IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v192y2026ics0304414925002583.html

The multivariate fractional Ornstein–Uhlenbeck process

Author

Listed:
  • Dugo, Ranieri
  • Giorgio, Giacomo
  • Pigato, Paolo

Abstract

Starting from the notion of multivariate fractional Brownian Motion introduced in [F. Lavancier, A. Philippe, and D. Surgailis. Covariance function of vector self-similar processes. Statistics & Probability Letters, 2009] we define a multivariate version of the fractional Ornstein–Uhlenbeck process. This multivariate Gaussian process is stationary, ergodic and allows for different Hurst exponents on each component. We characterize its correlation matrix and its short and long time asymptotics. Besides the marginal parameters, the cross correlation between one-dimensional marginal components is ruled by two parameters. We consider the problem of their inference, proposing two types of estimator, constructed from discrete observations of the process. We establish their asymptotic theory, in one case in the long time asymptotic setting, in the other case in the infill and long time asymptotic setting. The limit behavior can be asymptotically Gaussian or non-Gaussian, depending on the values of the Hurst exponents of the marginal components. The technical core of the paper relies on the analysis of asymptotic properties of functionals of Gaussian processes, that we establish using Malliavin calculus and Stein’s method. We provide numerical experiments that support our theoretical analysis and also suggest a conjecture on the application of one of these estimators to the multivariate fractional Brownian Motion.

Suggested Citation

  • Dugo, Ranieri & Giorgio, Giacomo & Pigato, Paolo, 2026. "The multivariate fractional Ornstein–Uhlenbeck process," Stochastic Processes and their Applications, Elsevier, vol. 192(C).
    • Handle: RePEc:eee:spapps:v:192:y:2026:i:c:s0304414925002583
    DOI: 10.1016/j.spa.2025.104814
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414925002583
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2025.104814?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Hansen, Lars Peter, 1982. "Large Sample Properties of Generalized Method of Moments Estimators," Econometrica, Econometric Society, vol. 50(4), pages 1029-1054, July.
    2. Gilles Zumbach, 2009. "Time reversal invariance in finance," Quantitative Finance, Taylor & Francis Journals, vol. 9(5), pages 505-515.
    3. Wang, Xiaohu & Xiao, Weilin & Yu, Jun, 2023. "Modeling and forecasting realized volatility with the fractional Ornstein–Uhlenbeck process," Journal of Econometrics, Elsevier, vol. 232(2), pages 389-415.
    4. Markus Bibinger & Jun Yu & Chen Zhang, 2025. "Modeling and Forecasting Realized Volatility with Multivariate Fractional Brownian Motion," Papers 2504.15985, arXiv.org.
    5. Yong Chen & Ying LI, 2021. "Berry-Esséen bound for the parameter estimation of fractional Ornstein-Uhlenbeck processes with the hurst parameter H∈(0,12)," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 50(13), pages 2996-3013, July.
    6. Josselin Garnier & Knut Sølna, 2018. "Option pricing under fast-varying and rough stochastic volatility," Annals of Finance, Springer, vol. 14(4), pages 489-516, November.
    7. Chiara Amorino & Arnaud Gloter, 2020. "Contrast function estimation for the drift parameter of ergodic jump diffusion process," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 47(2), pages 279-346, June.
    8. Marcus Cordi & Damien Challet & Serge Kassibrakis, 2021. "The market nanostructure origin of asset price time reversal asymmetry," Quantitative Finance, Taylor & Francis Journals, vol. 21(2), pages 295-304, February.
    9. Yaozhong Hu & David Nualart & Hongjuan Zhou, 2019. "Parameter estimation for fractional Ornstein–Uhlenbeck processes of general Hurst parameter," Statistical Inference for Stochastic Processes, Springer, vol. 22(1), pages 111-142, April.
    10. Arcones, Miguel A., 2000. "Distributional limit theorems over a stationary Gaussian sequence of random vectors," Stochastic Processes and their Applications, Elsevier, vol. 88(1), pages 135-159, July.
    11. Josselin Garnier & Knut Solna, 2015. "Correction to Black-Scholes formula due to fractional stochastic volatility," Papers 1509.01175, arXiv.org, revised Mar 2017.
    12. M.L. Kleptsyna & A. Le Breton, 2002. "Statistical Analysis of the Fractional Ornstein–Uhlenbeck Type Process," Statistical Inference for Stochastic Processes, Springer, vol. 5(3), pages 229-248, October.
    13. Yaroslav Eumenius-Schulz, 2020. "Spot estimation for fractional Ornstein–Uhlenbeck stochastic volatility model: consistency and central limit theorem," Statistical Inference for Stochastic Processes, Springer, vol. 23(2), pages 355-380, July.
    14. Giulia Livieri & Saad Mouti & Andrea Pallavicini & Mathieu Rosenbaum, 2018. "Rough volatility: Evidence from option prices," IISE Transactions, Taylor & Francis Journals, vol. 50(9), pages 767-776, September.
    15. Lavancier, Frédéric & Philippe, Anne & Surgailis, Donatas, 2009. "Covariance function of vector self-similar processes," Statistics & Probability Letters, Elsevier, vol. 79(23), pages 2415-2421, December.
    16. Pipiras,Vladas & Taqqu,Murad S., 2017. "Long-Range Dependence and Self-Similarity," Cambridge Books, Cambridge University Press, number 9781107039469, November.
    17. Carsten Chong & Marc Hoffmann & Yanghui Liu & Mathieu Rosenbaum & Gr'egoire Szymanski, 2022. "Statistical inference for rough volatility: Central limit theorems," Papers 2210.01216, arXiv.org, revised Jun 2024.
    18. Fabienne Comte & Eric Renault, 1998. "Long memory in continuous‐time stochastic volatility models," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 291-323, October.
    19. Giacomo Giorgio & Barbara Pacchiarotti & Paolo Pigato, 2023. "Short-Time Asymptotics for Non-Self-Similar Stochastic Volatility Models," Applied Mathematical Finance, Taylor & Francis Journals, vol. 30(3), pages 123-152, May.
    20. Bolko, Anine E. & Christensen, Kim & Pakkanen, Mikko S. & Veliyev, Bezirgen, 2023. "A GMM approach to estimate the roughness of stochastic volatility," Journal of Econometrics, Elsevier, vol. 235(2), pages 745-778.
    21. B. Podobnik & D. F. Fu & H. E. Stanley & P. Ch. Ivanov, 2007. "Power-law autocorrelated stochastic processes with long-range cross-correlations," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 56(1), pages 47-52, March.
    22. Jim Gatheral & Thibault Jaisson & Mathieu Rosenbaum, 2018. "Volatility is rough," Quantitative Finance, Taylor & Francis Journals, vol. 18(6), pages 933-949, June.
    23. 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.
    24. Hu, Yaozhong & Nualart, David, 2010. "Parameter estimation for fractional Ornstein-Uhlenbeck processes," Statistics & Probability Letters, Elsevier, vol. 80(11-12), pages 1030-1038, June.
    25. C. Bayer & P. K. Friz & A. Gulisashvili & B. Horvath & B. Stemper, 2019. "Short-time near-the-money skew in rough fractional volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 19(5), pages 779-798, May.
    26. El Mehdi Haress & Yaozhong Hu, 2021. "Estimation of all parameters in the fractional Ornstein–Uhlenbeck model under discrete observations," Statistical Inference for Stochastic Processes, Springer, vol. 24(2), pages 327-351, July.
    27. Carsten Chong & Marc Hoffmann & Yanghui Liu & Mathieu Rosenbaum & Gr'egoire Szymanski, 2022. "Statistical inference for rough volatility: Minimax Theory," Papers 2210.01214, arXiv.org, revised Feb 2024.
    28. Christian Bayer & Peter Friz & Jim Gatheral, 2016. "Pricing under rough volatility," Quantitative Finance, Taylor & Francis Journals, vol. 16(6), pages 887-904, June.
    29. Duan Wang & Boris Podobnik & Davor Horvati'c & H. Eugene Stanley, 2011. "Quantifying and Modeling Long-Range Cross-Correlations in Multiple Time Series with Applications to World Stock Indices," Papers 1102.2240, arXiv.org.
    30. Josselin Garnier & Knut Solna, 2017. "Option Pricing under Fast-varying and Rough Stochastic Volatility," Papers 1707.00610, arXiv.org, revised Apr 2018.
    31. Josselin Garnier & Knut Solna, 2018. "Optimal hedging under fast-varying stochastic volatility," Papers 1810.08337, arXiv.org, revised Mar 2020.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Othmane Zarhali & Emmanuel Bacry & Jean-Franc{c}ois Muzy, 2026. "From rough to multifractal multidimensional volatility: A multidimensional Log S-fBM model," Papers 2601.10517, arXiv.org.

    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. Ranieri Dugo & Giacomo Giorgio & Paolo Pigato, 2024. "The Multivariate Fractional Ornstein-Uhlenbeck Process," CEIS Research Paper 581, Tor Vergata University, CEIS, revised 28 Aug 2024.
    2. Ranieri Dugo & Giacomo Giorgio & Paolo Pigato, 2024. "Multivariate Rough Volatility," Papers 2412.14353, arXiv.org, revised Feb 2026.
    3. Li, Yicun & Teng, Yuanyang, 2023. "Statistical inference in discretely observed fractional Ornstein–Uhlenbeck processes," Chaos, Solitons & Fractals, Elsevier, vol. 177(C).
    4. Markus Bibinger & Jun Yu & Chen Zhang, 2025. "Modeling and Forecasting Realized Volatility with Multivariate Fractional Brownian Motion," Working Papers 202528, University of Macau, Faculty of Business Administration.
    5. Xiaohu Wang & Weilin Xiao & Jun Yu & Chen Zhang, 2025. "Maximum Likelihood Estimation of Fractional Ornstein-Uhlenbeck Process with Discretely Sampled Data," Working Papers 202527, University of Macau, Faculty of Business Administration.
    6. Shi, Shuping & Yu, Jun & Zhang, Chen, 2024. "On the spectral density of fractional Ornstein–Uhlenbeck processes," Journal of Econometrics, Elsevier, vol. 245(1).
    7. Wang, Xiaohu & Xiao, Weilin & Yu, Jun, 2023. "Modeling and forecasting realized volatility with the fractional Ornstein–Uhlenbeck process," Journal of Econometrics, Elsevier, vol. 232(2), pages 389-415.
    8. Bolko, Anine E. & Christensen, Kim & Pakkanen, Mikko S. & Veliyev, Bezirgen, 2023. "A GMM approach to estimate the roughness of stochastic volatility," Journal of Econometrics, Elsevier, vol. 235(2), pages 745-778.
    9. Takaishi, Tetsuya, 2025. "Multifractality and sample size influence on Bitcoin volatility patterns," Finance Research Letters, Elsevier, vol. 74(C).
    10. Angelini, Daniele & Bianchi, Sergio, 2023. "Nonlinear biases in the roughness of a Fractional Stochastic Regularity Model," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    11. Carsten Chong & Marc Hoffmann & Yanghui Liu & Mathieu Rosenbaum & Gr'egoire Szymanski, 2022. "Statistical inference for rough volatility: Central limit theorems," Papers 2210.01216, arXiv.org, revised Jun 2024.
    12. 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.
    13. Benjamin James Duthie, 2019. "Portfolio optimisation under rough Heston models," Papers 1909.02972, arXiv.org.
    14. Josselin Garnier & Knut Sølna, 2018. "Option pricing under fast-varying and rough stochastic volatility," Annals of Finance, Springer, vol. 14(4), pages 489-516, November.
    15. Alfeus, Mesias & Nikitopoulos, Christina Sklibosios, 2022. "Forecasting volatility in commodity markets with long-memory models," Journal of Commodity Markets, Elsevier, vol. 28(C).
    16. Huy N. Chau & Duy Nguyen & Thai Nguyen, 2024. "On short-time behavior of implied volatility in a market model with indexes," Papers 2402.16509, arXiv.org, revised Mar 2025.
    17. Alessandro Bondi & Sergio Pulido & Simone Scotti, 2022. "The rough Hawkes Heston stochastic volatility model," Papers 2210.12393, arXiv.org.
    18. Aditi Dandapani & Paul Jusselin & Mathieu Rosenbaum, 2019. "From quadratic Hawkes processes to super-Heston rough volatility models with Zumbach effect," Papers 1907.06151, arXiv.org, revised Jan 2021.
    19. Xiao, Weilin & Yu, Jun, 2019. "Asymptotic theory for rough fractional Vasicek models," Economics Letters, Elsevier, vol. 177(C), pages 26-29.
    20. Saad Mouti, 2023. "Rough volatility: evidence from range volatility estimators," Papers 2312.01426, arXiv.org, revised Sep 2024.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;
    ;
    ;

    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:eee:spapps:v:192:y:2026:i:c:s0304414925002583. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    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.