Approximation and estimation in Markov control processes under a discounted criterion

summary:We consider a class of discrete-time Markov control processes with Borel state and action spaces, and $\Re ^{k}$-valued i.i.d. disturbances with unknown density $\rho .$ Supposing possibly unbounded costs, we combine suitable density estimation methods of $\rho $ with approximation procedures of the optimal cost function, to show the existence of a sequence $\lbrace \hat{f}_{t}\rbrace $ of minimizers converging to an optimal stationary policy $f_{\infty }.