Stochastic Path-Dependent Hamilton-Jacobi-Bellman Equations and Controlled Stochastic Differential Equations with Random Path-Dependent Coefficients

TL;DR

This paper introduces the stochastic path-dependent Hamilton-Jacobi-Bellman equation, establishing its solutions and uniqueness, and extends stochastic calculus tools to path-dependent settings, advancing stochastic control theory.

Contribution

It formulates the SPHJB equation for path-dependent stochastic control problems and proves existence, uniqueness, and a new Itô-Kunita-Wentzell-Krylov formula in this context.

Findings

Value function is the viscosity solution of the SPHJB equation.

Uniqueness of viscosity solutions established for superparabolic cases.

A new Itô-Kunita-Wentzell-Krylov formula for path-dependent stochastic calculus.

Abstract

In this paper, we propose and study the stochastic path-dependent Hamilton-Jacobi-Bellman (SPHJB) equation that arises naturally from the optimal stochastic control problem of stochastic differential equations with path-dependence and measurable randomness. Both the notions of viscosity solution and classical solution are proposed, and the value function of the optimal stochastic control problem is proved to be the viscosity solution to the associated SPHJB equation. A uniqueness result about viscosity solutions is also given for certain superparabolic cases, while the uniqueness of classical solution is addressed for general cases. In addition, an It\^o-Kunita-Wentzell-Krylov formula is proved for the compositions of random fields and stochastic differential equations in the path-dependent setting.