一类随机泛函微分方程带随机步长的EM逼近的渐近稳定

马丽; 马瑞楠

doi:10.21656/1000-0887.390057

一类随机泛函微分方程带随机步长的EM逼近的渐近稳定

doi: 10.21656/1000-0887.390057

马丽,
马瑞楠

海南师范大学数学与统计学院, 海口 571158

基金项目: 国家自然科学基金(11861029)；海南省高等学校科学研究项目(重点项目)（Hnky2018ZD6）;海南省自然科学基金(面上项目)（118MS040）;海南省自然科学基金(创新研究团队项目)（2018CXTD338）

详细信息

作者简介:
马丽(1979—)，女，副教授，博士，硕士生导师(通讯作者. E-mail: malihnsd@163.com).

中图分类号: O211.62
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出版历程
- 收稿日期: 2018-02-06
- 修回日期: 2018-08-22
- 刊出日期: 2019-01-01

Almost Sure Asymptotic Stability of the Euler-Maruyama Method With Random Variable Stepsizes for Stochastic Functional Differential Equations

MA Li,
MA Ruinan

School of Mathematics and Statistics, Hainan Normal University, Haikou 571158, P.R.China

Funds: The National Natural Science Foundation of China(11861029)

摘要

摘要: 研究了一类带有限延迟的随机泛函微分方程的Euler-Maruyama(EM)逼近，给出了该方程的带随机步长的EM算法，得到了随机步长的两个特点：首先，有限个步长求和是停时；其次，可列无限多个步长求和是发散的.最终，由离散形式的非负半鞅收敛定理，得到了在系数满足局部Lipschitz条件和单调条件下，带随机步长的EM数值解几乎处处收敛到0.该文拓展了2017年毛学荣关于无延迟的随机微分方程带随机步长EM数值解的结果.
- 随机泛函微分方程 /
- 带随机步长的EM逼近 /
- 非负半鞅收敛定理 /
- 几乎处处稳定
Abstract: The Euler-Maruyama (EM) approximation to a class of stochastic functional differential equations was studied. First, a numerical approximation with the EM method with random variable stepsizes was defined, then two characteristics of the random variable stepsizes were got: the summation of finite stepsizes is a stopping time and the summation of countably infinite stepsizes diverges. Finally, with the theory of non-negative semi-martingale convergence in discrete time, it was proved that the numerical approximation converges to zero almost surely if the coefficients satisfy the local Lipschitz condition and the monotonic condition. The results generalize the corresponding results of MAO Xuerong in a previous literature, where the EM approximation to a class of stochastic differential equations was studied and the numerical solution was proved to converge to zero almost surely.

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参考文献(15)

[1]	RODKINA A, SCHURZ H. Almost sure asymptotic stability of drift-implicit θ-methods for bilinear ordinary stochastic differential equations in R1[J]. Journal of Computational and Applied Mathematics,2005,180(1): 13-31.
[2]	WU F, MAO X R, SZPRUCH L. Almost sure exponential stability of numerical solutions for stochastic delay differential equations[J]. Numerische Mathematik,2010,115(4): 681-697.
[3]	WU F, MAO X R, KLOEDEN P E. Almost sure exponential stability of the Euler-Maruyama approximations for stochastic functional differential equations[J]. Random Operators and Stochastic Equations,2011,19(2): 165-186.
[4]	WU F, MAO X R. Numerical solutions of neutral stochastic functional differential equations[J]. Society for Industrial and Applied Mathematics,2008,46(4): 1821-1841.
[5]	JI Y T, BAO J H, YUAN C G. Convergence rate of Euler-Maruyama scheme for SDDEs of neutral type[J/OL]. [2018-02-06]. https://arxiv.org/abs/1511.07703v2.
[6]	MAO X R, SHEN Y, YUAN C G. Almost surely asymptotic stability of neutral stochastic dely differential equations with Markovian switching[J]. Stochastic Processes and Their Applications,2008,118: 1385-1406.
[7]	TIAN J G, WANG H L, GUO Y F, et al. Numerical solutions to neutral stochastic delay differential equations with Poisson jumps under local Lipschitz condition[J]. Mathematical Problems in Engineering,2014,2014: 976183.
[8]	YU Z H. Almost surely asymptotic stability of exact and numerical solutions for neutral stochastic pantograph equations[J]. Abstract and Applied Analysis,2011,2011: 143079.
[9]	MAO X R. Stochastic Differential Equation and Application [M]. Chichester: Horwood Publising, 2007.
[10]	MAO X R. LaSalle-type theorems for stochastic differential delay equations[J]. Journal of Mathematical Analysis and Applications,1999,236(2): 350-369.
[11]	MAO X R. A note on the LaSalle-type theorems for stochastic differential delay equations[J]. Journal of Mathematical Analysis and Applications,2002,268(1): 125-142.
[12]	MAO X R. The LaSalle-type theorems for stochastic functional differential equations[J]. Nonlinear Studies,2000,7(2): 307-328.
[13]	MAO X R. Stochastic versions of the LaSalle-type theorems[J]. Journal of Differential Equations,1999,153: 175-195.
[14]	HIGHAM D J, MAO X R, YUAN C G. Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations[J]. SIAM Journal on Numerical Analysis,2007,45(2): 592-609.
[15]	LIU W, MAO X R. Almost sure stability of the Euler-Maruyama method with random variable stepsize for stochastic differential equations[J]. Numerical Algorithms,2017,74(2): 573-592.