Doubly Reflected BSDEs in the predictable setting

TL;DR

This paper introduces predictable doubly reflected backward stochastic differential equations (DRBSDEs) on non quasi-left continuous filtrations, establishing existence and uniqueness of solutions under Mokobodzki's condition using fixed point methods.

Contribution

It extends the theory of DRBSDEs to a broader setting with predictable barriers and general filtrations, providing new existence and uniqueness results.

Findings

Existence of solutions under Mokobodzki's condition

Uniqueness of solutions via generalized Itô's formula

Application of Picard iteration and Banach fixed point theorem

Abstract

In this paper, we introduce a specific kind of doubly reflected Backward Stochastic Differential Equations (in short DRBSDEs), defined on probability spaces equipped with general filtration that is essentially non quasi-left continuous, where the barriers are assumed to be predictable processes. We call these equations predictable DRBSDEs. Under a general type of Mokobodzki's condition, we show the existence of the solution (in consideration of the driver's nature) through a Picard iteration method and a Banach fixed point theorem. By using an appropriate generalization of It\^o's formula due to Gal'chouk and Lenglart, we provide a suitable a priori estimates which immediately implies the uniqueness of the solution.