Fake stationary rough Heston volatility: Microstructure-inspired foundations

TL;DR

This paper derives the asymptotic behavior of heavy-tailed Hawkes processes, showing they converge to a time-inhomogeneous rough CIR process, providing a foundation for the fake stationary rough Heston volatility model.

Contribution

It introduces a new limit theorem for Hawkes processes with heavy tails, linking microstructure models to rough fractional stochastic volatility models.

Findings

Convergence of Hawkes processes to rough fractional CIR processes.

Analysis of moment bounds and path regularity of the limiting equations.

Application to the fake stationary rough Heston volatility model.

Abstract

This paper investigates the asymptotic behavior of suitably time-modulated Hawkes processes with heavy-tailed kernels in a nearly unstable regime. We show that, under appropriate scaling, both the intensity processes and the rescaled Hawkes processes converge to a mean-reverting, time-inhomogeneous rough fractional square-root process and its integrated counterpart, respectively. In particular, when the original Hawkes process has a stationary first moment (constant marginal mean), the limiting process takes the form of a time-inhomogeneous rough fractional Cox-Ingersoll-Ross (CIR) equation with a constant mean-reversion parameter and a time-dependent diffusion coefficient. This class of equations is particularly appealing from a practical perspective, especially for the so-called $\textit{fake stationary rough Heston}$ model. We further investigate the properties of such limiting scaled time-inhomogeneous Volterra equations, including moment bounds, path regularity and maximal inequality in the $L^p$ setting for every $p>0$.