Comparison of viscosity solutions for a class of second order PDEs on the Wasserstein space

TL;DR

This paper establishes a comparison principle for viscosity solutions of second order PDEs on the Wasserstein space, relevant for stochastic control and filtering problems, using a novel Ishii's lemma adaptation.

Contribution

It introduces a new comparison result for viscosity solutions in Wasserstein space, applicable to McKean-Vlasov control and filtering equations, with a specialized proof technique.

Findings

Comparison result for viscosity solutions established

Applicable to McKean-Vlasov control problems

Novel Ishii's lemma for Wasserstein space

Abstract

We prove a comparison result for viscosity solutions of second order parabolic partial differential equations in the Wasserstein space. The comparison is valid for semisolutions that are Lipschitz continuous in the measure in a Fourier-Wasserstein metric and uniformly continuous in time. The class of equations we consider is motivated by Mckean-Vlasov control problems with common noise and filtering problems. The proof of comparison relies on a novel version of Ishii's lemma, which is tailor-made for the class of equations we consider.