  Hawkes process is a class of simple point processes that is self-exciting and
has clustering effect. The intensity of this point process depends on its
entire past history. It has wide applications in finance, insurance,
neuroscience, social networks, criminology, seismology, and many other fields.
In this paper, we study linear Hawkes process with an exponential kernel in the
asymptotic regime where the initial intensity of the Hawkes process is large.
We establish large deviations for Hawkes processes in this regime as well as
the regime when both the initial intensity and the time are large. We
illustrate the strength of our results by discussing the applications to
insurance and queueing systems.
