Mean Field Games with Singular Controls

TL;DR

This paper proves the existence of relaxed solutions for mean field games with singular controls and demonstrates how these solutions can be approximated by solutions with regular controls, using the Skorokhod M1 topology.

Contribution

It introduces a framework for establishing existence and approximation of solutions in MFGs with singular controls, leveraging the Skorokhod M1 topology.

Findings

Existence of relaxed solutions for MFGs with singular controls

Approximation of singular control solutions by regular control solutions

Use of Skorokhod M1 topology for analysis

Abstract

This paper establishes the existence of relaxed solutions to mean field games (MFGs for short) with singular controls. We also prove approximations of solutions results for a particular class of MFGs with singular controls by solutions, respectively control rules, for MFGs with purely regular controls. Our existence and approximation results strongly hinge on the use of the Skorokhod $M_1$ topology on the space of c\`adl\`ag functions.