Numerical evaluation of the transition probability of the simple birth-and-death process

Abstract

The simple birth-and-death process is a continuous-time Markov process that is commonly employed for describing changes over time of the size of a population. Application areas are for example in queueing theory, demography, epidemiology, or biology. Although the transition probability is available in closed form, its direct evaluation can be numerically unstable and prevent the use of the birth-and-death process in real settings, for example for maximum likelihood estimation. I will show under which conditions the transition probability can be numerically unstable and present an alternative representation in terms of the Gaussian hypergeometric function. I will also show how the hypergeometric function can be evaluated efficiently and accurately using a three-term recursive relation. I will finally give an example on how to use the hypergeometric representation to numerically find the maximum likelihood estimator.