IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v168y2021ics0167715220302236.html
   My bibliography  Save this article

A probabilistic approach to the Φ-variation of classical fractal functions with critical roughness

Author

Listed:
  • Han, Xiyue
  • Schied, Alexander
  • Zhang, Zhenyuan

Abstract

We consider Weierstraß and Takagi–van der Waerden functions with critical degree of roughness. In this case, the functions have vanishing pth variation for all p>1 but are also nowhere differentiable and hence not of bounded variation either. We resolve this apparent puzzle by showing that these functions have finite, nonzero, and linear Wiener–Young Φ-variation along the sequence of b-adic partitions, where Φ(x)=x∕−logx. For the Weierstraß functions, our proof is based on the martingale central limit theorem (CLT). For the Takagi–van der Waerden functions, we use the CLT for Markov chains if a certain parameter b is odd, and the standard CLT for b even.

Suggested Citation

  • Han, Xiyue & Schied, Alexander & Zhang, Zhenyuan, 2021. "A probabilistic approach to the Φ-variation of classical fractal functions with critical roughness," Statistics & Probability Letters, Elsevier, vol. 168(C).
  • Handle: RePEc:eee:stapro:v:168:y:2021:i:c:s0167715220302236
    DOI: 10.1016/j.spl.2020.108920
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167715220302236
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.spl.2020.108920?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. Hans Follmer & Alexander Schied, 2013. "Probabilistic aspects of finance," Papers 1309.7759, arXiv.org.
    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. Xiyue Han & Alexander Schied & Zhenyuan Zhang, 2022. "A Limit Theorem for Bernoulli Convolutions and the $$\Phi $$ Φ -Variation of Functions in the Takagi Class," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2853-2878, December.
    2. Xiyue Han & Alexander Schied, 2021. "The roughness exponent and its model-free estimation," Papers 2111.10301, arXiv.org, revised Jun 2024.

    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. Xiyue Han & Alexander Schied, 2021. "The roughness exponent and its model-free estimation," Papers 2111.10301, arXiv.org, revised Jun 2024.
    2. Albrecht, Peter & Huggenberger, Markus, 2017. "The fundamental theorem of mutual insurance," Insurance: Mathematics and Economics, Elsevier, vol. 75(C), pages 180-188.
    3. Yannick Limmer & Blanka Horvath, 2023. "Robust Hedging GANs," Papers 2307.02310, arXiv.org.
    4. Andrew L. Allan & Chong Liu & David J. Promel, 2021. "A C\`adl\`ag Rough Path Foundation for Robust Finance," Papers 2109.04225, arXiv.org, revised May 2023.
    5. Henry Chiu & Rama Cont, 2023. "A model‐free approach to continuous‐time finance," Mathematical Finance, Wiley Blackwell, vol. 33(2), pages 257-273, April.
    6. Stoyan V. Stoyanov & Svetlozar T. Rachev & Stefan Mittnik & Frank J. Fabozzi, 2019. "Pricing Derivatives In Hermite Markets," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(06), pages 1-27, September.
    7. Henry Chiu & Rama Cont, 2022. "A model-free approach to continuous-time finance," Papers 2211.15531, arXiv.org.
    8. Knispel, Thomas & Laeven, Roger J.A. & Svindland, Gregor, 2016. "Robust optimal risk sharing and risk premia in expanding pools," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 182-195.
    9. Alexander Schied, 2015. "On a class of generalized Takagi functions with linear pathwise quadratic variation," Papers 1501.00837, arXiv.org, revised Aug 2015.
    10. Sebastian Herrmann & Johannes Muhle-Karbe & Frank Thomas Seifried, 2016. "Hedging with Small Uncertainty Aversion," Papers 1605.06429, arXiv.org.
    11. Nicolas Perkowski & David J. Promel, 2014. "Local times for typical price paths and pathwise Tanaka formulas," Papers 1405.4421, arXiv.org, revised Apr 2015.
    12. Sebastian Herrmann & Johannes Muhle-Karbe & Frank Thomas Seifried, 2017. "Hedging with small uncertainty aversion," Finance and Stochastics, Springer, vol. 21(1), pages 1-64, January.

    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:stapro:v:168:y:2021:i:c:s0167715220302236. 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/622892/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.