Analysis of stochastic delay differential equations in the framework of conformable fractional derivatives

Muhammad Imran Liaqat; Fahim Ud Din; Wedad Albalawi; Kottakkaran Sooppy Nisar; Abdel-Haleem Abdel-Aty; Muhammad Imran Liaqat; Fahim Ud Din; Wedad Albalawi; Kottakkaran Sooppy Nisar; Abdel-Haleem Abdel-Aty

doi:10.3934/math.2024549

AIMS Mathematics

2024, Volume 9, Issue 5: 11194-11211. doi: 10.3934/math.2024549

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Analysis of stochastic delay differential equations in the framework of conformable fractional derivatives

1.
Abdus Salam School of Mathematical Sciences, Government College University, 68-B, New MuslimTown, Lahore 54600, Pakistan
2.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
3.
Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia
4.
Department of Physics, College of Sciences, University of Bisha, Bisha 61922, Saudi Arabia

Received: 19 January 2024 Revised: 04 March 2024 Accepted: 07 March 2024 Published: 21 March 2024
MSC : 34A07, 34A08, 60G22

In numerous domains, fractional stochastic delay differential equations are used to model various physical phenomena, and the study of well-posedness ensures that the mathematical models accurately represent physical systems, allowing for meaningful predictions and analysis. A fractional stochastic differential equation is considered well-posed if its solution satisfies the existence, uniqueness, and continuous dependency properties. We established the well-posedness and regularity of solutions of conformable fractional stochastic delay differential equations (CFrSDDEs) of order $ \gamma\in(\frac{1}{2}, 1) $ in $ \mathbb{L}^{\mathrm{p}} $ spaces with $ \mathrm{p}\geq2 $, whose coefficients satisfied a standard Lipschitz condition. More specifically, we first demonstrated the existence and uniqueness of solutions; after that, we demonstrated the continuous dependency of solutions on both the initial values and fractional exponent $ \gamma $. The second section was devoted to examining the regularity of time. As a result, we found that, for each $ \Phi\in(0, \gamma-\frac{1}{2}) $, the solution to the considered problem has a $ \Phi- $H$ \ddot o $lder continuous version. Lastly, two examples that highlighted our findings were provided. The two main elements of the proof were the Burkholder-Davis-Gundy inequality and the weighted norm.
- conformable fractional stochastic delay differential equations,
- well-posedness,
- regularity
Citation: Muhammad Imran Liaqat, Fahim Ud Din, Wedad Albalawi, Kottakkaran Sooppy Nisar, Abdel-Haleem Abdel-Aty. Analysis of stochastic delay differential equations in the framework of conformable fractional derivatives[J]. AIMS Mathematics, 2024, 9(5): 11194-11211. doi: 10.3934/math.2024549

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Abstract

In numerous domains, fractional stochastic delay differential equations are used to model various physical phenomena, and the study of well-posedness ensures that the mathematical models accurately represent physical systems, allowing for meaningful predictions and analysis. A fractional stochastic differential equation is considered well-posed if its solution satisfies the existence, uniqueness, and continuous dependency properties. We established the well-posedness and regularity of solutions of conformable fractional stochastic delay differential equations (CFrSDDEs) of order $ \gamma\in(\frac{1}{2}, 1) $ in $ \mathbb{L}^{\mathrm{p}} $ spaces with $ \mathrm{p}\geq2 $, whose coefficients satisfied a standard Lipschitz condition. More specifically, we first demonstrated the existence and uniqueness of solutions; after that, we demonstrated the continuous dependency of solutions on both the initial values and fractional exponent $ \gamma $. The second section was devoted to examining the regularity of time. As a result, we found that, for each $ \Phi\in(0, \gamma-\frac{1}{2}) $, the solution to the considered problem has a $ \Phi- $H$ \ddot o $lder continuous version. Lastly, two examples that highlighted our findings were provided. The two main elements of the proof were the Burkholder-Davis-Gundy inequality and the weighted norm.

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