Population viewpoint on Hawkes processes

TL;DR

This paper introduces a population-based perspective on linear Hawkes processes with general immigrants, using an age pyramid concept to analyze their properties and provide new distribution results.

Contribution

It presents a novel population viewpoint and the age pyramid framework for Hawkes processes, extending their analysis beyond exponential fertility functions.

Findings

Derived new distribution properties for linear Hawkes processes

Introduced the age pyramid concept to track all past events

Provided a pathwise construction of the process and population

Abstract

This paper focuses on a class of linear Hawkes processes with general immigrants. These are counting processes with shot noise intensity, including self-excited and externally excited patterns. For such processes, we introduce the concept of age pyramid which evolves according to immigration and births. The virtue if this approach that combines an intensity process definition and a branching representation is that the population age pyramid keeps track of all past events. This is used to compute new distribution properties for a class of linear Hawkes processes with general immigrants which generalize the popular exponential fertility function. The pathwise construction of the Hawkes process and its underlying population is also given.