IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v38y2025i1d10.1007_s10959-024-01392-8.html
   My bibliography  Save this article

Error Distribution for One-Dimensional Stochastic Differential Equations Driven By Fractional Brownian Motion

Author

Listed:
  • Kento Ueda

    (The University of Tokyo)

Abstract

This paper deals with asymptotic errors, limit theorems for errors between numerical and exact solutions of stochastic differential equations (SDEs) driven by one-dimensional fractional Brownian motion (fBm). The Euler–Maruyama, higher-order Milstein, and Crank–Nicolson schemes are among the most studied numerical schemes for SDE (fSDE) driven by fBm. Most previous studies of asymptotic errors have derived specific asymptotic errors for these schemes as main theorems or their corollaries. Even in the one-dimensional case, the asymptotic error was not determined for the Milstein or the Crank–Nicolson method when the Hurst exponent is less than or equal to 1/3 with a drift term. We obtain a new evaluation method for convergence and asymptotic errors. This evaluation method improves the conditions under which we can prove convergence of the numerical scheme and obtain the asymptotic error under the same conditions. We completely determine the asymptotic error of the Milstein method for arbitrary orders. In addition, we newly determine the asymptotic error of the Crank–Nicolson method for $$1/4

Suggested Citation

  • Kento Ueda, 2025. "Error Distribution for One-Dimensional Stochastic Differential Equations Driven By Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 38(1), pages 1-61, March.
    • Handle: RePEc:spr:jotpro:v:38:y:2025:i:1:d:10.1007_s10959-024-01392-8
    DOI: 10.1007/s10959-024-01392-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-024-01392-8
    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/s10959-024-01392-8?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.

    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:38:y:2025:i:1:d:10.1007_s10959-024-01392-8. 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.

    We have no bibliographic references for this item. You can help adding them by using 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.