MAXIMUM PRINCIPLE FOR STOCHASTIC OPTIMAL CONTROL PROBLEM WITH DISTRIBUTED DELAYS

This paper is concerned with a Pontryagin’s maximum principle for the stochastic optimal control problem with distributed delays given by integrals of not necessarily linear functions of state or control variables. By virtue of the duality method and the generalized anticipated backward stochastic differential equations, we establish a necessary maximum principle and a sufficient verification theorem. In particular, we deal with the controlled stochastic system where the distributed delays enter both the state and the control. To explain the theoretical results, we apply them to a dynamic advertising problem.