Scaling limits for supercritical nearly unstable Hawkes processes

TL;DR

This paper studies the long-term behavior of nearly unstable Hawkes processes, revealing how different rescaling methods lead to deterministic limits or stochastic Cox-Ingersoll-Ross-like processes, with applications in epidemiology and social networks.

Contribution

It provides a detailed analysis of the asymptotic limits of nearly unstable Hawkes processes, including new scaling results and convergence to known stochastic processes.

Findings

Rescaling determines deterministic or stochastic limits.

Processes converge to an integrated Cox-Ingersoll-Ross like process.

Results applicable to modeling COVID-19 and social networks.

Abstract

In this paper, we investigate the asymptotic behavior of nearly unstable Hawkes processes whose regression kernel has $L^1$ norm strictly greater than one and close to one as time goes to infinity. We find that,the scaling size determines the scaling behavior of the processes like in \cite{MR3313750}. Specifically,after suitable rescaling, the limit of the sequence of Hawkes processes is deterministic.And also with another appropriate rescaling, the sequence converges in law to an integrated Cox Ingersoll Ross like process. This theoretical result may apply to model the recent COVID19 in epidemiology and in social network.