Viscosity Solutions to Path-Dependent HJB Equation and Applications

TL;DR

This paper introduces viscosity solutions for path-dependent HJB equations, establishing the value functional as the unique solution and applying it to backward stochastic HJB equations in optimal control of path-dependent SDEs.

Contribution

It extends the viscosity solution framework to path-dependent HJB equations and proves the uniqueness of the value functional as the solution.

Findings

Value functional identified as unique viscosity solution.

Application to backward stochastic HJB equations demonstrated.

Framework applicable to path-dependent stochastic control problems.

Abstract

In this article, the notion of viscosity solution is introduced for the path-dependent Hamilton-Jacobi-Bellman (PHJB) equations associated with the optimal control problems for path-dependent stochastic differential equations. We identify the value functional of the optimal control problems as unique viscosity solution to the associated PHJB equations. Applications to backward stochastic Hamilton-Jacobi-Bellman equations are also given.