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A copula-based approximation to Markov chains

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Abstract

The Markov chain is well studied and widely applied in many areas. For some Markov chains, it is infeasible to obtain the explicit expressions of their corresponding finite-dimensional distributions and sometimes it is time-consuming for computation. In this paper, we propose an approximation method for Markov chains by applying the copula theory. For this purpose, we first discuss the checkerboard copula-based Markov chain, which is the Markov chain generated by the family of checkerboard copulas. This Markov chain has some appealing properties, such as self-similarity in copulas and having explicit forms of finite-dimensional distributions. Then we prove that each Markov chain can be approximated by a sequence of checkerboard copula-based Markov chains, and the error bounds of the approximate distributions are provided. Employing the checkerboard copula-based approximation method, we propose a sufficient condition for the geometric β-mixing of copula-based Markov chains. This condition allows copulas of Markov chains to be asymmetric. Finally, by applying the approximation method, analytical recurrence formulas are also derived for computing approximate distributions of both the first passage time and the occupation time of a Markov chain, and numerical results are listed to show the approximation errors.

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Acknowledgements

The first and the third authors were supported by the National Key R&D Program of China (Grants No. 2018YFA0703900) and National Natural Science Foundation of China (Grants No. 11671021). The second author was supported by National Natural Science Foundation of China (Grants Nos. 11761051 and 11561047) and the Natural Science Foundation of Jiangxi Province (Grants Nos. 20181BAB211003 and 20192BAB211006). The authors thank two anonymous referees for their careful reading of the manuscript and helpful comments that led to an improved version.

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Authors and Affiliations

  1. Department of Financial Mathematics, Peking University, Beijing, 100871, China

    Zhengyong Zhou & Jingping Yang

  2. School of Statistics, Jiangxi University of Finance and Economics, Nanchang, 330013, China

    Jiehua Xie

  3. Key Laboratory of Mathematical Economics and Quantitative Finance, Ministry of Education, Beijing, 100871, China

    Jingping Yang

Authors
  1. Zhengyong Zhou
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  2. Jiehua Xie
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  3. Jingping Yang
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Correspondence to Jiehua Xie.

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Zhou, Z., Xie, J. & Yang, J. A copula-based approximation to Markov chains. Sci. China Math. 65, 623–654 (2022). https://doi.org/10.1007/s11425-019-1687-2

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  • DOI: https://doi.org/10.1007/s11425-019-1687-2

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