Scaling limit for supercritical nearly unstable Hawkes processes with heavy tail

Abstract

In this paper, we establish the asymptotic behavior of {\it supercritical} nearly unstable Hawkes processes with a power law kernel. We find that, the Hawkes process in our context admits a similar equation to that in \cite{MR3563196} for {\it subcritical} case. In particular, the rescaled Hawkes process $(Z_{nt}^n/n^{2\alpha }{)}_{t\in [0,1]}$ converges in law to a kind of integrated fractional Cox Ingersoll Ross process with different coefficients from that in \cite{MR3563196}, as $n$ tends to infinity.

13 pages