On martingale problems with continuous-time mixing and values of zero-sum games without Isaacs condition

TL;DR

This paper studies zero-sum stochastic differential games with continuous-time mixed strategies, establishing the existence of a game value as the unique viscosity solution of a randomized Isaacs equation, without requiring the Isaacs condition.

Contribution

It introduces a framework for zero-sum games with continuous-time mixed strategies and proves the existence of a game value via martingale problem solutions, extending classical results.

Findings

The game admits a unique value as a viscosity solution.

Strategies based on past states and continuous randomization are effective.

The approach does not require the Isaacs condition for value existence.

Abstract

We consider a zero-sum stochastic differential game over elementary mixed feed-back 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.