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Mean-Square Stability of Split-Step Theta Milstein Methods for Stochastic Differential Equations

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  • Mahmoud A. Eissa
  • Haiying Zhang
  • Yu Xiao

Abstract

The fundamental analysis of numerical methods for stochastic differential equations (SDEs) has been improved by constructing new split-step numerical methods. In this paper, we are interested in studying the mean-square (MS) stability of the new general drifting split-step theta Milstein (DSS M) methods for SDEs. First, we consider scalar linear SDEs. The stability function of the DSS M methods is investigated. Furthermore, the stability regions of the DSS M methods are compared with those of test equation, and it is proved that the methods with are stochastically A-stable. Second, the nonlinear stability of DSS M methods is studied. Under a coupled condition on the drifting and diffusion coefficients, it is proved that the methods with can preserve the MS stability of the SDEs with no restriction on the step-size. Finally, numerical examples are given to examine the accuracy of the proposed methods under the stability conditions in approximation of SDEs.

Suggested Citation

  • Mahmoud A. Eissa & Haiying Zhang & Yu Xiao, 2018. "Mean-Square Stability of Split-Step Theta Milstein Methods for Stochastic Differential Equations," Mathematical Problems in Engineering, Hindawi, vol. 2018, pages 1-13, January.
    • Handle: RePEc:hin:jnlmpe:1682513
    DOI: 10.1155/2018/1682513
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    Cited by:

    1. Zin Thu Win & Mahmoud A. Eissa & Boping Tian, 2022. "Stochastic Epidemic Model for COVID-19 Transmission under Intervention Strategies in China," Mathematics, MDPI, vol. 10(17), pages 1-17, August.

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