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

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.