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pth moment asymptotic stability for neutral stochastic functional differential equations with Lévy processes

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  • Xu, Yan
  • He, Zhimin
  • Wang, Peiguang

Abstract

In this paper, we will consider a class of neutral stochastic functional differential equations with Lévy processes. Lévy processes contain a number of very important processes as special cases such as Brownian motion, the Poisson process, stable and self-decomposable processes and subordinators, and so on. But its sample paths are discontinuity, which makes the analysis more difficult. In this paper, we try to get over this difficulty. The contributions of this paper are as follows: (a) we will use Lyapunov functional method to study the pth moment asymptotic stability and almost sure asymptotic stability of neutral stochastic functional differential equations with Lévy processes; (b) under the result of (a), we will investigate two types of continuity of the solution: continuous in the pth moment and continuous in probability. Finally, we provide an example to illustrate the usefulness of the obtained results.

Suggested Citation

  • Xu, Yan & He, Zhimin & Wang, Peiguang, 2015. "pth moment asymptotic stability for neutral stochastic functional differential equations with Lévy processes," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 594-605.
    • Handle: RePEc:eee:apmaco:v:269:y:2015:i:c:p:594-605
    DOI: 10.1016/j.amc.2015.07.070
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    References listed on IDEAS

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    1. Xu, Yong & Pei, Bin & Guo, Guobin, 2015. "Existence and stability of solutions to non-Lipschitz stochastic differential equations driven by Lévy noise," Applied Mathematics and Computation, Elsevier, vol. 263(C), pages 398-409.
    2. Bao, Jianhai & Hou, Zhenting & Yuan, Chenggui, 2009. "Stability in distribution of neutral stochastic differential delay equations with Markovian switching," Statistics & Probability Letters, Elsevier, vol. 79(15), pages 1663-1673, August.
    3. Mao, Wei & Zhu, Quanxin & Mao, Xuerong, 2015. "Existence, uniqueness and almost surely asymptotic estimations of the solutions to neutral stochastic functional differential equations driven by pure jumps," Applied Mathematics and Computation, Elsevier, vol. 254(C), pages 252-265.
    4. Mao, Xuerong & Shen, Yi & Yuan, Chenggui, 2008. "Almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching," Stochastic Processes and their Applications, Elsevier, vol. 118(8), pages 1385-1406, August.
    5. You, Surong & Mao, Wei & Mao, Xuerong & Hu, Liangjian, 2015. "Analysis on exponential stability of hybrid pantograph stochastic differential equations with highly nonlinear coefficients," Applied Mathematics and Computation, Elsevier, vol. 263(C), pages 73-83.
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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).

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