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Stochastic fractional differential equations: Modeling, method and analysis

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  • Pedjeu, Jean-C.
  • Ladde, Gangaram S.

Abstract

By introducing a concept of dynamic process operating under multi-time scales in sciences and engineering, a mathematical model described by a system of multi-time scale stochastic differential equations is formulated. The classical Picard–Lindelöf successive approximations scheme is applied to the model validation problem, namely, existence and uniqueness of solution process. Naturally, this leads to the problem of finding closed form solutions of both linear and nonlinear multi-time scale stochastic differential equations of Itô–Doob type. Finally, to illustrate the scope of ideas and presented results, multi-time scale stochastic models for ecological and epidemiological processes in population dynamic are outlined.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:chsofr:v:45:y:2012:i:3:p:279-293
    DOI: 10.1016/j.chaos.2011.12.009
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    References listed on IDEAS

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    1. Ahmed, E. & Elgazzar, A.S., 2007. "On fractional order differential equations model for nonlocal epidemics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 379(2), pages 607-614.
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    1. 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.
    2. Yang, Zhiwei & Zheng, Xiangcheng & Zhang, Zhongqiang & Wang, Hong, 2021. "Strong convergence of a Euler-Maruyama scheme to a variable-order fractional stochastic differential equation driven by a multiplicative white noise," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    3. 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.
    4. 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).
    5. Ahmadova, Arzu & Mahmudov, Nazim I., 2021. "Strong convergence of a Euler–Maruyama method for fractional stochastic Langevin equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 429-448.
    6. Hendy, Ahmed S. & Zaky, Mahmoud A. & Suragan, Durvudkhan, 2022. "Discrete fractional stochastic Grönwall inequalities arising in the numerical analysis of multi-term fractional order stochastic differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 193(C), pages 269-279.
    7. Abdelhamid Mohammed Djaouti & Zareen A. Khan & Muhammad Imran Liaqat & Ashraf Al-Quran, 2024. "Existence, Uniqueness, and Averaging Principle of Fractional Neutral Stochastic Differential Equations in the L p Space with the Framework of the Ψ-Caputo Derivative," Mathematics, MDPI, vol. 12(7), pages 1-21, March.
    8. Zakaria Ali & Minyahil Abera Abebe & Talat Nazir, 2024. "Strong Convergence of Euler-Type Methods for Nonlinear Fractional Stochastic Differential Equations without Singular Kernel," Mathematics, MDPI, vol. 12(18), pages 1-36, September.

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