Feedback Optimal Control for Stochastic Volterra Equations with Completely Monotone Kernels

TL;DR

This paper develops a framework for optimal control of stochastic Volterra equations with completely monotone kernels, proving existence and uniqueness of solutions to the associated HJB equation using semigroup methods.

Contribution

It introduces a novel reformulation of stochastic Volterra equations into semilinear evolution equations and establishes the differentiability of the related BSDEs for control purposes.

Findings

Proved existence and uniqueness of mild solutions for the HJB equation.

Established differentiability of BSDEs with respect to initial data.

Applied semigroup methods to stochastic Volterra equations with monotone kernels.

Abstract

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.