Sobolev-Type Neutral Stochastic Differential Equations with Hilfer Fractional Derivative and Finite Delay

This study establishes the class of Sobolev-type neutral stochastic differential equations with Hilfer derivative and finite delay. Existence and uniqueness of the mild solution are then established in the mean-square sense via the Banach contraction principle, and a Faedo–Galerkin scheme is constructed whose approximate solutions are shown to converge, in mean square, to the unique mild solution. An application to a stochastically perturbed Sobolev-type filtering/diffusion model closes the paper.