Zero-sum stochastic differential games of generalized McKean–Vlasov type

We study zero-sum stochastic differential games where the state dynamics of the two players is governed by a generalized McKean–Vlasov (or mean-field) stochastic differential equation in which the distribution of both state and controls of each player appears in the drift and diffusion coefficients, as well as in the running and terminal payoff functions. We prove the dynamic programming principle (DPP) in this general setting, which also includes the control case with only one player, where it is the first time that DPP is proved for open-loop controls. We also show that the upper and lower value functions are viscosity solutions to a corresponding upper and lower Master Bellman–Isaacs equation. Our results extend the seminal work [15] of Fleming and Souganidis (1989) to the McKean–Vlasov setting.