Scaling limit for supercritical nearly unstable Hawkes processes with heavy tail
Liping Xu, An Zhang
TL;DR
This work analyzes supercritical nearly unstable Hawkes processes with a heavy-tail kernel, showing that the properly rescaled counting process converges to an integrated rough fractional diffusion with CIR-like characteristics. The authors derive the limiting dynamics via a careful study of the intensity's renewal structure, Mittag-Leffler-type kernels, and a Malthusian reparameterization, culminating in a coupled limit where the limit process X is the integral of a volatility process Y that satisfies a stochastic Volterra equation driven by Brownian noise. The results extend subcritical nearly unstable findings to the supercritical regime, connect to rough volatility phenomena in finance, and establish a rigorous limit with a unique law, providing a precise description of the asymptotic behavior of heavy-tailed Hawkes systems in this regime.
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 $(Z^n_{nt}/n^{2α})_{t\in[0,1]}$ 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.