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.
点击查看大图
计量
- 文章访问数: 750
- HTML全文浏览量: 99
- PDF下载量: 403
- 被引次数: 0