Scaling limits for supercritical nearly unstable Hawkes processes

We investigate the asymptotic behavior of nearly unstable Hawkes processes whose regression kernel has L¹ norm strictly greater than 1 and close to 1 as time goes to infinity. We find that the scaling size determines the scaling behavior of the processes as in Jaisson and Rosenbaum (2015). Specifically, after a suitable rescale of (aT-1)/TebTTx , the limit of the sequence of Hawkes processes is deterministic. Also, with another appropriate rescaling of 1/T² , the sequence converges in law to an integrated Cox–Ingersoll–Ross-like process. This theoretical result may apply to model the recent COVID-19 outbreak in epidemiology and phenomena in social networks.