IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2610.05675.html

A Superprocess-based Approach to Rough CIR Processes and Feller Random Measures

Author

Listed:
  • Gordan v{Z}itkovi'c

Abstract

Feller random measures generalize the Feller diffusion (the CIR process) by giving it memory. They arise as the scaling limits of nearly unstable Hawkes processes, and include the rough CIR process and its hyper-rough and discontinuous relatives. We show that every Feller random measure is the occupation measure of a Dawson--Watanabe superprocess whose spatial motion is a killed L\'evy subordinator, integrated over the branching time. This is a continuum analogue of the Hawkes--Oakes cluster representation. It yields existence, stability in the parameters, and stochastic equations driven by an explicit noise. The structure of this noise depends on whether the kernel has an atom at the origin. Without an atom, the noise is a Brownian motion time-changed by the distribution function of the measure. This leads to a martingale characterization and to sharp results on densities and their H\"older regularity. With an atom, the noise is a compensated inverse-Gaussian process time-changed by the compensator of that distribution function, and the measure is purely atomic.

Suggested Citation

  • Gordan v{Z}itkovi'c, 2026. "A Superprocess-based Approach to Rough CIR Processes and Feller Random Measures," Papers 2610.05675, arXiv.org.
    • Handle: RePEc:arx:papers:2610.05675
    as

    Download full text from publisher

    File URL: https://arxiv.org/pdf/2610.05675
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Eduardo Abi Jaber, 2021. "Weak existence and uniqueness for affine stochastic Volterra equations with L1-kernels," Post-Print hal-02412741, HAL.
    2. Forde, Martin & Fukasawa, Masaaki & Gerhold, Stefan & Smith, Benjamin, 2022. "The Riemann–Liouville field and its GMC as H→0, and skew flattening for the rough Bergomi model," Statistics & Probability Letters, Elsevier, vol. 181(C).
    3. Dawson, Donald A. & Fleischmann, Klaus, 1994. "A super-Brownian motion with a single point catalyst," Stochastic Processes and their Applications, Elsevier, vol. 49(1), pages 3-40, January.
    4. Omar El Euch & Mathieu Rosenbaum, 2019. "The characteristic function of rough Heston models," Mathematical Finance, Wiley Blackwell, vol. 29(1), pages 3-38, January.
    5. Christian Bayer & Fabian Andsem Harang & Paolo Pigato, 2020. "Log-modulated rough stochastic volatility models," Papers 2008.03204, arXiv.org, revised May 2021.
    6. Eduardo Abi Jaber & Elie Attal, 2025. "Simulating integrated Volterra square-root processes and Volterra Heston models via Inverse Gaussian," Papers 2504.19885, arXiv.org.
    7. Paul Jusselin & Mathieu Rosenbaum, 2020. "No‐arbitrage implies power‐law market impact and rough volatility," Mathematical Finance, Wiley Blackwell, vol. 30(4), pages 1309-1336, October.
    8. Yuri Luchko, 2021. "General Fractional Integrals and Derivatives with the Sonine Kernels," Mathematics, MDPI, vol. 9(6), pages 1-17, March.
    9. Eduardo Abi Jaber, 2021. "Weak existence and uniqueness for affine stochastic Volterra equations with L1-kernels," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-02412741, HAL.
    10. Serge Darolles & Christian Gourieroux & Joann Jasiak, 2006. "Structural Laplace Transform and Compound Autoregressive Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(4), pages 477-503, July.
    11. Masaaki Fukasawa & Tetsuya Takabatake & Rebecca Westphal, 2022. "Consistent estimation for fractional stochastic volatility model under high‐frequency asymptotics," Mathematical Finance, Wiley Blackwell, vol. 32(4), pages 1086-1132, October.
    12. Thibault Jaisson & Mathieu Rosenbaum, 2013. "Limit theorems for nearly unstable Hawkes processes," Papers 1310.2033, arXiv.org, revised Mar 2015.
    13. Aurélien Alfonsi, 2015. "Affine Diffusions and Related Processes: Simulation, Theory and Applications," Post-Print hal-03127212, HAL.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Paul P. Hager & Dorte Kreher, 2026. "Expanding the rough Heston model in $H$," Papers 2606.16619, arXiv.org.
    2. Eduardo Abi Jaber & Nathan De Carvalho, 2023. "Reconciling rough volatility with jumps," Papers 2303.07222, arXiv.org, revised Sep 2024.
    3. Yingli Wang & Yinhao Wu & Lingjiong Zhu, 2026. "Microstructural Foundation for the Rough Hawkes--Heston Model," Papers 2608.07709, arXiv.org.
    4. Yingli Wang & Wei Xu & Lingjiong Zhu, 2026. "A Hawkes Microfoundation for Multitype Inverse Gaussian Subordinators," Papers 2610.10525, arXiv.org.
    5. Gr'egoire Szymanski & Wei Xu, 2025. "Mean-Field Limits for Nearly Unstable Hawkes Processes," Papers 2501.11648, arXiv.org.
    6. Eduardo Abi Jaber, 2022. "The characteristic function of Gaussian stochastic volatility models: an analytic expression," Finance and Stochastics, Springer, vol. 26(4), pages 733-769, October.
    7. David J. Promel & David Scheffels, 2022. "Pathwise uniqueness for singular stochastic Volterra equations with H\"older coefficients," Papers 2212.08029, arXiv.org, revised Jul 2024.
    8. Alfonsi, Aurélien, 2025. "Nonnegativity preserving convolution kernels. Application to Stochastic Volterra Equations in closed convex domains and their approximation," Stochastic Processes and their Applications, Elsevier, vol. 181(C).
    9. Thomas Deschatre & Marc Hoffmann & Mathieu Rosenbaum, 2026. "Rainfall is rough," Papers 2607.27099, arXiv.org.
    10. Alessandro Bondi & Sergio Pulido & Simone Scotti, 2022. "The rough Hawkes Heston stochastic volatility model," Papers 2210.12393, arXiv.org.
    11. Jie, Lijuan & Luo, Liangqing & Zhang, Hua, 2024. "One-dimensional McKean–Vlasov stochastic Volterra equations with Hölder diffusion coefficients," Statistics & Probability Letters, Elsevier, vol. 205(C).
    12. repec:hal:wpaper:hal-03827332 is not listed on IDEAS
    13. Eduardo Abi Jaber & Camille Illand & Shaun (Xiaoyuan) Li, 2025. "Joint SPX & VIX calibration with Gaussian polynomial volatility models: Deep pricing with quantization hints," Mathematical Finance, Wiley Blackwell, vol. 35(2), pages 470-519, April.
    14. Prömel, David J. & Scheffels, David, 2023. "Stochastic Volterra equations with Hölder diffusion coefficients," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 291-315.
    15. Bondi, Alessandro & Livieri, Giulia & Pulido, Sergio, 2024. "Affine Volterra processes with jumps," Stochastic Processes and their Applications, Elsevier, vol. 168(C).
    16. Eduardo Abi Jaber, 2022. "The characteristic function of Gaussian stochastic volatility models: an analytic expression," Post-Print hal-02946146, HAL.
    17. Johannes Muhle-Karbe & Youssef Ouazzani Chahdi & Mathieu Rosenbaum & Gr'egoire Szymanski, 2026. "A unified theory of order flow, market impact, and volatility," Papers 2601.23172, arXiv.org, revised Feb 2026.
    18. Aur'elien Alfonsi & Guillaume Szulda, 2024. "On non-negative solutions of stochastic Volterra equations with jumps and non-Lipschitz coefficients," Papers 2402.19203, arXiv.org, revised Jul 2024.
    19. Eduardo Abi Jaber & Nathan de Carvalho, 2024. "Reconciling rough volatility with jumps," Post-Print hal-04295416, HAL.
    20. Florian Aichinger & Sascha Desmettre, 2025. "Pricing of geometric Asian options in the Volterra-Heston model," Review of Derivatives Research, Springer, vol. 28(1), pages 1-30, April.
    21. Saad Mouti, 2026. "Rough Volatility Across Assets," Papers 2608.16749, arXiv.org.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2610.05675. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: https://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.