Fluctuation limits for mean-field interacting nonlinear Hawkes processes

TL;DR

This paper studies the large-population limit of nonlinear Hawkes processes modeling neuron networks, proving a central limit theorem that characterizes fluctuations via a stochastic Volterra integral equation.

Contribution

It introduces a novel approach using the resolvent to represent fluctuations as Skorokhod mappings of martingales, enabling explicit error bounds.

Findings

Proves a functional central limit theorem for mean spike activity.

Characterizes fluctuations with a stochastic Volterra integral equation.

Provides a method to estimate approximation errors.

Abstract

We investigate the asymptotic behaviour of networks of interacting non-linear Hawkes processes modeling a homogeneous population of neurons in the large population limit. In particular, we prove a functional central limit theorem for the mean spike-activity thereby characterizing the asymptotic fluctuations in terms of a stochastic Volterra integral equation. Our approach differs from previous approaches in making use of the associated resolvent in order to represent the fluctutations as Skorokhod continuous mappings of weakly converging martingales. Since the Lipschitz properties of the resolvent are explicit, our analysis in principle also allows to derive approximation errors in terms of driving martingales.