@article{Liaqat2024, 
author = {Muhammad Imran Liaqat and Fahim Ud Din and Wedad Albalawi and Kottakkaran Sooppy Nisar and Abdel-Haleem Abdel-Aty},
title = {Analysis of stochastic delay differential equations in the framework of conformable fractional derivatives},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {5},
pages = {11194-11211},
keywords = {conformable fractional stochastic delay differential equations, well-posedness, regularity},
url = {https://www.sciopen.com/article/10.3934/math.2024549},
doi = {10.3934/math.2024549},
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    γ  ∈  (      1    2    ,  1  ) in              L                      p             spaces with        p    ≥  2, 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    γ. The second section was devoted to examining the regularity of time. As a result, we found that, for each    Φ  ∈  (  0  ,  γ  −      1    2    ), the solution to the considered problem has a    Φ  −H             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.}
}