Semi‐Markov Processes ‐ Further Results

Abstract

Abstract Semi‐Markov processes are useful generalizations of a class of stochastic processes commonly referred to as Markov jump processes in continuous time. In most of the applications of this class of processes in biostatistics and biology, the state space is chosen as a discrete set with finite or countable elements that have biological meaning. One of the properties of Markov processes is that it follows from the Markov property in continuous time that the sojourn time (holding time) in any state has an exponential distribution with a parameter depending on that state. Semi‐Markov processes are a generalization of Markov jump processes in that the distribution of the sojourn time in any state can, in principle, be arbitrary. Furthermore, in semi‐Markov processes the finite dimensional distributions of the sample paths of the process are modeled directly in terms of arbitrary distributions of sojourn times for each state and conditional distributions of a jump to another future state, given the present state. In many applications, this property makes it possible to set down a formula for the likelihood function of the data with relative ease. Briefly, the paper contains substantive examples and a description of the mathematical tools needed to work with semi‐Markov processes.