Trajectory composition of Poisson time changes and Markov counting systems

Abstract

Changing time of simple continuous-time Markov counting processes by independent unit-rate Poisson processes results in Markov counting processes for which we provide closed-form transition rates via composition of trajectories and with which we construct novel, simpler infinitesimally over-dispersed processes.

Modeling and Inference for Infectious Disease Dynamics: A Likelihood-Based Approach

“…Simultaneous transitions can occur under some conditions (see, e.g., Bretó, 2012a; Kozubowski and Podgórski, 2009; Lee and Whitmore, 1993) when time is changed by a stochastic process or clock (see, e.g., Bochner, 1949; Barndorff-Nielsen and Shiryaev, 2010). In particular, simultaneous transitions arise naturally when individual transitions are accumulated and released at once at discontinuity points of paths of the new time, as happens when time is changed by a Poisson clock (Bretó, 2014a) or by the gamma clock of Section 4.1. Although such simultaneous transitions are necessary for overdispersion of continuous-time Markov chains (Bretó and Ionides, 2011), overdispersion can be described in different terms, e.g., in terms of randomized transition rates, as in the following example.…”