Comparison for semi-continuous viscosity solutions for second order PDEs on the Wasserstein space

Abstract

In this paper, we prove a comparison result for semi-continuous viscosity solutions of a class of second-order PDEs in the Wasserstein space. This allows us to remove the Lipschitz continuity assumption with respect to the Fourier-Wasserstein distance in AriX: 2309.05040 and obtain uniqueness by directly working in the Wasserstein space. In terms of its application, we characterize the value function of a stochastic control problem with partial observation as the unique viscosity solution to its corresponding HJB equation. Additionally, we present an application to a prediction problem under partial monitoring, where we establish an upper bound on the limit of regret using our comparison principle for degenerate dynamics.

To appear in the Journal of Differential Equations. Keywords: Wasserstein space, second-order PDEs, viscosity solutions, comparison principle