Open Access
February 2016 Hawkes processes on large networks
Sylvain Delattre, Nicolas Fournier, Marc Hoffmann
Ann. Appl. Probab. 26(1): 216-261 (February 2016). DOI: 10.1214/14-AAP1089


We generalise the construction of multivariate Hawkes processes to a possibly infinite network of counting processes on a directed graph $\mathbb{G}$. 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 $N\rightarrow\infty$. 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 $N,T\rightarrow\infty$ and exhibit different possible behaviours.

We finally consider the case $\mathbb{G}=\mathbb{Z}^{d}$ 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 $\mathbb{Z}^{d}$ 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.


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Sylvain Delattre. Nicolas Fournier. Marc Hoffmann. "Hawkes processes on large networks." Ann. Appl. Probab. 26 (1) 216 - 261, February 2016.


Received: 1 March 2014; Revised: 1 November 2014; Published: February 2016
First available in Project Euclid: 5 January 2016

zbMATH: 1334.60082
MathSciNet: MR3449317
Digital Object Identifier: 10.1214/14-AAP1089

Primary: 60F05 , 60G55 , 60G57

Keywords: interacting particle systems , limit theorems , Mean-field approximations , multivariate Hawkes processes , Point processes , Stochastic differential equations

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.26 • No. 1 • February 2016
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