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

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.