Generalized Dynkin Games and Doubly Reflected BSDEs with Jumps

We introduce a generalized Dynkin game problem with non linear conditional expectation ${\cal E}$ induced by a Backward Stochastic Differential Equation (BSDE) with jumps. Let $\xi, \zeta$ be two RCLL adapted processes with $\xi \leq \zeta$. The criterium is given by \begin{equation*} {\cal J}_{\tau, \sigma}= {\cal E}_{0, \tau \wedge \sigma } \left(\xi_{\tau}\textbf{1}_{\{ \tau \leq \sigma\}}+\zeta_{\sigma}\textbf{1}_{\{\sigma<\tau\}}\right) \end{equation*} where $\tau$ and $ \sigma$ are stopping times valued in $[0,T]$. Under Mokobodski's condition, we establish the existence of a value function for this game, i.e. $\inf_{\sigma}\sup_{\tau} {\cal J}_{\tau, \sigma} = \sup_{\tau} \inf_{\sigma} {\cal J}_{\tau, \sigma}$. This value can be characterized via a doubly reflected BSDE. Using this characterization, we provide some new results on these equations, such as comparison theorems and a priori estimates. When $\xi$ and $\zeta$ are left upper semicontinuous along stopping times, we prove the existence of a saddle point. We also study a generalized mixed game problem when the players have two actions: continuous control and stopping. We then address the generalized Dynkin game in a Markovian framework and its links with parabolic partial integro-differential variational inequalities with two obstacles.


Introduction
The classical Dynkin game has been widely studied: see e.g.Bismut [4], Alario-Nazaret et al. [1], Kobylanski et al. [17].Let ξ, ζ be two Right Continuous Left-Limited (RCLL) adapted processes with ξ ≤ ζ and ξ T = ζ T a.s.The criterium is given, for each pair (τ, σ) of stopping times valued in [0, T ], by Under Mokobodski's condition, which states that there exists two supermartingales such that their difference is between ξ and ζ, there exists a value function for the Dynkin game, i.e. inf σ sup τ J τ,σ = sup τ inf σ J τ,σ .When ξ t < ζ t , t < T , and when ξ and ζ are also left upper semicontinuous, it is proved that there exists a saddle point.
Using a change of variable, these results can be generalized to the case of a criterium with an instantaneous reward process (g t ), of the form E τ ∧σ 0 g s ds + ξ τ 1 {τ ≤σ} + ζ σ 1 {σ<τ } .
(1.1) {eq1} In the Brownian case and when (ξ t ) and (ζ t ) are continuous processes, Cvitanić and Karatzas have established in [6] links between these Dynkin games and doubly reflected Backward stochastic differential equations with driver process (g t ) and barriers (ξ t ) and (ζ t ).
In this paper, we introduce a generalization of the classical Dynkin game problem to the case of g-conditional expectations.Nonlinear expectations induced by BSDEs have been introduced by S. Peng [19] in the Brownian framework .Given a Lipschitz driver g(t, y, z), a stopping time τ ≤ T and a square integrable F S -measurable random variable η, the associated conditional g-expectation process denoted by (E t,τ , 0 ≤ t ≤ τ ) is defined as the solution of the BSDE with driver g and terminal conditions (τ, η).The extension to the case with jumps is studied in [23] and [21].We consider the following generalized Dynkin game problem where the criterium is given, for each pair (τ, σ) of stopping times valued in [0, T ], by J τ,σ = E 0,τ ∧σ ξ τ 1 {τ ≤σ} + ζ σ 1 {σ<τ } where ξ, ζ are two RCLL adapted processes with ξ ≤ ζ.
When the driver g does not depend on the solution, that is, when it is given by a process (g t ), the criterium J τ,σ coincides with (1.1).It is well-known that in this case, under Mokobodski's condition, the value function for the Dynkin game problem can be characterized as the solution of the Doubly Reflected BSDE (DRBSDE) associated with driver process (g t ) and barriers (ξ t ) and (ζ t ) (see e.g.[6,14,18]).We generalize this result to the case of a non linear driver g depending on the solution.More precisely, under Mokobodski's condition, we prove that inf σ sup τ J τ,σ = sup τ inf σ J τ,σ and we characterize this common value function as the solution of the DRBSDE associated with driver g and barriers (ξ t ) and (ζ t ).Moreover, when ξ and ζ are left-upper semicontinuous along stopping times, we show that there exist saddle points.Note that, contrary to the previous existence results given in the case of classical Dynkin games, we do not assume the strict separability of the barriers.
Then, using the characterization of the solution of a DRBSDE as the value function of a generalized Dynkin game, we prove some results on DRBSDEs, such as a comparison (respectively strict comparison) theorem and a priori estimates, which complete those given in the previous literature.
Moreover, we study a generalized mixed game problem when the players have two actions: continuous control and stopping.The first (resp.second) player chooses a pair (u, τ ) (resp.(v, σ)) of control and stopping time, and aims to maximize (resp.minimize) the criterium.In the previous literature (see [3] and [14]), the criterium is given, for each quadruple (u, τ, v, σ) of controls and stopping times, by where Q u,v are a priori probability measures and c(t, u t , v t ) represents the instantenous reward associated with controls u, v.In this paper, we consider the following generalized mixed game problem.We are given a family of Lipchitz drivers g u,v and the criterium is defined by E u,v 0,τ ∧σ ξ τ 1 {τ ≤σ} + ζ σ 1 {σ<τ } , (1.3) {nouveau} where E u,v corresponds to the g u,v -conditional expectation.Note that in the case of linear drivers g u,v , the criterium (1.3) corresponds to a criterium of the form (1.2).In this particular case, when ξ and ζ are regular, Hamadène and Lepeltier have established some links between this mixed game problem and DRBSDEs (see [14]).In this paper, we generalize these results to the case of non linear expectations and irregular payoffs ξ and ζ.We provide some sufficient conditions which ensure the existence of a value function for the above generalized mixed game problem, and show that the common value function can be characterized as the solution of a DRBSDE.Under additional regularity assumptions on ξ and ζ, we show the existence of saddle points.Finally, we address the generalized Dynkin game in the Markovian case and its links with parabolic partial integro-differential variational inequalities (PIDVI) with two obstacles.
The paper is organized as follows.In Section 2 we introduce notation and definitions and provide some preliminary results.In Section 3, we consider a classical Dynkin game problem and study its links with a DRBSDE associated with a driver which does not depend on the solution.We also provide an existence result for this game problem under relatively weak assumptions on ξ and ζ.In Section 4, we introduce a generalized Dynkin game problem expressed in terms of g-conditional expectations.We prove the existence of a value function for this game problem.We show that the common value function can be characterized as the solution of a non linear DRBSDE with jumps and RCLL barriers ξ and ζ.We then study a generalized mixed game problem when the players have two actions: continuous control and stopping.In Section 5, we provide comparison theorems and a priori estimates for DRBSDEs with jumps and RCLL obstacles.In the Markovian case, relations between generalized Dynkin games and PIDVIs are studied in Section 6.We state that the value function of the generalized Dynkin game corresponds to a solution of a PIDVI in the viscosity sense.Under additional assumptions, we obtain an uniqueness result in the class of continuous and bounded functions.

Notation and definitions
Let (Ω, F , P ) be a probability space.Let W be a one-dimensional Brownian motion.Let (E, K) be a measurable space equipped with a σ-finite positive measure ν and let N(dt, de) be a Poisson random measure with compensator ν(de)dt.Let Ñ(dt, de) be its compensated process.Let IF = {F t , t ≥ 0} be the natural filtration associated with W and N.
For each T > 0, we use the following notation: L 2 (F T ) is the set of random variables ξ which are F T -measurable and square integrable; IH 2 is the set of real-valued predictable processes φ such that φ 2 ) is the set of real-valued non decreasing RCLL predictable processes A with A 0 = 0 and E(A 2 T ) < ∞ (resp.E(A T ) < ∞).We also introduce the following spaces.
• IH 2 ν is the set of processes l which are predictable, that is, measurable Moreover, T 0 is the set of stopping times τ such that τ ∈ [0, T ] a.s. and for each S in T 0 , we denote by T S the set of stopping times τ such that S ≤ τ ≤ T a.s.Definition 2.1 (Driver, Lipschitz driver) A function g is said to be a driver if A driver g is called a Lipschitz driver if moreover there exists a constant C ≥ 0 such that dP ⊗ dt-a.s., for each (y Recall that for each Lipschitz driver g, and each terminal condition ξ ∈ L 2 (F T ), there exists a unique solution (X, π, l) ∈ S 2 × IH 2 × IH 2 ν satisfying The solution is denoted by (X(ξ, T ), π(ξ, T ), l(ξ, T )).This result can be extended when the terminal time T is replaced by a stopping time τ ∈ T 0 and when ξ is replaced by a random variable η ∈ L 2 (F τ ).The solution X • (η, τ ) corresponds to the so-called g-conditional expectation of η, denoted by E •,τ (η).

Definition 2.2 Let
We say that the measures dA t and dA ′ t are mutually singular, and we write dA t ⊥ dA ′ t , if there exist D ∈ P such that: where for each t ∈ [0, T ], D t is the section at time t of D, that is, We introduce the DRBSDEs with jumps, for which the solution is constrained to stay between two given RCLL processes called barriers ξ ≤ ζ.Two nondecreasing processes A and A ′ are introduced in order to push the solution Y above ξ and below ζ, and this in a minimal way.This minimality property of A and A ′ is ensured by the so called Skorohod conditions (see condition (iii) below) together with the additional constraint dA t ⊥ dA ′ t (see condition (ii) below).Definition 2.3 (Doubly Reflected BSDEs with Jumps) Let T > 0 be a fixed terminal time and g be a Lipschitz driver.doublyLet ξ and ζ be two adapted RCLL processes with said to be a solution of the doubly reflected BSDE (DRBSDE) associated with driver g and barriers ξ, ζ if Remark 2.4 The above definition is not exactly the same as the one given in the previous literature, where A and A ′ are not constrained to satisfy dA t ⊥ dA ′ t .Note that when A and A ′ are not required to be mutually singular, they can simultaneously increase on {ξ t − = ζ t − }.
We introduce the following definition.Definition 2.1 A progressive process (φ t ) (resp.integrable) is said to be left-upper semicontinuous (l.u.s.c.) along stopping times (resp.along stopping times in expectation ) if for all τ ∈ T 0 and for each non decreasing sequence of stopping times (τ n ) such that τ n ↑ τ a.s., (2.6) {usc} Remark 2.5 Note that when (φ t ) is left-limited, then (φ t ) is left-upper semicontinuous (l.u.s.c.) along stopping times if and only if for all predictable stopping time τ ∈ T 0 , φ τ ≥ φ τ − a.s.
3 Classical Dynkin games and links with doubly reflected BSDEs with a driver process In this section, we are given a predictable process g = (g t ) in H 2 .Let ξ and ζ be two adapted processes only supposed to be RCLL with We state that the doubly reflected BSDE associated with the driver process (g t ) and the barriers ξ and ζ admits a unique solution (Y, Z, k(•), A, A ′ ), which is related to a classical Dynkin game problem defined below.Our results complete the previous works on classical Dynkin games and DRBSDEs ( see for e.g.[6], [11]).In particular, we provide an existence result of saddle points under weaker assumptions than those made in the previous literature.
For any S ∈ T 0 and any stopping times τ, σ ∈ T S , consider the gain (or payoff): We introduce the following RCLL adapted processes which depend on the process g: They satisfy the important property ξg T = ζg T = 0 a.s.Moreover, this change of variables allows us to get rid of the term g(t)dt, and thus to simplify the notation.Some more comments on this change of variables are given in Remark 7.1 in the Appendix.
For each RCLL adapted process φ = (φ t ) 0≤t≤T valued in R ∪ {+∞} with φ − ∈ S 2 , we denote by R(φ) the Snell envelope of φ, defined as the minimal RCLL supermartingale greater or equal to φ a.s.By the optimal stopping theory, R(φ) is equal to the value function of the optimal stopping problem associated with the reward φ.
We state the following lemma.
Lemma 3.2 There exists a unique pair of non-negative RCLL supermartingales (J g , J ′g ) valued in [0, +∞] satisfying J g T = J ′g T = 0 a.s. and the system A sketch of the proof is given in the Appendix.Using this lemma, we derive the following result.
Theorem 3.3 Let ξ and ζ be two adapted RCLL processes in S 2 with ζ T = ξ T a.s. and ξ t ≤ ζ t , 0 ≤ t ≤ T a.s.Suppose that J g , J ′ g ∈ S 2 .Let Y be the RCLL adapted process defined by

.12) {oY}
There exist ) is a solution of DRBSDE (2.5) associated with the driver process g(t).

Proof.
By assumption, J g and J ′ g are square integrable supermartingales.The process Y is thus well defined.By Lemma 3.2, we have J g T = J ′g T a.s.Hence, Y T = ξ T a.s.By the Doob-Meyer decomposition, there exist two square integrable martingales M and M ′ and two processes B and B ′ ∈ A 2 such that: Let us now show that B, B ′ satisfy the Skorohod conditions (2.5)(iii).By the optimal stopping theory (see e.g.Proposition B.1 in [16]), the process B c increases only when the value function J g is equal to the corresponding reward J ′ g + ξg .Now, t = 0 a.s.Moreover, for each predictable stopping time τ ∈ T 0 we have ).Moreover, H − H ′ = J − J ′ .Hence, we have H ≥ H ′ + ξg and H ′ ≥ H − ζg .By the minimality property of J, J ′ (see Lemma 3.2), we derive that From this theorem, we derive the following uniqueness and existence result for the DRB-SDE associated with the driver process (g t ), as well as the characterization of the solution as the value function of the above Dynkin game problem.We also show that if the associated non decreasing processes A and A ′ are continuous, then there exist saddle points for this game problem.Theorem 3.5 Let ξ and ζ be two adapted RCLL processes in S 2 with ζ T = ξ T a.s. and ξ t ≤ ζ t , 0 ≤ t ≤ T a.s.Suppose that J g t , J ′g t ∈ S 2 .The doubly reflected BSDE (2.5) associated with driver process g(t) admits a unique solution For each S ∈ T 0 , Y S is the common value function of the Dynkin game, that is Moreover, if the processes A, A ′ are continuous, then, for each S ∈ T 0 , the pair of stopping times (τ * s , σ * s ) defined by A short proof is given in the Appendix.Note that the uniqueness of the non decreasing RCLL processes A and A ′ holds because of the constraint dA t ⊥ dA ′ t (see the Appendix for details).
We now provide a sufficient condition on ξ and ζ for the existence of saddle points.By the last assertion of Theorem 3.5, it is sufficient to give a condition which ensures the continuity of A and A ′ .Theorem 3.7 (Existence of S-saddle points) Suppose that the assumptions of Th. 3.5 are satisfied and that ξ and −ζ are l.u.s.c.along stopping times.Let (Y, Z, k(.), A, A ′ ) be the solution of DRBSDE (2.5).The processes A and A ′ are then continuous.Also, for each S ∈ T 0 , the pair of stopping times (τ * S , σ * S ) defined by (3.15) is an S-saddle point.
Remark 3.8 The assumptions made on ξ and ζ are milder than the ones made in the literature where it is also supposed ξ t < ζ t , t < T a.s.( see e.g.[1], [6], [17]).
Proof.By the second assertion of Theorem 3.5, it is sufficient to prove that A and A ′ are continuous.Let τ ∈ T 0 be a predictable stopping time.Let us show ∆A τ = 0 a.s.By Remark 3.6, we have ∆A τ = (∆Y τ ) − a.s.Since dA t ⊥ dA ′ t , there exists D ∈ P such that: The last inequality follows from the inequality ξ τ − ≤ ξ τ a.s.(see Remark 2.5).Since ξ ≤ Y , we derive that ∆A τ ≤ 0 a.s.Hence, ∆A τ = 0 a.s., and this holds for each predictable stopping time τ .Consequently, A is continuous.Similarly, one can show that A ′ is continuous.
Since J g ≥ J ′ g + ξg and J ′ g ≥ J g − ζg , the condition J g ∈ S 2 is equivalent to the condition J ′ g ∈ S 2 .We now recall the definition of Mokobodski's condition.
Proof.Using the minimality property of J and J ′ given in Lemma 3.2, one can show that J g ∈ S 2 if and only if there exist two non-negative supermartingales

.17) {Mokobis}
Since this equivalence holds for all g ∈ IH 2 , in particular when g = 0, we get (ii) ⇔ (iii).It remains to show (i) ⇔ (ii) For this, it is sufficient to show that (3.16) is equivalent to (3.17).Suppose that (3.16) is satisfied.By setting .17) holds.It remains to prove that (iv) implies (i).Let (Y, Z, k, A, A ′ ) be the solution of the DRBSDE (2.5) associated with driver process (g t ).Let

Generalized Dynkin games and links with doubly reflected BSDEs with a non linear driver
In this section, we are given a Lipschitz driver g.

Generalized Dynkin games
In this section, we introduce a generalized Dynkin game expressed in terms of g-conditional expectations.
In order to ensure that the g-conditional expectation E is non decreasing, we make the following assumption.
, there exists a bounded predictable process where ψ ∈ L 2 ν , and such that For example, this assumption is satisfied if g is C 1 with respect to ℓ with ∇ ℓ g ≥ −1 and |∇ ℓ g| ≤ ψ, where ψ ∈ L 2 ν (see Lemma 7.2 in the Appendix).Moreover, the above assumption ensures the non decreasing property of E g by the comparison theorem for BSDEs with jumps (see Theorem 4.2 in [21]).In the case when in (4.18), we have γ t > −1, by the strict comparison theorem (see Theorem 4.4 in [21]), it follows that E g is strictly monotonous.
We now introduce the following game problem, which can be seen as a Dynkin game written in terms of g-conditional expectations.
For each τ, σ ∈ T 0 , the reward at time τ ∧ σ is given by the random variable Let S ∈ T 0 .For each τ ∈ T S and σ ∈ T S , the associated criterium is given by E S,τ ∧σ (I(τ, σ)), the g-conditional expectation of the reward is the solution of the BSDE associated with driver g, terminal time τ ∧ σ and terminal condition I(τ, σ), that is s (e) Ñ(ds, de); X τ,σ τ ∧σ = I(τ, σ).
At time S, the first (resp.second) player chooses a stopping time τ (resp.σ) greater than S, and looks for maximizing (resp.minimizing) the criterium.
For each stopping time S ∈ T 0 , the upper and lower value functions at time S are defined respectively by V (S) := ess inf We clearly have the inequality V (S) ≤ V (S) a.s.By definition, we say that there exists a value function at time S for the generalized Dynkin game if V (S) = V (S) a.s.
We now introduce the definition of an S-saddle point for this game problem.
S is called an S-saddle point for the generalized Dynkin game if for each (τ, σ) ∈ T 2 S we have We first provide a sufficient condition for the existence of an S-saddle point and for the characterization of the common value function as the solution of the DRBSDE.
By inequality (4.24) and the monotonicity property of E, we derive inequality (4.23).
Similarly, one can show that for each σ ∈ T S , we have: The pair (τ , σ) is thus an S-saddle point and Y S = V (S) = V (S) a.s.
Remark 4.5 When f does not depend on y, z, k, from the above proposition, one can derive a well-known sufficient condition of optimality for classical Dynkin game problems ( see e.g.Theorem 2.4 in [1] or Proposition 3.1.in [17]).
We now provide an existence result under an additional assumption.
Then, for each S ∈ T 0 , the pairs of stopping times (τ * S , σ * S ) and (τ S , σ S ) are S-saddle points for the generalized Dynkin game and We now see that it is not necessary to have the existence of an S-saddle point to ensure the existence of a common value function and its characterization as the solution of a DRBSDE.Theorem 4.8 (Existence of the value function) Suppose that g satisfies Assumption (4.1).Let ξ and ζ be RCLL adapted processes in S 2 such that ξ T = ζ T a.s. and ξ t ≤ ζ t , 0 ≤ t ≤ T a.s.Suppose that Mokobodski's condition is satisfied.Let (Y, Z, k, A, A ′ ) be the solution of the DRBSDE (2.5).There exists a value function for the generalized Dynkin game, and for each stopping time S ∈ T 0 , we have Proof.For each S ∈ T 0 and for each ε > 0, let τ ε S and σ ε S be the stopping times defined by
Lemma 4.9 where the last inequality follows from the definition of I(τ, σ).Hence, using (4.31) and the monotonicity property of E, we get

Generalized mixed game problems
We now consider a generalized mixed game problem when the players have two actions: continuous control and stopping.
Let S ∈ T 0 .For each quadruple (u, τ, v, σ) ∈ U × T S × V × T S , the criterium at time S is given by E u,v S,τ ∧σ (I(τ, σ)), where E u,v corresponds to the g u,v -conditional expectation.The first (resp.second) player chooses a pair (u, τ ) (resp.(v, σ)) of control and stopping time, and looks for maximizing (resp.minimizing) the criterium.We say that there exists a value function at time S for the game problem if V (S) = V (S) a.s.We now introduce the definition of an S-saddle point for this game problem.
We now show that when the obstacles are supposed to be l.u.s.c.along stopping times, there exist some saddle points for the above generalized mixed game problem.
Theorem 4.14 Let (g u,v ; (u, v) ∈ U ×V) be a family of Lipschitz drivers satisfying Assumptions (4.1).Let ξ and ζ be RCLL adapted processes in S 2 and l.u.s.c.along stopping times, such that ξ T = ζ T a.s. and ξ t ≤ ζ t , 0 ≤ t ≤ T a.s.Suppose that Mokobodski's condition is satisfied and that there exist controls u ∈ U and v ∈ V such that for each (u, v) ∈ U × V, where (Y, Z, k, A, A ′ ) corresponds to the solution of the DRBSDE (2.5) associated with driver g u,v .Consider the stopping times The quadruple (u, τ * S , v, σ * S ) is then an S-saddle point for the generalized mixed game problem (4.33)-(4.34),and we have Y S = V (S) = V (S) a.s.
Proof.By the last assertion of Theoreom 4.6, the process Moreover, by Theorem 4.6, A ′ σ * S = A ′ s a.s., which implies that: Similarly, one can show that for each v ∈ V, σ ∈ T S , we have: The quadruple (u, τ * S , v, σ * S ) is thus an S-saddle point and Y S = V (S) = V (S) a.s.
Under less restricted assumptions on the obstacles, we show that there exist a value function for the above game problem which can be characterized as the solution of a DRBSDE.Theorem 4.15 (Existence of the value function) Let (g u,v ; (u, v) ∈ U × V) be a family of drivers satisfying Assumptions (4.1) and which are uniformly Lipschitz with common Lipchitz constant C. Let ξ and ζ be RCLL adapted processes in S 2 such that ξ T = ζ T a.s. and ξ t ≤ ζ t , 0 ≤ t ≤ T a.s.Suppose that Mokobodski's condition is satisfied and that there exist controls u ∈ U and v ∈ V such that for each u ∈ U, v ∈ V: associated with where (Y, Z, k, A, A ′ ) corresponds to the solution of the DRBSDE (2.5) associated with driver g u,v .Then, there exists a value function for the generalized mixed game problem (4.33)-(4.34),and for each stopping time S ∈ T 0 , we have Proof.For each S ∈ T 0 and for each ε > 0, let τ ε S and σ ε S be the stopping times defined by ( see Lemma 4.9), we have: By Lemma 4.9, A ′ σ ε S = A ′ S a.s.which implies that: Hence, (Y t ) S≤t≤τ ∧σ ε is the solution of the BSDE associated with generalized driver "f (•)dt + dA t " and terminal condition Y τ ∧σ ε .By using Assumption (4.36), the inequality Y τ ∧σ ε ≥ I(τ, σ ε ) − ε and the comparison theorem for BSDEs with jumps, we obtain
Let ξ and ζ be RCLL adapted processes in S 2 such that ξ T = ζ T a.s. and ξ t ≤ ζ t , 0 ≤ t ≤ T a.s.Suppose that Mokobodski's condition is satisfied.Let us consider the associated generalized mixed game problem.Define for each (t, ω, y, z, k) the map g(t, ω, y, z, k) = sup

.37) {ff}
Since U and V are polish spaces, there exist some dense countable subsets U (resp.V ) of U (resp.V ).Since F is continuous with respect to u, v, the sup and the inf can be taken over U (resp.V ).Hence, g is a Lipchitz driver.
) 2 be the solution of the DRBSDE associated with driver g and obstacles ξ and ζ.By classical convex analysis, for each (t, ω) there exist (u * , v * ) ∈ (U, V ) such that Since the set of all (t, ω, u * , v * ) ∈ [0, T ] × Ω × U × V satisfying conditions (4.38) belongs to P × B(U) × B(V ), by applying the section theorem (see Section 81 in the Appendix of Ch.III in [7]), we get that there exists a pair of predictable process (u * , v * ) ∈ U × V such that dt ⊗ dP a.s., for all (u, v) ∈ U × V we have dt ⊗ dP a.s.: Hence, Assumption (4.35) is satisfied.By applying Theorems 4.15 and 4.14, we derive the following result: Proposition 4.17 There exists a value function for the above generalized mixed game problem (associated with the map F (t, u, v, y, z, k)).Let Y be the solution of the DRBSDE associated with obstacles ξ, ζ and the driver g defined by (4.37).For each stopping time S ∈ T 0 , we have Y S = V (S) = V (S) a.s.Suppose that ξ and ζ are l.u.s.c.along stopping times.Consider the stopping times is then an S-saddle point for this mixed game problem.
We give now an example for which the above proposition can be applied.
Example: Let us now consider the particular case when F takes the following form: By classical results on linear BSDEs (see [21]), the criterium can be written with Q u,v the probability measure which admits Z u,v T as density with respect to P , where (Z u,v t ) is the solution of the following SDE: The process c(t, u t , v t ) can be interpreted as an instantaneous reward associated with controls u, v.This linear model takes into account some ambiguity on the model via the probability measures Q u,v as well as some ambiguity on the instantaneous reward.This case corresponds to the classical mixed game problems studied in [3] and [21].The above proofs provide some alternative short proofs of their results.
Remark 5.1 Note that a comparison theorem has been provided in [5] in the case of jumps under stronger assumptions.Their proof is different and based on Ito's calculus. Proof.
We give a short proof based on the characterization of solutions of DRBSDEs (Theorem 4.8) via generalized Dynkin games.Let t ∈ [0, T ].For each τ, σ ∈ T t , let us denote by E i .,τ∧σ (I i (τ, σ)) the unique solution of the BSDE associated with driver g i , terminal time τ ∧ σ and terminal condition Since g 2 ≤ g 1 , and I 2 (τ, σ) ≤ I 1 (τ, σ), by the comparison theorem for BSDEs, the following inequality holds for each τ , σ in T t .Hence, by taking the essential supremum over τ in T t and the essential infimum over σ in T t , and by using Theorem 4.8, we get We now provide a strict comparison theorem, which had not been given in the literature even in the Brownian case.The first assertion addresses the particular case when the non decreasing processes are continuous and the second one deals with the general case.Theorem 5.2 (Strict comparison.)Suppose that the assumptions of Theorem 5.1 hold and that the driver g 1 satisfies Assumption 4.1 with and 2. Consider the case when A i , A ′ i , i = 1, 2 are not necessarily continuous.For i = 1, 2, define for each ε > 0, Proof.We adopt the same notation as in the proof of the comparison theorem.Suppose first that A i , A ′ i , i = 1, 2 are continuous.By Theorem 4.6, for i = 1, 2, (τ i , σ i ) is a saddle point for the game problem associated with g s. Now, we apply the strict comparison theorem for non reflected BSDEs with jumps (see [21], Th 4.4) for terminal time θ.Hence, we get Y 1 t = Y 2 t , S ≤ t ≤ θ a.s., as well as equality (7), which provides the desired result.
Consider now the general case.Let ε > 0. By Remark 4.10, (Y i t , S ≤ t ≤ τ ε i ∧ σ ε i ) is an E i martingale.Hence we have

and equality (7) holds on [S, τ
, dt ⊗ dP -a.s.By letting ε tend to 0, we obtain the desired result.
We now give an application of the above comparison theorem to a control game problem for DRBSDEs.

Some new estimates
Using the links between generalized Dynkin games and DRBSDEs (see Theorem 4.8), we prove the following estimates.
Suppose that for i = 1, 2, ξ i and ζ i satisfy Mokobodski's condition.Let g 1 , g 2 be Lipschitz drivers satisfying Assumption 4.1 with Lipschitz constant C > 0. For i = 1, 2, let Y i be the solution of the DRBSDE associated with driver g i , terminal time T and barriers Then for each t, we have: (5.41) {eqA.1} Remark 5.4 The constants η and β are universal, i.e. they do not depend on T , ξ 1 , ξ 2 , g 1 , g 2 .Note that in the previous literature, there does not exist any result providing estimates on DRBSDEs, even in the Brownian case.
We also provide the following estimate on the common value function Y of our generalized Dynkin game problem ((4.21) and (4.22)) (or equivalently the solution of the DRBSDE associated with driver g).Proposition 5.5 For each t, we have: e β(s−t) g(s, 0, 0, 0) 2 ds|F t ] a.s. (5.45) Proof.Let X τ,σ t be the solution of the BSDE associated with driver g, terminal time τ ∧ σ and terminal condition I(τ, σ).By applying inequality (5.42) with (5.46) {A.5} By using the same procedure as in the proof of Proposition 5.3, the result follows.
We now study the links between generalized Dynkin games (or equivalently DRBSDEs) and obstacle problems, which complete the results of this paper.

Relation with partial integro-differential variational inequalities (PIDVI)
We now restrict ourselves to the Markovian case.Let b : R → R , σ : R → R be continuous mappings, globally Lipschitz and β : R × E → R a measurable function such that for some non negative real C, and for all e ∈ E x s , t ≤ s ≤ T ) be the unique R-valued solution of the SDE with jumps: and set X t,x s = x for s ≤ t.We consider the DRBSDE associated with obstacles ξ t,x , ζ t,x of the following form: ξ t,x s := h 1 (s, X t,x s ), ζ t,x s := h 2 (s, X t,x s ), s < T , ξ t,x T = ζ t,x T := g(X t,x T ).We suppose that g ∈ C(R), h 1 , h 2 : [0, T ] × R → R are jointly continuous in t and x, and that g, h 1 , h 2 have at most polynomial growth with respect to x.Moreover, the obstacles ξ t,x s and ζ t,x s are supposed to satisfy Mokobodski's condition, which holds if for example h 1 and h 2 are C 1,2 .
We consider two functions γ and f satisfying Assumption 2.1 in [9].More precisely, we are given a map γ : R × ν → R be a map supposed to be continuous in t uniformly with respect to x, y, z, k, and continuous in x uniformly with respect to y, z, k.It is also supposed to be uniformly Lipschitz with respect to y, z, k, and such that f (t, x, 0, 0, 0) at most polynomial growth with respect to x.It also satisfies that for each t, x, y, z, k The driver is defined by f (s, X t,x s (ω), y, z, k).By Theorem 4.1, for each (t, x) ∈ [0, T ] × R, there exists an unique solution (Y t,x , Z t,x , K t,x , A t,x , A ′ t,x ) of the associated DRBSDE.Moreover, by definition, ξ t,x and −ζ t,x are l.u.s.c.along stopping times.It follows that the processes A t,x , A ′ t,x are continuous.We define: which is a deterministic quantity.In the following, the map u is called the value function of the generalized Dynkin game.By the a priori estimates (see Propositions 5.3 and 5.5) and the same arguments as those used in the proofs of Lemma 3.1 and Lemma 3.2 in [9], we derive that the value function u is continuous in (t, x) and has at most polynomial growth at infinity.
Following the same arguments as in the proof of Theorem 3.4 in [9], one can show that u is viscosity subsolution of (6.48).By symmetry, we derive that u is also a viscosity supersolution of (6.48), which yields the following result: Theorem 6.2 The value function u is a viscosity solution (i.e. both a viscosity sub-and supersolution) of the obstacle problem (6.48).
In the sequel, we suppose that E = R * and that the function ϕ is defined by ϕ(e) := 1∧|e| and is supposed to belong in L 2 ν .We also suppose that g, h 1 and h 2 are bounded, and that Assumption 4.1 in [9] holds.More precisely, (i) f (s, X t,x s (ω), y, z, k) := f s, X t,x s (ω), y, z, R * k(e)γ(X t,x s (ω), e)ν(de) 1 s≥t , where f : [0, T ] × R 4 → R is a map which is continuous with respect to t uniformly in x, y, z, k, and continuous with respect x uniformly in y, z, k.It is also uniformly Lipschitz with respect to y, z, k and the map f (t, x, 0, 0, 0) is uniformly bounded.The map k → f (t, x, y, z, k) is also non-decreasing, for all t ∈ [0, T ], x, y, z ∈ R. (ii) For each R > 0, there exists a continuous function m R : R x, u, v, p, l ∈ R, where r > 0. To simplify notation, in the sequel, f is denoted by f .The operator B has now the following form: Bφ(x) := R * (φ(x+β(x, e))−φ(x))γ(x, e)ν(de).Theorem 6.3 (Comparison principle) If U is a bounded viscosity subsolution and V is a bounded viscosity supersolution of the obstacle problem (6.48), then U(t, x) ≤ V (t, x), for each (t, x) ∈ [0, T ] × R.
Here, we have to consider four cases.1st case: there exists a subsequence of (t η ) such that t η = T for all η ( of this subsequence) 2nd case: there exists a subsequence of (t η ) such that t η = T and for all η belonging to this subsequence, there exist a subsequence of (x ǫ,η ) ǫ and a subsequence of (t ǫ,η ) ǫ , such that U(t ǫ,η , x ǫ,η ) − h 1 (t ǫ,η , x ǫ,η ) = 0. 3rd case: there exists a subsequence such that t η = T , and for all η belonging to this subsequence, there exist a subsequence of (y ǫ,η ) ǫ and a subsequence of (s ǫ,η ) ǫ , such that V (s ǫ,η , y ǫ,η ) − h 2 (s ǫ,η , y ǫ,η ) = 0.Last case: we are left with the case when, for a subsequence of η we have t η = T , and for all η belonging to this subsequence, there exist a subsequence of (x ǫ,η ) ǫ , (y ǫ,η ) ǫ , (t ǫ,η ) ǫ and (s ǫ,η ) ǫ such that The first, second and fourth case are identical to the three cases considered for reflected BSDEs (see [9]).The third one, which didn't appear in the case of reflected BSDEs, can be treated similarly to the second one.
We derive that there exists an unique solution of the obstacle problem (6.48) in the class of bounded continuous functions.

Appendix
Proof of Lemma 3.2: For completeness, we give a sketch of the proof, where we draw attention to the importance of the property ξg T = ζg T = 0 a.s.Set J (0) • = 0 and J ′(0) • = 0 and define recursively for each n ∈ N, the supermartingales: which belong to S 2 .For sake of simplicity, in the above definition we have omitted the exposant g in the definition of J (n) .Since ξg T = ζg T = 0 a.s., it follows that, for each n, J We have J (0) = 0 and J ′ (0) = 0. Let us prove recursively that for each n, J ′ (n) , J (n) are well defined and nonnegative.Suppose that J ′ (n) , J (n) are well defined and nonnegative.Then J (n+1) , J ′ (n+1) are well defined since (J
Proof of Theorem 3.5: We have already proved the existence.Let (Y, Z, k, A, A ′ ) be a solution of the DRBSDE associated with driver process g(t) and obstacles (ξ, ζ).Let us prove that it is unique.We first show the uniqueness of Y .For each S ∈ T 0 and for each ε > 0, let We also have that S) a.s.This equality holds of each stopping time S ∈ T 0 , which implies the uniqueness of Y .It remains to show the uniqueness of (Z, k, A, A ′ ).By the uniqueness of the decomposition of the semimartingale Y t + t 0 g(s)ds, there exists an unique square integrable martingale M and an unique square integrable finite variation RCLL adapted process α with α 0 = 0 such that dY t + g(t)dt = dM t − dα t .The martingale representation theorem applied to M ensures the uniqueness of the pair (Z, k) ∈ IH 2 × IH 2 ν .The uniqueness of the processes A, A ′ follows from the uniqueness of the canonical decomposition of an RCLL process with integrable variation (see Proposition 7.5) β into itself as follows.Given (U, V, l) ∈ IH 2 β , by Theorem 3.5 there exists a unique process (Y, Z, k) = Φ(U, V, l) solution of the DRBSDE associated with driver process g(s) = g(s, U s , V s , l s ).Note that (Y, Z, k) ∈ IH 2 β .Let A, A ′ be the associated non decreasing processes.Let us show that Φ is a contraction and hence admits a unique fixed point (Y, Z, k) in IH 2 β , which corresponds to the unique solution of DRBSDE (2.5).The associated finite variation process is then uniquely determined in terms of (Y, Z, k) and the pair (A, A ′ ) corresponds to the unique canonical decomposition of this finite variation process.Let (U 2 , V 2 , l 2 ) be another element of IH s ≤ 0 a.s.Consequently, the second and the third term of (7.54) are non positive.By using the Lipschitz property of g and the inequality 2Cyu ≤ 2C 2 y 2 + 1 2 u 2 , we get Choosing β = 6C 2 + 1, we deduce (Y , Z, k) 2 β ≤ 1 2 (U, V , l) 2 β .The last assertion of the theorem follows from Theorem 3.7.Theorem 4.2 in [21]) gives that Y σ ≥ X τ σ = E σ,τ (Y τ ) a.s. on {σ ≤ τ }.The case when A is non-increasing can be shown similarly.
Let us show the second assertion.Fix S ∈ T 0 .Since (Y t ) is a strong E-supermartingale, we derive that for each τ ∈ T S , we have Y S ≥ E S,τ (Y τ ) a.s.We thus get Y S ≥ ess sup By the characterization theorem (Theorem 3.3 in [22]) of the solution of a reflected BSDE (associated with an obstacle supposed to be only RCLL), the process (Y t ) coincides with the solution of the reflected BSDE associated with the RCLL obstacle (Y t ).The result follows.
We now show the following result on RCLL adapted processes with integrable total variation.Proposition 7.5 Let (Ω, F , P ) be a probability space equipped with a completed right-continuous filtration (F t ) 0≤t≤T .Let α = (α t ) 0≤≤T be a RCLL adapted process with integrable total variation, that is, E( There exists an unique pair (A, A ′ ) ∈ (A 1 ) 2 such that α = A − A ′ with dA t ⊥ dA ′ t .This decomposition is called the canonical decomposition of the process α.Moreover, if (B, B ′ ) ∈ (A 1 ) 2 satisfies α = B − B ′ , then dA t << dB t in the (probabilistic) sense, that is, for each K ∈ P with T 0 1 K dB t = 0 a.s., then T 0 1 K dA t = 0 a.s.
Proof.By classical results, the process α can be written as α = B − B ′ with B, B ′ ∈ A 1 .Let C t := B t + B ′ t .This process belongs to A 1 .For almost every ω, the measures dB • (ω) and dB ′ • (ω) on [0, T ] are absolutely continuous with respect to dC • (ω).By using the Radon-Nikodym Theorem for predictable RCLL non decreasing processes (see Th. 67, Chap.VI in [8]), there exist non negative predictable processes H and H ′ such that for each t ∈ [0, T ], .15) {qqq} is an S-saddle point for the Dynkin game problem associated with the gain I S .Remark 3.6 For each predictable stopping time τ ∈ T 0 , we have ∆A d τ = (∆Y τ ) − and ∆A ′ d τ = (∆Y τ ) + a.s.

Assumption 4 . 1 A
lipschitz driver g is said to satisfy Assumption 4.1 if for each process

Proposition 5 . 1 (
Control game problem for DRBSDEs) Suppose that the assumptions of Th. 4.15 hold.For each (u, v) ∈ U × V, let Y u,v be the solution of the DRBSDE (2.5) associated with driver g u,v .Then, for eachS ∈ T 0 , Y u,v S ≤ Y u,v S ≤ Y u,vS a.s.Proof.By using Assumption (4.35) and by applying the comparison theorem for DRBS-DEs (Th.5.1), we get that for each u ∈ U, Y u,v S ≤ Y u,v S a.s.Similarly, for all v ∈ V, we have Y u,v S ≤ Y u,v S a.s.Remark 5.2 From this result, it follows that, under the Assumption (4.35), the value function of the above control game problem for DRBSDEs coincides with the one associated with the generalized mixed game problem studied in Section 4.3.

Remark 7 . 1
We point out that the property ξg T = ζg T = 0 a.s.ensures that for each n, J
Let ξ and ζ be RCLL adapted processes in S 2 such that ξ T = ζ T a.s. and ξ t ≤ ζ t , 0 ≤ t ≤ T a.s.Suppose that Mokobodski's condition is satisfied.Let (Y, Z, k, A, A ′ ) be the solution of the DRBSDE (2.5).Suppose that A, A ′ are continuous (which is the case if ξ and −ζ are l.u.s.c.along stopping times).For each S ∈ T 0 , consider .s.The same properties hold for τ S , σ S .It remains to show that (τ S , σ S ) is an S-saddle point.By definition of τ S , σ S , we have A τ S = A S a.s. and A ′ σ S = A ′ S a.s.because A and A ′ are continuous and τ S , σ S are predictable stopping times.Moreover, since the continuous process A increases only on {Y t = ξ t }, we have Y τ S = ξ τ S a.s.Similarly, Y σ S = ζ σ S a.s.The result then follows from Lemma 4.4.
* S ≤ σ S and τ * S ≤ τ S a.s.Moreover, by Proposition 7.4 in the Appendix, (Y t , S ≤ t ≤ τ S ) is a strong E-submartingale and (Y t , S ≤ t ≤ σ S ) is a strong E-supermartingale.Proof.Let S ∈ T 0 .Since Y and ξ are right-continuous processes, we have Y σ * S = ξ σ * S and Y τ * S = ξ τ * S a.s.By definition of τ * S , for almost every ω, we have Y t (ω) > ξ t (ω) for each t ∈ [S(ω), τ * S (ω)[.Hence, since Y is solution of the DRBSDE, the continuous process A is constant on [S, τ * S ] a.s.because A is continuous.Similarly, the process A ′ is constant on [S, σ * S ] a.s.By Lemma 4.4, (τ * S , σ * S ) is an S-saddle point and Y S = V (S) = V (S) a.s.
S≤t≤τ ∧σ * S is the solution of the BSDE associated with generalized driver "g u,v (•)dt+ dA t " and terminal condition Y τ ∧σ * S .By using Assumption (4.35), the inequality Y τ ∧σ * S ≥ I(τ, σ * S ) and the comparison theorem for BSDEs with jumps, we obtain that for each u ∈ U: Y t , Z t , k t )dt + dA t − Z t dW t − E k t (e) Ñ(dt, de); S ≤ t ≤ σ * S , dt ⊗ dP a.s.Hence, (Y t ) s. , where the second inequality follows from the a priori estimates for BSDEs with jumps.Here, the constant K only depends on T and C, the common Lipschitz constant.Consequently, we get Y S ≥ ess inf Remark 4.16 Note that Theorem 4.15 still holds if g u,v is replaced by any Lipschitz driver g which satisfies (4.36).