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Volume 41, Issue 4
The Convergence of Truncated Euler-Maruyama Method for Stochastic Differential Equations with Piecewise Continuous Arguments Under Generalized One-Sided Lipschitz Condition

Yidan Geng, Minghui Song & Mingzhu Liu

DOI: 10.4208/jcm.2109-m2021-0116

J. Comp. Math., 41 (2023), pp. 663-682.

Published online: 2023-02

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  • Abstract

In this paper, we consider the stochastic differential equations with piecewise continuous arguments (SDEPCAs) in which the drift coefficient satisfies the generalized one-sided Lipschitz condition and the diffusion coefficient satisfies the linear growth condition. Since the delay term $t-[t]$ of SDEPCAs is not continuous and differentiable, the variable substitution method is not suitable. To overcome this difficulty, we adopt new techniques to prove the boundedness of the exact solution and the numerical solution. It is proved that the truncated Euler-Maruyama method is strongly convergent to SDEPCAs in the sense of $L^{\bar{q}}(\bar{q}\ge 2)$. We obtain the convergence order with some additional conditions. An example is presented to illustrate the analytical theory.

  • Keywords

Stochastic differential equations, Piecewise continuous argument, One-sided Lipschitz condition, Truncated Euler-Maruyama method.

  • AMS Subject Headings

65L05, 65L60

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

ydgeng@hit.edu.cn (Yidan Geng)

songmh@hit.edu.cn (Minghui Song)

mzliu@hit.edu.cn (Mingzhu Liu)

  • BibTex
  • RIS
  • TXT
@Article{JCM-41-663, author = {Geng , YidanSong , Minghui and Liu , Mingzhu}, title = {The Convergence of Truncated Euler-Maruyama Method for Stochastic Differential Equations with Piecewise Continuous Arguments Under Generalized One-Sided Lipschitz Condition}, journal = {Journal of Computational Mathematics}, year = {2023}, volume = {41}, number = {4}, pages = {663--682}, abstract = {

In this paper, we consider the stochastic differential equations with piecewise continuous arguments (SDEPCAs) in which the drift coefficient satisfies the generalized one-sided Lipschitz condition and the diffusion coefficient satisfies the linear growth condition. Since the delay term $t-[t]$ of SDEPCAs is not continuous and differentiable, the variable substitution method is not suitable. To overcome this difficulty, we adopt new techniques to prove the boundedness of the exact solution and the numerical solution. It is proved that the truncated Euler-Maruyama method is strongly convergent to SDEPCAs in the sense of $L^{\bar{q}}(\bar{q}\ge 2)$. We obtain the convergence order with some additional conditions. An example is presented to illustrate the analytical theory.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2109-m2021-0116}, url = {http://global-sci.org/intro/article_detail/jcm/21410.html} }
TY - JOUR T1 - The Convergence of Truncated Euler-Maruyama Method for Stochastic Differential Equations with Piecewise Continuous Arguments Under Generalized One-Sided Lipschitz Condition AU - Geng , Yidan AU - Song , Minghui AU - Liu , Mingzhu JO - Journal of Computational Mathematics VL - 4 SP - 663 EP - 682 PY - 2023 DA - 2023/02 SN - 41 DO - http://doi.org/10.4208/jcm.2109-m2021-0116 UR - https://global-sci.org/intro/article_detail/jcm/21410.html KW - Stochastic differential equations, Piecewise continuous argument, One-sided Lipschitz condition, Truncated Euler-Maruyama method. AB -

In this paper, we consider the stochastic differential equations with piecewise continuous arguments (SDEPCAs) in which the drift coefficient satisfies the generalized one-sided Lipschitz condition and the diffusion coefficient satisfies the linear growth condition. Since the delay term $t-[t]$ of SDEPCAs is not continuous and differentiable, the variable substitution method is not suitable. To overcome this difficulty, we adopt new techniques to prove the boundedness of the exact solution and the numerical solution. It is proved that the truncated Euler-Maruyama method is strongly convergent to SDEPCAs in the sense of $L^{\bar{q}}(\bar{q}\ge 2)$. We obtain the convergence order with some additional conditions. An example is presented to illustrate the analytical theory.

Geng , YidanSong , Minghui and Liu , Mingzhu. (2023). The Convergence of Truncated Euler-Maruyama Method for Stochastic Differential Equations with Piecewise Continuous Arguments Under Generalized One-Sided Lipschitz Condition. Journal of Computational Mathematics. 41 (4). 663-682. doi:10.4208/jcm.2109-m2021-0116
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