A copula-based approximation to Markov chains  

作　　者：Zhengyong Zhou Jiehua Xie Jingping Yang 

机构地区：[1]Department of Financial Mathematics,Peking University,Beijing 100871,China [2]School of Statistics,Jiangxi University of Finance and Economics,Nanchang 330013,China [3]Key Laboratory of Mathematical Economics and Quantitative Finance,Ministry of Education,Beijing 100871,China

出　　处：《Science China Mathematics》2022年第3期623-654,共32页中国科学：数学（英文版）

基　　金：supported by the National Key R&D Program of China(Grant No.2018YFA0703900);National Natural Science Foundation of China(Grant No.11671021);supported by National Natural Science Foundation of China(Grant Nos.11761051and 11561047);the Natural Science Foundation of Jiangxi Province(Grant Nos.20181BAB211003 and 20192BAB211006)。

摘　　要：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.

关 键 词：Markov chain COPULA checkerboard copula-based Markov chain ■-convergence β-mixing property 

分 类 号：O211.62[理学—概率论与数理统计]