Distributional Properties of the Mixture of Continuous-Time Absorbing Markov Chains Moving at Different Speeds

Abstract

This paper analyses the mixture of two continuous-time Markov chains with absorption on the same state space moving at different speeds, where the mixture occurs at a random time. Variety of associated distributional properties of the Markov mixture process are discussed, for example the transition matrix, the distribution of its lifetime and the corresponding forward intensity. Identities are given explicitly in terms of the Bayesian updates of switching probability and the intensity matrices of the underlying Markov chains despite the fact that the mixture process is not Markovian. They form nonstationary functions of time and duration and have two appealing features: the ability to capture heterogeneity and path dependence when conditioning on the available information (either full or partial) about the past history of the process until current time. In particular, the unconditional lifetime distribution forms a generalized mixture of phase-type distributions. The distribution has dense and closure properties under finite convex mixtures and finite convolutions. When the underlying Markov chains move at the same speed, in which case the mixture process reduces to a simple Markov chain, the heterogeneity and path dependence are removed, and the lifetime has the usual phase-type distribution, Neuts [Neuts MF (1975) Probability distributions of phase-type. Liber Amicorum Prof. Emiritus H. Florin (University of Louvain, Belgium), 173–206]. Some numerical examples are given to illustrate the main results.