IDEAS home Printed from https://ideas.repec.org/a/bpj/mcmeap/v29y2023i3p221-242n3.html
   My bibliography  Save this article

Total variation bound for Milstein scheme without iterated integrals

Author

Listed:
  • Yamada Toshihiro

    (Hitotsubashi University, Tokyo, Japan)

Abstract

The paper gives new results for the Milstein scheme of stochastic differential equations. We show that (i) the Milstein scheme holds as a weak approximation in total variation sense and is given by second-order polynomials of Brownian motion without using iterated integrals under non-commutative vector fields; (ii) the accuracy of the Milstein scheme is better than that of the Euler–Maruyama scheme in an asymptotic sense. In particular, we prove dTV⁢(XTε,X¯Tε,Mil,(n))≤C⁢ε2/nd_{\mathrm{TV}}(X_{T}^{\varepsilon},\bar{X}_{T}^{\varepsilon,\mathrm{Mil},(n)})\leq C\varepsilon^{2}/n and dTV⁢(XTε,X¯Tε,EM,(n))≤C⁢ε/nd_{\mathrm{TV}}(X_{T}^{\varepsilon},\bar{X}_{T}^{\varepsilon,\mathrm{EM},(n)})\leq C\varepsilon/n, where dTVd_{\mathrm{TV}} is the total variation distance, XεX^{\varepsilon} is a solution of a stochastic differential equation with a small parameter 𝜀, and X¯ε,Mil,(n)\bar{X}^{\varepsilon,\mathrm{Mil},(n)} and X¯ε,EM,(n)\bar{X}^{\varepsilon,\mathrm{EM},(n)} are the Milstein scheme without iterated integrals and the Euler–Maruyama scheme, respectively. In computational aspect, the scheme is useful to estimate probability distribution functions by a simple simulation without Lévy area computation. Numerical examples demonstrate the validity of the method.

Suggested Citation

  • Yamada Toshihiro, 2023. "Total variation bound for Milstein scheme without iterated integrals," Monte Carlo Methods and Applications, De Gruyter, vol. 29(3), pages 221-242, September.
    • Handle: RePEc:bpj:mcmeap:v:29:y:2023:i:3:p:221-242:n:3
    DOI: 10.1515/mcma-2023-2007
    as

    Download full text from publisher

    File URL: https://doi.org/10.1515/mcma-2023-2007
    Download Restriction: For access to full text, subscription to the journal or payment for the individual article is required.
    File URL: https://libkey.io/10.1515/mcma-2023-2007?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:bpj:mcmeap:v:29:y:2023:i:3:p:221-242:n:3. 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: Peter Golla (email available below). General contact details of provider: https://www.degruyter.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.