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Hawkes processes on large networks

Delattre, Sylvain; Fournier, Nicolas; Hoffmann, Marc (2016), Hawkes processes on large networks, The Annals of Applied Probability, 26, 1, p. 216-261. 10.1214/14-AAP1089

Type
Article accepté pour publication ou publié
External document link
https://hal.archives-ouvertes.fr/hal-01102806
Date
2016
Journal name
The Annals of Applied Probability
Volume
26
Number
1
Publisher
Institute of Mathematical Statistics
Published in
Paris
Pages
216-261
Publication identifier
10.1214/14-AAP1089
Metadata
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Author(s)
Delattre, Sylvain
Fournier, Nicolas
Hoffmann, Marc
Abstract (EN)
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.
Subjects / Keywords
Stochastic differential equations; Limit theorems; Multivariate Hawkes processes; Point processes; Mean-field approximations.; Interacting particle systems

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