Some limit theorems for Hawkes processes and application to financial statistics

Emmanuel Bacry (), Sylvain Delattre, Marc Hoffmann () and Jean-François Muzy ()
Additional contact information
Emmanuel Bacry: CMAP - Centre de Mathématiques Appliquées de l'Ecole polytechnique - Inria - Institut National de Recherche en Informatique et en Automatique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique
Sylvain Delattre: LPMA - Laboratoire de Probabilités et Modèles Aléatoires - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche Scientifique
Marc Hoffmann: CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CNRS - Centre National de la Recherche Scientifique
Jean-François Muzy: SPE - Laboratoire « Sciences pour l’Environnement » (UMR CNRS 6134 SPE) - CNRS - Centre National de la Recherche Scientifique - Università di Corsica Pasquale Paoli [Université de Corse Pascal Paoli]

Post-Print from HAL

Abstract: Abstract In the context of statistics for random processes, we prove a law of large numbers and a functional central limit theorem for multivariate Hawkes processes observed over a time interval [ 0 , T ] when T ? ? . We further exhibit the asymptotic behaviour of the covariation of the increments of the components of a multivariate Hawkes process, when the observations are imposed by a discrete scheme with mesh ? over [ 0 , T ] up to some further time shift ? . The behaviour of this functional depends on the relative size of ? and ? with respect to T and enables to give a full account of the second-order structure. As an application, we develop our results in the context of financial statistics. We introduced in Bacry et al. (2013) [7] a microscopic stochastic model for the variations of a multivariate financial asset, based on Hawkes processes and that is confined to live on a tick grid. We derive and characterise the exact macroscopic diffusion limit of this model and show in particular its ability to reproduce the important empirical stylised fact such as the Epps effect and the lead?lag effect. Moreover, our approach enables to track these effects across scales in rigorous mathematical terms.

Keywords: Point; processes (search for similar items in EconPapers)
Date: 2013
References: Add references at CitEc
Citations: View citations in EconPapers (50)

Published in Stochastic Processes and their Applications, 2013, 123 (7), pp.2475 - 2499. ⟨10.1016/j.spa.2013.04.007⟩

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-01313994

DOI: 10.1016/j.spa.2013.04.007

Access Statistics for this paper

More papers in Post-Print from HAL
Bibliographic data for series maintained by CCSD ().