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Volume 40, Issue 2
Convergence and Mean-Square Stability of Exponential Euler Method for Semi-Linear Stochastic Delay Integro-Differential Equations

Haiyan Yuan

DOI: 10.4208/jcm.2010-m2019-0200

J. Comp. Math., 40 (2022), pp. 177-204.

Published online: 2022-01

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  • Abstract

In this paper, the numerical methods for semi-linear stochastic delay integro-differential equations are studied. The uniqueness, existence and stability of analytic solutions of semi-linear stochastic delay integro-differential equations are studied and some suitable conditions for the mean-square stability of the analytic solutions are also obtained. Then the numerical approximation of exponential Euler method for semi-linear stochastic delay integro-differential equations is constructed and the convergence and the stability of the numerical method are studied. It is proved that the exponential Euler method is convergent with strong order $\frac{1}{2}$ and  can keep the mean-square exponential stability of the analytical solutions under some restrictions on the step size. In addition, numerical experiments are presented to confirm the theoretical results.

  • Keywords

Semi-linear stochastic delay integro-differential equation, Exponential Euler method, Mean-square exponential stability, Trapezoidal rule.

  • AMS Subject Headings

60H35, 65C20, 65C30, 65L20

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

yhy82_47@163.com (Haiyan Yuan)

  • BibTex
  • RIS
  • TXT
@Article{JCM-40-177, author = {Yuan , Haiyan}, title = {Convergence and Mean-Square Stability of Exponential Euler Method for Semi-Linear Stochastic Delay Integro-Differential Equations}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {40}, number = {2}, pages = {177--204}, abstract = {

In this paper, the numerical methods for semi-linear stochastic delay integro-differential equations are studied. The uniqueness, existence and stability of analytic solutions of semi-linear stochastic delay integro-differential equations are studied and some suitable conditions for the mean-square stability of the analytic solutions are also obtained. Then the numerical approximation of exponential Euler method for semi-linear stochastic delay integro-differential equations is constructed and the convergence and the stability of the numerical method are studied. It is proved that the exponential Euler method is convergent with strong order $\frac{1}{2}$ and  can keep the mean-square exponential stability of the analytical solutions under some restrictions on the step size. In addition, numerical experiments are presented to confirm the theoretical results.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2010-m2019-0200}, url = {http://global-sci.org/intro/article_detail/jcm/20183.html} }
TY - JOUR T1 - Convergence and Mean-Square Stability of Exponential Euler Method for Semi-Linear Stochastic Delay Integro-Differential Equations AU - Yuan , Haiyan JO - Journal of Computational Mathematics VL - 2 SP - 177 EP - 204 PY - 2022 DA - 2022/01 SN - 40 DO - http://doi.org/10.4208/jcm.2010-m2019-0200 UR - https://global-sci.org/intro/article_detail/jcm/20183.html KW - Semi-linear stochastic delay integro-differential equation, Exponential Euler method, Mean-square exponential stability, Trapezoidal rule. AB -

In this paper, the numerical methods for semi-linear stochastic delay integro-differential equations are studied. The uniqueness, existence and stability of analytic solutions of semi-linear stochastic delay integro-differential equations are studied and some suitable conditions for the mean-square stability of the analytic solutions are also obtained. Then the numerical approximation of exponential Euler method for semi-linear stochastic delay integro-differential equations is constructed and the convergence and the stability of the numerical method are studied. It is proved that the exponential Euler method is convergent with strong order $\frac{1}{2}$ and  can keep the mean-square exponential stability of the analytical solutions under some restrictions on the step size. In addition, numerical experiments are presented to confirm the theoretical results.

Yuan , Haiyan. (2022). Convergence and Mean-Square Stability of Exponential Euler Method for Semi-Linear Stochastic Delay Integro-Differential Equations. Journal of Computational Mathematics. 40 (2). 177-204. doi:10.4208/jcm.2010-m2019-0200
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