Summary Statistics for Endpoint-Conditioned Continuous-Time Markov Chains

Abstract

Continuous-time Markov chains are a widely used modelling tool. Applications include DNA sequence evolution, ion channel gating behaviour, and mathematical finance. We consider the problem of calculating properties of summary statistics (e.g. mean time spent in a state, mean number of jumps between two states, and the distribution of the total number of jumps) for discretely observed continuous-time Markov chains. Three alternative methods for calculating properties of summary statistics are described and the pros and cons of the methods are discussed. The methods are based on (i) an eigenvalue decomposition of the rate matrix, (ii) the uniformization method, and (iii) integrals of matrix exponentials. In particular, we develop a framework that allows for analyses of rather general summary statistics using the uniformization method.