多项Caputo分数阶随机微分方程的Euler-Maruyama方法

doi:10.12286/jssx.j2020-0765

计算数学 ›› 2022, Vol. 44 ›› Issue (3): 354-367.DOI: 10.12286/jssx.j2020-0765

多项Caputo分数阶随机微分方程的Euler-Maruyama方法

霍振阳, 张静娜, 黄健飞

扬州大学数学科学学院, 扬州 225002

收稿日期:2020-12-15 出版日期:2022-07-14 发布日期:2022-08-03
通讯作者: 黄健飞,Email:jfhuang@lsec.cc.ac.cn.
基金资助:
国家自然科学基金项目(11701502,11871065)和江苏省自然科学基金项目(BK20201427)资助.

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霍振阳, 张静娜, 黄健飞. 多项Caputo分数阶随机微分方程的Euler-Maruyama方法[J]. 计算数学, 2022, 44(3): 354-367.

Huo Zhengyang, Zhang Jingna, Huang Jianfei. EULER-MARUYAMA SCHEME FOR MULTI-TERM CAPUTO FRACTIONAL STOCHASTIC DIFFERENTIAL EQUATIONS[J]. Mathematica Numerica Sinica, 2022, 44(3): 354-367.

EULER-MARUYAMA SCHEME FOR MULTI-TERM CAPUTO FRACTIONAL STOCHASTIC DIFFERENTIAL EQUATIONS

Huo Zhengyang, Zhang Jingna, Huang Jianfei

College of Mathematical Sciences, Yangzhou University, Yangzhou 225002, China

Received:2020-12-15 Online:2022-07-14 Published:2022-08-03

本文主要研究了一类多项Caputo分数阶随机微分方程的Euler-Maruyama (EM)方法，并证明了其强收敛性.具体地，我们首先构造了求解多项Caputo分数阶随机微分方程初值问题的EM方法，然后证明分数阶导数的指标满足$\frac{1}{2}<\alpha_{1}<\alpha_{2}<\cdots<\alpha_{m}<1$时，该方法是$\alpha_{m}-\alpha_{m-1}$阶强收敛的.文末的数值试验验证了理论结果的正确性.

关键词： 分数阶随机微分方程, 多项Caputo导数, Euler-Maruyama方法, 强收敛性

In this paper, we study the Euler-Maruyama (EM) method for a class of multi-term Caputo fractional stochastic differential equations, and prove its strong convergence. Specifically, we first construct the EM method for the initial value problem of multi-term Caputo fractional stochastic differential equations, and then we prove that the method is $\alpha_{m}-\alpha_{m-1}$ order strong convergence, where $\alpha_{i},i=1,\cdots,m$, is the fractional order with $\frac{1}{2}<\alpha_{1}<\alpha_{2}<\cdots<\alpha_{m}<1$. Finally, numerical experiments are given to support the theoretical results of our EM method.

Key words: fractional stochastic differential equations, multi-term Caputo differential operators, Euler-Maruyama scheme, strong convergence

MR(2010)主题分类:

65C30
60H10

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摘要

多项Caputo分数阶随机微分方程的Euler-Maruyama方法

EULER-MARUYAMA SCHEME FOR MULTI-TERM CAPUTO FRACTIONAL STOCHASTIC DIFFERENTIAL EQUATIONS

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