Lp Solutions of BSDEs with Stochastic Lipschitz Condition

We are concerned with the solutions of a special class of backward stochastic differential equations which are driven by a Brownian motion, where the uniform Lipschitz continuity is replaced by a stochastic one. We prove the existence and uniqueness of the solution in L p with p > 1 .


Introduction
In this paper, we study backward stochastic differential equations (BSDEs for short) of the form where τ is a bounded stopping time for the filtration F.
Since the first result about the solutions in L 2 was obtained by Pardoux and Peng [1], some related results have been generalized.Moreover, for mathematical interest, many people have studied the results of existence and uniqueness in L p .Let us mention that when the generator is uniformly Lipschitz continuous, a result of El Karoui et al. [2] provides the existence of a solution when the data ξ and { f (t,0,0)} t∈[0,T] are in L p for p ∈ (1,∞).But in many applications, Lipschitz condition is too restrictive to be assumed.Consequently, we are interested in replacing the Lipschitz condition with a weaker one and we always assume that τ is bounded.In this field, in [3], Briand and Carmona have discussed the L p solutions for BSDEs with polynomial growth generators and then in [4], Briand et al. generalized the result.Now let us mention that the pricing problem of an American claim is equivalent to solving the BSDE where r(t) is the interest rate and θ(t) is the risk premium vector.In general, both of them may be unbounded, therefore the results mentioned above may be invalid.
In this paper, we try to get the existence and uniqueness result of L p (p > 1) solutions for BSDEs with stochastic Lipschitz condition, which was introduced by El Karoui and Huang [5].We have to mention that Bender and Kohlman have discussed BSDEs with stochastic Lipschitz condition and by strengthening the integrability conditions on the generator and the terminal value, they got a wellposedness result in L 2 in [6].We also strengthen the integrability conditions both on the data (ξ, f ) and on the solutions, but we do not use the contraction mapping theorem which plays a key role in [6] any longer.Instead, just like the work in [4], we construct a sequence of special BS-DEs which have unique solutions in L 2 , and then prove that the sequence of their solutions converge in L p .However, now it is not constants that control the generator.On the other hand, noting that the maturity of an American claim is bounded in general, we assume the stopping time is bounded in this paper.
The paper is organized as follows.In Section 2, we introduce the assumptions, some notations including some spaces, which are different from the standard spaces used before.In Section 3, some useful a priori estimates are given.The main result of this paper, an existence and uniqueness theorem in L p , is obtained in Section 4.

Definition and notations.
First of all, W = {W t } t≥0 is a standard Brownian motion with values in R d defined on some complete probability space (Ω,Ᏺ,P).F = {Ᏺ t } t≥0 augmented by all P-null-sets is the natural filtration of W, which satisfies the usual conditions.
For convenience in writing and reading, we always consider the space L 2p where p > 1/2 instead of the space L p where p > 1.
The standard inner product of R m is denoted by We study the following BSDE: where τ is a stopping time for filtration F. Now we can introduce the appropriate spaces.

Assumptions on data (ξ, f
).Now we make the following assumptions.For β > 0, (A1) τ is a stopping time for the filtration F and P-a.s., τ ≤ T < ∞, where T is a positive constant; (A2) there are two nonnegative F-adapted processes μ(t) and γ(t) such that ∀(y, z, y ,z We refer to (A2) as the stochastic Lipschitz condition.

A priori estimates
The goal of this section is to study some estimates concerning solutions to the BSDE (2.
Firstly, we recall the result of Bender and Kohlmann [6, Theorem 3].
Since Theorem 3.1 demands that β is large enough, we can always assume that Moreover, letting (A6) holds, by Lemma 2.2, the unique pair (Y ,Z) in Theorem 3.1 is an (a,β)-solution of BSDE (2.1).Now we give a basic estimate concerning the solution.
Let us turn to the existence part.