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Asymptotic mean-square stability of weak second-order balanced stochastic Runge–Kutta methods for multi-dimensional Itô stochastic differential systems

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  • Rathinasamy, Anandaraman
  • Nair, Priya

Abstract

In this paper, the linear asymptotic mean-square stability of the weak second-order stochastic Runge–Kutta methods for multi-dimensional Itô stochastic differential equations due to Rößler (2009) and Tang and Xiao (2017) are obtained. Further, we have developed the stochastic Runge–Kutta methods for multi-dimensional Itô stochastic differential equations to the balanced stochastic Runge–Kutta methods by using the control functions. We carry out a linear stability analysis of the class of stochastic Runge–Kutta methods for the linear test equations with multiplicative noise thereby providing an explicit structure of stability matrices. Some comparisons and illustrations shows that there is an improvement in the stability and error analysis of these new balanced stochastic Runge–Kutta methods comparatively to stochastic Runge–Kutta methods and thus conforming the obtained theoretical results.

Suggested Citation

  • Rathinasamy, Anandaraman & Nair, Priya, 2018. "Asymptotic mean-square stability of weak second-order balanced stochastic Runge–Kutta methods for multi-dimensional Itô stochastic differential systems," Applied Mathematics and Computation, Elsevier, vol. 332(C), pages 276-303.
    • Handle: RePEc:eee:apmaco:v:332:y:2018:i:c:p:276-303
    DOI: 10.1016/j.amc.2018.03.065
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    1. Buckwar, Evelyn & Sickenberger, Thorsten, 2011. "A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(6), pages 1110-1127.
    2. Haghighi, A. & Hosseini, S.M., 2014. "Analysis of asymptotic mean-square stability of a class of Runge–Kutta schemes for linear systems of stochastic differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 105(C), pages 17-48.
    3. Yin, Zhengwei & Gan, Siqing, 2015. "An error corrected Euler–Maruyama method for stiff stochastic differential equations," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 630-641.
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    Cited by:

    1. Tan, Jianguo & Chen, Yang & Men, Weiwei & Guo, Yongfeng, 2021. "Positivity and convergence of the balanced implicit method for the nonlinear jump-extended CIR model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 195-210.
    2. Avaji, M. & Jodayree Akbarfam, A. & Haghighi, A., 2019. "Stability analysis of high order Runge–Kutta methods for index 1 stochastic differential-algebraic equations with scalar noise," Applied Mathematics and Computation, Elsevier, vol. 362(C), pages 1-1.

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