Modelling volatility using a non-homogeneous martingale model for processes with constant mean on count data

In this article a non-homogeneous martingale model is proposed to model volatility in a
stochastic time series of count data with constant mean. The approach is derived from a general non-
homogeneous birth-and-death process, in which the mean and the variance of population size can vary
as a function of time. This model can be important in modelling early warning signals that there is
going to be a change of state in a complex system. The net reproduction ratio obtained from fitting a
non-homogeneous birth–death model can be used as an additional tool to compare this model with a
model where there is no change in the mean over the observation period. These models and procedures
are illustrated with quarterly Methicillin resistant staphylococcus aureus prevalence data registered
since 2001 from three Acute Trusts of hospitals of the National Health Service in Great Britain.