Stochastic Optimal Control with Delay in the Control II: Verification Theorem and Optimal Feedbacks

TL;DR

This paper establishes a verification theorem and constructs optimal feedback controls for stochastic delay control problems, building on previous work solving the associated HJB equation with partial smoothing.

Contribution

It provides a rigorous verification theorem and demonstrates the existence of optimal feedback controls for stochastic control problems with delays in the control.

Findings

Verification theorem for stochastic control with delay

Existence of optimal feedback controls

Solution framework based on regular HJB equation

Abstract

We consider a stochastic optimal control problem governed by a stochastic differential equation with delay in the control. Using a result of existence and uniqueness of a sufficiently regular mild solution of the associated Hamilton-Jacobi-Bellman (HJB) equation, see the companion paper "Stochastic Optimal Control with Delay in the Control I: solving the HJB equation through partial smoothing ", we solve the control problem by proving a Verification Theorem and the existence of optimal feedback controls.