dB t < < dA t , dD t < < dC t and dD t < < dC t .

Existence of a (minimal) solution of the coupled system of RBSDEs
Let f = (f t ) ∈ IH 2 be a driver process (as above). We show the existence of a solution to the system (3.12) under Mokobodzki's condition. To do that, we use Picard's iterations.
We set X 0 = 0 and X 0 = 0, and we define recursively, for each n ∈ N, the processes: We see, by induction, that the processes X n and X n are well-defined; moreover, X n and X n are strong supermartingales in S 2 . For the sake of simplicity, we have omitted the dependence on f in the notation for X n and X n .

Proposition 3.2.
Assume that the admissible pair (ξ, ζ) satisfies Mokobodzki's condition. The sequences of optional processes (X n · ) n∈N and (X n · ) n∈N defined above are nondecreasing.
The limit processes are nonnegative strong supermartingales in S 2 satisfying the system (3.12) of coupled RBSDEs. Moreover, X f · , X f · are the smallest processes in S 2 satisfying system (3.12). The processes X f , X f are also characterized as the minimal strong supermartingales in S 2 satisfying the inequalitiesξ f ≤ X f − X f ≤ζ f .
The proof is given in the Appendix. In the following theorem we summarize some of the properties established so far. (ii) The system (3.12) of coupled RBSDEs admits a solution.
(iii) The DRBSDE (3.1) with driver process f has a solution.
Proof. The implication (i) ⇒ (ii) has been just proved (by using Picard's iterations). The equivalence between (ii) and (iii) has been established in Proposition 3.3. We have noticed that the implication (iii) ⇒ (i) holds (in the general case of a Lipschitz driver f ) in Remark 3.5.

Uniqueness of the solution
Let us now investigate the question of uniqueness of the solution to the DRBSDE with driver process (f t ) ∈ IH 2 . To this purpose, we first state a lemma which will be used in the sequel.
be a solution to the DRBSDE associated with driver process f = (f t ) ∈ IH 2 (resp.f = (f t ) ∈ IH 2 ) and with a pair of admissible obstacles (ξ, ζ). There exists c > 0 such that for all ε > 0, for all β ≥ 1 The proof, which relies on Gal'chouk-Lenglart's formula (cf. Corollary A.2 in [18]), is given in the Appendix.
We prove below the uniqueness of the solution to the DRBSDE associated with the driver process (f t ) and with the admissible pair of barriers (ξ, ζ) satisfying Mokobodzki's condition.
Proof. Theorem 3.4 yields the existence of a solution. It remains to show the uniqueness.
Let (Y, Z, k, A, C, A , C ) be a solution of the DRBSDE associated with the driver process (f t ) and the barriers ξ and ζ. By the a priori estimates (cf. Lemma 3.5), we derive the uniqueness of (Y, Z, k). By Remark 3.3, we have ∆C t = (Y t+ − Y t ) − for all t a.s. and ∆C t = (Y t+ − Y t ) + for all t a.s. , which implies the uniqueness of the purely discontinuous processes C and C . Moreover, since (Y, Z, k, A, C, A , C ) satisfies the equation (3.1), it follows that the process A − A can be expressed in terms of Y, C, C , the integral of the driver process (f t ) with respect to the Lebesgue measure, and the stochastic integrals of Z and k with respect to W andÑ , respectively, which yields the uniqueness of the finite variation process A − A . Now, since dA t ⊥ dA t , the nondecreasing processes A and A correspond to the (unique) canonical decomposition of this finite variation process, which ends the proof.
Using the minimality property of (X f , X f ) (cf. Proposition 3.2), together with the uniqueness property of the solution of the DRBSDE (3.1) with driver process f = (f t ) and Proposition 3.3, we derive that the limit processes X f and X f defined by (3.15) can be written in terms of the solution of the DRBSDE. . Proposition 3.7 (Identification of X f and X f ). Let X f and X f be the strong supermartingales defined by (3.15). We have a.s.
where A, C, A and C are the four last coordinates of the solution of the DRBSDE (3.1) associated with barriers ξ and ζ, and driver process f = (f t ). Moroever, we have The proof is given in the Appendix.

The case of a general Lipschitz driver f (t, y, z, k )
We now prove the existence and uniqueness of the solution to the DRBSDE from Definition 3.1 in the case of a general Lipschitz driver.
The proof, which relies on the estimates provided in Lemma 3.5 and a fixed point theorem, is given in the Appendix.

Doubly reflected BSDEs with irregular barriers and E f -Dynkin games with irregular rewards
The purpose of this section is to connect our DRBSDE with irregular barriers to a zero-sum game problem between two "stoppers" whose pay-offs are irregular and are assessed by non-linear f -expectations.
In the "classical" case where f ≡ 0 (or, more generally, where f is a given process (f t ) ∈ H 2 ), this topic has been first studied in [6] in the case of continuous barriers, and in [21] and [22] in the case of right-continuous barriers. The case of right-continuous barriers and a general Lipschitz driver f has been studied in [8].

The case where ξ is right upper-semicontinuous and ζ is right lower-semicontinuous
In this subsection we focus on the case where ξ is right upper-semicontinuous (right u.s.c.) and ζ is right lower-semicontinuous (right l.s.c.). We interpret the solution of our Doubly Reflected BSDE in terms of the value process of a suitably defined zero-sum game problem on stopping times with (non-linear) f -expectations. Let ξ ∈ S 2 and ζ ∈ S 2 be two optional processes (which are not necessarily non negative).
We consider a game problem with two players where each of the players' strategy is a stopping time in T 0 and the players payoffs are defined in terms of the given processes ξ and ζ. More precisely, if the first agent chooses τ ∈ T 0 as his/her strategy and the second agent chooses σ ∈ T 0 , then, at time τ ∧ σ (when the game ends), the pay-off (or reward) is I(τ, σ), where I(τ, σ) := ξ τ 1 τ ≤σ + ζ σ 1 σ<τ .
The associated criterion (from time 0 perspective) is defined as the f -evaluation of the pay-off, that is, by E f 0,τ ∧σ [I(τ, σ)]. The first agent aims at choosing a stopping time τ ∈ T 0 which maximizes the criterion. The second agent has the antagonistic objective of choosing a strategy σ ∈ T 0 which minimizes the criterion.
As is usual in stochastic control, we embed the above (game) problem in a dynamic setting, by considering the game from time θ onwards, where θ runs through T 0 . From the perspective of time θ (where θ ∈ T 0 is given), the first agent aims at choosing a strategy τ ∈ T θ such that E f θ,τ ∧σ [I(τ, σ)] be maximal. The second agent has the antagonistic objective of choosing σ ∈ T θ such that E f θ,τ ∧σ [I(τ, σ)] be minimal.
The following notions will be used in the sequel: We will now show the existence of saddle points under an additional regularity assumption on the barriers. Let (Y, Z, k, A, A , C, C ) be the solution of the DRBSDE (3.1). For each θ ∈ T 0 , we introduce the following stopping times: (4.12) and hold. We assume moreover that ξ is left u.s.c.and ζ is left l.s.c.along stopping times. Then, for each θ ∈ T 0 , the pairs of stopping times (τ * θ , σ * θ ) and (τ θ , σ θ ), defined by (4.12) and (4.13), are saddle points at time θ for the E f -Dynkin game.
Proof. The proof of the theorem is given in the Appendix.

Classical Dynkin game with irregular rewards
In this paragraph, we consider the particular case where f ≡ 0, that is, the case where the f -expectation reduces to the classical linear expectation. Let (ξ, ζ) be an admissible pair of barriers satisfying Mokobodzki's condition and such that ξ is right u.s.c.and ζ are right l.s.c. (as in Theorem 4.5). Let θ ∈ T 0 . For τ ∈ T θ and σ ∈ T θ , it holds E  We thus recover the classical Dynkin game on stopping times (with linear expectations) recalled in the introduction (cf., e.g., [4] and [1]). In [1], it has been shown that this classical Dynkin game has a value. From our Theorem 4.5, we derive an infinitesimal characterization of the value of this game. From Theorem 4.6, we derive the existence of saddle points under the additional regularity assumption of the reward processes.
DRBSDEs and E f -Dynkin games: beyond right-continuity Corollary 4.1. There exists a process Y ∈ S 2 which aggregates the common value function, i.e., Y is such that for all θ ∈ T 0 , Y θ = V (θ) = V (θ) a.s. Moreover, the process Y is equal to the first component of the solution (Y, Z, k, A, A , C, C ) of the DRBSDE (3.1) associated with driver f = 0 and with barriers ξ and ζ.

Game options
In this paragraph, we briefly discuss how the results of this section can be applied to the problem of pricing of game options in some market models with imperfections. We recall that a game option is a financial instrument which gives the buyer the right to exercise at any stopping time τ ∈ T and the seller the right to cancel at any stopping time σ ∈ T . If the buyer exercises at time τ before the seller cancels, then the seller pays to the buyer the amount ξ τ ; if the seller cancels at time σ before the buyer exercises, the seller pays to the buyer the amount ζ σ at the cancellation time σ. The difference ζ − ξ ≥ 0 corresponds to a penalty which the seller pays to the buyer in the case of an early cancellation of the contract. Thus, if the seller chooses a cancellation time σ and the buyer chooses an exercise time τ , the former pays to the latter the payoff I(τ, σ) (defined in (1.1)) at time τ ∧ σ. In the seminal paper [14], Kifer relates the problem of pricing of game options in a frictionless complete market model to the theory of "classical" Dynkin games (with " classical" linear expectations). Since Kifer's work [14], it is well-known that if the market model is complete and if the processes ξ and ζ are right-continuous and satisfy Mokobodzki's condition, then the price of the game option (up to a discount factor) is equal to the common value of the classical Dynkin game from equation (1.2), where the expectation is taken under the unique martingale measure of the model. Let us also recall that, in a complete market model, the expectation under the unique martingale measure corresponds (up to discounting) to the pricing functional for European-type options. In market models with imperfections however, pricing rules for European-type options are in general no longer linear (cf, e.g. the notion of non linear pricing system introduced in [13] or the notion of pricing rule introduced in [29]). In a large class of market models with imperfections, European options can be priced via an f -expectation/evaluation, where f is a nonlinear driver in which the imperfections are encoded (cf. [13] where also several concrete examples of imperfections are provided). In such a framework, the problem of pricing of game options has been considered in [9]: when ξ and ζ are right-continuous and satisfy Mokobodzki's condition, the common value of the E f -Dynkin game from equation (1.3) is shown to be equal to the "seller's price" of the game option (cf. Theorem 3.12 in [9]). Using Theorem 4.5 and Proposition 3.1 of the present paper, we can show that the result of [9] can be generalized to the case where the assumption of right-continuity is replaced by the weaker assumption of right upper-semicontinuity of ξ and right lower-semicontinuity of ζ.

The general irregular case
In this subsection (ξ, ζ) is an admissible pair of barriers satisfying Mokobodzki's condition. Contrary to the previous subsection, here we do not make any regularity assumptions on the pair (ξ, ζ). In this general case, we will interpret the DRBSDE with a pair of obstacles (ξ, ζ) in terms of the value of "an extension" of the zero-sum game of the previous subsection over a larger set of "stopping strategies" than the set of stopping times T 0 . To this purpose we introduce the following notion of stopping system. By taking H = Ω in the above definition, we see that the notion of a stopping system generalizes that of a stopping time (in the usual sense).

Remark 4.9.
A stopping system is an example of divided stopping time (from the French "temps d'arrêt divisé") in the sense of [10] or [1].
We denote by S 0 the set of all stopping systems; for a stopping time θ ∈ T 0 , we denote by S θ the set of stopping systems ρ = (τ, H) such that such that θ ≤ τ .
For an optional right-limited process φ and a stopping system ρ = (τ, H), we define φ ρ by In the particular case where ρ = (τ, Ω), we have φ ρ = φ τ , so the notation is consistent.
Note that when φ is right-limited, we have φ u ρ = φ l ρ = φ ρ . Moreover, in the particular case where ρ = (τ, Ω), we have φ u ρ = φ l ρ = φ τ , so the notation is consistent. With the help of the above definitions and notation we formulate an extension of the zero-sum game problem from Subsection 4.1 where the set of "stopping strategies" of the agents is the set of stopping systems. More precisely, for two stopping systems ρ = (τ, H) ∈ S 0 and δ = (σ, G) ∈ S 0 , we define the pay-off I(ρ, δ) by I(ρ, δ) := ξ u ρ 1 τ ≤σ + ζ l δ 1 σ<τ . (4.15) We note that, by definition, I(ρ, δ) is an F τ ∧σ -measurable random variable. As in the previous subsection, the pay-off is assessed by an f -expectation, where f is a Lipschitz driver. Let θ ∈ T 0 be a stopping time  in a similar manner, by replacing the set of stopping times T θ by the set of stopping systems S θ . We will refer to this game problem as "extended" E f -Dynkin game (over the set of stopping systems). We will show that, for any θ ∈ T 0 , the "extended" E f -Dynkin where the last inequality follows from the definition of I(ρ ε θ , δ). By using the inequality (4.24) and the nondecreasingness of E f , we derive ∧σ [I(ρ ε θ , δ)] + Lε a.s. , (4.25) where the last inequality follows from an estimate on BSDEs (cf. Proposition A.4 in [36]).
Using similar arguments, it can be shown that together with (4.25), leads to the desired inequalities (4.23).
In the following theorem we show that the "extended" E f -Dynkin game has a value which coincides with the first component of the DRBSDE with irregular barriers.
s. The proof is thus complete.

Two useful corollaries
Using the characterization of the solution of the nonlinear DRBSDE as the value function of the "extended" E f -Dynkin game (over the set of stopping systems) from Theorem 4.9, we now establish a comparison theorem and a priori estimates with universal constants (i.e. depending only on the terminal time T and the common Lipschitz constant K) for DRBSDEs with completely irregular barriers.
be the solution of the DRBSDE associated with driver f i and barriers ξ i , ζ i . Assume that ξ 2 ≤ ξ 1 and ζ 2 ≤ ζ 1 and f 2 (t, Y 2 in this inequality, and using the characterization of the solution of the DRBSDE with obstacles (ξ, ζ) as the value function of the "extended" E f -Dynkin game (cf. Theorem 4.9), we obtain: Since this inequality holds for each θ ∈ T 0 , we get Y 2 ≤ Y 1 .
Step 2: We now place ourselves under the assumptions of the theorem (which are weaker than those made in Step 1). Letf be the process defined byf t : A is a nondecreasing right-continuous predictable process with A 0 = 0 and such that C is a nondecreasing right-continuous adapted purely discontinuous process with C 0− = 0 and such that (X τ − ξ τ )(C τ − C τ − ) = 0 a.s. for all τ ∈ T 0 . .

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DRBSDEs and E f -Dynkin games: beyond right-continuity We introduce the following operator:  [19], the process X is equal to the value function of the optimal stopping problem with payoff ξ, that is for each stopping time θ, we have Hence, by classical results of Optimal Stopping Theory, the process Ref [ξ] = X is equal to the Snell envelope of the process ξ, that is, the smallest strong supermartingale greater than or equal to ξ. Using this property, we easily derive the three assertions of the lemma. Remark 6.15. We recall that the nondecreasing limit of a sequence of strong supermartingales is a strong supermartingale (which can be easily shown by the Lebesgue theorem for conditional expectations).
We now show a monotone convergence result for the operator Ref .
Lemma 6.3. Let (ξ n ) be a sequence of processes belonging to S 2 , supposed to be nondecreasing, i.e., such that for each n ∈ IN , ξ n ≤ ξ n+1 . Let ξ := lim n→+∞ ξ n . If ξ ∈ S 2 , then Ref Indeed, if J, J are nonnegative strong supermartingales in S 2 satisfyingξ f ≤ J −J ≤ζ f , then, using the same arguments as above, we derive that X f ≤ J and X f ≤ J . From this minimality property, it follows that (X f , X f ) is also characterized as the minimal solution of the system (3.12) of coupled RBSDEs.
Let us first consider the sum of the first and the second term on the r.h.s. of equality (6.7). By applying the inequality 2ab ≤ ( a ε ) 2 + ε 2 b 2 , valid for all (a, b) ∈ R 2 , we get: a.s. for all t ∈ e βsf 2 (s)ds.
For the third term (resp. the fourth term) on the r.h.s. of (6.7) it can be shown that, a.s. for all t ∈ [0, T ], +2 ]t,T ] e βsỸ s− dÃ s ≤ 0 (resp. −2 ]t,T ] e βsỸ s− dÃ s ≤ 0) The proof uses property (3.3) of the definition of the DRBSDE and the properties Y ≥ ξ, Y ≥ ξ (resp. Y ≤ ζ,Ȳ ≤ ζ) ; the details are similar to those in the case of RBSDE (with one lower obstacle) (cf., for instance, the proof of Lemma 3.2 in [18]).
For the eighth and the ninth terms on the r.h.s. of (6.7) we show that, a.s. for all  where, in order to obtain the equality, we have used the fact that the processes A,Ā, A , andĀ jump only at predictable stopping times, and N (·, de) jumps only at totally inaccessible stopping times (cf. also Remark 3.3).
By adding the term ]t,T ] e βs ||k s || 2 ν ds on both sides of inequality (6.8) and by using the above computation, we derive that almost surely, for all t ∈ [0, T ], e βs E (2Ỹ s−ks (e) +k 2 s (e))Ñ (ds, de).