HAWKES PROCESSES ON LARGE NETWORKS - Université Pierre et Marie Curie Accéder directement au contenu
Article Dans Une Revue The Annals of Applied Probability Année : 2016

HAWKES PROCESSES ON LARGE NETWORKS

Résumé

We generalise the construction of multivariate Hawkes processes to a possibly infinite network of counting processes on a directed graph G. The process is constructed as the solution to a system of Poisson driven stochastic differential equations, for which we prove pathwise existence and uniqueness under some reasonable conditions. We next investigate how to approximate a standard N -dimensional Hawkes process by a simple inhomogeneous Poisson process in the mean-field framework where each pair of individuals interact in the same way, in the limit N → ∞. In the so-called linear case for the interaction, we further investigate the large time behaviour of the process. We study in particular the stability of the central limit theorem when exchanging the limits N, T → ∞ and exhibit different possible behaviours. We finally consider the case G = Z d with nearest neighbour interactions. In the linear case, we prove some (large time) laws of large numbers and exhibit different behaviours, reminiscent of the infinite setting. Finally we study the propagation of a single impulsion started at a given point of Z d at time 0. We compute the probability of extinction of such an impulsion and, in some particular cases, we can accurately describe how it propagates to the whole space. Mathematics Subject Classification (2010): 60F05, 60G55, 60G57.
Fichier principal
Vignette du fichier
DFH.pdf (466.68 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-01102806 , version 1 (13-01-2015)

Identifiants

  • HAL Id : hal-01102806 , version 1

Citer

Sylvain Delattre, Nicolas Fournier, Marc Hoffmann. HAWKES PROCESSES ON LARGE NETWORKS. The Annals of Applied Probability, 2016. ⟨hal-01102806⟩
441 Consultations
989 Téléchargements

Partager

Gmail Facebook X LinkedIn More