Value in mixed strategies for zero-sum stochastic differential games without Isaacs condition

TL;DR

This paper introduces a new concept of mixed strategies for zero-sum stochastic differential games without Isaacs condition, proving the existence of a game value as the limit of approximate values and characterizing it via a PDE.

Contribution

It develops a novel framework for mixed strategies in stochastic differential games without Isaacs condition, establishing the existence and characterization of the game value.

Findings

The lower and upper value functions converge uniformly to a unique value function.

The value function is characterized as the viscosity solution of a Hamilton-Jacobi-Bellman-Isaacs equation.

A new concept of mixed strategies with nonanticipative controls is introduced.

Abstract

In the present work, we consider 2-person zero-sum stochastic differential games with a nonlinear pay-off functional which is defined through a backward stochastic differential equation. Our main objective is to study for such a game the problem of the existence of a value without Isaacs condition. Not surprising, this requires a suitable concept of mixed strategies which, to the authors' best knowledge, was not known in the context of stochastic differential games. For this, we consider nonanticipative strategies with a delay defined through a partition $\pi$ of the time interval $[0,T]$. The underlying stochastic controls for the both players are randomized along $\pi$ by a hazard which is independent of the governing Brownian motion, and knowing the information available at the left time point $t_{j-1}$ of the subintervals generated by $\pi$, the controls of Players 1 and 2 are conditionally independent over $[t_{j-1},t_j)$. It is shown that the associated lower and upper value functions $W^{\pi}$ and $U^{\pi}$ converge uniformly on compacts to a function $V$, the so-called value in mixed strategies, as the mesh of $\pi$ tends to zero. This function $V$ is characterized as the unique viscosity solution of the associated Hamilton-Jacobi-Bellman-Isaacs equation.