Nonlinear Hawkes Processes

TL;DR

This paper surveys nonlinear Hawkes processes, establishes limit theorems and large deviations, and explores different regimes and properties, advancing understanding of their long-term behavior and applications.

Contribution

It provides new limit theorems, large deviation principles, and regime classifications for nonlinear Hawkes processes, including cases with random marks.

Findings

Established central limit theorem and large deviations for nonlinear Hawkes processes.

Categorized nonlinear Hawkes processes into regimes with distinct asymptotic behaviors.

Analyzed limit theorems for linear Hawkes processes with random marks.

Abstract

The Hawkes process is a simple point process that has long memory, clustering effect, self-exciting property and is in general non-Markovian. The future evolution of a self-exciting 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 process-level, i.e. level-3 large deviations for nonlinear Hawkes processes. The level-1 large deviation principle holds as a result of the contraction principle. We also provide an alternative variational formula for the rate function of the level-1 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, sub-critical, critical, super-critical 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.