IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v30y2017i3d10.1007_s10959-016-0667-7.html
   My bibliography  Save this article

On the Equivalence of Probability Spaces

Author

Listed:
  • Daniel Alpay

    (Ben-Gurion University of the Negev)

  • Palle Jorgensen

    (University of Iowa)

  • David Levanony

    (Ben Gurion University of the Negev)

Abstract

For a general class of Gaussian processes W, indexed by a sigma-algebra $${\mathscr {F}}$$ F of a general measure space $$(M,{\mathscr {F}}, \sigma )$$ ( M , F , σ ) , we give necessary and sufficient conditions for the validity of a quadratic variation representation for such Gaussian processes, thus recovering $$\sigma (A)$$ σ ( A ) , for $$A\in {\mathscr {F}}$$ A ∈ F , as a quadratic variation of W over A. We further provide a harmonic analysis representation for this general class of processes. We apply these two results to: (i) a computation of generalized Ito integrals and (ii) a proof of an explicit and measure-theoretic equivalence formula, realizing an equivalence between the two approaches to Gaussian processes, one where the choice of sample space is the traditional path space, and the other where it is Schwartz’ space of tempered distributions.

Suggested Citation

  • Daniel Alpay & Palle Jorgensen & David Levanony, 2017. "On the Equivalence of Probability Spaces," Journal of Theoretical Probability, Springer, vol. 30(3), pages 813-841, September.
    • Handle: RePEc:spr:jotpro:v:30:y:2017:i:3:d:10.1007_s10959-016-0667-7
    DOI: 10.1007/s10959-016-0667-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-016-0667-7
    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/s10959-016-0667-7?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. Alpay, Daniel & Attia, Haim & Levanony, David, 2010. "On the characteristics of a class of Gaussian processes within the white noise space setting," Stochastic Processes and their Applications, Elsevier, vol. 120(7), pages 1074-1104, July.
    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. Daniel Alpay & Palle Jorgensen, 2022. "mu-Brownian Motion, Dualities, Diffusions, Transforms, and Reproducing Kernel Hilbert Spaces," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2757-2783, December.

    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. Daniel Alpay & Palle Jorgensen, 2022. "mu-Brownian Motion, Dualities, Diffusions, Transforms, and Reproducing Kernel Hilbert Spaces," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2757-2783, December.
    2. Alpay, Daniel & Salomon, Guy, 2013. "Non-commutative stochastic distributions and applications to linear systems theory," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 2303-2322.

    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:30:y:2017:i:3:d:10.1007_s10959-016-0667-7. 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.