Compound Markov counting processes and their applications to modeling infinitesimally over-dispersed systems

TL;DR

This paper introduces an infinitesimal dispersion index for Markov counting processes, characterizes over-dispersion in such processes, and develops compound models for applications where traditional moment constraints do not apply.

Contribution

It defines a new dispersion index for Markov counting processes and characterizes infinitesimal over-dispersion, extending modeling capabilities for complex stochastic systems.

Findings

Infinitesimal over-dispersion occurs in compound processes with jumps of multiple units.

Simple processes are characterized by jumps of one unit, even under over-dispersion.

Constructed models include multivariate over-dispersed compartment and queueing systems.

Abstract

We propose an infinitesimal dispersion index for Markov counting processes. We show that, under standard moment existence conditions, a process is infinitesimally (over-) equi-dispersed if, and only if, it is simple (compound), i.e. it increases in jumps of one (or more) unit(s), even though infinitesimally equi-dispersed processes might be under-, equi- or over-dispersed using previously studied indices. Compound processes arise, for example, when introducing continuous-time white noise to the rates of simple processes resulting in Levy-driven SDEs. We construct multivariate infinitesimally over-dispersed compartment models and queuing networks, suitable for applications where moment constraints inherent to simple processes do not hold.