We study singular perturbations of a class of stochastic control problems with unbounded data. We construct an effective Hamiltonian and prove the convergence of the value function to the solution of a limit (effective) Cauchy problem with a parabolic equation of HJB type. We use methods of probability, viscosity solutions and homogenization. We finally represent the solution of the limit HJB equation as the value function of a control problem described by stochastic functional inclusions and prove a weak convergence result for the singularly perturbed trajectories. Some applications are also discussed.
Keywords: Singular perturbations, stochastic control, viscosity solutions, Hamilton-Jacobi-Bellman equations, invariant measures.