Lifting of Volterra processes: optimal control in UMD Banach spaces

TL;DR

This paper develops a method to solve stochastic control problems involving Volterra processes by lifting them into infinite-dimensional UMD Banach spaces, enabling the use of dynamic programming for optimal control.

Contribution

It introduces a novel lifting approach to transform non-Markovian Volterra control problems into Markovian problems in UMD Banach spaces, facilitating solution characterization.

Findings

Equivalent infinite-dimensional Markovian formulation of the original problem

Characterization of optimal controls via dynamic programming

Demonstration of the method's applicability to Volterra processes

Abstract

We study a stochastic control problem for a Volterra-type controlled forward equation with past dependence obtained via convolution with a deterministic kernel. To be able to apply dynamic programming to solve the problem, we lift it to infinite dimensions and we formulate a UMD Banach-valued Markovian problem, which is shown to be equivalent to the original finite-dimensional non-Markovian one. We characterize the optimal control for the infinite dimensional problem and show that this also characterizes the optimal control for the finite dimensional problem.