BSDEs with two RCLL Reflecting Obstacles driven by a Brownian Motion and Poisson Measure and related Mixed Zero-Sum Games

TL;DR

This paper investigates BSDEs with two RCLL reflecting obstacles driven by Brownian motion and Poisson measure, establishing existence, uniqueness, and applications to mixed zero-sum games with jumps.

Contribution

It introduces new existence and uniqueness results for BSDEs with two RCLL obstacles driven by both Brownian motion and Poisson jumps, without assuming obstacle difference supermartingales.

Findings

Proved existence and uniqueness under separated barriers and Lipschitz generator.

Established the value existence for related mixed zero-sum differential-integral games.

Extended BSDE theory to include jumps and general obstacle conditions.

Abstract

In this paper we study Backward Stochastic Differential Equations with two reflecting right continuous with left limits obstacles (or barriers) when the noise is given by Brownian motion and a Poisson random measure mutually independent. The jumps of the obstacle processes could be either predictable or inaccessible. We show existence and uniqueness of the solution when the barriers are completely separated and the generator uniformly Lipschitz. We do not assume the existence of a difference of supermartingales between the obstacles. As an application, we show that the related mixed zero-sum differential-integral game problem has a value.