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Stability of the drift-implicit and double-implicit Milstein schemes for nonlinear SDEs

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  • Yao, Jinran
  • Gan, Siqing

Abstract

This paper examines the stability of numerical solutions of nonlinear stochastic differential equations (SDEs) with non-global Lipschitz continuous coefficients. Two implicit Milstein schemes, called drift-implicit Milstein scheme and double-implicit Milstein scheme, are considered to simulate the underlying SDEs. It is proved that the schemes can preserve the stability and contractivity in mean square of the underlying systems.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:339:y:2018:i:c:p:294-301
    DOI: 10.1016/j.amc.2018.07.026
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    References listed on IDEAS

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    1. Kahl Christian & Schurz Henri, 2006. "Balanced Milstein Methods for Ordinary SDEs," Monte Carlo Methods and Applications, De Gruyter, vol. 12(2), pages 143-170, April.
    2. Mahmoud A. Eissa & Boping Tian, 2017. "Lobatto-Milstein Numerical Method in Application of Uncertainty Investment of Solar Power Projects," Energies, MDPI, vol. 10(1), pages 1-19, January.
    3. G. N. Milstein & Eckhard Platen & H. Schurz, 1998. "Balanced Implicit Methods for Stiff Stochastic Systems," Published Paper Series 1998-1, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
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    3. Liu, Yufen & Cao, Wanrong & Li, Yuelin, 2022. "Split-step balanced θ-method for SDEs with non-globally Lipschitz continuous coefficients," Applied Mathematics and Computation, Elsevier, vol. 413(C).
    4. 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).

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