Copula-based Markov Process

Starting from a bivariate copula family, we investigate the existence of a Markov process whose temporal dependence is modeled by the given copula family. Due to that the transition function plays a core role for constructing a Markov process, a transition function should be defined from a copula family. For this purpose, the modified partial Dini derivatives of a bivariate copula are defined and applied for defining transition probabilities, and some properties of the modified partial Dini derivatives are proved. A necessary and sufficient condition for the family of the defined transition probabilities to be a transition function is provided. Given a bivariate copula family, a sufficient condition for the existence of a Markov process is provided, where the Markov process has a transition function generated by the modified partial Dini derivatives of the bivariate copula family and the temporal dependence of the Markov process is modeled by the given copula family. The resulting Markov process is named as the copula-based Markov process. Moreover, under some assumptions the consistency of the bivariate copula family under the * product operation is necessary and sufficient for the existence of a Markov process. In terms of copulas, some criteria are provided for a copula-based Markov process to be path right-continuous with left limits or path continuous, and a necessary and sufficient condition for a time-homogeneous copula-based Markov process to be a Feller process is obtained. It is interesting that a Markov process with the transition function generated by the modified partial Dini derivatives of FGM copulas is not a Feller process. Finally, paths of some typical copula-based Markov processes are simulated to show the importance of fitting the copula method into the framework of stochastic processes. It is a joint work with Jun Fang, Fan Jiang and Yong Liu.