Stochastic Perron's method and elementary strategies for zero-sum differential games

TL;DR

This paper introduces a simplified approach to zero-sum differential games using the Stochastic Perron Method with elementary feedback strategies, establishing the existence and uniqueness of the game value via viscosity solutions.

Contribution

It develops the Stochastic Perron Method within a new framework of elementary feedback strategies, simplifying the analysis of zero-sum differential games and proving the game value's characterization.

Findings

The upper and lower values are characterized as unique viscosity solutions of the Isaacs equations.

The method provides a new proof of the Dynamic Programming Principle for these games.

Under the Isaacs condition, the game has a well-defined value over elementary strategies.

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, feed-back strategies (as in [CR09] or [PZ12]) as opposed to the Elliott-Kalton formulation prevalent in the literature. The class of feed-back 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 sub-solution of the upper Isaacs equation dominating the upper value of the game, and a viscosity super-solution 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 for the lower value and lower Isaacs equation. A version of the Dynamic Programming Principle is obtained as a by-product. If the Isaacs condition holds, the game has a value over elementary (pure) strategies.