Rough fractional diffusions as scaling limits of nearly unstable heavy tailed Hawkes processes

TL;DR

This paper studies the long-term behavior of nearly unstable heavy-tailed Hawkes processes and shows they converge to a fractional Cox-Ingersoll-Ross process, revealing links to rough volatility in finance.

Contribution

It establishes the convergence of certain heavy-tailed Hawkes processes to a fractional CIR process, highlighting the impact of tail behavior on the limit process.

Findings

Convergence to fractional CIR process for heavy-tailed Hawkes processes

Identification of Hurst parameter as alpha minus one-half

Connection to rough volatility in financial markets

Abstract

We investigate the asymptotic behavior as time goes to infinity of Hawkes processes whose regression kernel has $L^1$ norm close to one and power law tail of the form $x^{-(1+\alpha)}$, with $\alpha\in(0,1)$. We in particular prove that when $\alpha\in(1/2,1)$, after suitable rescaling, their law converges to that of a kind of integrated fractional Cox-Ingersoll-Ross process, with associated Hurst parameter $H=\alpha-1/2$. This result is in contrast to the case of a regression kernel with light tail, where a classical Brownian CIR process is obtained at the limit. Interestingly, it shows that persistence properties in the point process can lead to an irregular behavior of the limiting process. This theoretical result enables us to give an agent-based foundation to some recent findings about the rough nature of volatility in financial markets.