Maximum principle for stochastic optimal control problem of finite state forward-backward stochastic difference systems

Abstract

In this paper, we study the maximum principle for stochastic optimal control problems of forward-backward stochastic difference systems (FBS∆Ss) where the uncertainty is modeled by a discrete time, finite state process, rather than white noises. Two types of FBS∆Ss are investigated. The first one is described by a partially coupled forward-backward stochastic difference equation (FBS∆E) and the second one is described by a fully coupled FBS∆E. By adopting an appropriate representation of the product rule and an appropriate formulation of the backward stochastic difference equation (BS∆E), we deduce the adjoint difference equation. Finally, the maximum principle for this optimal control problem with the control domain being convex is established.