Existence, Uniqueness and Comparison Results for BSDEs with L\'evy Jumps in an Extended Monotonic Generator Setting

TL;DR

This paper establishes existence, uniqueness, and comparison results for one-dimensional BSDEs with Lévy jumps under an extended monotonicity condition, and introduces an approximation method for solutions involving jumps of varying sizes.

Contribution

It generalizes previous results on BSDEs with jumps by relaxing conditions and introduces a new approximation technique for Lévy jump processes.

Findings

Proved existence and uniqueness of solutions under extended monotonicity.

Established a comparison theorem for BSDEs with Lévy jumps.

Developed an approximation method converging to the original BSDE solutions.

Abstract

We show existence of a unique solution and a comparison theorem for a one-dimensional backward stochastic differential equation with jumps that emerge from a L\'evy process. The considered generators obey a time-dependent extended monotonicity condition in the y-variable and have linear time-dependent growth. Within this setting, the results generalize those of Royer (2006), Yin and Mao (2008) and, in the $L^2$-case with linear growth, those of Kruse and Popier (2016). Moreover, we introduce an approximation technique: Given a BSDE driven by Brownian motion and Poisson random measure, we consider BSDEs where the Poisson random measure admits only jumps of size larger than $1/n$. We show convergence of their solutions to those of the original BSDE, as $n \to \infty.$ The proofs only rely on It\^o's formula and the Bihari-LaSalle inequality and do not use Girsanov transforms.