# Mean field limits for interacting Hawkes processes in a diffusive regime

## TL;DR

This paper studies the behavior of large systems of interacting Hawkes processes in a diffusive regime, showing their intensities converge to a CIR-type diffusion and the processes themselves converge to a limit point process.

## Contribution

It establishes the mean field limit of Hawkes processes with stochastic intensities driven by Poisson measures, revealing their convergence to a diffusion process.

## Key findings

- Hawkes process intensities converge to CIR-type diffusions as N grows.
- The Hawkes processes converge in distribution to a limit point process.
- Analytical techniques based on generator convergence are used for proofs.

## Abstract

We consider a sequence of systems of Hawkes processes having mean field interactions in a diffusive regime. The stochastic intensity of each process is a solution of a stochastic differential equation driven by N independent Poisson random measures. We show that, as the number of interacting components N tends to infinity, this intensity converges in distribution in Skorohod space to a CIR-type diffusion. Moreover, we prove the convergence in distribution of the Hawkes processes to the limit point process having the limit diffusion as intensity. To prove the convergence results, we use analytical technics based on the convergence of the associated infinitesimal generators and Markovian semigroups.