On the ergodicity of certain Markov chains in random environments

TL;DR

This paper investigates the long-term behavior of Markov chains in stationary random environments, establishing convergence to a limiting distribution, estimating convergence speed, and applying results to models like stochastic volatility in finance.

Contribution

It extends ergodic theory to Markov chains in complex random environments, covering models previously considered intractable, including Gaussian-modulated difference equations.

Findings

Convergence of the process laws to a limiting distribution in weighted total variation.

Estimation of the convergence speed under combined small set and drift conditions.

Application of results to stochastic volatility models in mathematical finance.

Abstract

We study the ergodic behaviour of a discrete-time process $X$ which is a Markov chain in a stationary random environment. The laws of $X_t$ are shown to converge to a limiting law in (weighted) total variation distance as $t\to\infty$. Convergence speed is estimated and an ergodic theorem is established for functionals of $X$. Our hypotheses on $X$ combine the standard "small set" and "drift" conditions for geometrically ergodic Markov chains with conditions on the growth rate of a certain "maximal process" of the random environment. We are able to cover a wide range of models that have heretofore been untractable. In particular, our results are pertinent to difference equations modulated by a stationary Gaussian process. Such equations arise in applications, for example, in discretized stochastic volatility models of mathematical finance.