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

TL;DR

This paper develops a dynamic programming principle and viscosity solution framework for zero-sum stochastic differential games involving generalized McKean-Vlasov dynamics, extending classical results to mean-field settings.

Contribution

It introduces the first proof of the dynamic programming principle for open-loop controls in McKean-Vlasov games and links value functions to Master Bellman-Isaacs equations.

Findings

DPP established for generalized McKean-Vlasov games

Value functions are viscosity solutions to Master Bellman-Isaacs equations

Extension of Fleming and Souganidis' work to mean-field context

Abstract

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 of Fleming and Souganidis [15] to the McKean-Vlasov setting.