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

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.