Existence Results for Semilinear Stochastic Differential Equations with State-Dependent Delay and an Illustrative Example

This paper studies the existence and uniqueness of mild solutions to a class of semilinear stochastic differential equations with state-dependent delay, driven by an infinite-dimensional Wiener process. The drift and diffusion coefficients are assumed to satisfy Lipschitz and linear-growth conditions, and the delay function is assumed to be Lipschitz continuous with respect to the phase-space norm. Because the delay depends on the unknown solution, the associated solution operator is not automatically a contraction on the space of continuous adapted processes: the delay-mismatch terms can be bounded only if the trajectories are regular in time. We therefore work on a closed set of trajectories with a common Lipschitz-in-time modulus and a mean-square size bound, and we assume explicitly that the solution operator maps this set into itself. Under this assumption, the Banach fixed-point theorem and Itô-type moment estimates yield a local existence-and-uniqueness theorem, with uniqueness understood within the chosen class. We give explicit sufficient conditions for the invariance when the diffusion vanishes, show that the invariance cannot hold in this class for a non-degenerate diffusion, and illustrate the result with a two-dimensional example.