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

doi:10.12286/jssx.2014.1.65

计算数学 ›› 2014, Vol. 36 ›› Issue (1): 65-74.DOI: 10.12286/jssx.2014.1.65

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

赵桂华¹, 李春香², 孙波¹

1. 江苏科技大学数理学院, 江苏镇江 212000;
2. 中国人民解放军第二军医大学数理教研室, 上海 200433

收稿日期:2013-04-25 出版日期:2014-02-15 发布日期:2014-02-12
基金资助:
江苏省自然科学基金青年基金项目（BK20130472）、江苏科技大学博士启动基金（35050903）及校管科研课题项目（633051205）.

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赵桂华, 李春香, 孙波. 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[J]. 计算数学, 2014, 36(1): 65-74.

Zhao Guihua, Li Chunxiang, Sun Bo. ALMOST SURE AND MOMENT EXPONENTIAL STABILITIES OF EULER-MARUYAMA METHOD FOR STOCHASTIC DIFFERENTIAL EQUATIONS WITH JUMPS[J]. Mathematica Numerica Sinica, 2014, 36(1): 65-74.

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

Zhao Guihua¹, Li Chunxiang², Sun Bo¹

1. Department of Mathematics, Jiangsu University of Science and Technology, Zhenjiang 212003, Jiangsu, China;
2. Department of Mathematics and Physics, Second Military Medical University, Shanghai 200433, China

Received:2013-04-25 Online:2014-02-15 Published:2014-02-12

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.

关键词： 带跳随机微分方程, Euler-Maruyama 方法, 几乎处处指数稳定性, 均方指数稳定性

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 meansquare exponential stability well under global Lipschtiz condition. Finally, some examples are provided to illustrate the results.

Key words: stochastic differential equation with jumps, Euler-Maruyama method, almost sure exponential stability, mean-square exponential stability

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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

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