Large Deviations for Markovian Nonlinear Hawkes Processes

In the 2007 paper, Bordenave and Torrisi [1] proves the large deviation principles for Poisson cluster processes and in particular, the linear Hawkes processes. In this paper, we prove first a large deviation principle for a special class of nonlinear Hawkes process, i.e. 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 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.