The adapted solution and comparison theorem for backward stochastic differential equations with Poisson jumps and applications
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.
backward stochastic differential equation
BSDE backward stochastic differential equation
scientific article; zbMATH DE number 5313888
scientific article