Mean-field limit of generalized Hawkes processes

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:127

Article

Chevallier, Julien¹ 

[1] Univ Nice Sophia Antipolis, CNRS, LJAD, UMR 7351, F-06100 Nice, France

关键词: Hawkes process;    Mean-field approximation;    Interacting;    particle systems;    Renewal equation;    Neural network;   

DOI  :  10.1016/j.spa.2017.02.012

来源: Elsevier

【 摘 要 】

We generalize multivariate Hawkes processes mainly by including a dependence with respect to the age of the process, i.e. the delay since the last point. Within this class, we investigate the limit behaviour, when n goes to infinity, of a system of n mean field interacting age-dependent Hawkes processes. We prove that such a system can be approximated by independent and identically distributed age dependent point processes interacting with their own mean intensity. This result generalizes the study performed by Delattre et al. (2016). In continuity with Chevallier et al. (2015), the second goal of this paper is to give a proper link between these generalized Hawkes processes as microscopic models of individual neurons and the age-structured system of partial differential equations introduced by Pakdaman et al. (2010) as macroscopic model of neurons. (C) 2017 Elsevier B.V. All rights reserved.

【 授权许可】

Free