Hawkes processes on large networks

Abstract : 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=Zd 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 Zd 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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Article dans une revue
Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2016, 26 (1), pp.216-261. <10.1214/14-AAP1089>
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Contributeur : Philippe Macé <>
Soumis le : jeudi 21 janvier 2016 - 11:33:49
Dernière modification le : jeudi 20 octobre 2016 - 11:36:59

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S. Delattre, N. Fournier, M. Hoffmann. Hawkes processes on large networks. Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2016, 26 (1), pp.216-261. <10.1214/14-AAP1089>. <hal-01259910>

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