Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-05-14T18:22:39.874Z Has data issue: false hasContentIssue false
  • English
  • Français

Population viewpoint on Hawkes processes

Part of: Stochastic processes Markov 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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adamopoulos, L. (1975).Some counting and interval properties of the mutually-exciting processes.J. Appl. Prob. 12,78–86.Google Scholar
Aït-Sahalia, Y.,Cacho-Diaz, J. and Laeven, R. J. A. (2010).Modeling financial contagion using mutually exciting jump processes. Working Paper 15850, National Bureau of Economic Research CrossRefGoogle Scholar
Bensusan, H.,Boumezoued, A.,El Karoui, N. and Loisel, S. (2015).Bridging the gap from microsimulation practice to population models: a survey. (Preliminary version as Chapter 2 of A. Boumezoued Doctoral thesis.). Available at https://tel.archives-ouvertes.fr/tel-01307921/document.Google Scholar
Brémaud, P. and Massoulié, L. (1996).Stability of nonlinear Hawkes processes.Ann. Prob. 24,1563–1588.CrossRefGoogle Scholar
Brémaud, P. and Massoulié, L. (2002).Power spectra of general shot noises and Hawkes point processes with a random excitation.Adv. Appl. Prob. 34,205–222.CrossRefGoogle Scholar
Çinlar, E. (2011).Probability and Stochastics (Graduate Texts Math.261).Springer,New York.Google Scholar
Da Fonseca, J. and Zaatour, R. (2014).Hawkes process: fast calibration, application to trade clustering, and diffusive limit.J. Futures Markets 34,548–579.Google Scholar
Daley, D. J. and Vere-Jones, D. (2008).An Introduction to the Theory of Point Processes, Vol. II,General Theory and Structure.2nd edn.Springer,New York.Google Scholar
Dassios, A. and Zhao, H. (2011).A dynamic contagion process.Adv. Appl. Prob. 43,814–846.CrossRefGoogle Scholar
Delattre, S.,Fournier, N. and Hoffmann, M. (2014).High dimensional Hawkes processes. Preprint. Available at http://arxiv.org/abs/1403.5764.Google Scholar
Errais, E.,Giesecke, K. and Goldberg, L. (2010).Affine point processes and portfolio credit risk.SIAM J. Financial Math. 1,642–665.Google Scholar
Fournier, N. and Méléard, S. (2004).A microscopic probabilistic description of a locally regulated population and macroscopic approximations.Ann. Appl. Prob. 14,1880–1919.Google Scholar
Grigelionis, B. (1971).The representation of integer-valued random measures as stochastic integrals over the Poisson measure.Litovsk. Mat. Sb. 11,93–108.Google Scholar
Hardiman, S. J.,Bercot, N. and Bouchaud, J.-P. (2013).Critical reflexivity in financial markets: a Hawkes process analysis. Preprint. Available at http://arxiv.org/abs/1302.1405.Google Scholar
Harris, T. E. (1963).The Theory of Branching Processes.Springer,Berlin.CrossRefGoogle Scholar
Hawkes, A. G. (1971).Spectra of some self-exciting and mutually exciting point processes.Biometrika 58,83–90.Google Scholar
Hawkes, A. G. and Oakes, D. (1974).A cluster process representation of a self-exciting process.J. Appl. Prob. 11,493–503.CrossRefGoogle Scholar
Jovanović, S.,Hertz, J. and Rotter, S. (2014).Cumulants of Hawkes point processes. Preprint. Available at http://arxiv.org/abs/1409.5353.Google Scholar
Kerstan, J. (1964).Teilprozesse Poissonscher prozesse. In Trans. Third Prague Conf. Information Theory, Statist. Decision Functions, Random Processes (Liblice, 1962),Publishing House of the Czech Academy of Sciences,Prague, pp.377–403.Google Scholar
Lewis, P. A. and Shedler, W. G. S. (1979).Simulation of nonhomogeneous Poisson processes by thinning.Naval Res. Logistics Quart. 26,403–413.CrossRefGoogle Scholar
Massoulié, L. (1998).Stability results for a general class of interacting point processes dynamics, and applications.Stoch. Process. Appl. 75,1–30.Google Scholar
Oakes, D. (1975).The Markovian self-exciting process.J. Appl. Prob. 12,69–77.CrossRefGoogle Scholar
Ogata, Y. (1981).On Lewis' simulation method for point processes.IEEE Trans. Inf. Theory 27,23–31.CrossRefGoogle Scholar
Rambaldi, M.,Pennesi, P. and Lillo, F. (2014).Modeling FX market activity around macroeconomic news: a Hawkes process approach. Preprint. Available at http://arxiv.org/abs/1405.6047.Google Scholar
Saichev, A. I. and Sornette, D. (2011).Generating functions and stability study of multivariate self-excited epidemic processes.Europ. Physical J. B 83,271–282.CrossRefGoogle Scholar
Tran, V. C. (2006).Modèles particulaires stochastiques pour des problèmes d'évolution adaptative et pour l'approximation de solutions statistiques. Doctoral Thesis, Université Paris X – Nanterre. Available at http://tel.archives-ouvertes.fr/tel-00125100.Google Scholar
Tran, V. C. (2008).Large population limit and time behaviour of a stochastic particle model describing an age-structured population.ESAIM Prob. Statist. 12,345–386.Google Scholar
Wheatley, S.,Filimonov, V. and Sornette, D. (2014).Estimation of the Hawkes process with renewal immigration using the EM algorithm. Preprint. Available at http://arxiv.org/abs/1407.7118.Google Scholar