Feedback Optimal Control for Stochastic Volterra Equations with   Completely Monotone Kernels
  In this paper we are concerned with a class of stochastic Volterra integro-differential problems 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 provide other interesting result and require a precise descriprion of the properties of the generated semigroup. The first main result of the paper is the proof of existence and uniqueness of a mild solution for the corresponding Hamilton-Jacobi-Bellman (HJB) equation. The main technical point consists in the differentiability of the BSDE associated with the reformulated equation with respect to its initial datum x. 
Fulvia Confortola, Elisa Mastrogiacomo
math.PR math.OC