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非自治刚性随机微分方程正则EM分裂方法的收敛性和稳定性

余妍妍, 代新杰, 肖爱国   

  1. 湘潭大学数学与计算科学学院, 湘潭 411105
  • 收稿日期:2020-10-22 出版日期:2022-02-14 发布日期:2022-02-14
  • 基金资助:
    国家自然科学基金(12071403)和湖南省教育厅科学研究项目(21A0108)资助

余妍妍, 代新杰, 肖爱国. 非自治刚性随机微分方程正则EM分裂方法的收敛性和稳定性[J]. 计算数学, 2022, 44(1): 19-33.

Yu Yanyan, Dai Xinjie, Xiao Aiguo. CONVERGENCE AND STABILITY OF THE CANONICAL EM SPLITTING METHOD FOR NONAUTONOMOUS STIFF STOCHASTIC DIFFERENTIAL EQUATIONS[J]. Mathematica Numerica Sinica, 2022, 44(1): 19-33.

CONVERGENCE AND STABILITY OF THE CANONICAL EM SPLITTING METHOD FOR NONAUTONOMOUS STIFF STOCHASTIC DIFFERENTIAL EQUATIONS

Yu Yanyan, Dai Xinjie, Xiao Aiguo   

  1. School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China
  • Received:2020-10-22 Online:2022-02-14 Published:2022-02-14
本文研究了数值求解非自治随机微分方程的正则Euler-Maruyama分裂(CEMS)方法,该方程的漂移项系数带有刚性且允许超线性增长,扩散项系数满足全局Lipschitz条件.首先,证明了CEMS方法的强收敛性及收敛速度.其次,证明了在适当条件下CEMS方法是均方稳定的.进一步,利用离散半鞅收敛定理,研究了CEMS方法的几乎必然指数稳定性.结果表明,CEMS方法在漂移系数的刚性部分满足单边Lipschitz条件下可保持几乎必然指数稳定性.最后通过数值实验,检验了CEMS方法的有效性并证实了我们的理论结果.
This paper studies the canonical Euler——Maruyama splitting (CEMS) method for numerically solving non-autonomous stochastic differential equations. The drift coefficient of the equation is stiff and allows super-linear growth, and the diffusion coefficient satisfies the global Lipschitz condition. First, we prove the CEMS method is strongly convergent and discuss the convergence rate. Second, it is proved that the CEMS method is stable in mean square sense under mild conditions. Further, using the discrete semi-martingale convergence theorem, the almost surely exponential stability of the CEMS method is studied. The results show that the CEMS method can preserve almost surely exponential stability when the stiff part of drift cofficient satisfies the one-sided Lipschitz condition. Finally, numerical experiments verify the effectiveness of the CEMS method and confirm our theoretical results.

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