HAWKES PROCESSES ON LARGE NETWORKS

• 成果类型: Article
• 署名作者: Delattre, Sylvain; Fournier, Nicolas; Hoffmann, Marc
• 署名单位: Universite Paris Cite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Sorbonne Universite; Universite Paris Cite; Universite PSL; Universite Paris-Dauphine
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/14-AAP1089
• 发表日期: 2016
• 页码: 216-261
• 关键词: point-processes; Poisson processes; large deviations; MODEL; equilibrium; simulation; STABILITY; DYNAMICS

摘要

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 -> infinity. 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 -> infinity 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 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.