On some perturbation problems in stochastic control

We consider a class of Markov processes X?? = (x?? (t), y??(t)) where y?? (t) is a "fast component" and x?? (t) the "slow component". The generator of X?? is given by -A?? = -(Ax + 1/?? By). In this paper we study the behaviour of the optimal cost function for optimal stopping and impulse control when ?? ?? 0.