REFLECTED SOLUTIONS OF BACKWARD DOUBLY SDES DRIVEN BY BROWNIAN MOTION AND POISSON RANDOM MEASURE

. We consider backward doubly stochastic diﬀerential equations (BDSDEs in short) driven by a Brownian motion and an independent Pois- son random measure. We give suﬃcient conditions for the existence and the uniqueness of solutions of equations with Lipschitz generator which is, ﬁrst, standard and then depends on the values of a solution in the past. We also prove comparison theorem for reﬂected BDSDEs.

1. Introduction. Motivated by the probabilistic interpretation of solutions to a class of quasilinear parabolic differential equations (PDEs in short), Pardoux and Peng [42] introduced non linear BSDEs. Since then, BSDEs have been intensively developed with great interest and encountered in many fields of applied mathematics such as finance, stochastic games and optimal control (see [20], [23], [27], [30] for instance, and so on).
As a variation of BSDEs, a BSDE with one reflecting obstacle (say lower obstacle L) K is non decreasing, continuous, K 0 = 0, T 0 (Y t − L t ) dK t = 0, (1) was first studied by El Karoui et al. [19]. Here W = (W t ) t≤T is a Brownian motion, f : Ω × [0, T ] × R × R d −→ R is the generator and the terminal value ξ is F T -measurable where for any t ≤ T , F t = σ{B s , s ≤ t}; the adaptation is with respect to (F t ) t≤T . The second condition in 1 says that the first component Y of the solution is forced to stay above L. The role of K is to push Y upwards in order to keep it above L in a minimal way, which leads to the third condition in 1. Note that the usual BSDEs may be considered as a special case of RBSDEs with L = −∞ (and K ≡ 0).

MONIA KAROUF
In [19], El Karoui et al. proved that if f is Lipschitz continuous in (y, z) and both the terminal value ξ and lower barrier L are square integrable, then the solution of the reflected BSDE 1 exists and is unique. They make use of two methods, penalization and Snell envelope.
Based on [19], Cvitanic and Karatzas [9] extended the research of BSDEs to those with two reflecting barriers. The solution of such a RBSDE has to stay between two prescribed continuous processes L and U called lower and upper barriers. Precisely a solution of that equation, associated with (ξ, f, L, U ) is a quadruplet of adapted processes (Y t , Z t , K + t , K − t ) t≤T with values in R 1+d+1+1 which mainly satisfies: The process K + (resp. K − ) is continuous non-decreasing and its role is to keep Y above L (resp. under U ). Moreover they act only when necessary. This type of equation is a powerful tool in zero-sum mixed game problems [28] and in American game options [10].
In [9], Cvitanic and Karatzas showed that a doubly reflected BSDE admits a unique solution if, on one hand, the generator of such a RBSDE is Lipschitz and, on the other hand, either the barriers are regular or the so called Mokobodski condition is satisfied (there exists a quasimartingale between the two barriers).
Some of further efforts to RBSDEs with two barriers can be found in [1,28,29] and the references therein. All the above papers on doubly reflected BSDEs assumed either Mokobodski's condition or the aforementioned regularity condition. However Mokobodski's condition is a bit troublesome since it is difficult to verify in practice. On the other hand, the regularity of one of the barriers is somewhat restrictive. So in [24] Hamadène and Hassani removed these conditions and showed that if the barriers are continuous and completely separated, meaning L t < U t , ∀t ≤ T , then a solution of 2 exists and its unique. Later the case of discontinuous barriers has been also studied in Hamadène et al. [25] where they pasted local solutions to form a unique solution of a reflected BSDE with two distinct obstacles. Since then, the complete separation of the obstacles has been postulated by most of the subsequent papers including [6,18,26,32] as well as [5].
After Pardoux and Peng [42], these authors introduced the theory of BSDEs in [43] which brought forward a new kind of BSDEs, that is a class of backward doubly stochastic differential equations with two different directions of stochastic integrals, i.e., equations involving both a standard (forward) Itȏ stochastic integral dW and a backward Itô stochastic integral dB. That is, BDSDEs are stochastic differential equations of the form where ξ, f and g are the given data of the problem. In [43], Pardoux and Peng proved the existence and uniqueness, and also, discussed the probabilistic representation of solution of quasilinear stochastic PDEs. Many researchers worked in this area (see, e.g., [3], [7], [8], [41], [44], [55], and the references therein).
Recently, one barrier reflected BDSDEs was introduced by Bahlali et al. in [2], that is, for t ∈ [0, T ], They established the existence and uniqueness of solutions of reflected BDSDEs 3 under the uniformly Lipschitz condition on the coefficients. In the case when the coefficient f is only continuous they established the existence of maximal and minimal solutions.
Backward doubly SDEs driven by a Brownian motion and a Poisson process with Lipschitzian coefficient on a fixed time interval were studied by Sun an Lu [50]. Then several authors investigated successfully in weakening Lipschitz assumptions (see among others [40], [52]). The assumption usually satisfied by the drift is replaced by a rather smooth one which ensures existence and uniqueness result. Inspired by the method developed in [52], Sow [49] extended Wang and Huang's result to BDSDE with jumps and proved a large derivation principle of such family of equations. Then in [48], the author generalizes the result established in [49] to BDSDE associated to a random Poisson measure. Using the solvability of the equation in the case of Lipschitzian coefficients and an efficient iterative procedure, the author proved existence and uniqueness with coefficient satisfying rather weaker conditions.
As further extensions of BSDEs are the ones with time delayed generators (time delayed BSDEs). Those equations were first introduced by Delong and Imkeller [15,16], the dynamics of which is given by where the generator f at time s depends arbitrarily on the past values of a solution (Y s , Z s ) = (Y (s + u), Z(s + u)) −T ≤u≤0 . In Delong and Imkeller [15], the authors established the existence and uniqueness of a solution for time delayed BSDEs. Also, in [16], they proved the existence and uniqueness as well as the Malliavin differentiability of the solution for time delayed BSDEs driven by Brownian motions and Poisson measures. Dos Reis et al. [17] extended the results of Delong and Imkeller [15,16] in L p -spaces. Then, Luo et al. [38] studied BDSDEs with time delayed generators. Based on an extension of the existence result of [38], Lu et al. [37] showed the existence and uniqueness of the solution of multivalued time delayed BDSDEs by means of Yosida approximation. At present, BSDEs with time delayed generators are widely recognized to provide a useful and efficient tool for studying problems in different mathematical fields, such as mathematical finance, stochastic control and game theory (see, e.g., [14], [45], [46], [47]).
The first connection between reflected BSDEs and time delayed equation 4 was made by Zhou and Ren in [54] where it is proved, under the specific assumptions of the reflected case and the delayed BSDE, that there exists a unique solution of a RBSDE with time delayed generators. Karouf [35] extended the results of [54] to the wider class of barrier processes which are right continuous with left limits (càdlàg). Later on, the author studied in [36] time delayed RBSDEs with two reflecting barriers driven by a Brownian noise and by a Brownian and Poisson noise. Very recently, Mansouri et al. [39] considered reflected BDSDEs with time delayed generators.
In this paper, we carry on the study of BDSDEs with two reflecting barriers driven by a Brownian motion and an independent Poisson process. More precisely, we consider equations of the following form whereμ is the compensated measure associated with µ. Firstly, we show the existence of a solution when f is Lipschitz and under the well-known Mokobodski hypothesis. In a second part, we deal with the problem of existence of a solution for the same equation when the generator f depends on the past value of the solution. The paper is organized as follows. In Section 2 we formulate the problem and set up the main assumptions. In Subsection 3.1 and 3.2, we address the question of existence of the solution of the reflected BSDEs with two reflecting barriers when the barrier processes have either only inaccessible or predictable jumps. For both cases, we first deal with the case when the coefficient f does not depend on (y, z, v). Then, in order to obtain the result for generators depending on (y, z, v), we introduce a contraction mapping in an appropriate Banach space of processes which has a fixed point, which in turn provides the unique solution of the BDSDEs. We end the Subsection 3.1 by proving a comparison theorem. In Section 4, we consider the problem of reflected BDSDEs with time delayed generators. Using the results of Section 3, we show the existence and uniqueness of the solution for a time horizon small enough or for a Lipschitz generator with a constant small enough. The end of this section investigates the concepts of comparison principle. Finally, in the appendix, some definitions and important results about the Snell envelope and optimal stopping are presented.

2.
Setting of the problem. Let (Ω, F, P) be a complete probability space and T > 0 be a fixed terminal time. We suppose that {F t } t≥0 is generated by the following three mutually independent processes -a l-dimensional Brownian motion

RBDSDES DRIVEN BY BROWNIAN MOTION AND POISSON RANDOM MEASURE 5575
F η t = F η 0,t . Note that the collection (F t ) 0≤t≤T is neither increasing nor decreasing, so it does not constitute a classic filtration. Let P (respectivelyP) be the σ-algebra of (F t )-progressively measurable (respectively predictable) sets on Ω × [0, T ]. For any β > 0, we consider the following spaces of processes: ) the space of (F t )-adapted continuous (respectively càdlàg) non-decreasing processes K = (K t ) t≤T such that K 0 = 0 and E[K 2 T ] < ∞; • T s is the set of (F t )-stopping times τ such that s ≤ τ ≤ T , P-a.s. for s ≤ T ; • for a given càdlàg process π = (π t ) t≤T let π t − = lim s t π s , 0 ≤ t ≤ T , We also denote by B β,s the equivalent norm in the space B β with the following definition We are now given a set of five data ξ, g, f , L and U which satisfy along with this paper the Assumptions (H1)-(H3) below: (H1) A random variable ξ which belongs to L 2 (Ω, F T , P).
(H2) The jointly measurable functions f : are such that: (i) for all (y, z, v) ∈ R×R d ×R, g(., y, z, v) ∈ H 2 β (R) and f (., y, z, v) ∈ H 2 β (R); (ii) there exist constants C > 0 and 0 < α < 1 such that for any (w, (H3) Two barriers L := (L t ) t≤T and U := (U t ) t≤T processes of S 2 β (R) which satisfy: Next let us give the following result which is an extension of the well-known Itô formula and whose proof is given in [50, Lemma 2.2, p. 76].

MONIA KAROUF
Then, for any function φ ∈ C 2 (R, R), In particular, for any β > 0 3. The case of non-delayed coefficients. In this section, we consider BSDEs with two reflecting barriers when the noise is driven by a Brownian motion and an independent Poisson random measure. We first assume that the jumping times of the barriers are totally inaccessible stopping times which roughly speaking means that they are not predictable (cf. Appendix Definition 5.1). In this case the involved increasing processes K ± are continuous. If this latter condition is not satisfied and especially if the barriers are càdlàg then Y could have predictable jumps and the processes K ± would be no longer continuous.

RBDSDE with totally inaccessible jump times in barriers.
3.1.1. Existence and uniqueness. We are going to show the existence of a solution for reflected BDSDEs such that the jumps of the barriers occur only at inaccessible stopping times. This subsection mainly discusses the following BDSDE:

called a solution for the BDSDE with jumps and two reflecting barriers if
The inequalities and the integral condition in iii) are called the barriers constraints and the minimality respectively.
For instance, equation 6 may not have a solution. Actually, if for example, L and U coincide and L is not a semimartingale then we cannot find a quintuplet which satisfies the relation ii). Therefore, in order to obtain a solution we are led to assume the Mokobodski condition: (Mk) There exist two non-negative supermartingales in S 2 We firstly present the following existence theorem when f and g do not depend on (y, z, v), i.e., f (t, y, z, v) ≡ f (t) and g(t, y, z, v) ≡ g(t).
Theorem 3.2. Assume that Assumptions (H1)-(H3) and (Mk) hold. Then the two barriers reflected BDSDE 6 has a solution (Y, Z, V, The processes K ± can be chosen singular and then they are unique. Proof. In its main steps, the proof is classic (see, e.g., [9] or [28]).
Since h and h are non-negative supermartingales, H and Θ are also non-negative supermartingales of S 2 β (R) and verify H T = Θ T = 0. On the other hand, through (Mk), it follows that for any t ≤ T , Now let us consider the sequences (N ± n ) n≥0 of processes defined recursively as follows: N ± 0 = 0, and for n ≥ 0, , where R is the Snell envelope operator (see Proposition 2 in the appendix). Using 7 and induction we can easily verify that (see [9]): ) converges pointwise to a supermartingale N + (resp. N − ) (see, e.g., [34], pp. 21). In addition N + and N − belong to S 2 β (R) and satisfy (see Proposition 3): Further the Doob-Meyer decompositions of N ± imply the existence of càdlàg martingales (M ± t ) t≤T and G t -adapted non-decreasing processes ( , thanks to the predictable dual projection of nondecreasing processes (see, e.g., [13], pp. 221), we have E[(K ± T ) 2 ] < ∞ and (M ± t ) t≤T belong also to S 2 β (R). Therefore, an obvious extension of Itô's martingale representation theorem (see [33]) yields the existence of two processes ( Let us denote by K ±,d (respectively, K ±,c ) the purely discontinuous (respectively, continuous) part of K ± . By arguments similar to the ones in Proposition 4.1 in [24] we show that So K +,d − K −,d = 0, and actually in Definition 3.1 the terms K + t and K − t are continuous processes. Next, for t ≤ T let us set where the processes η t and ρ t belong to H 2 (R d ) and L 2 (μ, R) respectively and satisfy: Obviously for any t ≤ T we have This implies that (Y, Z, V, K + , K − ) solves 6.

RBDSDES DRIVEN BY BROWNIAN MOTION AND POISSON RANDOM MEASURE 5579
For the term martingales thanks to the Burkholder-Davis-Gundy and Young inequalities. Then taking the expectation in 8, we obtain Finally, in a classic way, we can show that if K ± are chosen to be singular then we have also K ± = K ± (see Definition 5.2 and Remark 4.1, page 138 of [24]).
Next with the help of this result we will be able to prove that the BDSDE 6 has a solution in the case when the functions f and g depend on (y, z, v). Actually we have: The processes K ± can be chosen singular and then they are unique.
Now for (Y , Z , V ) ∈ B β , we define in the same way the quintuplet (Ȳ ,Z ,V ,K + , Noting that T t e βs δȲ s − (dδK + s − dδK − s ) ≤ 0 and taking the expectation in 9, we obtain The term including f can be enlarged via Assumption (H2), and 2bd ≤ b 2 + 1 d 2 , > 0, as follows: Putting the previous inequality into 10 and using Assumption (H2)  Hence, Thus, since 1+α 2 < 1, the map φ is a strict contraction on (B β , . B β ). Therefore it has a unique fixed point (Y, Z, V ) which (with the associated process K ± ) is the solution of the BDSDE with two reflecting barriers associated to (ξ, f, g, L, U ). The proof of existence is now complete.
Let us focus on uniqueness. If (Y , Z , V , Actually to prove this claim, we just need to argue as in the proof of uniqueness of Theorem 3.2. Proof. First let us point that if (X t ) t≤T is an R-valued càdlàg semi-martingale and if X + t := max{X t , 0} then, using Tanaka's formula, [13] page 349), hence, δV s (e)μ(ds, de).
Next relation 11 with δY yields: ∀t ≤ T Thus, noting that ξ 1 ≤ ξ 2 , by virtue of the previous three inequalities we can get that

RBDSDES DRIVEN BY BROWNIAN MOTION AND POISSON RANDOM MEASURE 5583
But f 1 ≤ f 2 , then we have Thus going back to 12 and using the fact that g verifies (H2), we get Finally taking the expectation above, and since the last three terms are martingales, we obtain Consequently, choosing 0 < < 1−α C and using Gronwall's lemma, we get E[(δY + t ) 2 ] = 0 for any t ≤ T , i.e., Y 1 ≤ Y 2 , which is the desired result.

3.2.
RBDSDEs with arbitrary jumping times in barriers. The second building block of this section consists in considering reflected BDSDEs with jumps when the barrier processes are only càdlàg. No further conditions are required on the nature of their jump times. The jump times are in fact arbitrary; they can be either predictable or inaccessible. The difficulty here lies in the fact that since the barriers L and U are allowed to have jumps then the process Y has too and then the reflecting processes K + and K − are no longer continuous but only càdlàg. Therefore the setting of the problem is not the same as in 6.
Throughout this subsection, Mokobodski's condition is in force. We moreover assume that the barriers L and U and their left limits are separated, meaning that they satisfy the following assumption: (H4) For any t < T , L t < U t and L t − < U t − . Definition 3.5. We say that a quintuplet (Y, Z, V, K + , K − ) of processes is a solution of the RBDSDE (ξ, f, g, L, U ) with jumps and two reflecting càdlàg barriers if if K ±,d is the purely discontinuous part of K ± then K ±,d is P − measurable, and for any t ≤ T, ∆K +,d Firstly, we are going to focus on the uniqueness of the solution of 13. Then we have: Proof. Let (Y, Z, V, K ± ) and (Y , Z , V , K ± ) be two solutions of 13. Firstly let us note that by the classical discussion (see [31], page 274), we have Therefore,

RBDSDES DRIVEN BY BROWNIAN MOTION AND POISSON RANDOM MEASURE 5585
Following Assumption (H2), elementary inequality, we have and Hence, Henceforth from Gronwall's lemma and the right continuity of δY , we obtain Since L < U , we conclude that K −,c = K −,c and then K +,c = K +,c , whence the uniqueness of the solution of 13.
The following theorem is the main result of this subsection.
Theorem 3.6. Suppose that Assumptions (H1)-(H4) and (Mk) hold. Then, there exists a unique process which is a solution to the BDSDE 13.
Proof. The proof will be split into two steps. The first step deals with the case where the functions f and g are independent of the variables (y, z, v) under Assumptions (H1), (H2), (H4) and (Mk), and the second step considers the general case.
Step 1. We assume that f and g are independent of (y, z, v).
The proof of the existence and the uniqueness is analogous to that of Theorem 3.2, the only difference being in the fact that in the latter part of the proof of Step 1 we follows [31] instead of [24] to prove that We now establish a comparison theorem for solutions of BDSDEs with two càdlàg reflecting barriers.
be the solutions of the two barriers RBDSDE 13 for i = 1, 2. If f 1 and g are independent of v, ξ 1 ≤ ξ 2 and for t ≤ T , The main idea is to make use of Itô-Tanaka's formula. Therefore using Section 3.1 notations and applying formula 11 with δY t for t ∈ [0, T ], From the definition of a solution of reflected BDSEs with two reflecting barriers, we have Actually the first term is null since when the continuous K 1+,c (resp. purely discontinuous K 1+,d ) increases, we then have Arguing in the same way we obtain that the second term is null too. Therefore, since ξ 1 ≤ ξ 2 , we have s − ] |δg(s)| 2 ds

RBDSDES DRIVEN BY BROWNIAN MOTION AND POISSON RANDOM MEASURE 5587
Now, in view of the assumptions on f 1 , f 2 and g, the remainder of the proof runs as the proof of Theorem 3.4 (see the reasoning following 12).
By virtue of Theorem 3.6, the following corollary follows immediately.

Corollary 1. Under (H1)-(H2) and
(H3') L = (L t ) t≤T is a process of S 2 β (R) and L T ≤ ξ, P-a.s. with arbitrary jumping times, the RBDSDE with jumps and one reflecting càdlàg lower barrier associated with (ξ, f, g, L) has a unique solution, meaning that there exists a unique quadruplet (Y, Z, V, K) such that 4. The case of time-delayed coefficients. We now address the problem of reflected BDSDEs with jumps and time delayed generators. Once again we consider the case when the barrier processes have totally inaccessible jump times as well as the case when the barrier processes have general jump times.
We will also need the following lemma.

Lemma 4.1. Suppose that (H2) is in force and
We then have, for some constantC = Ca, Proof. We only prove 17, 18 can be proved in the same way. R). Firstly note that for some processes (Y t ) t∈[0,T ] , (Z t ) t∈[0,T ] and (V t ) t∈[0,T ] satisfying appropriate integrability conditions, definition 15 yields

RBDSDES DRIVEN BY BROWNIAN MOTION AND POISSON RANDOM MEASURE 5589
So, Assumption (H2) and Jensen's inequality, imply that Then, Commuting the order of integration (Tonelli lemma) yields for then the reflected BDSDE 20 has at most one solution Moreover if the processes are singular then we have also K ± = K ± .
Coming back to 23 and taking the supremum over t ∈ [0, T ] then the expectation, we obtain Hence, pulling 26, 27 and 28 into 25 we get Using once again Lemma 4.1, we get e βs (δV s (e)) 2 λ(de) ds .
Then from 24 we obtain Since by conditions 21 and 22 we can choose it is not difficult to see that Henceforth, E[sup t∈[0,T ] e βt |δY t | 2 ] ≤ 0 and then Y = Y , Z = Z , V = V and K + − K − = K + − K − . Finally, in a classic way, we can show that if K ± are chosen to be singular then we have also K ± = K ± . The proof is complete.
We are now in position to give the main theorem of this section.

RBDSDES DRIVEN BY BROWNIAN MOTION AND POISSON RANDOM MEASURE 5593
Let (Y , Z , V ) be another element of B β and φ(Y , Z , V ) = (Ỹ ,Z ,Ṽ ), then using Itô's formula we get, for any t ≤ T and > 0,

< < β.
Hence φ is a strict contraction on B β with the norm 5. Therefore φ admits a unique fixed point (Y, Z, V ), meaning φ(Y, Z, V ) = (Y, Z, V ), and then, with the associated processes K + and K − , the quintuplet (Y, Z, V, K + , K − ) is a solution of the reflected BDSDE associated with (ξ, f, g, L, U ). Remark that this one is unique if K ± are chosen to be singular.
The following corollary is a particular case with only one barrier.
Assume further that T, C, C 1 and θ are chosen as in Theorem 4.4 and such that conditions 29 and 30 hold. Then, the BDSDE with one lower barrier and time delayed generator associated with (ξ, f, g, L) admits a unique solution, so there exists a unique quadruplet of processes (Y t , Z t , V t , K t ) t≤T such that 16 is satisfied and    i) Y ∈ S 2 β (R), Z ∈ H 2 β (R), V ∈ L 2 β (μ, R) and K ∈ S 2 ci (R), ii) ∀t ≤ T, Y t ≥ L t and T 0 (Y t − L t ) dK t = 0.
We next deal with existence of the solution of the reflected BDSDE with general jumps and time delayed generators. Theorem 3.2 gives the existence in the case when the functions f and g do not depend on (y, z, v). For the general case of the generators, the proof can be obtained similarly as the one of Theorem 4.4, and we omit it here. Theorem 4.5. Assume (H4) and all the assumptions of Theorem 4.4 hold. Then the BDSDE with two reflecting càdlàg barriers and time delayed generators associated with (ξ, f, g, L, U ) has a unique solution, meaning there exists a unique quintuplet of processes (Y t , Z t , V t , K + t , K − t ) t≤T which satisfies 16 and items i)-iii)-iv) of Definition 3.5.
A direct consequence of Theorem 4.5 is the following corollary.

Comparison principle.
This section is devoted to the study of the comparison theorem of the time delayed BDSDEs with two reflecting barriers which have only inaccessible jumps.
2) Let ν 1 , ν 2 be two signed measures on (Ω, A). We say that they are mutually singular (or more simply that they are singular or that ν 1 is singular to ν 2 or that ν 2 is singular to ν 1 ), denoted ν 1 ⊥ν 2 , if there are disjoint sets A, B ∈ A with Ω = A ∪ B for which ν 1 (A) = ν 2 (B) = 0. In other words, two signed measures are mutually singular if there is a set in A which is null for one of them and its complement is null for the other.