IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v120y2010i2p223-254.html
   My bibliography  Save this article

Discretizing the fractional Lévy area

Author

Listed:
  • Neuenkirch, A.
  • Tindel, S.
  • Unterberger, J.

Abstract

In this article, we give sharp bounds for the Euler discretization of the Lévy area associated to a d-dimensional fractional Brownian motion. We show that there are three different regimes for the exact root mean square convergence rate of the Euler scheme, depending on the Hurst parameter H[set membership, variant](1/4,1). For H 3/4 the exact rate is n-1. Moreover, we also show that a trapezoidal scheme converges (at least) with the rate n-2H+1/2. Finally, we derive the asymptotic error distribution of the Euler scheme. For H 3/4 the limit distribution is of Rosenblatt type.

Suggested Citation

  • Neuenkirch, A. & Tindel, S. & Unterberger, J., 2010. "Discretizing the fractional Lévy area," Stochastic Processes and their Applications, Elsevier, vol. 120(2), pages 223-254, February.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:2:p:223-254
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(09)00182-3
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    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. Neuenkirch, Andreas, 2008. "Optimal pointwise approximation of stochastic differential equations driven by fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 118(12), pages 2294-2333, December.
    2. Baudoin, Fabrice & Coutin, Laure, 2007. "Operators associated with a stochastic differential equation driven by fractional Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 117(5), pages 550-574, May.
    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. Ferreiro-Castilla, Albert & Utzet, Frederic, 2011. "Lévy area for Gaussian processes: A double Wiener-Itô integral approach," Statistics & Probability Letters, Elsevier, vol. 81(9), pages 1380-1391, September.

    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. Peter Kloeden & Andreas Neuenkirch & Raffaella Pavani, 2011. "Multilevel Monte Carlo for stochastic differential equations with additive fractional noise," Annals of Operations Research, Springer, vol. 189(1), pages 255-276, September.
    2. Sreekar Vadlamani, 2010. "Fractional Brownian Flows," Journal of Theoretical Probability, Springer, vol. 23(1), pages 257-276, March.
    3. Hufei Li & Shaojuan Ma, 2023. "The Evolution of Probability Density Function for Power System Excited by Fractional Gaussian Noise," Mathematics, MDPI, vol. 11(13), pages 1-16, June.
    4. Bardina, X. & Nourdin, I. & Rovira, C. & Tindel, S., 2010. "Weak approximation of a fractional SDE," Stochastic Processes and their Applications, Elsevier, vol. 120(1), pages 39-65, January.
    5. Qi Feng & Jianfeng Zhang, 2021. "Cubature Method for Stochastic Volterra Integral Equations," Papers 2110.12853, arXiv.org, revised Jul 2023.
    6. Kęstutis Kubilius & Aidas Medžiūnas, 2022. "Pathwise Convergent Approximation for the Fractional SDEs," Mathematics, MDPI, vol. 10(4), pages 1-16, February.
    7. Yuzuru Inahama, 2010. "A Stochastic Taylor-Like Expansion in the Rough Path Theory," Journal of Theoretical Probability, Springer, vol. 23(3), pages 671-714, September.
    8. Yaozhong Hu & Samy Tindel, 2013. "Smooth Density for Some Nilpotent Rough Differential Equations," Journal of Theoretical Probability, Springer, vol. 26(3), pages 722-749, September.
    9. Baudoin, Fabrice & Ouyang, Cheng, 2011. "Small-time kernel expansion for solutions of stochastic differential equations driven by fractional Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 121(4), pages 759-792, April.
    10. Alexandra Chronopoulou & Samy Tindel, 2013. "On inference for fractional differential equations," Statistical Inference for Stochastic Processes, Springer, vol. 16(1), pages 29-61, April.
    11. Song, Jian & Tindel, Samy, 2022. "Skorohod and Stratonovich integrals for controlled processes," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 569-595.
    12. Passeggeri, Riccardo, 2020. "On the signature and cubature of the fractional Brownian motion for H>12," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1226-1257.

    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:120:y:2010:i:2:p:223-254. 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.