Ergodic Mean-Field Games of Singular Control with Regime-Switching (Extended Version)

TL;DR

This paper develops a theoretical framework for stationary mean-field games involving singular control and regime-switching, establishing existence, uniqueness, and approximate Nash equilibria for large populations.

Contribution

It introduces a novel approach linking ergodic singular control problems with regime switching to Dynkin games, proving existence and uniqueness of mean-field equilibria.

Findings

Proves existence and uniqueness of stationary mean-field equilibrium.

Establishes that the mean-field equilibrium approximates Nash equilibria in large N-player games.

Introduces a new Dynkin game characterization for singular control with regime switching.

Abstract

This paper studies a class of stationary mean-field games of singular stochastic control with regime-switching. The representative agent adjusts the dynamics of a Markov-modulated It\^o-diffusion via a two-sided singular stochastic control and faces a long-time-average expected profit criterion. The mean-field interaction is of scalar type and it is given through the stationary distribution of the population. Via a constructive approach, we prove the existence and uniqueness of the stationary mean-field equilibrium. Furthermore, we show that this realizes a symmetric $\varepsilon_N$-Nash equilibrium for a suitable ergodic $N$-player game with singular controls. The proof hinges on the characterization of the optimal solution to the representative player's ergodic singular stochastic control problem with regime switching in terms of an auxiliary Dynkin game, which is of independent interest and appears here for the first time.