Almost Sure Asymptotic Stability of the Euler-Maruyama Method With Random Variable Stepsizes for Stochastic Functional Differential Equations
-
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.
-
关键词:
- 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. -
[1] RODKINA A, SCHURZ H. Almost sure asymptotic stability of drift-implicit θ-methods for bilinear ordinary stochastic differential equations in R1[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. 期刊类型引用(19)
1. 马丽,孙芳芳. 非Lipschitz条件下高维McKean-Vlasov随机微分方程解的存在唯一性. 应用数学和力学. 2023(10): 1272-1290 . 本站查看
2. 周军,张健,杨顺枫. Internet路由器随机建模与收敛性分析. 应用数学和力学. 2022(02): 207-214 . 本站查看
3. 刘芝秀,吕凤姣,李运通. 隐变量对EM算法的影响. 安徽师范大学学报(自然科学版). 2022(03): 221-226 . 百度学术
4. 梁青. 一类带扰动的随机脉冲泛函微分方程解的渐近性. 应用数学和力学. 2022(09): 1034-1044 . 本站查看
5. 廖晓花. 大型稀疏线性代数方程组的通用性迭代解法. 宁夏师范学院学报. 2021(01): 11-15 . 百度学术
6. 刘欣. 基于蚁群算法的医疗人力资源应急调度设计. 信息技术. 2021(02): 142-146 . 百度学术
7. 孟红军,徐校会,袁国军. 分数阶微分方程组边值问题的可解性分析. 宁夏师范学院学报. 2021(04): 20-25 . 百度学术
8. 李光洁,杨启贵. G-Brown运动驱动的非线性随机时滞微分方程的稳定化. 应用数学和力学. 2021(08): 841-851 . 本站查看
9. 吴恒飞,张宗标. 约束矩阵方程线性约束逼近解交替投影方法研究. 廊坊师范学院学报(自然科学版). 2021(03): 8-11+25 . 百度学术
10. 郑明亮. 时滞Lagrange系统的Lie对称性与守恒量研究. 应用数学和力学. 2021(11): 1161-1168 . 本站查看
11. 张艳芬. 多层线性规划过程折中最优解计算方法研究. 兰州文理学院学报(自然科学版). 2020(04): 23-27 . 百度学术
12. 孙玉涛,司凤山,崔迪. 基于最优边界划分的多层次复杂系统参数辨识方法. 佳木斯大学学报(自然科学版). 2020(04): 133-136+141 . 百度学术
13. 王玮. 非线性方程组解法在梯度投影约束最优化问题中的应用. 宁夏师范学院学报. 2020(04): 5-10 . 百度学术
14. 张纪强. 常微分方程的数值解析的实践与应用. 宁夏师范学院学报. 2020(04): 101-106 . 百度学术
15. 缪彩花. 一类二阶线性复微分方程的亚纯解分析. 宁夏师范学院学报. 2020(10): 27-32 . 百度学术
16. 张海侠. 用正交函数求解光纤陀螺误差的数学模型分析. 激光杂志. 2020(12): 27-31 . 百度学术
17. 梁静. 基于微分中值定理的基本不等式证明方法. 长春师范大学学报. 2020(12): 10-15 . 百度学术
18. 刘夏瑜,牛哲斌. 短距离游泳身体机能疲劳极限监控方法研究. 赤峰学院学报(自然科学版). 2019(08): 101-104 . 百度学术
19. 江慧敏. Banach空间中不适定线性算子的广义概率范数. 安阳师范学院学报. 2019(05): 1-4+7 . 百度学术
其他类型引用(2)
-
计量
- 文章访问数: 1140
- HTML全文浏览量: 214
- PDF下载量: 409
- 被引次数: 21