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Search Results (3)

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Keywords = Itô–Doob stochastic systems

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28 pages, 621 KB  
Article
Averaging Principle for Itô–Doob Fractional Stochastic Systems in mth Moment
by Muhammad Imran Liaqat and Ramy M. Hafez
Axioms 2026, 15(4), 262; https://doi.org/10.3390/axioms15040262 - 4 Apr 2026
Viewed by 542
Abstract
This study presents results on the well-posedness, Ulam–Hyers stability, and mth moment averaging principle for the Itô–Doob fractional stochastic system within the framework of η-Caputo fractional derivatives. We demonstrate well-posedness using the fixed-point approach. A generalized Grönwall inequality is employed to [...] Read more.
This study presents results on the well-posedness, Ulam–Hyers stability, and mth moment averaging principle for the Itô–Doob fractional stochastic system within the framework of η-Caputo fractional derivatives. We demonstrate well-posedness using the fixed-point approach. A generalized Grönwall inequality is employed to establish sufficient conditions for Ulam–Hyers stability. Furthermore, we establish the averaging principle that facilitates obtaining a simplified averaged system from the original complex, multiple time-scale system. Finally, numerical simulations using the Euler–Maruyama method are provided to support the theoretical findings. Full article
(This article belongs to the Special Issue Numerical Analysis, Approximation Theory and Related Topics)
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14 pages, 277 KB  
Article
Finite-Time Stability for a Class of Fractional Itô–Doob Stochastic Time Delayed Systems
by Wissam Ghoul, Hussien Albala, Hamid Boulares, Faycal Bouchelaghem and Abdelkader Moumen
Fractal Fract. 2025, 9(11), 683; https://doi.org/10.3390/fractalfract9110683 - 23 Oct 2025
Viewed by 850
Abstract
This paper addresses the finite-time stability of a class of fractional Itô–Doob stochastic systems with time delays. Novel stability criteria are established using a combination of Gronwall-type, Hölder’s, and Burkholder–Davis–Gundy (BDG) inequalities, thereby generalizing classical integer-order stability theory to the fractional domain. Furthermore, [...] Read more.
This paper addresses the finite-time stability of a class of fractional Itô–Doob stochastic systems with time delays. Novel stability criteria are established using a combination of Gronwall-type, Hölder’s, and Burkholder–Davis–Gundy (BDG) inequalities, thereby generalizing classical integer-order stability theory to the fractional domain. Furthermore, the analysis uniquely integrates stochastic perturbations and time delays, providing a comprehensive framework for systems exhibiting both memory and randomness. The effectiveness of the proposed approach is demonstrated through a numerical example of a three-dimensional stochastic delayed system with fractional dynamics. Full article
(This article belongs to the Special Issue Stochastic Models, Fractional Calculus and Non-Local Operators: Theoretical Results and Applications)
14 pages, 294 KB  
Article
Proportional Itô–Doob Stochastic Fractional Order Systems
by Abdellatif Ben Makhlouf, Lassaad Mchiri, Hakeem A. Othman, Hafedh M. S. Rguigui and Salah Boulaaras
Mathematics 2023, 11(9), 2049; https://doi.org/10.3390/math11092049 - 26 Apr 2023
Cited by 4 | Viewed by 1321
Abstract
In this article, we discuss the existence and uniqueness of proportional Itô–Doob stochastic fractional order systems (PIDSFOS) by using the Picard iteration method. The paper provides new results using the proportional fractional integral and stochastic calculus techniques. We have shown the convergence of [...] Read more.
In this article, we discuss the existence and uniqueness of proportional Itô–Doob stochastic fractional order systems (PIDSFOS) by using the Picard iteration method. The paper provides new results using the proportional fractional integral and stochastic calculus techniques. We have shown the convergence of the solution of the averaged PIDSFOS to that of the standard PIDSFOS in the context of the mean square and also in probability. One example is given to illustrate our results. Full article
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