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Convergence rate of the truncated Milstein method of stochastic differential delay equations

Author

Listed:
  • Zhang, Wei
  • Yin, Xunbo
  • Song, M.H.
  • Liu, M.Z.

Abstract

This paper is concerned with the strong convergence of highly nonlinear stochastic differential delay equations (SDDEs) without the linear growth condition. On the one hand, these nonlinear SDDEs do not have explicit solutions, therefore implementable numerical methods for such SDDEs are required. On the other hand, the implicit Euler methods are known to converge strongly to the exact solution of such SDDEs. However, they require additional computational efforts. In this article, we propose the truncated Milstein method which is an explicit method under the local Lipschitz condition plus Khasminskii-type condition, study its pth monent boundedness (p is a parameter in Khasminskii-type condition) and show that its rate of strong convergence is close to one.

Suggested Citation

  • Zhang, Wei & Yin, Xunbo & Song, M.H. & Liu, M.Z., 2019. "Convergence rate of the truncated Milstein method of stochastic differential delay equations," Applied Mathematics and Computation, Elsevier, vol. 357(C), pages 263-281.
    • Handle: RePEc:eee:apmaco:v:357:y:2019:i:c:p:263-281
    DOI: 10.1016/j.amc.2019.04.001
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    Citations

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    Cited by:

    1. Xuewen Tan & Pengpeng Liu & Wenhui Luo & Hui Chen, 2022. "Analysis of a Class of Predation-Predation Model Dynamics with Random Perturbations," Mathematics, MDPI, vol. 10(18), pages 1-12, September.
    2. Liu, Yulong & Niu, Yuanling & Cheng, Xiujun, 2022. "Convergence and stability of the semi-tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients," Applied Mathematics and Computation, Elsevier, vol. 414(C).
    3. Bao, Zhenyu & Tang, Jingwen & Shen, Yan & Liu, Wei, 2021. "Equivalence of pth moment stability between stochastic differential delay equations and their numerical methods," Statistics & Probability Letters, Elsevier, vol. 168(C).

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