A maximum principle for Markov-modulated SDEs of mean-field type and mean-field game

TL;DR

This paper develops a stochastic maximum principle for controlling Markov-modulated mean-field SDEs, analyzes their solutions, and applies the results to approximate Nash equilibria in mean-field games with Markov modulation.

Contribution

It introduces a maximum principle for Markov-modulated mean-field SDEs and extends mean-field game theory to include Markov chain influences.

Findings

Established existence and uniqueness of solutions for Markov-modulated mean-field SDEs.

Proved propagation of chaos for the associated particle system.

Derived approximate Nash equilibria in the Markov-modulated mean-field game.

Abstract

In this paper, we analyze mean-field game modulated by finite states markov chains. We first develop a sufficient stochastic maximum principle for the optimal control of a Markov-modulated stochastic differential equation (SDE) of mean-field type whose coefficients depend on the state of the process, some functional of its law as well as variation of time and sample. As coefficients are perturbed by a Markov chain and thus random, to study such SDEs, we analyze existence and uniqueness of solutions of a class of mean-field type SDEs whose coefficients are random Lipschitz as well as the property of propagation of chaos for associated interacting particles system with method parallel to existing results as a byproduct. We also solve approximate Nash equilibrium for the Markov-modulated mean-field game by mean-field theory.