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Perfect simulation of Hawkes processes

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Jesper Møller  and
Jakob G. Rasmussen
Jesper Møller*
Affiliation:
Aalborg University
Jakob G. Rasmussen*
Affiliation:
Aalborg University
*
∗ Postal address: Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, DK-9220 Aalborg, Denmark.
∗ Postal address: Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, DK-9220 Aalborg, Denmark.
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Abstract

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Our objective is to construct a perfect simulation algorithm for unmarked and marked Hawkes processes. The usual straightforward simulation algorithm suffers from edge effects, whereas our perfect simulation algorithm does not. By viewing Hawkes processes as Poisson cluster processes and using their branching and conditional independence structures, useful approximations of the distribution function for the length of a cluster are derived. This is used to construct upper and lower processes for the perfect simulation algorithm. A tail-lightness condition turns out to be of importance for the applicability of the perfect simulation algorithm. Examples of applications and empirical results are presented.

Keywords

Approximate simulationdominated coupling from the pastedge effectexact simulationmarked Hawkes processmarked point processperfect simulationpoint processPoisson cluster processthinningupper processlower process

MSC classification

Primary: 60G55: Point processes
Secondary: 68U20: Simulation
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 37 , Issue 3 , September 2005 , pp. 629 - 646
Copyright
Copyright © Applied Probability Trust 2005 

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