$L^p$ Solutions of Backward Stochastic Differential Equations with Jumps

TL;DR

This paper establishes the existence and uniqueness of $L^p$ solutions for multi-dimensional backward stochastic differential equations with jumps, even when the generator lacks Lipschitz continuity, by approximation techniques.

Contribution

It introduces a method to prove $L^p$ solvability for BSDEJs with non-Lipschitz generators using mollifier-based approximation and stability analysis.

Findings

Proves existence of unique $L^p$ solutions for BSDEJs with jumps.

Extends solvability results to generators that are not Lipschitz continuous.

Employs mollifier approximation to handle non-Lipschitz generators.

Abstract

Given $p \in (1, 2)$, we study $L^p$-solutions of a multi-dimensional backward stochastic differential equation with jumps (BSDEJ) whose generator may not be Lipschitz continuous in $(y,z)-$variables. We show that such a BSDEJ with a p-integrable terminal data admits a unique $L^p$ solution by approximating the monotonic generator by a sequence of Lipschitz generators via convolution with mollifiers and using a stability result.