Infinite horizon stochastic optimal control for Volterra equations with completely monotone kernels

The aim of the paper is to study an optimal control problem on infinite horizon for an infinite dimensional integro-differential equation with completely monotone kernels, where we assume that the noise enters the system when we introduce a control. We start by reformulating the state equation into a semilinear evolution equation which can be treated by semigroup methods. The application to optimal control provides other interesting results and requires a precise description of the properties of the generated semigroup. The main tools consist in studying the differentiability of the forward–backward system with infinite horizon corresponding with the reformulated problem and the proof of existence and uniqueness of mild solutions to the corresponding Hamilton Jacobi Bellman (HJB) equation.