Existence, uniqueness and comparison results for BSDEs with Lévy jumps in an extended monotonic generator setting

Christel Geiss; Alexander Steinicke

doi:10.1186/s41546-018-0034-y

Article Contents

2018, Volume 3: 9. Doi: 10.1186/s41546-018-0034-y

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Existence, uniqueness and comparison results for BSDEs with Lévy jumps in an extended monotonic generator setting

Christel Geiss¹ and
Alexander Steinicke^2, ,

1. University of Jyvaskyla, Department of Mathematics and Statistics, P. O. Box 35, 40014 Jyvaskyla, Finland;
2. Department of Mathematics and Information Technology, Montanuniversitaet Leoben, Leoben, Austria

Alexander Steinicke, alexander.steinicke@unileoben.ac.at

Received: January 03, 2018

Revised: November 28, 2018

Published: September 2020

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Abstract

We show that the comparison results for a backward SDE with jumps established in Royer (Stoch. Process. Appl 116:1358-1376, 2006) and Yin and Mao (J. Math. Anal. Appl 346:345-358, 2008) hold under more simplified conditions. Moreover, we prove existence and uniqueness allowing the coefficients in the linear growth- and monotonicity-condition for the generator to be random and time-dependent. In the L²-case with linear growth, this also generalizes the results of Kruse and Popier (Stochastics 88:491-539, 2016). For the proof of the comparison result, we introduce an approximation technique:Given a BSDE driven by Brownian motion and Poisson random measure, we approximate it by BSDEs where the Poisson random measure admits only jumps of size larger than 1/n.

Keywords:

Backward stochastic differential equation,
Lévy process,
comparison theorem,
existence and uniqueness.

Mathematics Subject Classification: 60H10.

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