Existence, Uniqueness and Comparison Results for BSDEs with Lévy 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 Lévy 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 Itô's formula and the Bihari-LaSalle inequality and do not use Girsanov transforms.

Version 3 is the final, reviewed version as published in Probability, Uncertainty and Quantitative Risk

References in corpus (1)

• On Skorohod spaces as universal sample path spaces

Cited by in corpus (4)

• On the monotone stability approach to BSDEs with jumps: Extensions, concrete criteria and examples
• Existence, Uniqueness and Malliavin Differentiability of Lévy-driven BSDEs with locally Lipschitz Driver
• The Global Maximum Principle for Progressive Optimal Control of Partially Observed Forward-Backward Stochastic Systems with Random Jumps
• $L^p$-Solutions and Comparison Results for Lévy Driven BSDEs in a Monotonic, General Growth Setting