A Generalized Existence Theorem Of BSDEs And RRBSDEs

One-dimensional backward stochastic differential equations (BSDEs) are equations of the following type:where(B_s)_{0≤t≤T} is a standard d-dimensional Brownian motion on a probability space(Ω,F,P), with {F_t,0≤t≤T} the standard Brownian filtration. The functiong:Ω×[0,t]×R×R^d→R is generally called a coefficient of (1.1), T theterminal time, and the R-valued F_T-adapted random variableξa terminal condition; (g, T,ξ) are the parameters of (1.1). A solution is a couple (y.,z.)of processes adapted to filtration {F_t,(0≤t≤T}.Nonlinear BSDEs were first introduced by Pardoux and Peng (1990), who proved the existence and uniqueness of a solution under suitable assumptions on g andξ, the most important of which are the Lipschitz continuity of g and the square integrability of f. Since then, BSDEs have been studied intensively. In particular, many efforts have been made to relax the assumption on the generator g; for instance, Lepeltier and San Martin ( 1997) have proved the existence of a solution for (1) when g is only continuous with linear growth, and Kobylanski (2000) obtained the existence and uniqueness of a solution when g is continuous and has a quadratic growth in z and the terminal condition is bounded. In their proofs, the comparison theorem plays an important role.Karoui, Kapoudjian, Pardoux, Peng and Quenez introduced in 1997 the notion of reflected BSDE (RBSDE in short) on one lower barrier [1]: the solution is forced to remain abovea continuous process, which is considered as the lower barrier. More precisely, a solutionfor such equation associated to a coefficient g, a terminal valueξ, a continuous barrier L, is a triple(Y_t,Z_t,K_t) adapted processes valued on R^1+d+1 , which satisfies a squareintegrability conditionFurthermore, the process(K_t)_{0≤t≤T}is non decreasing, continuous, and therole of (K_t)_{0≤t≤T} is to push upward the state process in a minimal way, to keep itabove L. They proved existence and uniqueness of a solution when f is Lipschitz in (y, z) uniformly in (t,ω).Cvitanic and Karatzas (1995) studied the backward stochastic differential equation with two barriers. In this case, a solution Y has remain between the lower boundary L and upper boundary U, almost surely. This is achieved by the cumulative action of two continuous, increasing reflecting processes。In this paper, authors proves the existence and uniqueness of the solution, under certain condition ofξ, L and U, and Lipschitz condition of generator g。More recently, Jia (2006) obtained a generalized existence theorem for one-dimensional BSDEs where the coefficient is left-Lipschitz in y (may be discontinuous) and Lipschitz in z. Shiqiu Zheng and Shengwu Zhou (2008) consider the existence of solutions of one-dimensional RBSDEs with two obstacles, under these assumptions. Jia and Xu (2006) obtained some necessary and sufficient conditions for the uniqueness for solution of one-dimensional BSDEs where the coefficient is uniformly continuous in z.In this paper, we will obtain generalized existence theorems for one-dimensional BSDEs and RBSDEs where the coefficient is left-Lipschitz in y and uniformly continuous in z.