Generalized BSDEs with random time horizon in a progressively enlarged filtration

TL;DR

This paper investigates generalized backward stochastic differential equations with a non-stopping time random horizon, introducing a reduction method to establish existence results without restrictive assumptions on the random time.

Contribution

It extends the theory of BSDEs to more general random horizons without specific assumptions, including cases with jumps in the driver and the martingale.

Findings

Established existence of solutions for BSDEs with non-stopping time horizons.

Extended results to reflected BSDEs with jumps in the driver.

Provided a reduction approach applicable to a broad class of random times.

Abstract

We study generalized backward stochastic differential equations (BSDEs) up to a random time horizon $\vartheta$, which is not a stopping time, under minimal assumptions regarding the properties of $\vartheta$. In contrast to existing works in this area, we do not impose specific assumptions on the random time $\vartheta$ and we study the existence of solutions to BSDEs and reflected BSDEs with a random time horizon through the method of reduction. In addition, we also examine BSDEs and reflected BSDEs with a l\`adl\`ag driver where the driver is allowed to have a finite number of common jumps with the martingale part.