带Caputo导数的变分数阶随机微分方程的Euler-Maruyama方法

刘家惠; 邵林馨; 黄健飞

doi:10.21656/1000-0887.430250

带Caputo导数的变分数阶随机微分方程的Euler-Maruyama方法

doi: 10.21656/1000-0887.430250

扬州大学数学科学学院，江苏扬州 225002

基金项目:

江苏省自然科学基金项目 BK20201427

国家自然科学基金项目 11701502

国家自然科学基金项目 11871065

详细信息

作者简介:
刘家惠(1998—), 女, 硕士生(E-mail: 965440574@qq.com)

通讯作者:
黄健飞(1983—), 男, 副教授, 博士(通讯作者. E-mail: jfhuang@lsec.cc.ac.cn)

中图分类号: O211.5;O241.8
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- 文章访问数: 390
- HTML全文浏览量: 113
- PDF下载量: 60
- 被引次数: 0
出版历程
- 收稿日期: 2022-08-04
- 修回日期: 2022-11-29
- 刊出日期: 2023-06-01

An Euler-Maruyama Method for Variable Fractional Stochastic Differential Equations With Caputo Derivatives

School of Mathematical Sciences, Yangzhou University, Yangzhou, Jiangsu 225002, P.R.China

摘要

摘要: 该文构造了Euler-Maruyama(EM)方法求解一类带Caputo导数的变分数阶随机微分方程. 首先, 证明了该方程的适定性; 然后, 详细推导出对应的EM方法, 并对该方法进行了强收敛性的分析, 通过使用EM方法的连续形式证明了其强收敛阶为β-0.5, 其中β是Caputo导数的阶数，且满足0.5 < β < 1. 最后, 通过数值实验验证了理论分析结果的正确性.
- 变分数阶随机微分方程 /
- Caputo导数 /
- Euler-Maruyama方法 /
- 强收敛性
Abstract: A Euler-Maruyama (EM) method was constructed to solve a class of variable fractional stochastic differential equations with Caputo derivatives. Firstly, the well-posedness of the equation was proved. Then, the corresponding EM method was derived in detail, and the strong convergence of the method was analyzed. By means of the continuous form of the EM method, its strong convergence order was proved to be β-0.5, where β is the order of the Caputo derivative and 0.5 < β < 1. Numerical experiments verify the correctness of the theoretical results.
- variable fractional stochastic differential equation /
- Caputo derivative /
- Euler-Maruyama method /
- strong convergence

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表 1 β=0.9时，EM方法的误差与收敛阶

Table 1. Errors and convergence orders of the EM method for β=0.9

h	α₁=0.2, α₂=0.6		α₁=0.6, α₂=0.2
h	error e_h	convergence order n_co	error e_h	convergence order n_co
1/32	0.180 197	-	0.180 329	-
1/64	0.135 958	0.406	0.135 998	0.407
1/128	0.101 704	0.419	0.101 716	0.419
1/256	0.076 984	0.402	0.076 986	0.402

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表 2 β=0.8时，EM方法的误差与收敛阶

Table 2. Errors and convergence orders of the EM method for β=0.8

h	α₁=0.2, α₂=0.5		α₁=0.5, α₂=0.2
h	error e_h	convergence order n_co	error e_h	convergence order n_co
1/32	0.251 476	-	0.251 654	-
1/64	0.202 993	0.310	0.203 053	0.310
1/128	0.162 630	0.320	0.162 650	0.320
1/256	0.131 342	0.308	0.131 333	0.309

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表 3 β=0.7时，EM方法的误差与收敛阶

Table 3. Errors and convergence orders of the EM method for β=0.7

h	α₁=0.1, α₂=0.5		α₁=0.5, α₂=0.1
h	error e_h	convergence order n_co	error e_h	convergence order n_co
1/32	0.343 325	-	0.343 776	-
1/64	0.296 312	0.212	0.296 487	0.214
1/128	0.254 176	0.221	0.254 244	0.222
1/256	0.218 904	0.216	0.223 625	0.185

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