Path-dependent Hamilton-Jacobi-Bellman equations related to controlled stochastic functional differential systems

TL;DR

This paper develops a framework for stochastic optimal control of systems governed by stochastic functional differential equations, establishing the connection between the value function and a path-dependent HJB equation using viscosity solutions.

Contribution

It introduces a novel approach to derive the Path-dependent HJB equation for controlled stochastic functional differential systems and proves the value function is its viscosity solution.

Findings

Established the dynamic programming principle for the system.

Proved the value function is a viscosity solution of the Path-dependent HJB equation.

Extended stochastic control theory to functional differential systems.

Abstract

In this paper, a stochastic optimal control problem is investigated in which the system is governed by a stochastic functional differential equation. In the framework of functional It\^o calculus, we build the dynamic programming principle and the related Path-dependent Hamilton-Jacobi-Bellman (HJB) equation. We prove that the value function is the viscosity solution of the Path-dependent HJB equation.