A note on existence and uniqueness for solutions of multidimensional reflected BSDEs

In this note, we provide an innovative and simple approach for proving the existence of a unique solution for multidimensional reﬂected BSDEs associated to switching problems. Getting rid of a monotonicity assumption on the driver function, this approach simpliﬁes and extends the recent results of Hu & Tang [4] or Hamadene & Zhang [3].


Introduction
The theory of Backward Stochastic Differential Equations (BSDEs for short) offers a large number of applications in the field of stochastic control or mathematical finance. Recently, Hu and Tang [4] introduced and studied a new type of BSDE, constrained by oblique reflections, and associated to optimal switching problems. Via heavy arguments, Hamadène and Zhang [3] generalized the form of these multidimensional reflected BS-DEs. They allow for the consideration of more general oblique constraints, as well as drivers depending on the global solution of the BSDE. Unfortunately, their framework requires the driver function to be increasing with respect to the components of the global 1 solution of the BSDE. In the context of linear reflections of switching type, we are able to get rid of this limiting monotonicity assumption. We provide in this note a new method to prove existence and uniqueness for such type of BSDEs. We follow the classical scheme introduced for e.g. in [2], which consists in proving that a well chosen operator is a contraction for a given norm. This is done via the introduction of a convenient one dimensional dominating BSDE and the use of a standard comparison theorem. The rest of this note is organized as follows. In Section 2, we present our main result, namely Theorem 2.1. In Section 3, we give its proof which requires several intermediary results.

Framework and main result
Let T > 0 be a given time horizon and (Ω, F, P) be a stochastic basis supporting a q-dimensional Brownian motion W , q ≥ 1. F = (F t ) t≤T is the completed filtration generated by the Brownian motion W , and P denotes the σ−algebra on [0, T ] × Ω generated by F−progressively measurable processes. In the following, we shall omit the dependence on ω ∈ Ω when it is clearly given by the context.
We introduce the following spaces of processes: • S 2 (resp. S 2 c ) is the set of R−valued, adapted and càdlàg 1 (resp. continuous) processes • H 2 is the set of R q −valued, progressively measurable process (Z t ) 0≤t≤T such that • K 2 (resp. K 2 c ) is the subset of nondecreasing processes (K t ) 0≤t≤T ∈ S 2 (resp. (K t ) 0≤t≤T ∈ S 2 c ), starting from K 0 = 0.
We are given a matrix valued continuous process C = (C ij ) 1≤i,j≤d such that C ij ∈ S 2 c , for i, j ∈ I, where I := {1, . . . , d}, d ≥ 2, and satisfying the following structure condition (2.1) 1 French acronym for right continuous with left limits.

2
In "switching problems", the quantity C interprets as a cost of switching. As in [4] or [3] in a more general framework, this assumption makes instantaneous switching irrelevant. We introduce the family of random closed convex sets (C t ) 0≤t≤T associated to C: and the oblique projection operator P onto C defined by We are also given a terminal random variable We then consider the following system of reflected BSDEs: We also assume that The existence and uniqueness of a solution to the system (2.2) has already been derived with the addition of one of the following assumptions: We now introduce the following weaker assumption: • (Hf ) For i ∈ I, the i−th component of the random function f depends on y and the i−th column of the variable z: The main contribution of this note is the following result.

Proof of Theorem 2.1
The proof divides in three steps. First, we slightly generalize the switching representation result presented in Theorem 3.1 of [4], allowing for the consideration of random driver and costs. Second, we introduce a Picard type operator associated to the BSDE (2.2) of interest. Finally, via the introduction of a convenient dominating BSDE and the use of a comparison argument, we prove that this operator is a contraction for a well chosen norm.

The optimal switching representation property
We first give a key representation property for the solution of (2.2) under the following assumption: • (H3) For i ∈ I, the i−th component of the random driver f depends only on y i and the i−th column of the variable z: Observe that (H1) =⇒ (H3) =⇒ (H2). We first provide the existence of a solution to (2.2) under Assumption (H3). Proof. Since (H3) is stronger than (H2), the existence of a solution is provided by Theorem 3.2 in [3]. Due to the the particular form of switching oblique reflections and for sake of completeness, we decide to report here a simplified sketch of proof.
We use Picard iteration. Let (Y .,0 , Z .,0 ) ∈ [S 2 × H 2 ] d be the solution to the following BSDE without reflection: For i ∈ I and n ≥ 1, define recursively Y i,n as the first component of the unique solution (see Theorem 5.2 in [1]) of the reflected BSDE For any i ∈ I, one easily verifies by induction that the sequence (Y i,n ) n∈N is nondecreasing and upper bounded byȲ the first component of the unique solution tō For any i ∈ I, by Peng's monotonic limit Theorem (see Theorem 3.6 in [5]), the limiṫ Y i of (Y i,n ) n is a càdlàg process and there exists ( In order to prove that (Ẏ ,Ż,K) is the minimal solution, we then introduce, in the spirit of [6] and for any i ∈ I, the smallest f i -supermartingaleỸ i with lower barrier max j =i {Ẏ j − C ij } defined as the solution of For i ∈ I, we directly deduce from (3.2) thatẎ i ≥Ỹ i , and, since (Y .,n ) n∈N is increasing, a direct comparison argument leads to Y i,n ≤Ỹ i , for n ∈ N. Therefore, we getẎ =Ỹ so thatẎ satisfies (2.2).
It only remains to prove thatẎ is continuous. We look towards a contradiction and suppose on the contrary the existence of a measurable subset B of Ω satisfying P(B) > 0 and such that We fix ω ∈ B and deduce from (3.3) that ∆Ẏ for some i 1 (ω) = i 0 (ω). This leads directly tȯ < 0. The exact same argument leads to the existence of an integer i 2 (ω) ∈ I such that i 2 (ω) = i 1 (ω) anḋ t(ω) < 0. Since I is finite, we get, after repeating this argument at most d times, the existence of a finite sequence (i k (ω)) 0≤k≤n(ω) such that i n(ω) (ω) = i 0 (ω) anḋ for any k < n(ω). Therefore, we have Furthermore, using recursively the structural condition (2.1) (iii), we easily compute .
Since i n(ω) (ω) = i 0 (ω), we deduce from (2.1) (iii) and (i) that Since (3.4) and (3.5) hold true for any ω ∈ B, this contradicts P(B) > 0 and concludes the proof. ✷ Under (H3), a slight generalization of Theorem 3.1 in [4] allows to represent the process (Ẏ i ) i∈I as the value process of an optimal switching problem on a family of onedimensional BSDEs. More precisely, we consider the set of admissible 2 strategies A consisting in the sequence a = (θ k , α k ) k≥0 with • (θ k ) k≥0 a nondecreasing sequence of stopping times valued in [0, T ], and such that there exists an integer valued random variable N a , F T measurable and θ N a = T ,