Mean field limits for interacting Hawkes processes in a diffusive regime

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 the Skorokhod 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.

Automated summary

This paper considers a system of Hawkes processes with mean field interactions in a diffusive regime. Each process has a stochastic intensity that is a solution of a stochastic differential equation driven by N independent Poisson random measures. The paper proves that as the number of interacting components N goes to infinity, the intensity converges in distribution in the Skorokhod space to a CIR-type diffusion. Furthermore, the paper also demonstrates the convergence in distribution of the Hawkes processes to a limit point process with the limit diffusion as intensity. To prove these convergence results, analytical techniques based on the convergence of the associated infinitesimal generators and Markovian semigroups are used.