GENERALIZED BSDE DRIVEN BY A LÉVY PROCESS

A linear version of backward stochastic differential equations (BSDEs) was first studied by Bismut [4] as the adjoint processes in the maximum principal of stochastic control. Pardoux and Peng in [20] introduced the notion of nonlinear BSDE. Since then, the interest in BSDEs has increased. Indeed, BSDEs provide connection with mathematical finance [10], stochastic control [11], and stochastic game [9]. On the other hand, this class of BSDEs is a powerful tool to give probabilistic formulas for solution of partial differential equations (see [18, 19]). Given a Brownian motion (Wt)0≤t≤T , we denote by ( t)0≤t≤T its natural filtration. Consider the nonlinear BSDE:


Introduction
A linear version of backward stochastic differential equations (BSDEs) was first studied by Bismut [4] as the adjoint processes in the maximum principal of stochastic control.Pardoux and Peng in [20] introduced the notion of nonlinear BSDE.Since then, the interest in BSDEs has increased.
Indeed, BSDEs provide connection with mathematical finance [10], stochastic control [11], and stochastic game [9].On the other hand, this class of BSDEs is a powerful tool to give probabilistic formulas for solution of partial differential equations (see [18,19]).
Given a Brownian motion (W t ) 0≤t≤T , we denote by (Ᏺ t ) 0≤t≤T its natural filtration.Consider the nonlinear BSDE: where ξ is an Ᏺ T -measurable random variable that will become certain only at the terminal time T, and f is a progressively measurable process.
In [20], the authors showed that there exists a unique Ᏺ t -adapted process (Y ,Z) solution of the BSDE (1.1), when the coefficient f is Lipschitz in y and z, ξ is square integrable.
Many existence and uniqueness results have been proved in relaxing the Lipschitz condition of the coefficient.For instance, Peng introduced for the first time monotone coefficient in [22], see also [2,6,8].In the one-dimensional case, Lepeltier and San Martin [13] described the BSDEs with a continuous coefficient, and Kobylanski [12] studied those with a coefficient which is quadratic in z.
Further, other settings of BSDEs have been introduced.Pardoux and Zhang [21] introduced a new class of BSDEs, which involves the integral with respect to a continuous increasing process.This kind of equations is called generalized BSDEs.
In [20,21], the main ingredient is the classical martingale representation theorem.In [16], Nualart and Schoutens proved a martingale representation theorem for Lévy processes, then in [17] they established the existence and uniqueness of solution for BSDEs associated with Lévy process.Bahlali et al. [1] showed the same result for the BSDEs driven by a Brownian motion and the martingales of Teugels associated with an independent Lévy process, having a Lipschitz or a locally Lipschitz coefficient.
The aim of this paper is to study the one-dimensional generalized BSDE driven by a Brownian motion and the martingales of Teugels associated to a pure jump-independent Lévy process.We prove existence and uniqueness of the solution when the coefficient verifies some conditions of Lipschitz.In this setting, we deal with both constant and random terminal times.If the coefficient is left continuous, increasing, and bounded, we prove the existence of a solution.As an application, we give a probabilistic interpretation for large class of partial differential integral equations (PDIEs) with Neumann (nonlinear) boundary condition.
The rest of the paper is organized as follows.In Section 2, we introduce some notations.In Section 3, we prove the existence and uniqueness of the solution of the generalized BSDE when the coefficient is monotone in y and uniformly Lipschitz in z and u.Section 4 is devoted to study the case where the coefficient is left continuous in y, increasing, and bounded.Finally, we give in Section 5 a probabilistic interpretation of PDIE with Neumann boundary condition, and we introduce some examples of PDIE.
For every λ ∈ R, μ ≥ 0, every increasing process (A t ) t and every Hilbert space H, we denote (i . We put L t − = lim s t L s and ΔL t = L t − L t − .We define the so-called power-jump processes L (1)  t = L t and t − m i t, called Teugels martingales.We associated with the Lévy process (L t ) t the family of processes (H (i) ) i≥1 defined by The coefficients a i j correspond to the orthonormalization of the polynomials 1,X,X 2 ,... with respect to the measure π defined by π(dx) = x 2 ν(dx).We set for i ≥ 1, (2.4) The martingales H (i) are strongly orthogonal (i.e., H (i) H ( j) is a martingale ⇔ [H (i) ,H ( j) ] is a martingale) and H (i) ,H ( j) t = δ i j q i t, where q i = i j,k=1 a i j a ik m j+k (for more details, see [16]).
Let us give the data (ξ, f ,g,A) defined by (i) a terminal value ξ ∈ L 2 (Ω,F T ,P), (ii) a map f : a continuous one-dimensional increasing Ᏺ t -progressively measurable process (A t ) t∈[0,T] satisfying A 0 = 0.In the following, C denotes a generic constant, that may take different values from line to line.

Generalized BSDEs driven by a Lévy process on a finite interval
In this section, we propose to show the existence and uniqueness of the solution of generalized BSDE driven by a Brownian motion and independent Lévy process (GBSDEL).
Given the data (ξ, f ,g,A), we introduce for all t ∈ [0,T] the GBSDEL: We assume that for some constant α ∈ R, β < 0, μ ≥ 0, and K > 0, some adapted processes The objective of this section is to prove the next results.We want next to state an analogous result in the case where the terminal time is replaced by a stopping time τ.More precisely, we consider the BSDE: We assume that ξ is an Ᏺ τ -measurable, and that for some λ > 2α + 4K for all i ≥ 1, the process ( satisfies an analogous GBSDEL with f and g replaced by f (t, y,z,u) = e μAt f t,e −μAt y,e −μAt z,e −μAt u , g(t, y) = e μAt g t,e −μAt y − μy. (3.10) Hence, if g satisfies (3.v) with a possibly nonnegative β, we can always choose μ such that g satisfies (3.v) with a strictly negative β.

Preliminary estimates and uniqueness.
We first establish a priori estimate on the solution.

Existence result of GBSDEL on fixed finite time interval.
We first prove existence and uniqueness result under an additional assumption.We suppose that for all y, y ,z ∈ R and u ∈ 2 , dt × dP a.e.
(3.ix) Proof.First let us assume that the map f does not depend on (y,z,v).Using the martingale representation theorem, we can prove that the following GBSDE: has a unique solution that verifies (3.3).Now, define the sequence (Y n ,Z n ,U n ) as follows.
(Y 0 ,Z 0 ,U 0 ) = (0,0,0), and (Y n+1 ,Z n+1 ,U n+1 ) is the unique solution of the BSDE: (3.23) We will prove that (Y n ,Z n ,U n ) is a Cauchy sequence in the Banach space Ᏼ 2 λ,μ .Note that for some (λ,μ), we can show that To simplify, we put for n ≥ m ≥ 1 and 0 Consequently, the sequence (Y n ,Z n ,U n ) converges in the Banach space Ᏼ 2 λ,μ to a process (Y ,Z,U), that is not difficult to show that verifies (3.1).
We now establish existence and uniqueness for (3.1) under the conditions (3.i)-(3.viii).First, we need the following proposition. 2), there exists a unique progressively measurable process (Y t ,Z t ,U t ) 0≤t≤T solution of Proof.The proof is very similar to that of [21,Proposition 1.8].
To simplify, we put f (s, y) = f (s, y, Z s , U s ).Notice that f (s, y) satisfies the following.
We approximate f and g by f n and g n such that (i) for each n, f n and g n are uniformly Lipschitz in y, (ii) f n satisfies (3.iii) and (3.vi) , and g n satisfies (3.v) and (3.vi) with fixed constants α, β, K and fixed process {( ϕ t ) t ,(ψ t ) t } 0≤t≤T satisfying (3.vii) and (3.vii).For each n, there exists a unique progressively measurable process ( Y n , Z n , U n ) solution of (3.1), such that (3.29) Defining V n t = f n (t, Y n t ) and W n t = g(t, Y n t ), we deduce from the above and from our assumptions that (3.30) From weak convergence along a subsequence, we conclude that there exists a progressively measurable process (Y t ,Z t ,U t , V t , W t ) 0≤t≤T verifying Finally, we can show that V t = f (t,Y t ) and W t = g(t,Y t ).
Let X and X be two progressively measurable processes such that e αt Y n t − X t g n t, Y n t − g n t,X t dA t ≤ 0. (3.32) (3.37) Mohamed El Otmani 11 We choose X , divide them by ε and let ε → 0 to conclude.
Proof of Theorem 3.2.We construct a mapping Φ from Ᏼ 2 into itself, which to ( Y , Z, U) associates (Y ,Z,U) = Φ( Y , Z, U) solution of (3.28).Our aim is to show that Φ admits a unique fixed point.
It follows that Φ has a unique fixed point solution of the GBSDEL (3.1).
Existence.In view of Theorem 3.3, by using the BSDE with data (ξ n ,I [0,τ] f ,I [0,τ] g,A), for each n ≥ 0, we construct the sequence (Y n t ,Z n t ,U n t ) 0≤t≤T solution of the following GBSDEL: We suppose that Z n t = 0, U n t = 0 for all t > τ.In fact, (3.46) is equivalent to e λr+μAr f (r,0,0,0) 2 dr + g(r,0) 2 dA r . (3.48) We now prove that (Y n ,Z n ,U n ) is a Cauchy sequence in the Banach space Ᏼ 2 λ,μ .We adopt the notations which are in the proof of Theorem 3.7.
(i) For ( (3.59) In view of (3.56), the right-hand side tends to 0 as m goes to infinity, and one concludes that (Y n ,Z n ,U n ) is a Cauchy sequence for the Ᏼ 2 λ,μ norm.Its limit is the solution of (3.4) and it satisfies (3.7).

GBSDEL with a left-continuous coefficient
In this section, we study the GBSDEL with continuous coefficient.We present a comparison theorem when the coefficient is uniformly Lipschitz and we prove existence of a solution when the coefficient is left continuous, increasing, and bounded.
Theorem 4.1.Suppose that ξ 1 ≤ ξ 2 , f 1 (t, y,z) ≤ f 2 (t, y,z), and g 1 (t, y) ≤ g 2 (t, y) for all (t, y,z) three progressively measurable processes such that has a unique solution, and we can write that We denote by In view of the above notations, we get   We note by the Burkhölder-Davis-Gundy inequality that To finish with, we write The right-hand side goes to 0 as n tends to infinity.We conclude that for all t ∈ [0,T], This completes the proof of the theorem.
Remark 4.4.We obtain the same result if we suppose that (i) f is right continuous, decreasing, and bounded, (ii) f is continuous with linear growth in y independent of z (see [13] for approximation).

Application to PDIE
In this section, we study the link between generalized BSDE driven by Lévy process and a class of partial differential integral equations with Neumann boundary condition.We suppose that the process L has bounded jump (without lost of generality, we suppose that sup t |ΔL t | ≤ 1).Then, for all p = 1,2,3, .. from which we get the result of the theorem.
We next consider some examples of PDIEs.
• 0 a pure jumps process, then by [23, Theorem 26, page 75], the last term is equal to 0. So, if (Y t ,Z t ,U t ) satisfies (3.1), then Y t ,Z t ,U t = e μAt Y t ,e μAt Z t ,e μAt U t(3.9) X t )| 2 dA t → 0 as n → +∞, we can write .38) Denote by δX s = X s − X s for X = Y , Y , Z, Z, U, and U.