Probability distribution of the waiting time in the stress release model: the Gompertz distribution

Abstract

This communication aims to show some results on the distribution of the waiting time of a self-correcting point process that is known in seismology as the stress release model, and in actuarial sciences as the classical risk model. By assuming this model and conditional on the history of the process up to time $$t$$ , the waiting time of the next event from $$t$$ follows a Gompertz distribution, in which the shape parameter is history dependent. This reveals some new features of the model: (1) Just after a major event, a high probability can be attributed to very short waiting times, and not exclusively to long ones; (2) By updating the history while waiting for the next event, the expected waiting time is postponed in time and its variance decreases. We illustrate the application of a stress release model to a sequence of Italian earthquakes. A Bayesian approach is proposed to consider the uncertainty of the model parameters.