Fake stationary rough Heston volatility: Microstructure-inspired foundations

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 $p>0$.

💡 Research Summary

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This paper studies the asymptotic behavior of time‑modulated Hawkes processes with heavy‑tailed kernels in a nearly unstable regime, and shows that under a suitable joint scaling of time and intensity the processes converge to a time‑inhomogeneous rough fractional Cox‑Ingersoll‑Ross (CIR) dynamics. The authors start from a general Hawkes model defined on the whole real line, allowing a non‑zero random initial intensity (Z_0(t)) that captures the impact of events occurring before time zero. They assume a baseline intensity (\mu_T(t)) that is constant on the negative half‑line and a family of kernels (\phi_T) whose (L^1) norm approaches one as the observation horizon (T) tends to infinity, i.e. the “nearly unstable” condition.

The main contribution is a functional limit theorem: after rescaling time by (T) and normalising the intensity by a factor (c_T), the intensity process (\Lambda_T^*) converges in law (Skorokhod (J_1) topology) to a stochastic Volterra equation of the form

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