Stochastic Perron's Method and Elementary Strategies for Zero-Sum Differential Games

Abstract

We develop here the stochastic Perron method in the framework of two-player zero-sum differential games. We consider the formulation of the game where both players play, symmetrically, feedback strategies as opposed to the Elliott--Kalton formulation prevalent in the literature. The class of feedback strategies we use is carefully chosen so that the state equation admits strong solutions and the technicalities involved in the stochastic Perron method carry through in a rather simple way. More precisely, we define the game over elementary strategies, which are well motivated by intuition. Within this framework, the stochastic Perron method produces a viscosity subsolution of the upper Isaacs equation dominating the upper value of the game, and a viscosity supersolution of the upper Isaacs equation lying below the upper value of the game. Using a viscosity comparison result we obtain that the upper value is the unique and continuous viscosity solution of the upper Isaacs equation. An identical statement holds true for the lower value and the lower Isaacs equation. A version of the dynamic programming principle is obtained as a byproduct. If the Isaacs condition is satisfied, the game has a value over elementary (pure) strategies.