Scaling limits for supercritical nearly unstable Hawkes processes

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.