A Stochastic Maximum Principle for Partially Observed Stochastic Control Systems with Delay

This paper deals with partially-observed optimal control problems for the state governed by stochastic differential equation with delay. We develop a stochastic maximum principle for this kind of optimal control problems using a variational method and a filtering technique. Also, we establish a sufficient condition without assumption of the concavity. Two examples that shed light on the theoretical results are established in the paper. In particular, in the example of an optimal investment problem with delay, its numerical simulation shows the effect of delay via a discretization technique for forward–backward stochastic differential equations (FBSDEs) with delay and anticipate terms.