  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, neuroscience and many
other fields. In this paper, we study the large deviations for nonlinear Hawkes
processes. The large deviations for linear Hawkes processes has been studied by
Bordenave and Torrisi. In this paper, we prove first a large deviation
principle for a special class of nonlinear Hawkes processes, that is, a
Markovian Hawkes process with nonlinear rate and exponential exciting function,
and then generalize it to get the result for sum of exponentials exciting
functions. We then provide an alternative proof for the large deviation
principle for a linear Hawkes process. Finally, we use an approximation
approach to prove the large deviation principle for a special class of
nonlinear Hawkes processes with general exciting functions.
