Limit theorems for Markovian Hawkes processes with a large initial intensity

Abstract: Hawkes process is a simple point process 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, social networks, criminology, seismology, and many other fields. In this paper, we study the linear Hawkes process with an exponential exciting function in the asymptotic regime where the initial intensity of the Hawkes process is large. We derive limit theorems under this asymptotic regime as well as the regime when both the initial intensity and the time are large.