Fluctuations for mean-field interacting age-dependent Hawkes processes

TL;DR

This paper establishes a functional central limit theorem for mean-field age-dependent Hawkes processes, describing the fluctuations of empirical measures around their deterministic limits as the number of processes grows large.

Contribution

It extends previous law of large numbers results by proving a central limit theorem involving Gaussian-driven stochastic differential equations for fluctuations.

Findings

Fluctuation process converges to a Gaussian-driven SDE

Provides a detailed description of the fluctuations at scale n^{-1/2}

Enhances understanding of stochastic behavior in age-dependent Hawkes processes

Abstract

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 +$\infty$) 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).