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Volume 33, Issue 6
Strong Predictor-Corrector Approximation for Stochastic Delay Differential Equations

Yuanling Niu, Chengjian Zhang & Kevin Burrage

DOI: 10.4208/jcm.1507-m4505

J. Comp. Math., 33 (2015), pp. 587-605.

Published online: 2015-12

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  • Abstract

This paper presents a strong predictor-corrector method for the numerical solution of stochastic delay differential equations (SDDEs) of Itô-type. The method is proved to be mean-square convergent of order min{$1/2, \hat{p}$} under the Lipschitz condition and the linear growth condition, where $\hat{p}$ is the exponent of Hölder condition of the initial function. Stability criteria for this type of method are derived. It is shown that for certain choices of the flexible parameter $p$ the derived method can have a better stability property than more commonly used numerical methods. That is, for some $p$, the asymptotic MS-stability bound of the method will be much larger than that of the Euler-Maruyama method. Numerical results are reported confirming convergence properties and comparing stability properties of methods with different parameters $p$. Finally, the vectorised simulation is discussed and it is shown that this implementation is much more efficient.

  • Keywords

Strong predictor-corrector approximation, Stochastic delay differential equations, Convergence, Mean-square stability, Numerical experiments, Vectorised simulation.

  • AMS Subject Headings

65C30, 60H35.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

yuanlingniu@csu.edu.cn (Yuanling Niu)

cjzhang@mail.hust.edu.cn (Chengjian Zhang)

kevin.burrage@gmail.com (Kevin Burrage)

  • BibTex
  • RIS
  • TXT
@Article{JCM-33-587, author = {Niu , YuanlingZhang , Chengjian and Burrage , Kevin}, title = {Strong Predictor-Corrector Approximation for Stochastic Delay Differential Equations}, journal = {Journal of Computational Mathematics}, year = {2015}, volume = {33}, number = {6}, pages = {587--605}, abstract = {

This paper presents a strong predictor-corrector method for the numerical solution of stochastic delay differential equations (SDDEs) of Itô-type. The method is proved to be mean-square convergent of order min{$1/2, \hat{p}$} under the Lipschitz condition and the linear growth condition, where $\hat{p}$ is the exponent of Hölder condition of the initial function. Stability criteria for this type of method are derived. It is shown that for certain choices of the flexible parameter $p$ the derived method can have a better stability property than more commonly used numerical methods. That is, for some $p$, the asymptotic MS-stability bound of the method will be much larger than that of the Euler-Maruyama method. Numerical results are reported confirming convergence properties and comparing stability properties of methods with different parameters $p$. Finally, the vectorised simulation is discussed and it is shown that this implementation is much more efficient.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1507-m4505}, url = {http://global-sci.org/intro/article_detail/jcm/9862.html} }
TY - JOUR T1 - Strong Predictor-Corrector Approximation for Stochastic Delay Differential Equations AU - Niu , Yuanling AU - Zhang , Chengjian AU - Burrage , Kevin JO - Journal of Computational Mathematics VL - 6 SP - 587 EP - 605 PY - 2015 DA - 2015/12 SN - 33 DO - http://doi.org/10.4208/jcm.1507-m4505 UR - https://global-sci.org/intro/article_detail/jcm/9862.html KW - Strong predictor-corrector approximation, Stochastic delay differential equations, Convergence, Mean-square stability, Numerical experiments, Vectorised simulation. AB -

This paper presents a strong predictor-corrector method for the numerical solution of stochastic delay differential equations (SDDEs) of Itô-type. The method is proved to be mean-square convergent of order min{$1/2, \hat{p}$} under the Lipschitz condition and the linear growth condition, where $\hat{p}$ is the exponent of Hölder condition of the initial function. Stability criteria for this type of method are derived. It is shown that for certain choices of the flexible parameter $p$ the derived method can have a better stability property than more commonly used numerical methods. That is, for some $p$, the asymptotic MS-stability bound of the method will be much larger than that of the Euler-Maruyama method. Numerical results are reported confirming convergence properties and comparing stability properties of methods with different parameters $p$. Finally, the vectorised simulation is discussed and it is shown that this implementation is much more efficient.

Niu , YuanlingZhang , Chengjian and Burrage , Kevin. (2015). Strong Predictor-Corrector Approximation for Stochastic Delay Differential Equations. Journal of Computational Mathematics. 33 (6). 587-605. doi:10.4208/jcm.1507-m4505
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