Fake stationary rough Heston volatility: Microstructure-inspired foundations

Emmanuel Gnabeyeu (), Gilles Pagès () and Mathieu Rosenbaum ()
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Emmanuel Gnabeyeu: SU - Sorbonne Université
Gilles Pagès: SU - Sorbonne Université
Mathieu Rosenbaum: Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres

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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, timeinhomogeneous 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 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 $p>0$.

Keywords: Hawkes Processes; Regularity; Mittag-Leffler Functions; Fourier-Laplace Transforms; Fractional Stochastic Differential Equations; Skorokhod Topology; Limit Theorems; Scaling Limits (search for similar items in EconPapers)
Date: 2026-04-09
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