Multiscale Stochastic Control With Domain Restriction in the Slow Variable

Abstract We investigate multiscale stochastic optimal control problems in the infinite-horizon regime with constraints on the slow dynamics. The associated multiscale Hamilton-Jacobi-Bellman (HJB) equation is a fully nonlinear degenerate elliptic partial differential equation with a Neumann boundary condition in the slow dynamics. The analysis combines singular perturbation techniques with the theory of viscosity solutions. These two techniques allow for the identification of the effective HJB equation and for establishing the uniform convergence on a compact set of the multiscale optimal value functions to the effective optimal value function. The resulting effective HJB equation characterizes an effective stochastic optimal control problem posed on the slow dynamics and subject to state constraints. An example illustrating the application of the developed framework is also presented.