Another Approach to the Existence of Value Functions of Stochastic Differential Games

Abstract

The existence of value functions for general two-player, zero-sum stochastic differential games has been obtained by Fleming and Souganidis. In this paper we present a new approach to this problem. We prove optimality inequalities of dynamic programming for viscosity sub- and supersolutions of the associated Bellman–Isaacs equations. These inequalities are well known for deterministic differential games but are new for stochastic differential games. It then easily follows that value functions are the unique viscosity solutions of the Bellman–Isaacs equations and satisfy the principle of dynamic programming. The results presented here are not the same as those of Fleming and Souganidis because we work with different reference spaces and the independence of value functions of the choice of reference spaces is not clear to us.