On Zero-Sum Stochastic Differential Games

Abstract

We generalize the results of Fleming and Souganidis (1989) on zero sum stochastic differential games to the case when the controls are unbounded. We do this by proving a dynamic programming principle using a covering argument instead of relying on a discrete approximation (which is used along with a comparison principle by Fleming and Souganidis). Also, in contrast with Fleming and Souganidis, we define our pay-off through a doubly reflected backward stochastic differential equation. The value function (in the degenerate case of a single controller) is closely related to the second order doubly reflected BSDEs.

Key Words: Zero-sum stochastic differential games, Elliott-Kalton strategies, dynamic programming principle, stability under pasting, doubly reflected backward stochastic differential equations, viscosity solutions, obstacle problem for fully non-linear PDEs, shifted processes, shifted SDEs, second-order doubly reflected backward stochastic differential equations