Discounted cost optimality problem: stability with respect to weak metrics

We find inequalities to estimate the stability (robustness) of a discounted cost optimization problem for discrete-time Markov control processes on a Borel state space. The one stage cost is allowed to be unbounded. Unlike the known results in this area we consider a perturbation of transition probabilities measured by the Kantorovich metric, closely related to the weak convergence. The results obtained make possible to estimate the vanishing rate of the stability index when approximation is made through empirical measures.