Nonlinear Hawkes Processes
Abstract
The Hawkes process is a simple point process that has long memory, clustering effect, selfexciting property and is in general nonMarkovian. The future evolution of a selfexciting point process is influenced by the timing of the past events. There are applications in finance, neuroscience, genome analysis, seismology, sociology, criminology and many other fields. We first survey the known results about the theory and applications of both linear and nonlinear Hawkes processes. Then, we obtain the central limit theorem and processlevel, i.e. level3 large deviations for nonlinear Hawkes processes. The level1 large deviation principle holds as a result of the contraction principle. We also provide an alternative variational formula for the rate function of the level1 large deviations in the Markovian case. Next, we drop the usual assumptions on the nonlinear Hawkes process and categorize it into different regimes, i.e. sublinear, subcritical, critical, supercritical and explosive regimes. We show the different time asymptotics in different regimes and obtain other properties as well. Finally, we study the limit theorems of linear Hawkes processes with random marks.
 Publication:

arXiv eprints
 Pub Date:
 April 2013
 arXiv:
 arXiv:1304.7531
 Bibcode:
 2013arXiv1304.7531Z
 Keywords:

 Mathematics  Probability
 EPrint:
 218 pages, 9 figures