Mean Field Games with Singular Controls of Bounded Velocity

Abstract

This paper studies a class of mean field games (MFGs) with singular controls of bounded velocity. By relaxing the absolute continuity of the control process, it generalizes the MFG framework of Lasry and Lions and Huang, Malhame, and Caines. It provides a unique solution to the MFG with explicit optimal control policies and establishes the e-Nash equilibrium of the corresponding N-player game. Finally, it analyzes a particular MFG with explicit solutions in a systemic risk model originally formulated by Carmona, Fouque, and Sun in a regular control setting.