Schauder estimates and classical solutions of the Dirichlet problem for stochastic parabolic equations

We study second-order stochastic parabolic equations in a cylindrical domain with homogeneous Dirichlet boundary conditions. Under a natural compatibility condition on the gradient-type noise, we establish global Schauder estimates in stochastic Hölder spaces for the Dirichlet problem. The coefficients and free terms are assumed to be Hölder continuous in the spatial variables, while only their boundary traces are required to be Hölder in time. As a consequence, we obtain existence and uniqueness of quasi-classical solutions in stochastic Hölder spaces, and further derive pathwise classical solvability in Hölder classes.