Fluctuations for mean-field interacting age-dependent Hawkes processes

Julien Chevallier (1)

1 AGM - UMR 8088 - Analyse, Géométrie et Modélisation

The propagation of chaos and associated law of large numbers for mean-field interacting age-dependent Hawkes processes (when the number of processes n goes to +∞) being granted by the study performed in (Chevallier, 2015), the aim of the present paper is to prove the resulting functional central limit theorem. It involves the study of a measure-valued process describing the fluctuations (at scale n −1/2) of the empirical measure of the ages around its limit value. This fluctuation process is proved to converge towards a limit process characterized by a limit system of stochastic differential equations driven by a Gaussian noise instead of Poisson (which occurs for the law of large numbers limit).

Mots clés: Hawkes process; central limit theorem; interacting particle systems; stochastic partial differential equation; neural network

Domaines: Probabilités [math.PR]

Julien Chevallier. Fluctuations for mean-field interacting age-dependent Hawkes processes. Electronic Journal of Probability, 2017, 22 (42), ⟨10.1214/17-EJP63⟩. ⟨hal-01393373v2⟩