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Population viewpoint on Hawkes processes

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  10 June 2016

Alexandre Boumezoued
Alexandre Boumezoued*
Affiliation:
Paris 6 University
*
* Current address: Milliman, 14 Rue Pergolèse, 75016 Paris, France. Email address: alexandre.boumezoued@milliman.com
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Abstract

In this paper we focus 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 the age pyramid which evolves according to immigration and births. The virtue of 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 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.

Keywords

Hawkes processbranchingimmigrationage pyramidnonstationarityLaplace transformthinningPoisson point measure

MSC classification

Primary: 60G55: Point processes
Secondary: 60G57: Random measures 60G44: Martingales with continuous parameter 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 2 , June 2016 , pp. 463 - 480
Copyright
Copyright © Applied Probability Trust 2016 

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