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This paper deals with partially-observed optimal control problems for thestate governed by stochastic differential equation with delay. We develop astochastic maximum principle for this kind of optimal control problems using avariational method and a filtering technique. Also, we establish a sufficientcondition without assumption of the concavity. Two examples shed light on thetheoretical results are established in the paper. In particular, in the exampleof an optimal investment problem with delay, its numerical simulation shows theeffect of delay via a discretization technique for forward-backward stochasticdifferential equations (FBSDEs) with delay and anticipate terms.

A stochastic maximum principle for partially observed stochastic control systems with delay