, Volume 42, Issue 4, pp 989-1020
Date: 26 Oct 2012

Value function of differential games without Isaacs conditions. An approach with nonanticipative mixed strategies

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Abstract

In the present paper we investigate the problem of the existence of a value for differential games without Isaacs condition. For this we introduce a suitable concept of mixed strategies along a partition of the time interval, which are associated with classical nonanticipative strategies (with delay). Imposing on the underlying controls for both players a conditional independence property, we obtain the existence of the value in mixed strategies as the limit of the lower as well as of the upper value functions along a sequence of partitions which mesh tends to zero. Moreover, we characterize this value in mixed strategies as the unique viscosity solution of the corresponding Hamilton–Jacobi–Isaacs equation.