Backward linear quadratic stochastic optimal control problems and nonzero sum differential games

In R. Buckdahn, Li, Peng [6], the authors obtained mean-field backward stochastic Differential equations (BSDEs) in a natural way as a limit of some highly dimensional system of forward and backward SDEs, corresponding to a great number of particles. In this paper, firstly, we prove that there exists a unique solution of fully coupled MF-FBSDE. Then we use the solutions of mean-field forward-backward stochastic differential equations to get the explicit form of the optimal control for linear quadratic optimal control problem and the open-loop Nash equilibrium point of nonzero sum differential games.

1 Introduction

FBSDEs can be encountered in stochastic optimization problem, especially when one studies the stochastic maximum principle. It also has useful applications in mathematical finance, especially in derivative security pricing models that involve large investors. The FBSDE with terminal coupled can be encountered in the classical optimal control problems and game problems of SDEs. Using probability method, Hu and Peng [15], Peng and Wu [11] obtained the existence and uniqueness result under some monotone assumptions. Nonlinear backward stochastic differential equations (BSDEs) in the framework of Brownian motion were originally introduced by Pardoux and Peng [10]. They proved the existence and the uniqueness of the solution when the coefficient is Lipschitz in and the terminal condition is square integrable. The backward linear quadratic stochastic control problems which have important applications in mathematical finance have been studied by Lim and Zhou [14] using square complete technique with deterministic coefficients. This paper is concerned with the game problem of a MF-FBSDE.