Maximum Principle for Backward Doubly Stochastic Control Systems with Applications

Abstract

We investigate the optimal control problems for backward doubly stochastic control systems. As a necessary condition of the optimal control we obtain a stochastic maximum principle. We found that the deduced stochastic Hamiltonian system exactly corresponds to a type of time-symmetric forward-backward stochastic differential equations, which was first introduced by Peng and Shi [C. R. Math. Acad. Sci. Paris, 336 (2003), pp. 773–778]. Applying the stochastic maximum principle to doubly stochastic linear quadratic problems, we obtain the unique optimal control. The existence and uniqueness of the solution is also obtained for a type of generalized forward-backward doubly stochastic differential equations derived from the maximum principle under some suitable assumptions. In particular, the result can be used in backward doubly stochastic linear quadratic optimal control problems. An example is given to illustrate the theoretical results.

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