Optimal control of stochastic delay differential equations: Optimal feedback controls

TL;DR

This paper develops a framework for optimal control of stochastic delay differential equations using viscosity solutions of Hamilton-Jacobi-Bellman equations, enabling the construction of optimal feedback controls with applications to advertising.

Contribution

It introduces a verification theorem for optimal controls in stochastic delay systems using partial regularity of the value function, advancing control theory in infinite-dimensional spaces.

Findings

Established a verification theorem for stochastic delay control problems.

Applied the theory to stochastic optimal advertising models.

Demonstrated the use of viscosity solutions in feedback control construction.

Abstract

In this manuscript, we study optimal control problems for stochastic delay differential equations using the dynamic programming approach in Hilbert spaces via viscosity solutions of the associated Hamilton-Jacobi-Bellman equations. We show how to use the partial $C^{1,\alpha}$-regularity of the value function established in \cite{defeo_federico_swiech} to obtain optimal feedback controls. The main result of the paper is a verification theorem which provides a sufficient condition for optimality using the value function. We then discuss its applicability to the construction of optimal feedback controls. We provide an application to stochastic optimal advertising problems.