On Martingale Problems with Continuous-Time Mixing and Values of Zero-Sum Games without the Isaacs Condition

Abstract

We consider a zero-sum stochastic differential game over elementary mixed feedback strategies. These are strategies based only on the knowledge of the past state, randomized continuously in time from a sampling distribution which is kept constant in between some stopping rules. Once both players choose such strategies, the state equation admits a unique solution in the sense of the martingale problem of Stroock and Varadhan. We show that the game defined over martingale solutions has a value, which is the unique continuous viscosity solution of the randomized Isaacs equation.