Geiss Christel, Steinicke Alexander
1University of Jyvaskyla, Department of Mathematics and Statistics, P.O. Box 35, Jyvaskyla, 40014 Finland.
2Department of Mathematics and Information Technology, Montanuniversitaet Leoben, Leoben, Austria.
Probab Uncertain Quant Risk. 2018;3(1):9. doi: 10.1186/s41546-018-0034-y. Epub 2018 Dec 28.
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 -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/.
我们表明，Royer（《随机过程及其应用》116: 1358 - 1376, 2006）以及Yin和Mao（《数学分析与应用杂志》346: 345 - 358, 2008）中建立的关于带跳倒向随机微分方程的比较结果在更简化的条件下成立。此外，我们证明了存在性和唯一性，允许生成元的线性增长和单调性条件中的系数是随机的且依赖于时间。在具有线性增长的 - 情形下，这也推广了Kruse和Popier（《随机过程》88: 491 - 539, 2016）的结果。对于比较结果的证明，我们引入一种逼近技术：给定一个由布朗运动和泊松随机测度驱动的倒向随机微分方程，我们用泊松随机测度仅允许大小大于1/ 的跳跃的倒向随机微分方程来逼近它。
