Viscosity Solutions to Second Order Path-Dependent Hamilton-Jacobi-Bellman Equations in Hilbert Spaces

TL;DR

This paper introduces a new concept of viscosity solutions for second order path-dependent Hamilton-Jacobi-Bellman equations in Hilbert spaces, establishing their uniqueness and consistency with classical solutions.

Contribution

It defines viscosity solutions for PHJB equations in infinite-dimensional spaces and proves their uniqueness and stability, linking them to optimal control problems.

Findings

Value functional is the unique viscosity solution.

Viscosity solutions are consistent with classical solutions.

The notion satisfies stability properties.

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 evolution equations in Hilbert spaces. 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.