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

TL;DR

This paper develops a theoretical framework using singular perturbation analysis and viscosity solutions to solve constrained multiscale stochastic control problems, demonstrating convergence and applicability through examples.

Contribution

It introduces a novel approach for handling state constraints in multiscale stochastic control via singular perturbation and viscosity solutions, with proven convergence results.

Findings

Established convergence of multiscale value functions.

Applied the framework to illustrative examples.

Demonstrated effectiveness in constrained settings.

Abstract

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.