  We consider ergodic backward stochastic differential equations, in a setting
where noise is generated by a countable state uniformly ergodic Markov chain.
We show that for Lipschitz drivers such that a comparison theorem holds, these
equations admit unique solutions. To obtain this result, we show by coupling
and splitting techniques that uniform ergodicity estimates of Markov chains are
robust to perturbations of the rate matrix, and that these perturbations
correspond in a natural way to EBSDEs. We then consider applications of this
theory to Markov decision problems with a risk-averse average reward criterion.
