Maximum principle for forward-backward doubly stochastic control systems and applications*
School of Mathematics, Shandong University,
Jinan 250100, P.R. China. email@example.com
2 Laboratoire de Mathématiques, Université de Bretagne Occidentale, 29285 Brest Cedex, France.
Revised: 4 January 2010
Revised: 14 August 2010
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.
Mathematics Subject Classification: 93E20 / 60H10
Key words: Maximum principle / stochastic optimal control / forward-backward doubly stochastic differential equations / spike variations / variational equations / stochastic partial differential equations / nonzero sum stochastic differential game
L. Zhang has been partially supported by Marie Curie Initial Training Network (ITN) project: “Deterministic and Stochastic Controlled System and Application”, FP7-PEOPLE-2007-1-1-ITN, No. 213841-2. Y. Shi has been partially supported by National Natural Science Foundation of China Grants 10771122 and 11071145, Natural Science Foundation of Shandong Province of China Grant Y2006A08, Independent Innovation Foundation of Shandong University Grant 2010JQ010 and National Basic Research Program of China (973 Program, No. 2007CB814900).
© EDP Sciences, SMAI, 2010