Hawkes processes as competing hazards models and a simulation algorithm

In this paper, we establish a competing hazards interpretation of Hawkes processes. Linking the excitation functions to proper hazard functions for waiting times offers a useful construction of the Hawkes process, relying on the properties of the corresponding hazard probability distributions. We leverage this link to propose a Gillespie-type simulation algorithm for the Hawkes process. Based on quantitative and qualitative comparison studies, our simulation algorithm overcomes the drawbacks encountered by existing methods, allowing general forms of the excitation function while maintaining good performance. An example of application to earthquakes data illustrates the use of our method.