Irregular barrier reflected BDSDEs with general jumps under stochastic Lipschitz and linear growth conditions

In this paper, a solution is given to reflected backward doubly stochastic differential equations when the barrier is not necessarily right-continuous, and the noise is driven by two independent Brownian motions and an independent Poisson random measure. The existence and uniqueness of the solution is shown, firstly when the coefficients are stochastic Lipschitz, and secondly by weakening the conditions on the stochastic growth coefficient.


Introduction
Nonlinear backward stochastic differential equations (BSDEs in short) were first introduced by Pardoux and Peng [27] with the uniform Lipschitz condition under which they proved the celebrated existence and uniqueness result. Since then, the theory of BSDEs has been intensively developed in the last years. The great interest in this the-ory comes from its connections with many other fields of research, such as mathematical finance [12,11], stochastic control and stochastic games [10] and partial differential equations [28]. After Pardoux and Peng introduced the theory of BSDEs, they considered [29] a new kind of BSDEs, that is a class of backward doubly stochastic differential equations (BDSDEs in short) with two different directions of stochastic integrals with respect to two independent Brownian motions. They proved the existence and uniqueness of solutions to BDSDEs under uniform Lipschitz conditions on the coefficients.
In the setting of reflected BSDEs (resp. BDSDEs), an additional nondecreasing process is added in order to keep the solution above a certain lower-boundary process, called barrier (or obstacle), and to do this in a minimal fashion. The reflected BSDEs (RBSDEs in short) were introduced by El Karoui et al. [13], again under the uniform Lipschitz condition on the coefficients. The authors of [13] proved the existence and uniqueness results in the case of a Brownian filtration and a continuous barrier. The reflected BDSDEs (RBDSDEs in short) were introduced by Bahlali et al. [6] where the authors studied the case of RBDSDEs with continuous coefficients, and proved the existence and uniqueness of the solution.
To the best of our knowledge, the paper by Grigorova et al. [14] is the first one which studied RBSDEs in the case where the barrier is not necessarily rightcontinuous (just right upper semi-continuous). The authors of [14] studied the existence and uniqueness result under the Lipschitz assumption on the coefficients in a filtration that supports a Brownian motion and an independent Poisson random measure. Later, several authors have studied the RBSDEs following Grigorova et al. [14] (see e.g. [1-3, 17, 20, 23]). Recently, Berrhazi et al. [7] discussed the case of RBDSDE with a right upper semi-continuous barrier under Lipschitz coefficients.
Our aim in this paper is to extend the work on RBDSDEs with jumps (RBDS-DEJs in short) to the case of an irregular barrier (which is assumed to be not necessarily right-continuous). The specificity of such equations lies in the fact that the two independent Brownian motions are coupled with an independent Poisson random measure. We'll prove the existence and uniqueness of the solution to such equations under the so-called stochastic Lipschitz coefficients. The interest in this last condition is based on the fact that, unfortunately, in many applications, the usual Lipschitz conditions cannot be satisfied. For example, the pricing of the American claim is equivalent to solving the linear RBDSE where ξ t is the amount received from the seller at time t, r t is the interest rate process and θ t is the risk premium process. The additional process K is needed for this problem because there exists no replicating strategy for the option. We have to use a super-replicating strategy with a consumption process K. The minimality condition on K just states that we only invest money in the portfolio when V t > ξ t . Here both r t and θ t are not bounded in general. So, it is not possible to solve the RBSDE (1) by the result of El Karoui et al. [13]. Thus, in order to study more general RBSDEs (resp. RBDSDEs), one needs to relax the uniform Lipschitz conditions on the coefficients. To this direction, several attempts have been done. Among others, we refer to [4,5,9,15,[21][22][23][24] for the case of BSDEs, and [16,25,26,30] for BDSDEs. In our paper, we use a generalization of the Doob-Meyer decomposition called the Mertens decomposition. This decomposition is used for strong optional supermartingales which are not necessarily right-continuous. We also use some tools from the optimal stopping theory, as well as a generalization of the Itô formula to the case of a strong optional supermartingale called the Gal'chouk-Lenglart formula due to Lenglart [19].
The paper is organized as follows. In Section 2, we give some notations, assumptions and main contributions needed in this paper. In Section 3, we prove the existence and uniqueness of the solution to RBDSDEJs with a stochastic Lipschitz coefficients (f, g) and an irregular barrier ξ, and we also give a comparison theorem for solutions. Section 4 is devoted to prove the existence of a minimal solution to RBDSDEJs under a stochastic growth coefficient f .

Definitions and preliminary results
Let 0 < T < +∞ be a non-random horizon time, Ω be a non-empty set, F be a σ-algebra of sets of Ω and P be a probability measure defined on F . The triple (Ω, F , P) defines a probability space which is assumed to be complete. We assume there are three mutually independent processes: We consider the family (F t ) t≤T given by where for any process Here N denotes the class of P-null sets of F . Note that the family (F t ) t≤T does not constitute a classical filtration.
For an integer k ≥ 1, | . | and ., . stand for the Euclidian norm and the inner product in R k , T [t,T ] denotes the set of stopping times τ such that τ ∈ [t, T ] and P denotes the σ-algebra of F t -predictable sets of Ω × [0, T ].
For every F t -measurable process (a t ) t≤T , we define an increasing process (A t ) t≤T by setting A t = t 0 a 2 s ds. For every β > 0, we consider the following sets (where E denotes the mathematical expectation with respect to the probability measure P): For a làdlàg (limited from right and left) process (Y t ) t≤T , we denote by: and (ξ t ) t≤T be an optional process which is assumed to be right upper semi-continuous and limited from left. The process (ξ t ) t≤T will be called irregular barrier. We are interested in the following RBDSDEJs associated with parameters (f, g, ξ): (2) Here Θ s stands for the triple (Y s , Z s , U s ).
Let us consider the filtration (G t ) t≤T given by G t = F W t ∨F B T ∨F µ t , 0 ≤ t ≤ T which is assumed to be right-continuous and quasi-left-continuous, and make precise the notion of solution to RBDSDEJ (2). Definition 1. Let ξ be an irregular barrier. A process (Y, Z, U, K, C) is called a solution to RBDSDEJ associated with parameters (f, g, ξ), if it satisfies the system (2) and Remark 2.1. We note that a process (Y, Z, U, K, C) ∈ B 2 β (R k ) × S 2 (R k ) × S 2 (R k ) satisfies the equation (2) if and only if Remark 2.2. If (Y, Z, U, K, C) is a solution to RBDSDEJ (2), then ∆C t = Y t − Y t+ for all t ≤ T outside an evanescent set. It follows that Y t ≥ Y t+ for all t ≤ T , which implies that Y is necessarily right upper semi-continuous. Moreover, the t≤T is a strong supermartingale. Actually, by using Hölder's inequality and the stochastic Lipschitz condition on f (below), we obtain, for each τ ∈ T [0,T ] , Since K and C are nondecreasing processes, and Y t + t 0 f (s, Θ s )ds t≤T is a G tadapted process then the observation follows.
Remark 2.3. In our framework the filtration is quasi-left-continuous, martingales have only totally inaccessible jumps and Y has two type of left-jumps: totally inaccessible jumps which stem from stochastic integral with respect to µ, and predictable jumps which come from the predictable jumps of the irregular barrier ξ. The latter are the source of the predictability of K. Moreover, the processes K and µ do not have jumps in common.
Remark 2.4 (The particular case of a right-continuous barrier). If the barrier ξ is Moreover, the right-continuity of Y combined with the fact that ∆C t = Y t −Y t+ give C t = C t− for all t ≤ T . As C is right-continuous, purely discontinuous and such that C 0− = 0, we deduce C = 0. Thus, we recover the usual formulation of RBDSDEJs with a right-continuous barrier.
with Y being a làdlàg process, and let a coefficient g(.) ∈ M 2 β (R k×ℓ ). Then Proof. Using the left-continuity of trajectories of the process Y s− , we have On the other hand, we have |Y t− | 2 ≤ ess sup Then for all , we get the finite expectation. Since the process t 0 e βAs Y s , Z s dW s t≤T is adapted, it is a martingale.
By the same arguments, Let us recall some results from the general theory of optional processes, which will be useful in the sequel.
Theorem 2.6 (Mertens decomposition). LetỸ be a strong optional supermartingale of class (D). There exists a unique uniformly integrable martingale (càdlàg) N , a unique nondecreasing right-continuous predictable process K with K 0 = 0 and E|K T | 2 < +∞, and a unique nondecreasing right-continuous adapted purely dis- The proof is established in chapter VI, Theorem 99, [8] for the case of a nondecreasing process which is not necessarily right-continuous nor left-continuous. The proof is established in [23].
where M k is a (càdlàg) local martingale, R k is a right-continuous process of finite variation such that R k 0 = 0 and O k is a left-continuous process of finite variation which is purely discontinuous and such that O k 0 = 0. Let F be a twice continuously differentiable function on R n . Then, almost surely, for all t ≥ 0, where D k denotes the differentiation operator with respect to the k-th coordinate, and M k,c denotes the continuous part of M k .
and O are as in Theorem 2.9. Then, almost surely, for all t ≥ 0, Proof. To prove the corollary, it suffices to apply the change of variables formula from Theorem 2.9 with F (X, Y ) = XY 2 for X t = e βAt .
and we have, for any β > 0 and t ≤ T , 3 Reflected BDSDEJs with stochastic Lipschitz coefficients

Existence and uniqueness of solution
Before proving the existence and uniqueness, let us establish the corresponding result in the case where the coefficients f and g do not depend on the variables Y , Z and U . So we consider the RBDSDEJ, (3) where K = K c + K d (continuous + purely discontinuous part) is a nondecreasing right-continuous predictable process with K 0 = 0 and C is a nondecreasing rightcontinuous predictable purely discontinuous process with C 0− = 0. Moreover, the irregular barrier ξ satisfies (A1.4) and the coefficients (f, g) satisfy the following condition: Let us prove an a priori estimate of the solution in the following lemma.
Then there exists a constant κ(β) depending on β such that for all β > 1 From Remark 2.3, the processes K and µ do not have jumps in common, but K jumps at predictable stopping times and µ jumps only at totally inaccessible stopping times.
Then we can note that On the other hand, by using the Skorokhod and minimality conditions on K and C we can show that Y s− , dK Furthermore we have Y s , dC s = Y s , ∆C s , and by the same arguments, we have, for all s ≤ T , Moreover, by using the fact that the inequality (4) becomes Taking the expectation on the both sides of the inequality (5) and using Proposition 2.5, we get, for all β > 1, On the other hand, by taking the essential supremum over τ ∈ T [0,T ] and then the expectation on both sides of inequality (5) we obtain From the Burkhölder-Davis-Gundy inequality, there exists a universal constant c such that and 2E ess sup Consequently, The desired result is obtained by combining the estimates (6) and (7) for β > 1.
In the following, we state the existence and uniqueness result for the solution to RBDSDEJ (3).
and Y (ν) by The process ξ t + t 0 f (s)ds + t 0 g(s)dB s t≤T is progressively measurable. Therefore, the family ( Y (ν)) ν∈T [0,T ] is a supermartingale family. This observation with the Remark b. page 435 in [8] ensures the existence of a strong optional super- On the other hand, almost all trajectories of the strong optional supermartingale are làdlàg, then the làdlàg optional process Now, it remains to show that the candidate Y ∈ S 2 β (R k ). Using the Jensen's, Young's and Hölder's inequalities respectively, we obtain Taking the essential supremum over ν ∈ T [0,T ] on the above sides and using the Doob's martingale inequality, we conclude that E ess sup where κ ′ (β) is a positive constant depending on β. It follows that Y ∈ S 2 β (R k ). Note that the strong optional supermartingale Y is of class (D) (i.e. the set of all random variables Y ν , for each finite stopping time ν, is uniformly integrable). Then by the Mertens decomposition (see Theorem 2.6), there exists a uniformly integrable martingale (càdlàg) N , a nondecreasing right-continuous predictable process K (with K 0 = 0) such that E|K T | 2 < +∞ and a nondecreasing right-continuous adapted purely discontinuous process C (with C 0− = 0) such that E|C T | 2 < +∞, with the following equality: By an extension of Itô's martingale representation Theorem, there exists a unique pair of predictable processes (Z, U ) ∈ M 2 (R k×d ) × L 2 (R k ) such that with Y T = Y (T ) = ξ T and Y τ = Y (τ ) ≥ ξ τ a.s for all τ ∈ T [0,T ] . Next, let us focus on the Skorokhod and minimality conditions. Since ∆ + Y τ = 1 {Y τ =ξτ } ∆ + Y τ a.s.(see Remark A.4 in [14]), from (8) we have ∆C τ = −∆ + Y τ a.s., then ∆C τ = 1 {Y τ =ξτ } ∆C τ a.s. It follows that the minimality condition on C is satisfied. Further, due to a result from the optimal stopping theory (see Proposition B.11 in [18]), for each predictable stopping time τ , we have s. Then the process K satisfies the Skorokhod condition. Thus, we found a process (Y , Z, U, K, C) which satisfies the RBDSDEJ (3). Now, it remains to show that (Y , Z, U, K, C) ∈ B 2 β (R k ) × S 2 (R k ) × S 2 (R k ). Indeed, let K t := K t + C t− be the Mertens process associated with Y . By the definition of Y ν , we see that From Corollary 2.8, there exists a positive constant c such that where c(β) is a positive constant depending on β. K is nondecreasing, and it implies that E ess sup On the other hand, from Lemma 2.11 we have From Remark 2.3, the processes K and µ do not have jumps in common, but K jumps at predictable stopping times and µ jumps only at totally inaccessible stopping times, then we can write Here we have used also the Skorokhod and minimality conditions on K and C. Next, by taking the expectation on both sides of above inequality, we get Proof. Given (y, z, u) ∈ B 2 β (R k ), we define f (t) = f (t, y t , z t , u t ) and g(t) = g(t, y t , z t , u t ). Let us show that f and g satisfy (A1.5). From the assumptions (A1.1) and (A1.2), we have s |y s | 2 + a 2 s |z s | 2 + a 2 s u s This implies that f and g satisfy (A1.5) since (y, z, u) ∈ B 2 β (R k ) and in view of the assumption (A1.3). Hence the result follows from Proposition 3.2.
Proof. (i) Existence. Our strategy in the proof of existence is to use the Picard approximate sequence. To this end, we consider the sequence (Θ n ) n≥0 := (Y n , Z n , U n ) n≥0 ∈ B 2 β (R k ) defined recursively by Y 0 = Z 0 = U 0 = 0 and for any n ≥ 1, In the sequel, we shall show that (Y n , Z n , U n ) n≥0 is a Cauchy sequence in the Banach space B 2 β (R k ). We define ℜ n+1 = ℜ n+1 − ℜ n for ℜ ∈ {Y, Z, U, K, C}, We derive that for any n ≥ 1 the process (Y n+1 , Z n+1 , U n+1 , K n+1 , C n+1 ) satisfies the following equation From Remark 2.3, the processes K n+1 and µ do not have jumps in common, but On the other hand, by using the Skorokhod and minimality conditions on K n+1 and C n+1 we can show that Y Plugging these inequalities in (10), and taking the expectation in both side, we deduce that, for any β > 0 and ε > 0, .
Hence, choosing ε > 0 such that ε + α < 1, we deduce that (Y n , Z n , U n ) n≥1 is a Cauchy sequence in the Banach space A 2 β (R k ). It remains to show that (Y n ) n≥1 is a Cauchy sequence in S 2 β (R k ). To this end, we define for any integers n, m ≥ 1 ℜ n,m = ℜ n − ℜ m for ℜ ∈ {Y, Z, U, K, C}, and ∀h ∈ {f, g} , h n,m Then it is readily seen that Applying Lemma 2.11 to (11), and taking the essential supremum over τ ∈ T [0,T ] and then the expectation on both sides we get But, for any ε > 0, Moreover, by the Burkhölder-Davis-Gundy inequality, there exists a universal constant c such that and 2E ess sup Since (Y n , Z n , U n ) n≥1 is a Cauchy sequence in A 2 β (R k ), we deduce that (Y n ) n≥1 is a Cauchy sequence in S 2 β (R k ). Hence, (Y n , Z n , U n ) n≥1 is a Cauchy sequence in the Banach space B 2 β (R k ), so it converges in B 2 β (R k ) to a limit Θ = (Y, Z, U ). Now let us show that (Y, Z, U ), with the additional Mertens process (K, C), is a solution to RBDSDEJ (2).
Plugging these inequalities in (19) and taking expectation, we obtain Moreover,