Probabilistic Interpretation for Systems of Isaacs Equations with Two Reflecting Barriers

TL;DR

This paper studies zero-sum stochastic differential games with two reflecting barriers, establishing the deterministic nature of value functions and their characterization as unique viscosity solutions of associated Isaacs equations, using penalization methods.

Contribution

It introduces new estimates for RBSDEs with two barriers and demonstrates the approximation of viscosity solutions of Isaacs equations with barriers by penalized equations.

Findings

Value functions are deterministic despite initial randomness.

Unique viscosity solutions correspond to lower and upper Bellman-Isaacs equations.

New sharper estimates for RBSDEs with two barriers.

Abstract

In this paper we investigate zero-sum two-player stochastic differential games whose cost functionals are given by doubly controlled reflected backward stochastic differential equations (RBSDEs) with two barriers. For admissible controls which can depend on the whole past and so include, in particular, information occurring before the beginning of the game, the games are interpreted as games of the type "admissible strategy" against "admissible control", and the associated lower and upper value functions are studied. A priori random, they are shown to be deterministic, and it is proved that they are the unique viscosity solutions of the associated upper and the lower Bellman-Isaacs equations with two barriers, respectively. For the proofs we make full use of the penalization method for RBSDEs with one barrier and RBSDEs with two barriers. For this end we also prove new estimates for RBSDEs with two barriers, which are sharper than those in [18]. Furthermore, we show that the viscosity solution of the Isaacs equation with two reflecting barriers not only can be approximated by the viscosity solutions of penalized Isaacs equations with one barrier, but also directly by the viscosity solutions of penalized Isaacs equations without barrier.