带跳随机微分方程的Euler-Maruyama方法的几乎处处指数稳定性和矩稳定性

ALMOST SURE AND MOMENT EXPONENTIAL STABILITIES OF EULER-MARUYAMA METHOD FOR STOCHASTIC DIFFERENTIAL EQUATIONS WITH JUMPS

本文首先研究了一维带跳随机微分方程的指数稳定性,并证明Euler-Maruyama(EM)方法保持了解析解的稳定性.其次,研究了多维带跳随机微分方程的稳定性,证明若系数满足全局Lipchitz条件,则EM方法能够很好地保持解析解的几乎处处指数稳定性、均方指数稳定性.最后,给出算例来支持所得结论的正确性.

First, the exponential stability for a scalar stochastic differential equation with jumps （SDEwJs） is studied. And, we show that Euler-Maruyama （EM） method reproduces the exponential stability of analytical solutions. Then, we study the stability for n-dimension SDEwJs. We show that EM method recovers almost sure exponential stability and mean- square exponential stability well under global Lipschtiz condition. Finally, some examples are provided to illustrate the results.