Viscosity Solutions to First Order Path-Dependent HJB Equations

TL;DR

This paper introduces a new notion of viscosity solutions for first order path-dependent HJB equations, establishing their uniqueness, consistency with classical solutions, and stability, thus advancing the analysis of path-dependent control problems.

Contribution

It defines viscosity solutions for path-dependent HJB equations and proves their uniqueness and consistency with classical solutions.

Findings

The value functional is the unique viscosity solution.

The proposed viscosity solutions are consistent with classical solutions.

The notion satisfies a stability property.

Abstract

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