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

• Published in: Probability, uncertainty and quantitative risk Vol. 3; no. 1; pp. 9 - 33
• Main Authors: Geiss, Christel, Steinicke, Alexander
• Format: Journal Article
• Language: English
• Published: Singapore Springer Singapore 28.12.2018; American Institute of Mathematical Sciences
• Subjects: Brownian motion; Mathematics; Mathematics and Statistics; Probability Theory and Stochastic Processes; Time dependence; Uniqueness; 60H10; Backward stochastic differential equation; existence and uniqueness; Lévy process; comparison theorem
• ISSN: 2367-0126; 2095-9672; 2367-0126
• DOI: 10.1186/s41546-018-0034-y

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 L 2 -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 .