BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH OBLIQUE REFLECTION AND LOCAL LIPSCHITZ DRIFT

It was mainly during the last decade that the theory of backward stochastic differential equations took shape as a distinct mathematical discipline. This theory has found a wide field of applications as in stochastic optimal control and stochastic games (see Hamadène and Lepeltier [9]), in mathematical finance via the theory of hedging and nonlinear pricing theory for imperfect markets (see El Karoui et al.[6]). Backward stochastic differential equations also appear to be a powerful tool for constructing Γ−martingales on manifolds (see Darling [4]). These kind of equations provide probabilistic formulae for solutions to partial differential equations (see Pardoux and Peng [14]). Consider the following linear backward stochastic differential equation { −dYs = [Ysβs + Zsγs]ds− ZsdBs, 0 ≤ s ≤ T YT = ξ. (1.1)


Introduction
It was mainly during the last decade that the theory of backward stochastic differential equations took shape as a distinct mathematical discipline.This theory has found a wide field of applications as in stochastic optimal control and stochastic games (see Hamadène and Lepeltier [9]), in mathematical finance via the theory of hedging and nonlinear pricing theory for imperfect markets (see El Karoui et al. [6]).Backward stochastic differential equations also appear to be a powerful tool for constructing Γ−martingales on manifolds (see Darling [4]).These kind of equations provide probabilistic formulae for solutions to partial differential equations (see Pardoux and Peng [14]).
Consider the following linear backward stochastic differential equation As is well known, equation (1.1) was first introduced by Bismut [1,2] when he was studying the adjoint equation associated with the stochastic maximum principle in optimal stochastic control.It is used in the context of mathematical finance as the model behind the Black and Scholes formula for the pricing and hedging option.

The development of general backward stochastic differential equation (BSDE in short)
−dY s = f (s, Y s , Z s )ds − Z s dB s , 0 ≤ s ≤ T Y T = ξ begins with the paper of Pardoux and Peng [14].Since then, BSDEs have been intensively studied.For example, BSDE with reflecting barrier have been studied among others by El Karoui et al. [5], Cvitanic and Karatzas [3], Matoussi [12] and Hamadène et al. [10] in the one dimensional case.The higher dimensional one has been considered by Gegout-Petit and Pardoux [8] for reflection in a convex domain.The multivalued context can be found in Pardoux and Rascanu [15], N'zi and Ouknine [13], Hamadène and Ouknine [11] and Essaky et al [7].These works concern the case of normal reflection at the boundary.In the last two decades, thanks to the numerous applications in queuing theory, the deterministic as well as stochastic Skorokhod problem (in a convex polyhedron with oblique reflection at the boundary) has been studied by many authors.Recently, S. Ramasubramanian [16] has considered reflected backward stochastic differential equations (RBSDE's) in an orthant with oblique reflection at the boundary.He has established the existence and uniqueness of the solution under a uniform spectral radius condition on the reflection matrix (plus of course, a Lipschitz continuity condition on the coefficient).
The aim of this article is to weaken the Lipschitz condition on the drift to a locally Lipchitz one.The paper is organized as follows.In section 2, we introduce the underlying assumptions and state the main result.Section 3 is devoted to the proof of the main result.

Assumptions and Formulation of the Main Result
Let B = {B(t) = (B 1 (t), ..., B d (t)) : t ≥ 0} be a d− dimensional standard Brownian motion defined on a probability space (Ω, F, P ) and let {F t } be the natural filtration generated by B, with F 0 containing all P −null sets. Let We are given the following: • T > 0 is a terminal time; • ξ is an F T −measurable, bounded, G−valued random variable;

Definition 2.1:
ively measurable integrable processes is said to solve RBSDE (ξ, b, R) if the following hold: (ii) for every i = 1, ..., d, and 0 ≤ t ≤ T, We make the following assumptions on the coefficients b, R.
Remark 2.1: In view of (A3), there exists constants a j , 1 ≤ j ≤ d and 0 < α < 1 such that Let H stands for the space of all {F t } −progressively measurable, continuous pairs of processes where ϕ t (g) denotes the total variation of g over [t, T ] and θ > 0 is a fixed constant which will be chosen suitably later.
For (Y, K) , ( Y , K) ∈ H, we define the metric Let H denote the collection of all (Y, K) ∈ H such that there exists an {F t } −progressively measurable process Since H is a closed subset of H, ( H, d) is a complete metric space.We consider the norm y = a i |y i | which is equivalent to the Euclidean norm in R d .So, we may assume that the local Lipschitz continuity in (A1) and Lipschitz continuity in (A2) are with respect to this norm.
Before stating our main result, let us remark that if Therefore, using integration by parts in (2.1), we have For every z ∈ M d (R), we put .
Let H denote the space of all F t -progressively measurable processes Z = (Z ij ) 1≤i,j≤d such that endowed with the norm It is clear that H is a Banach space.Now, we state our main result: Theorem 2.1: Assume (A1)-(A3).Let ξ be a bounded, F T −measurable G−valued random variable.Then there is a unique couple ((Y, K), Z) ∈ H×H solving RBSDE (ξ, b, R).

Proof of the Main Result
The proof of Theorem 2.1 needs some preliminary lemmas.
Lemma 3.1: Let b be a process satisfying assumption (A1).Then there exists a sequence of processes b n such that Proof: Let ψ n be a sequence of smooth functions with support in the ball B (0, n + 1) such that sup ψ n = 1.It not difficult to see that the sequence (b n ) n≥1 of truncated functions defined by b n = bψ n , satisfies all the properties quoted above.
In view of Ramasubramanian [16], there exists a unique couple of processes {((Y n (t), We formulate some uniform estimates for the processes {((Y n (t), K n (t)), Z n (t)) : t ≥ 0} in the following way.

Lemma 3.2: Assume (A1)-(A3).
Then there exists a constant C, such that for every n ≥ 1 Proof:Let the triple (Y n , K n , Z n ) be the unique solution of RBSDE (ξ, b n , R).We have for every i = 1, ..., d, and 0 Since applying Theorem 3.2 [16] and using integration by parts, we obtain We know that for every (t, ω, y) and every i = j, Let us note that Multiplying (3.2) by a i and adding leads to In view of the inequality Hence inequality (3.1) is proved.Now, we shall prove the convergence of the sequence (Y n , K n , Z n ) n≥1 .

Theorem 3.1: Assume (A1)-(A3). Then there exists ((Y, K), Z) ∈ H×H such that
where Proof: It follows from the same idea used in the proof of inequality (3.1) that For an arbitrary number N > 1, let L N be the Lipschitz constant of b in the ball B(0, N).We put It follows that It not difficult to check that Since b i is L N a i −locally Lipschitz, we get In view of the Lipschitz condition on R and the boundedness of D n j (t), we obtain that there exists C 1 > 0 such that Now, from the boundness of R, we have By virtue of (3.3)-(3.6),we deduce that Multiplying (3.7) by a i , adding and using i =j a i v ij ≤ αa j , we obtain Choosing θ large enough such that where We have By virtue of (3.1), there exists C > 0 such that (3.9) Passing to the limit on n, m and N in (3.9 ), we deduce that (Y Multiplying (3.10) by a i and adding leads to the existence of C > 0 such that Passing to the limit on m, n, we deduce that (Z n ) n≥1 is a Cauchy sequence in the Banach H. Since H is a Banach space, we put Multiplying (3.12) by a i and adding, we get By virtue of (3.1), we deduce that there exists C > 0 such that Passing to the limit on n, N , completes the proof of Lemma 3.5.
n , K n ) n∈N is a Cauchy sequence in H. Since H is a Banach space, we set If we return to the equation satisfied by the triple (Y n , K n , Z n ) n∈N and use Itô's formula, we have