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Carathéodory approximations and stability of solutions to non-Lipschitz stochastic fractional differential equations of Itô-Doob type

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  • Abouagwa, Mahmoud
  • Liu, Jicheng
  • Li, Ji

Abstract

The existence and uniqueness theorem of solutions provides an effective tool for the model validation of both deterministic and stochastic equations. The objective of this paper is to establish the existence and uniqueness of solutions for a class of Itô-Doob stochastic fractional differential equations under non-Lipschitz condition which is weaker than Lipschitz one and contains it as a special case. The solution is constructed with the aid of Carathéodory approximation. Moreover, the continuous dependence of solutions on the initial value is investigated in view of the stability of solutions in the sense of mean square. Finally, an example is given to illustrate the theory.

Suggested Citation

  • Abouagwa, Mahmoud & Liu, Jicheng & Li, Ji, 2018. "Carathéodory approximations and stability of solutions to non-Lipschitz stochastic fractional differential equations of Itô-Doob type," Applied Mathematics and Computation, Elsevier, vol. 329(C), pages 143-153.
    • Handle: RePEc:eee:apmaco:v:329:y:2018:i:c:p:143-153
    DOI: 10.1016/j.amc.2018.02.005
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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. Pedjeu, Jean-C. & Ladde, Gangaram S., 2012. "Stochastic fractional differential equations: Modeling, method and analysis," Chaos, Solitons & Fractals, Elsevier, vol. 45(3), pages 279-293.
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    Cited by:

    1. Luo, Danfeng & Tian, Mengquan & Zhu, Quanxin, 2022. "Some results on finite-time stability of stochastic fractional-order delay differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 158(C).
    2. Abdellatif Ben Makhlouf & Lassaad Mchiri & Hakeem A. Othman & Hafedh M. S. Rguigui & Salah Boulaaras, 2023. "Proportional Itô–Doob Stochastic Fractional Order Systems," Mathematics, MDPI, vol. 11(9), pages 1-14, April.
    3. Kahouli, Omar & Ben Makhlouf, Abdellatif & Mchiri, Lassaad & Rguigui, Hafedh, 2023. "Hyers–Ulam stability for a class of Hadamard fractional Itô–Doob stochastic integral equations," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).

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