  The Hawkes process is a model for counting the number of arrivals to a system
which exhibits the self-exciting property - that one arrival creates a
heightened chance of further arrivals in the near future. The model, and its
generalizations, have been applied in a plethora of disparate domains, though
two particularly developed applications are in seismology and in finance. As
the original model is elegantly simple, generalizations have been proposed
which: track marks for each arrival, are multivariate, have a spatial
component, are driven by renewal processes, treat time as discrete, and so on.
This paper creates a cohesive review of the traditional Hawkes model and the
modern generalizations, providing details on their construction, simulation
algorithms, and giving key references to the appropriate literature for a
detailed treatment.
