Significant results in the $\mathrm{p}$th moment for Hilfer fractional stochastic delay differential equations

Well-posedness is crucial in studying fractional stochastic differential equations, as it ensures that solutions are mathematically sound and applicable to practical situations. A well-formulated model satisfies the essential requirements for solutions, such as existence, uniqueness, and stability concerning various parameters. Using fixed-point theory, we prove that the solution to stochastic fractional delay differential equations with the Hilfer fractional operator exists, is unique, and continuously depends on the initial values and the fractional derivative. Additionally, we establish a smoothness theorem for the solution and demonstrate that the solution of the original system converges to the averaged system in the $\mathrm{p}$th moment. Last, to support our theoretical findings, we provide examples and graphical illustrations. The primary tools used in our proofs include the Burkholder-Davis-Gundy inequality, Jensen's inequality, and Hölder's inequality.