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

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 $\mathrm{p}\ge 2$, 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 $\gamma$. The second section was devoted to examining the regularity of time. As a result, we found that, for each $\mathrm{\Phi }\in (0,\gamma -\frac{1}{2})$, the solution to the considered problem has a $\mathrm{\Phi }-$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.