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

Published online by Cambridge University Press:  08 November 2010

Liangquan Zhang  and
Yufeng Shi
Liangquan Zhang
Affiliation:
School of Mathematics, Shandong University, Jinan 250100, P.R. China. [email protected] Laboratoire de Mathématiques, Université de Bretagne Occidentale, 29285 Brest Cedex, France.
Yufeng Shi
Affiliation:
School of Mathematics, Shandong University, Jinan 250100, P.R. China. [email protected]
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Abstract

The maximum principle for optimal control problems of fully coupledforward-backward doubly stochastic differential equations (FBDSDEs in short)in the global form is obtained, under the assumptions that the diffusioncoefficients do not contain the control variable, but the control domainneed not to be convex. We apply our stochastic maximum principle (SMP inshort) to investigate the optimal control problems of a class of stochasticpartial differential equations (SPDEs in short). And as an example of theSMP, we solve a kind of forward-backward doubly stochastic linear quadraticoptimal control problems as well. In the last section, we use the solutionof FBDSDEs to get the explicit form of the optimal control for linearquadratic stochastic optimal control problem and open-loop Nash equilibriumpoint for nonzero sum stochastic differential games problem.

Keywords

Maximum principlestochastic optimal controlforward-backward doubly stochastic differential equationsspike variationsvariational equationsstochastic partial differential equationsnonzerosum stochastic differential game
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 4 , October 2011 , pp. 1174 - 1197
Copyright
© EDP Sciences, SMAI, 2010

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