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On the Ergodicity of Certain Markov Chains in Random Environments

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  • Balázs Gerencsér

    (Alfréd Rényi Institute of Mathematics and Eötvös Loránd University)

  • Miklós Rásonyi

    (Alfréd Rényi Institute of Mathematics and Eötvös Loránd University)

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 $$X_t$$ X t are shown to converge to a limiting law in (weighted) total variation distance as $$t\rightarrow \infty $$ t → ∞ . Convergence speed is estimated, and an ergodic theorem is established for functionals of X. Our hypotheses on X combine the standard “drift” and “small set” 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 intractable. In particular, our results are pertinent to difference equations modulated by a stationary (Gaussian) process. Such equations arise in applications such as discretized stochastic volatility models of mathematical finance.

Suggested Citation

  • Balázs Gerencsér & Miklós Rásonyi, 2023. "On the Ergodicity of Certain Markov Chains in Random Environments," Journal of Theoretical Probability, Springer, vol. 36(4), pages 2093-2125, December.
    • Handle: RePEc:spr:jotpro:v:36:y:2023:i:4:d:10.1007_s10959-023-01256-7
    DOI: 10.1007/s10959-023-01256-7
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    References listed on IDEAS

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    1. R. Cont, 2001. "Empirical properties of asset returns: stylized facts and statistical issues," Quantitative Finance, Taylor & Francis Journals, vol. 1(2), pages 223-236.
    2. Lovas, Attila & Rásonyi, Miklós, 2021. "Markov chains in random environment with applications in queuing theory and machine learning," Stochastic Processes and their Applications, Elsevier, vol. 137(C), pages 294-326.
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