VALUE IN MIXED STRATEGIES FOR ZERO-SUM STOCHASTIC DIFFERENTIAL GAMES WITHOUT ISAACS CONDITION

摘要

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 Wπ and Uπ 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.