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Exponential mean-square stability of the θ-method for neutral stochastic delay differential equations with jumps

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  • Haoyi Mo
  • Xueyan Zhao
  • Feiqi Deng

Abstract

The exponential mean-square stability of the θ-method for neutral stochastic delay differential equations (NSDDEs) with jumps is considered. With some monotone conditions, the trivial solution of the equation is proved to be exponentially mean-square stable. If the drift coefficient and the parameters satisfy more strengthened conditions, for the constrained stepsize, it is shown that the θ-method can preserve the exponential mean-square stability of the trivial solution for θ ∈ [0, 1]. Since θ-method covers the commonly used Euler–Maruyama (EM) method and the backward Euler–Maruyama (BEM) method, the results are valid for the above two methods. Moreover, they can adapt to the NSDDEs and the stochastic delay differential equations (SDDEs) with jumps. Finally, a numerical example illustrates the effectiveness of the theoretical results.

Suggested Citation

  • Haoyi Mo & Xueyan Zhao & Feiqi Deng, 2017. "Exponential mean-square stability of the θ-method for neutral stochastic delay differential equations with jumps," International Journal of Systems Science, Taylor & Francis Journals, vol. 48(3), pages 462-470, February.
    • Handle: RePEc:taf:tsysxx:v:48:y:2017:i:3:p:462-470
    DOI: 10.1080/00207721.2016.1186245
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    References listed on IDEAS

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    1. Chen, Lin & Wu, Fuke, 2012. "Almost sure exponential stability of the θ-method for stochastic differential equations," Statistics & Probability Letters, Elsevier, vol. 82(9), pages 1669-1676.
    2. Jianguo Tan & Hongli Wang & Yongfeng Guo, 2012. "Existence and Uniqueness of Solutions to Neutral Stochastic Functional Differential Equations with Poisson Jumps," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-20, July.
    3. Li, Xiuping & Cao, Wanrong, 2015. "On mean-square stability of two-step Maruyama methods for nonlinear neutral stochastic delay differential equations," Applied Mathematics and Computation, Elsevier, vol. 261(C), pages 373-381.
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    Cited by:

    1. Wan, Fangzhe & Hu, Po & Chen, Huabin, 2020. "Stability analysis of neutral stochastic differential delay equations driven by Lévy noises," Applied Mathematics and Computation, Elsevier, vol. 375(C).
    2. Li, Guangjie & Yang, Qigui, 2021. "Stability analysis of the θ-method for hybrid neutral stochastic functional differential equations with jumps," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).

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