Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Poisson approximation of point processes with stochastic intensity, and application to nonlinear Hawkes processes

Giovanni Luca Torrisi

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Abstract

We give a general inequality for the total variation distance between a Poisson distributed random variable and a first order stochastic integral with respect to a point process with stochastic intensity, constructed by embedding in a bivariate Poisson process. We apply this general inequality to first order stochastic integrals with respect to a class of nonlinear Hawkes processes, which is of interest in queueing theory, providing explicit bounds for the Poisson approximation, a quantitative Poisson limit theorem, confidence intervals and asymptotic estimates of the moments.

Résumé

Nous donnons une inégalité générale pour la distance en variation totale entre une variable de Poisson aléatoire et une intégrale stochastique par rapport à un processus ponctuel avec une intensité stochastique, construite par plongement dans un processus de Poisson bivarié. Nous appliquons cette inégalité générale aux intégrales stochastiques par rapport à une classe de processus de Hawkes non linéaires, ce qui a un intérêt en théorie des files d’attente, en fournissant des bornes explicites pour l’approximation Possonienne, ainsi qu’un théorème limite Poissonien quantitatif et des intervalles de confiance et estimatées asymptotiques des moments.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 2 (2017), 679-700.

Dates
Received: 23 December 2014
Revised: 2 October 2015
Accepted: 10 November 2015
First available in Project Euclid: 11 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1491897741

Digital Object Identifier
doi:10.1214/15-AIHP730

Mathematical Reviews number (MathSciNet)
MR3634270

Zentralblatt MATH identifier
1367.60020

Subjects
Primary: 60F05: Central limit and other weak theorems 60G55: Point processes

Keywords
Chen–Stein’s method Clark–Ocone formula Confidence interval Erlang loss system Hawkes process Malliavin’s calculus Poisson approximation Stochastic intensity

Citation

Torrisi, Giovanni Luca. Poisson approximation of point processes with stochastic intensity, and application to nonlinear Hawkes processes. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 2, 679--700. doi:10.1214/15-AIHP730. https://projecteuclid.org/euclid.aihp/1491897741


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