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

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.