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

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