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Fake stationary rough Heston volatility: Microstructure-inspired foundations

Emmanuel Gnabeyeu, Gilles Pagès, Mathieu Rosenbaum

TL;DR

This work analyzes time-modulated, heavy-tailed Hawkes processes in a near-instability regime and proves that, under a joint time-space scaling, the intensity converges to a time-inhomogeneous rough fractional square-root process $\Lambda^*$ while the Hawkes count converges to its integrated version. In the stationary marginal-mean case, the limit reduces to a time-inhomogeneous rough fractional CIR dynamics with constant mean reversion and a time-dependent diffusion, providing a probabilistic microstructure foundation for the fake stationary rough Heston model. The authors establish $\mathcal{C}$-tightness and moment bounds for the limiting Volterra diffusion, derive Hölder regularity, and prove uniqueness in law for the limit, connecting fractional stochastic calculus with Hawkes scaling limits. The results yield a rigorous pathwise and distributional description of rough volatility in a microstructure-inspired setting, with potential implications for volatility fitting and derivative pricing across horizons. The approach unifies near-unstable Hawkes limits with stabilized Volterra dynamics, offering a robust framework for fake stationary rough Heston models and their statistical properties.

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 $p>0$.

Fake stationary rough Heston volatility: Microstructure-inspired foundations

TL;DR

This work analyzes time-modulated, heavy-tailed Hawkes processes in a near-instability regime and proves that, under a joint time-space scaling, the intensity converges to a time-inhomogeneous rough fractional square-root process while the Hawkes count converges to its integrated version. In the stationary marginal-mean case, the limit reduces to a time-inhomogeneous rough fractional CIR dynamics with constant mean reversion and a time-dependent diffusion, providing a probabilistic microstructure foundation for the fake stationary rough Heston model. The authors establish -tightness and moment bounds for the limiting Volterra diffusion, derive Hölder regularity, and prove uniqueness in law for the limit, connecting fractional stochastic calculus with Hawkes scaling limits. The results yield a rigorous pathwise and distributional description of rough volatility in a microstructure-inspired setting, with potential implications for volatility fitting and derivative pricing across horizons. The approach unifies near-unstable Hawkes limits with stabilized Volterra dynamics, offering a robust framework for fake stationary rough Heston models and their statistical properties.

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 model. We further investigate the properties of such limiting scaled time-inhomogeneous Volterra equations, including moment bounds, path regularity and maximal inequality in the setting for every .
Paper Structure (19 sections, 15 theorems, 149 equations)

This paper contains 19 sections, 15 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Hawkes processes: background and related work
  3. Our contribution
  4. Time-dependent fractional square root processes as limits of heavy-tailed and nearly unstable Hawkes processes
  5. Assumptions and heuristic computations for the convergence of heavy-tailed nearly unstable Hawkes processes
  6. A priori estimates and $\mathcal{C}$-tightness
  7. Dynamics of the limit points and regularity
  8. Characterization of the limiting processes and moment control
  9. Regularity and maximal inequality
  10. Proofs of the main results
  11. Proofs of Propositions \ref{['prop:Lhawkes_tightness']} and \ref{['prop:tightIntensity']}
  12. Proofs of Theorems \ref{['Thm:Limit']} and \ref{['Thm:regul']}
  13. Analytical tools and properties of Hawkes processes with initial states
  14. Preliminaries: solvent core of a Borel function and Wiener-Hopf equations.
  15. A summary about Hawkes processes with initial state condition
  16. ...and 4 more sections

Key Result

Proposition 2.1

Under Assumption assum:Lhawkes, and denoting $\mathcal{L}(f) :=L_{f}$ the Laplace transform for any functional f, we have the following claims where $f_{\alpha, \lambda}$ is the Mittag-Leffler density function defined in Example Ex:frackernel and $R_{\alpha, \lambda}(t)$ denotes the corresponding re

Theorems & Definitions (16)

  • Proposition 2.1: The limits for the integrated rescaled resolvent function
  • Proposition 2.2: Stability result for the density $m^{T}(ds)$
  • Corollary 2.3: Uniform almost sure convergences
  • Proposition 2.4
  • Proposition 2.5: A priori estimates
  • Proposition 2.6: $\mathcal{C}$-tightness of the process $X^T = \left( \tilde{\mathcal{I}}^{\Lambda^{*T}}, \tilde{N}^{*T}, \tilde{M}^{*T}, \tilde{M}^{T} \right)$
  • Proposition 2.7: $\mathcal{C}$-tightness of $(\Lambda^{*T})_{T > 0}$
  • Theorem 3.1: Characterization of the limits of the scaled nearly unstable Hawkes processes
  • Theorem 3.2: Regularity and maximal inequality for inhomogeneous Volterra fractional diffusion
  • Lemma 4.1
  • ...and 6 more