Stochastic Differential Games of Mean-Field Dynamics and Second-Order Bellman–Isaacs Equations on the Wasserstein Space

Abstract

This paper concerns two-player zero-sum stochastic differential games with nonanticipative strategies against closed-loop controls in the case where the coefficients of mean-field stochastic differential equations and cost functional depend on the joint distribution of the state and the control. In our game, both the (lower and upper) value functions and the (lower and upper) second-order Bellman–Isaacs equations are defined on the Wasserstein space \({\cal P}_{2}({\mathbb R}^{n})\) which is an infinite dimensional space. The dynamic programming principle for the value functions is proved. If the (upper and lower) value functions are smooth enough, we show that they are the classical solutions to the second-order Bellman–Isaacs equations. On the other hand, the classical solutions to the (upper and lower) Bellman–Isaacs equations are unique and coincide with the (upper and lower) value functions. As an illustrative application, the linear quadratic case is considered. Under the Isaacs condition, the explicit expressions of optimal closed-loop controls for both players are given. Finally, we introduce the intrinsic notion of viscosity solution of our second-order Bellman–Isaacs equations, and characterize the (upper and lower) value functions as their viscosity solutions.