Viscosity solutions for obstacle problems on Wasserstein space

TL;DR

This paper extends the characterization of mean field optimal stopping problems on Wasserstein space by establishing a viscosity solution framework under weaker regularity conditions, including existence, stability, and comparison results.

Contribution

It introduces a viscosity solution concept for obstacle problems on Wasserstein space and proves key properties without requiring smoothness of the value function.

Findings

Established a viscosity solution framework for obstacle problems on Wasserstein space.

Proved existence, stability, and comparison principles for the viscosity solutions.

Extended previous characterizations to weaker regularity settings.

Abstract

This paper is a continuation of our accompanying paper [Talbi, Touzi and Zhang (2021)], where we characterized the mean field optimal stopping problem by an obstacle equation on the Wasserstein space of probability measures, provided that the value function is smooth. Our purpose here is to establish this characterization under weaker regularity requirements. We shall define a notion of viscosity solutions for such equation, and prove existence, stability, and comparison principle.