Ergodic Backward Stochastic Difference Equations

TL;DR

This paper studies ergodic backward stochastic difference equations driven by finite state Markov chains, establishing existence, uniqueness, and comparison results, and applies the theory to ergodic control problems.

Contribution

It introduces a novel approach using Nummelin splitting to prove ergodicity and solve ergodic backward stochastic difference equations in discrete time.

Findings

Proved existence and uniqueness of solutions.

Established a comparison theorem.

Applied results to ergodic control problems.

Abstract

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.