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Exponential stability of the split-step θ-method for neutral stochastic delay differential equations with jumps

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  • Mo, Haoyi
  • Deng, Feiqi
  • Zhang, Chaolong

Abstract

The exponential mean-square stability of the split-step θ-method for neutral stochastic delay differential equations (NSDDEs) with jumps is considered. New conditions for jumps are proposed to ensure the exponential mean-square stability of the trivial solution. If the drift coefficient satisfies the linear growth condition, it is shown that the split-step θ-method can reproduce the exponential mean-square stability of the trivial solution for the constrained stepsize. Then by applying the Chebyshev inequality and the Borel–Cantelli lemma, the almost sure exponential stability of both the trivial solution and the numerical solution can be obtained. Since split-step θ-method covers Euler–Maruyama (EM) method and split-step backward Euler (SSBE) method, the conclusions are valid for these two methods. Moreover, they can adapt to the NSDDEs and the SDDEs with jumps. Finally, a numerical example illustrates the effectiveness of the theoretical results.

Suggested Citation

  • Mo, Haoyi & Deng, Feiqi & Zhang, Chaolong, 2017. "Exponential stability of the split-step θ-method for neutral stochastic delay differential equations with jumps," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 85-95.
    • Handle: RePEc:eee:apmaco:v:315:y:2017:i:c:p:85-95
    DOI: 10.1016/j.amc.2017.06.034
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    References listed on IDEAS

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    1. 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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