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Convergence and stability of the semi-tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients

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  • Liu, Yulong
  • Niu, Yuanling
  • Cheng, Xiujun

Abstract

A new explicit stochastic scheme of order 1 is proposed for solving commutative stochastic differential equations (SDEs) with non-globally Lipschitz continuous coefficients. The proposed method is a semi-tamed version of Milstein scheme to solve SDEs with the drift coefficient consisting of non-Lipschitz continuous term and globally Lipschitz continuous term. It is easily implementable and achieves higher strong convergence order. A stability criterion for this method is derived, which shows that the stability condition of the numerical methods and that of the solved equations keep uniform. Compared with some widely used numerical schemes, the proposed method has better performance in inheriting the mean square stability of the exact solution of SDEs. Numerical experiments are given to illustrate the obtained convergence and stability properties.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:apmaco:v:414:y:2022:i:c:s0096300321007645
    DOI: 10.1016/j.amc.2021.126680
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    References listed on IDEAS

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    1. Yao, Jinran & Gan, Siqing, 2018. "Stability of the drift-implicit and double-implicit Milstein schemes for nonlinear SDEs," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 294-301.
    2. 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.
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