Maximum principle for forward-backward doubly stochastic control systems and applications^*

¹ School of Mathematics, Shandong University, Jinan 250100, P.R. China. yfshi@sdu.edu.cn
² Laboratoire de Mathématiques, Université de Bretagne Occidentale, 29285 Brest Cedex, France.

Received: 30 June 2009
Revised: 4 January 2010
Revised: 14 August 2010

Abstract

The maximum principle for optimal control problems of fully coupled forward-backward doubly stochastic differential equations (FBDSDEs in short) in the global form is obtained, under the assumptions that the diffusion coefficients do not contain the control variable, but the control domain need not to be convex. We apply our stochastic maximum principle (SMP in short) to investigate the optimal control problems of a class of stochastic partial differential equations (SPDEs in short). And as an example of the SMP, we solve a kind of forward-backward doubly stochastic linear quadratic optimal control problems as well. In the last section, we use the solution of FBDSDEs to get the explicit form of the optimal control for linear quadratic stochastic optimal control problem and open-loop Nash equilibrium point for nonzero sum stochastic differential games problem.