Fake stationary rough Heston volatility: Microstructure-inspired foundations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918332995731456 |
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| author | Gnabeyeu, Emmanuel Pagès, Gilles Rosenbaum, Mathieu |
| author_facet | Gnabeyeu, Emmanuel Pagès, Gilles Rosenbaum, Mathieu |
| contents | 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 $p>0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_11032 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fake stationary rough Heston volatility: Microstructure-inspired foundations Gnabeyeu, Emmanuel Pagès, Gilles Rosenbaum, Mathieu Probability 33E12, 45D05, 60G17, 60G22, 60G55, 91G80 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 $p>0$. |
| title | Fake stationary rough Heston volatility: Microstructure-inspired foundations |
| topic | Probability 33E12, 45D05, 60G17, 60G22, 60G55, 91G80 |
| url | https://arxiv.org/abs/2602.11032 |