in the forward equation (see also [13] for the existence of solution to one-dimensional FBSDEs with bounded Lipschitz coe cients and non-degenerate di usion in the forward equation). In [14], Hu and Peng introduced the monotonicity condition, under which the FBSDEs can be solved, and their main idea is to use the method of continuation.
FBSDEs are encountered when one applies the stochastic maximum principle to optimal stochastic control problems (see, e.g. [11]). Such equations are also encountered in ÿnance (see, e.g. [10,5,3]), as well as in the potential theory (see, e.g. [12]). More recently, one can apply such FBSDEs to study the homogenization and singular perturbations of certain quasilinear parabolic PDEs with periodic structures (see [4,2]).
However, as far as we know, in the works concerning the existence and uniqueness of solutions to FBSDEs, the coe cients b; ; g and h are always assumed to be Lipschitz. The objective of this paper is to study the existence and uniqueness of solution to FBSDEs with non-Lipschitz coe cients.
We shall establish an existence and uniqueness result for the FBSDEs over an arbitrarily prescribed time duration, without the Lipschitz condition on the coe cients b; ; g and h; instead, we assume a kind of "monotonicity" condition as is assumed by [14]. Also, we will give the "deterministic" version of our result which gives the existence and uniqueness of solution to a two-point boundary value problem. We think that this result is of independent interest in the deterministic two-point boundary value problem.
This paper is inspired by the recent books of Da Prato and Zabczyk [6,7], where the authors have used the Yosida approximations for monotone and continuous functions to study the existence and uniqueness of solution to inÿnite-dimensional stochastic equations with non-Lipschitz coe cients.
This paper is organized as follows: In Section 2, we give the formulation of the problem and our standing assumptions; in Section 3, we recall the Yosida approximations for monotone and continuous functions from [6,7]; in Section 4, we give our main result about the existence and uniqueness of solution to FBSDEs (1) and (2); in Section 5, we give the "deterministic" version of our result; and in the last section, we apply the same method to generalize the result of Dellacherie et al. [9] and that of [12].