On Optimal Control Problem for Backward Stochastic Doubly Systems

We are going to study an approach of optimal control problems where the state equation is a backward doubly stochastic differential equation, and the set of strict (classical) controls need not be convex and the diffusion coefficient and the generator coefficient depend on the terms control. The main result is necessary conditions as well as a sufficient condition for optimality in the form of a relaxed maximum principle. 1. Introduction In 1994, Pardoux and Peng [1] considered a new kind of BSDE, that is a class of backward doubly stochastic differential equations (BDSDEs in short) with two different directions of stochastic integrals, that is, the equations involve both a standard (forward) stochastic It？ integral and a backward stochastic It？ integral . More precisely, they dealt with the following BDSDE: They proved that if and are uniform Lipschitz, then (1), for any square integrable terminal value , has a unique solution in the interval . They also showed that BDSDEs can produce a probabilistic representation for solutions to some quasi-linear stochastic partial differential equations. Since this first existence and uniqueness result, many papers have been devoted to existence and/or uniqueness result under weaker assumptions. Among these papers, we can distinguish two different classes: Scalar BDSDEs and multidimensional BDSDEs. In the first case, one can take advantage of the comparison theorem: we refer to Shi et al. [2] who weakened the uniform Lipschitz assumptions to linear growth and continuous conditions by virtue of a comparison theorem introduced by them. They obtained the existence of solutions to BDSDEs but without uniqueness. In this spirit, let us mention the contributions of N’zi and Owo [3], which dealt with discontinuous coefficients. For multidimensional BDSDE, there is no comparison theorem, and to overcome this difficulty, a monotonicity assumption on the generator in the variable is used. This appears in the works of Peng and Shi [4] which have introduced a class of forward backward doubly stochastic differential equations, under the Lipschitz condition and monotonicity assumption. Unfortunately, the uniform Lipschitz condition cannot be satisfied in many applications. More recently, N’zi and Owo [5] established existence and uniqueness result under non-Lipschitz assumptions. Moreover the authors apply his theory to solve the financial model of cash flow valuation. In this paper, we study a stochastic control problem where the system is governed by a nonlinear backward doubly stochastic differential equation (BDSDE) of the