Viscosity solutions to HJB equations associated with optimal control problem for McKean-Vlasov SDEs

TL;DR

This paper develops a viscosity solution framework for HJB equations on Wasserstein space to address optimal control problems for McKean-Vlasov SDEs, establishing a comparison principle using advanced variational methods.

Contribution

It introduces a novel viscosity solution theory for HJB equations on Wasserstein space, incorporating Mortensen's derivative and overcoming compactness issues with Borwein-Preiss variational principle.

Findings

Established a comparison principle for viscosity solutions.

Developed a new approach using Mortensen's derivative.

Addressed compactness challenges in Wasserstein space.

Abstract

This work concerns the optimal control problem for McKean-Vlasov SDEs. In order to characterize the value function, we develop the viscosity solution theory for Hamilton-Jacobi-Bellman (HJB) equations on the Wasserstein space using Mortensen's derivative. In particular, a comparison principle for viscosity solution is established. Our approach is based on Borwein-Preiss variational principle to overcome the loss of compactness for bounded sets in the Wasserstein space.