Singular perturbations in stochastic optimal control with unbounded data

We study singular perturbations of a class of two-scale stochastic control systems with unbounded data. The assumptions are designed to cover some relaxation problems for deep neural networks. We construct e ective Hamiltonian and initial data and prove the convergence of the value function to the solution of a limit (e ective) Cauchy problem for a parabolic equation of HJB type. We use methods of probability, viscosity solutions and homogenization.