Stability and approximation of invariant measures of Markov chains in random environments

We consider finite-state Markov chains driven by stationary ergodic invertible processes representing random environments. Our main result is that the invariant measures of Markov chains in random environments (MCREs) are stable under a wide variety of perturbations. We prove stability in the sense of convergence in probability of the invariant measure of the perturbed MCRE to the original invariant measure. Our approach makes no assumptions on the transition matrix functions representing the Markov chains except measurability with respect to the random environment. We also develop a new numerical scheme to construct rigorous approximations of the invariant measures, which converge in probability as the resolution of the scheme increases. This numerical approach is illustrated with an example of a random walk in a random environment. MSC 2010 subject classifications: Primary 60J; secondary 37H.