Viscosity Solutions to Second Order Path-Dependent Hamilton-Jacobi-Bellman Equations and Applications

TL;DR

This paper introduces a new concept of viscosity solutions for second order path-dependent Hamilton-Jacobi-Bellman equations, establishing their uniqueness, stability, and applicability to stochastic control problems.

Contribution

It defines viscosity solutions for PHJB equations, proves their uniqueness and stability, and connects them to classical solutions and applications in stochastic control.

Findings

Value functional is the unique viscosity solution.

Viscosity solutions are consistent with classical solutions.

Applications to backward stochastic HJB equations are demonstrated.

Abstract

In this article, a notion of viscosity solutions is introduced for second order path-dependent Hamilton-Jacobi-Bellman (PHJB) equations associated with optimal control problems for path-dependent stochastic differential equations. We identify the value functional of optimal control problems as unique viscosity solution to the associated PHJB equations. We also show that our notion of viscosity solutions is consistent with the corresponding notion of classical solutions, and satisfies a stability property. Applications to backward stochastic Hamilton-Jacobi-Bellman equations are also given.