Stochastic Differential Games with Reflection and Related Obstacle Problems for Isaacs Equations

TL;DR

This paper studies zero-sum stochastic differential games with reflection, establishing the dynamic programming principle and proving the value functions as unique viscosity solutions of associated obstacle Hamilton-Jacobi-Bellman-Isaacs equations, with new estimates for RBSDEs.

Contribution

It introduces a novel approach to stochastic differential games with reflection, proving the dynamic programming principle and the uniqueness of viscosity solutions for related Isaacs equations with obstacles.

Findings

Established the dynamic programming principle for reflected stochastic differential games.

Proved the value functions are unique viscosity solutions of obstacle Hamilton-Jacobi-Bellman-Isaacs equations.

Derived a sharper estimate for Reflected Backward Stochastic Differential Equations (RBSDEs).

Abstract

In this paper we first investigate zero-sum two-player stochastic differential games with reflection with the help of theory of Reflected Backward Stochastic Differential Equations (RBSDEs). We will establish the dynamic programming principle for the upper and the lower value functions of this kind of stochastic differential games with reflection in a straight-forward way. Then the upper and the lower value functions are proved to be the unique viscosity solutions of the associated upper and the lower Hamilton-Jacobi-Bellman-Isaacs equations with obstacles, respectively. The method differs heavily from those used for control problems with reflection, it has its own techniques and its own interest. On the other hand, we also prove a new estimate for RBSDEs being sharper than that in El Karoui, Kapoudjian, Pardoux, Peng and Quenez [7], which turns out to be very useful because it allows to estimate the $L^p$-distance of the solutions of two different RBSDEs by the $p$-th power of the distance of the initial values of the driving forward equations. We also show that the unique viscosity solution of the approximating Isaacs equation which is constructed by the penalization method converges to the viscosity solution of the Isaacs equation with obstacle.