IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v36y2023i1d10.1007_s10959-023-01235-y.html
   My bibliography  Save this article

Moderate Deviation Principle for Multiscale Systems Driven by Fractional Brownian Motion

Author

Listed:
  • Solesne Bourguin

    (Boston University)

  • Thanh Dang

    (Boston University)

  • Konstantinos Spiliopoulos

    (Boston University)

Abstract

In this paper, we study the moderate deviations principle (MDP) for slow–fast stochastic dynamical systems where the slow motion is governed by small fractional Brownian motion (fBm) with Hurst parameter $$H\in (1/2,1)$$ H ∈ ( 1 / 2 , 1 ) . We derive conditions on the moderate deviations scaling and on the Hurst parameter H under which the MDP holds. In addition, we show that in typical situations the resulting action functional is discontinuous in H at $$H=1/2$$ H = 1 / 2 , suggesting that the tail behavior of stochastic dynamical systems perturbed by fBm can have different characteristics than the tail behavior of such systems that are perturbed by standard Brownian motion.

Suggested Citation

  • Solesne Bourguin & Thanh Dang & Konstantinos Spiliopoulos, 2023. "Moderate Deviation Principle for Multiscale Systems Driven by Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 36(1), pages 1-57, March.
    • Handle: RePEc:spr:jotpro:v:36:y:2023:i:1:d:10.1007_s10959-023-01235-y
    DOI: 10.1007/s10959-023-01235-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-023-01235-y
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10959-023-01235-y?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 search for a different version of it.

    References listed on IDEAS

    as
    1. Fouque,Jean-Pierre & Papanicolaou,George & Sircar,Ronnie & Sølna,Knut, 2011. "Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives," Cambridge Books, Cambridge University Press, number 9780521843584.
    2. Hu, Yaozhong & Nualart, David & Song, Xiaoming, 2008. "A singular stochastic differential equation driven by fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2075-2085, October.
    3. José Luís Silva & Mohamed Erraoui & El Hassan Essaky, 2018. "Mixed Stochastic Differential Equations: Existence and Uniqueness Result," Journal of Theoretical Probability, Springer, vol. 31(2), pages 1119-1141, June.
    4. Giacomo Ascione & Yuliya Mishura & Enrica Pirozzi, 2021. "Fractional Ornstein-Uhlenbeck Process with Stochastic Forcing, and its Applications," Methodology and Computing in Applied Probability, Springer, vol. 23(1), pages 53-84, March.
    5. Dupuis, Paul & Spiliopoulos, Konstantinos, 2012. "Large deviations for multiscale diffusion via weak convergence methods," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1947-1987.
    6. Masaaki Fukasawa, 2017. "Short-time at-the-money skew and rough fractional volatility," Quantitative Finance, Taylor & Francis Journals, vol. 17(2), pages 189-198, February.
    7. Freidlin, Mark I. & Sowers, Richard B., 1999. "A comparison of homogenization and large deviations, with applications to wavefront propagation," Stochastic Processes and their Applications, Elsevier, vol. 82(1), pages 23-52, July.
    8. Jim Gatheral & Thibault Jaisson & Mathieu Rosenbaum, 2018. "Volatility is rough," Quantitative Finance, Taylor & Francis Journals, vol. 18(6), pages 933-949, June.
    9. 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.
    10. Klebaner, F. C. & Liptser, R., 1999. "Moderate deviations for randomly perturbed dynamical systems," Stochastic Processes and their Applications, Elsevier, vol. 80(2), pages 157-176, April.
    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. Jacquier, Antoine & Pannier, Alexandre, 2022. "Large and moderate deviations for stochastic Volterra systems," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 142-187.
    2. 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.
    3. Peter K. Friz & Paul Gassiat & Paolo Pigato, 2022. "Short-dated smile under rough volatility: asymptotics and numerics," Quantitative Finance, Taylor & Francis Journals, vol. 22(3), pages 463-480, March.
    4. Florian Bourgey & Stefano De Marco & Peter K. Friz & Paolo Pigato, 2023. "Local volatility under rough volatility," Mathematical Finance, Wiley Blackwell, vol. 33(4), pages 1119-1145, October.
    5. Qinwen Zhu & Gregoire Loeper & Wen Chen & Nicolas Langrené, 2021. "Markovian approximation of the rough Bergomi model for Monte Carlo option pricing," Post-Print hal-02910724, HAL.
    6. Eduardo Abi Jaber, 2019. "Lifting the Heston model," Post-Print hal-01890751, HAL.
    7. Josselin Garnier & Knut Solna, 2018. "Optimal hedging under fast-varying stochastic volatility," Papers 1810.08337, arXiv.org, revised Mar 2020.
    8. Eduardo Abi Jaber, 2018. "Lifting the Heston model," Papers 1810.04868, arXiv.org, revised Nov 2019.
    9. Antoine Jacquier & Alexandre Pannier, 2020. "Large and moderate deviations for stochastic Volterra systems," Papers 2004.10571, arXiv.org, revised Apr 2022.
    10. Paul Hager & Eyal Neuman, 2020. "The Multiplicative Chaos of $H=0$ Fractional Brownian Fields," Papers 2008.01385, arXiv.org.
    11. Aur'elien Alfonsi & Ahmed Kebaier, 2021. "Approximation of Stochastic Volterra Equations with kernels of completely monotone type," Papers 2102.13505, arXiv.org, revised Mar 2022.
    12. Qinwen Zhu & Grégoire Loeper & Wen Chen & Nicolas Langrené, 2021. "Markovian Approximation of the Rough Bergomi Model for Monte Carlo Option Pricing," Mathematics, MDPI, vol. 9(5), pages 1-21, March.
    13. Eduardo Abi Jaber, 2018. "Lifting the Heston model," Working Papers hal-01890751, HAL.
    14. Masaaki Fukasawa, 2020. "Volatility has to be rough," Papers 2002.09215, arXiv.org.
    15. Masaaki Fukasawa & Tetsuya Takabatake & Rebecca Westphal, 2019. "Is Volatility Rough ?," Papers 1905.04852, arXiv.org, revised May 2019.
    16. Antoine Jacquier & Konstantinos Spiliopoulos, 2018. "Pathwise moderate deviations for option pricing," Papers 1803.04483, arXiv.org, revised Dec 2018.
    17. Martin Keller-Ressel & Martin Larsson & Sergio Pulido, 2018. "Affine Rough Models," Papers 1812.08486, arXiv.org.
    18. 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.
    19. Alfeus, Mesias & Nikitopoulos, Christina Sklibosios, 2022. "Forecasting volatility in commodity markets with long-memory models," Journal of Commodity Markets, Elsevier, vol. 28(C).
    20. Marc Mukendi Mpanda & Safari Mukeru & Mmboniseni Mulaudzi, 2020. "Generalisation of Fractional-Cox-Ingersoll-Ross Process," Papers 2008.07798, arXiv.org, revised Jul 2022.

    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:spr:jotpro:v:36:y:2023:i:1:d:10.1007_s10959-023-01235-y. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.