  We consider ergodic backward stochastic differential equations in a discrete
time setting, where noise is generated by a finite state Markov chain. We show
existence and uniqueness of solutions, along with a comparison theorem. To
obtain this result, we use a Nummelin splitting argument to obtain ergodicity
estimates for a discrete time Markov chain which hold uniformly under suitable
perturbations of its transition matrix. We conclude with an application of this
theory to a treatment of an ergodic control problem.
