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Peano Theorems for Pedjeu–Ladde-Type Multi-Time Scale Stochastic Differential Equations Driven by Fractional Noises

Author

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  • Arcady Ponosov

    (Department of Mathematics, Norwegian University of Life Sciences, 1432 Aas, Norway
    These authors contributed equally to this work.)

  • Lev Idels

    (Department of Mathematics, Vancouver Island University, 900 Fifth St., Nanaimo, BC V9S 5S5, Canada
    These authors contributed equally to this work.)

Abstract

This paper examines fractional multi-time scale stochastic functional differential equations that, in addition, are driven by fractional noises. Based on a specially crafted fixed-point principle for the so-called “local operators”, we prove a Peano-type theorem on the existence of weak solutions, that is, those defined on an extended stochastic basis. To encompass all commonly used particular classes of fractional multi-time scale stochastic models, including those with random delays and impulses at random times, we consider equations with nonlinear random Volterra operators rather than functions. Some crucial properties of the associated integral operators, needed for the proofs of the main results, are studied as well. To illustrate major findings, several existence theorems, generalizing those known in the literature, are offered, with the emphasis put on the most popular examples such as ordinary stochastic differential equations driven by fractional noises, fractional stochastic differential equations with variable delays and fractional stochastic neutral differential equations.

Suggested Citation

  • Arcady Ponosov & Lev Idels, 2025. "Peano Theorems for Pedjeu–Ladde-Type Multi-Time Scale Stochastic Differential Equations Driven by Fractional Noises," Mathematics, MDPI, vol. 13(2), pages 1-21, January.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:2:p:204-:d:1563665
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    References listed on IDEAS

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    1. Guangjun Shen & Ruidong Xiao & Xiuwei Yin, 2020. "Averaging principle and stability of hybrid stochastic fractional differential equations driven by Lévy noise," International Journal of Systems Science, Taylor & Francis Journals, vol. 51(12), pages 2115-2133, September.
    2. Zhi Wang & Litan Yan, 2013. "Stochastic Volterra Equation Driven by Wiener Process and Fractional Brownian Motion," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-8, November.
    3. 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.
    4. Zhang, Yinghan & Yang, Xiaoyuan, 2015. "Fractional stochastic Volterra equation perturbed by fractional Brownian motion," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 20-36.
    5. Fan, Xiliang & Huang, Xing & Suo, Yongqiang & Yuan, Chenggui, 2022. "Distribution dependent SDEs driven by fractional Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 151(C), pages 23-67.
    6. Anh, P.T. & Doan, T.S. & Huong, P.T., 2019. "A variation of constant formula for Caputo fractional stochastic differential equations," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 351-358.
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