• 论文 • 上一篇    下一篇

随机延迟微分方程分裂步单支θ方法的强收敛性

张维, 王文强   

  1. 湘潭大学数学与计算科学学院, 湘潭 411105
  • 收稿日期:2017-06-07 出版日期:2018-06-15 发布日期:2018-06-12
  • 基金资助:

    国家自然科学基金(11571373,11271311).

张维, 王文强. 随机延迟微分方程分裂步单支θ方法的强收敛性[J]. 数值计算与计算机应用, 2018, 39(2): 135-149.

Zhang Wei, Wang Wenqiang. STRONG CONVERGENCE OF THE SPLIT-STEP ONE-LEG θ METHODS FOR STOCHASTIC DIFFERENTIAL EQUATIONS[J]. Journal of Numerical Methods and Computer Applications, 2018, 39(2): 135-149.

STRONG CONVERGENCE OF THE SPLIT-STEP ONE-LEG θ METHODS FOR STOCHASTIC DIFFERENTIAL EQUATIONS

Zhang Wei, Wang Wenqiang   

  1. School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China
  • Received:2017-06-07 Online:2018-06-15 Published:2018-06-12
当扩散项系数gx,y)关于变量xy满足全局Lipschitz条件,而漂移项系数fx,y)关于变量x满足单边Lipschitz条件,变量y满足全局Lipschitz条件时,本文建立了随机延迟微分方程分裂步单支θ方法的有界性和收敛性,并证明了当数值方法的参数θ满足1/2≤θ≤1时,分裂步单支θ方法对于这类随机延迟微分方程是强收敛的,并在现有文献的基础上将该方法从随机常微分方程推广到随机延迟微分方程.文末的数值试验验证了理论结果的正确性.
This paper establishes the boundedness, convergence of the split-step one-leg theta methods(SSOLTM) for stochastic delay differential equations, When the diffusion obeys the global Lipschitz in both x and y, but the drift f(x, y) satisfies one-sided Lipschitz condition in x and globally Lipschitz in y. In this paper, SSOLTM are shown to be mean-square convergent for such SDDEs if the method parameter satisfies 1/2 ≤ θ ≤ 1. At the same time, we extend the numerical approach from stochastic ordinary differential equations to stochastic delay differential equations on the basis of the existing literature. Finally, the obtained results are supported by numerical experiments.

MR(2010)主题分类: 

()
[1] Wang X, Gan S. B-convergence of split-step one-leg theta methods for stochastic differential equations[J]. J. Appl. Math. Comput., 2012, 38:489-503.

[2] Zhang H, Gan S, Hu L. The split-step backward Euler method for linear stochastic delay differential equations[J]. J. Comput. Appl. Math., 2009, 225:558-568.

[3] Yue C. High-order split-step theta methods for non-autonomous stochastic differential equations with non-globally Lipschitz continuous coefficients[J]. Math. Meth. Appl. Sci., 2016, 39:2380- 2400.

[4] Wanner G, Hairer E. Solving ordinary differential equations Π:Stiff and Differential-Algebraic Problems. second ed., Spring-Verlag, Berlin, 1996.

[5] Wang X, Gan S. The improved split-step backward Euler method for stochastic differential delay equations[J]. Int. J. Comput. Math., 2011, 88:2359-2378.

[6] Mao X, Sabanis S. Numerical solutions of stochastic differential delay equations under local Lipschitz condition[J]. J. Comput. Appl. Math., 2003, 151:215-227.

[7] Higham D J, Mao X, Stuart A M. Strong convergence of Euler-type methods for nonlinear stochastic differential equations[J]. SIAM J. Numer. Anal., 2002, 40:1041-1063.

[8] Mao X, Sabanis S. Numerical solutions of stochastic differential delay equations with under local Lipschitz condition[J]. J. Comput. Appl. Math., 2003, 151:215-227.

[9] Yue C, Huang C, Jiang F. Strong convergence of split-step theta methods for non-autonomous stochastic differential equations[J]. Int. J. Comput. Math., 2014, 10:2260-2275.

[10] Zong X, Wu F, Huang C. Theta schemes for SDDEs with non-globally Lipschitz continuous coefficients[J]. J. Comput. Appl. Math., 2015, 278:258-277.

[11] Higham D J. An Algorithmic introduction to numerical simulation of stochastic differential equations[J]. SIAM review., 2001, 43:525-546.

[12] Liu M, Cao W, Fan Z. Convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equations[J]. J. Comput. Appl. Math., 2004, 170:255-268.

[13] Mao X. Exponential stability of equidistant Euler-Maruyama approximations of stochastic differential delay equations[J]. J. Comput. Appl. Math., 2007, 200:297-316.

[14] Higham D J, Kloeden P E. Numerical methods for nonlinear stochastic differential equations with jumps[J]. Numer. Math., 2005, 101:101-119.
[1] 王文强,黄山,李寿佛. 非线性随机延迟微分方程半隐式Euler方法的均方稳定性[J]. 数值计算与计算机应用, 2008, 29(1): 73-80.
阅读次数
全文


摘要