Singular Perturbation in Multiscale Stochastic Control Problems with Domain Restriction in the Slow Variable

We study a multiscale stochastic optimal control problem subject to state constraints on the slow variable. To address this class of problems, we develop a rigorous theoretical framework based on singular perturbation analysis, tailored to settings with constrained dynamics. Our approach relies on the theory of viscosity solutions for degenerate Hamilton-Jacobi-Bellman equations with Neumann-type boundary conditions. We also establish the convergence of the multiscale value functions in the infinite-horizon regime. Finally, we present two illustrative examples that highlight the applicability and effectiveness of the proposed framework.