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.
Allan, A. L., & Cohen, S. N. (2016). Ergodic backward stochastic difference equations. Stochastics: An International Journal of Probability and Stochastic Processes, 88(8), 1207-1239. https://doi.org/10.1080/17442508.2016.1224881