SIGNIFICANT FINDINGS CONCERNING A SPECIFIC CLASS OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH FRACTIONAL-ORDER

Muhammad Imran Liaqat, Hossein Jafari and Saleh Alshammari
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Muhammad Imran Liaqat: Abdus Salam School of Mathematical Sciences, Government College University, 68-B, New Muslim Town, Lahore 54600, Pakistan
Hossein Jafari: ��Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences (SIMATS ), Chennai 602105, Tamil Nadu, India‡Department of Mathematical Sciences, University of South Africa, UNISA0003, South Africa§Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 110122, Taiwan
Saleh Alshammari: �Department of Mathematics, College of Science, University of Hail, Hail 2240, Saudi Arabia

FRACTALS (fractals), 2025, vol. 33, issue 08, 1-16

Abstract: Many real-world systems exhibit memory effects, where the current state depends not only on past states but also on delayed responses. Neutral stochastic fractional differential equations, which incorporate neutral terms to represent these delayed responses, offer a more accurate modeling framework than traditional stochastic differential equations. In this study, we establish significant results, including the existence and uniqueness of solutions for stochastic differential equations with neutral terms within the framework of Caputo–Hadamard fractional derivatives. Additionally, we derive results for the critical concept of the averaging principle. These findings are established in the pth moment, generalizing the case for p = 2. Using contraction mapping techniques, we first prove the existence and uniqueness of solutions for a specific class of fractional-order stochastic differential equations. Next, we validate the averaging principle through inequality and interval translation methods. Finally, illustrative examples are provided to clarify and support the theoretical results.

Keywords: Stochastic Differential Equations; Caputo–Hadamard Derivatives; Brownian Motion; Averaging Principle (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1142/S0218348X2540153X

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