Infinite Horizon Stochastic Delay Evolution Equations in Hilbert Spaces and Stochastic Maximum Principle

In this paper, a class of infinite horizon optimal control problems is established, where the state equation is given by a stochastic delay evolution equation (SDEE), and the corresponding adjoint equation is given by an anticipated backward stochastic evolution equation (ABSEE). Firstly, we extend the form of Ito formula. After that, we establish the priori estimate for the solution to ABSEEs, and then the existence and uniqueness results of ABSEEs on in finite horizon are obtained. Finally, we establish necessary and sufficient conditions of stochastic maximum principle for infinite horizon optimal control problem in the form of Pontryagin’s maximum principle.