The adapted solution and comparison theorem for backward stochastic differential equations with Poisson jumps and applications
type: Item
instance of: scholarly article
title: The adapted solution and comparison theorem for backward stochastic differential equations with Poisson jumps and applications
zbMATH Open document ID: 1157.60061
author: Juliang Yin
author: Xuerong Mao
published in: Journal of Mathematical Analysis and Applications
publication date: 2008-08-19T00:00:00Z
full work available at URL: 165redactYin Mao JMAA.pdf
review text: The authors of the present paper study \(n\)-dimensional backward stochastic differential equations (BSDEs) driven by a Brownian motion and an independent Poisson random measure. The terminal time of the BSDE is a stopping time and the driving coefficient is supposed to satisfy assumptions which are weaker than the Lipschitz condition. So the paper combines difficulties which have been formerly studied by different authors in various papers. The authors of the present paper prove the existence and the uniqueness for this type of BSDE, generalizing earlier results on BSDEs with jumps by \textit{R. Situ} [Stochastic Processes Appl. 66, 209--236 (1997; Zbl 0890.60049)] and by \textit{G. Barles}, \textit{R. Buckdahn} and \textit{E. Pardoux} [Stochastic Anal. Appl. 60, No. 1--2, 57--83 (1997; Zbl 0878.60036)]. For the case of one-dimensional BSDEs with jumps a comparison theorem is proved. For this the authors assume the driving coefficient of one of the both BSDEs to depend linearly on the integrand \(U\) of the integral with respect to the compensated Poisson random measure. On the other hand, the factor in front of \(U\) in this driving coefficient is only supposed to be greater or equal to \(-1\). After the comparison result is applied by the authors to prove the existence and the uniqueness of a minimal solution for one-dimensional BSDs with continuous driving coefficient. Finally, the authors show that their BSDE gives a stochastic interpretation to parabolic integral-partial differential equations.
reviewed by: Rainer Buckdahn
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60H20
zbMATH DE Number: 5313888
zbMATH Keywords: backward stochastic differential equation
zbMATH Keywords: BSDE backward stochastic differential equation
zbMATH Keywords: BSDE with jumps
zbMATH Keywords: comparison theorem
zbMATH Keywords: Feynman-Kac formula
zbMATH Keywords: viscosity solution
zbMATH Keywords: PDIE solution
zbMATH Keywords: PDIE
MaRDI profile type: Publication
OpenAlex ID: W2045068311
cites work: Backward stochastic differential equations and integral-partial differential equations
cites work: User’s guide to viscosity solutions of second order partial differential equations
cites work: Backwards SDE with random terminal time and applications to semilinear elliptic PDE
cites work: Filtration-consistent nonlinear expectations and related \(g\)-expectations
cites work: Q4344073
cites work: Backward Stochastic Differential Equations in Finance
cites work: Q3992729
cites work: Backward stochastic differential equations with continuous coefficient
cites work: Comparison theorem for solutions of backward stochastic differential equations with continuous coefficient
cites work: Adapted solutions of backward stochastic differential equations with non- Lipschitz coefficients
cites work: Q4338975
cites work: Q4357511
cites work: Forward-backward stochastic differential equations and quasilinear parabolic PDEs
cites work: Probabilistic interpretation for systems of quasilinear parabolic partial differential equations
cites work: A Generalized dynamic programming principle and hamilton-jacobi-bellman equation
cites work: Backward stochastic differential equations with jumps and related nonlinear expectations
cites work: Q3702221
cites work: On solutions of backward stochastic differential equations with jumps and applications
cites work: Necessary Conditions for Optimal Control of Stochastic Systems with Random Jumps
cites work: On Solutions of Forward‐Backward Stochastic Differential Equations with Poisson Jumps
cites work: Hilbert space-valued forward-backward stochastic differential equations with Poisson jumps and applications
DOI: 10.1016/J.JMAA.2008.05.072
Recommended article: D-solutions of BSDEs with Poisson jumps
Recommended article: Backward stochastic differential equations with jumps and their actuarial and financial applications. BSDEs with jumps
Recommended article: Existence, uniqueness and comparison results for BSDEs with Lévy jumps in an extended monotonic generator setting
Recommended article: Fully coupled FBSDE with Brownian motion and Poisson process in stopping time duration
Recommended article: BSDE driven by Poisson point processes with discontinuous coefficient
Recommended article: Q1580627
Recommended article: Q4528472
Recommended article: Q5439447
Recommended article: On solutions of backward stochastic differential equations with jumps and applications
description: scientific article; zbMATH DE number 5313888