In this paper, we introduce and study the optimality conditions for stochastic control problems of nonlinear backward doubly stochastic differential equations. Necessary and sufficient optimality conditions, where the control domain is convex and the coefficients depend explicitly on the variable control, are proved. The results are stated in the form of weak stochastic maximum principle, and under additional hypotheses, we give these results in the global form. This is the first version of the stochastic maximum principle that covers the backward-doubly systems.
We consider control problems for systems driven by linear backward stochastic differential equations (BSDEs). We prove the existence of strict optimal controls under the convexity of the control domain as well as the cost functional. Our approach is based on strong convergence techniques for the associated linear BSDEs. Moreover, we establish necessary as well as sufficient conditions of optimality, satisfied by an optimal strict control. The proof of this result is based on the convex optimization principle.
Abstract. In this paper we study the stochastic maximum principle for a control problem in infinite dimensions. This problem is governed by a fully coupled forward-backward doubly stochastic differential equation (FBDSDE) driven by two cylindrical Wiener processes on separable Hilbert spaces and a Poisson random measure. We allow the control variable to enter in all coefficients appearing in this system. Existence and uniqueness of the solutions of FBDSDEs and an extended martingale representation theorem are provided as well.
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