Stochastic optimal control for backward stochastic partial differential systems

This paper studies optimal controls for a class of backward stochastic partial differential systems in the abstract evolution form. Under the assumption of a convex control domain, necessary and sufficient conditions for an admissible control to be optimal are derived in the form of stochastic maximum principles by means of a convex variation method and a duality technique. As an application, the optimal control for a linear backward stochastic evolution equation (BSEE) with quadratic cost criteria (called BSEELQ problem) is discussed, and the corresponding optimal control is characterized via the stochastic Hamilton system which is a linear full-coupled forward–backward stochastic evolution equation (FBSEE) and consists of the state equation, the adjoint equation and the dual presentation of the optimal control.