Stochastic maximum principle for delayed backward doubly stochastic control systems

In this paper, we investigate a class of doubly stochastic optimal control problems that the state trajectory is described by backward doubly stochastic differential equations with time delay. By means of martingale representation theorem and contraction mapping principle, the existence and uniqueness of solution for the delayed backward doubly stochastic differential equation can be guaranteed. When the control domain is convex, we deduce a stochastic maximum principle as a necessary condition of the optimal control by using classical variational technique. At the same time, under certain assumptions, a sufficient condition of optimality is obtained by using the duality method. In the last section, we give the explicit form of the optimal control for delayed doubly stochastic linear quadratic optimal control problem by our stochastic maximal principle