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

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 $π$ of the time interval $[0,T]$. The underlying stochastic controls for the both players are randomized along $π$ 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 $π$, 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 $W^π$ and $U^π$ converge uniformly on compacts to a function $V$, the so-called value in mixed strategies, as the mesh of $π$ tends to zero. This function $V$ is characterized as the unique viscosity solution of the associated Hamilton-Jacobi-Bellman-Isaacs equation.