IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v11y1998i2d10.1023_a1022679721638.html
   My bibliography  Save this article

Local Times and Related Properties of Multidimensional Iterated Brownian Motion

Author

Listed:
  • Yimin Xiao

Abstract

Let {W(t), t∈R} and {B(t), t≥0} be two independent Brownian motions in R with W(0) = B(0) = 0 and let $$Y(t) = W(B(t)){\text{ }}(t \geqslant 0)$$ be the iterated Brownian motion. Define d-dimensional iterated Brownian motion by $$X(t) = (X_1 (t),...,X_d (t)){\text{ }}(t \geqslant 0)$$ where X 1, X d are independent copies of Y. In this paper, we investigate the existence, joint continuity and Hölder conditions in the set variable of the local time $$L = \{ L(x,B):x \in {\text{R}}^d ,B \in B({\text{R}}_{\text{ + }} )\}$$ of X(t), where $$B({\text{R}}_{\text{ + }} )$$ is the Borel σ-algebra of R +. These results are applied to study the irregularities of the sample paths and the uniform Hausdorff dimension of the image and inverse images of X(t).

Suggested Citation

  • Yimin Xiao, 1998. "Local Times and Related Properties of Multidimensional Iterated Brownian Motion," Journal of Theoretical Probability, Springer, vol. 11(2), pages 383-408, April.
    • Handle: RePEc:spr:jotpro:v:11:y:1998:i:2:d:10.1023_a:1022679721638
    DOI: 10.1023/A:1022679721638
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1023/A:1022679721638
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1023/A:1022679721638?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. Hu, Y. & Shi, Z., 1995. "The Csörgo-Révész modulus of non-differentiability of iterated Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 58(2), pages 267-279, August.
    2. Shi, Z., 1995. "Lower limits of iterated Wiener processes," Statistics & Probability Letters, Elsevier, vol. 23(3), pages 259-270, May.
    3. Bertoin, Jean, 1996. "Iterated Brownian motion and stable() subordinator," Statistics & Probability Letters, Elsevier, vol. 27(2), pages 111-114, April.
    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. Yueyun Hu, 1999. "Hausdorff and Packing Measures of the Level Sets of Iterated Brownian Motion," Journal of Theoretical Probability, Springer, vol. 12(2), pages 313-346, April.

    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. Csáki, Endre & Csörgo, Miklós & Földes, Antónia & Révész, Pál, 1997. "On the occupation time of an iterated process having no local time," Stochastic Processes and their Applications, Elsevier, vol. 70(2), pages 199-217, October.
    2. Yueyun Hu, 1999. "Hausdorff and Packing Measures of the Level Sets of Iterated Brownian Motion," Journal of Theoretical Probability, Springer, vol. 12(2), pages 313-346, April.
    3. Nane, Erkan, 2009. "Laws of the iterated logarithm for a class of iterated processes," Statistics & Probability Letters, Elsevier, vol. 79(16), pages 1744-1751, August.
    4. Csáki, Endre & Csörgo, Miklós & Földes, Antónia & Révész, Pál, 2011. "On the local time of random walk on the 2-dimensional comb," Stochastic Processes and their Applications, Elsevier, vol. 121(6), pages 1290-1314, June.
    5. Liu, Jin V., 2013. "On Chung’s law of the iterated logarithm for the Brownian time Lévy’s area process," Statistics & Probability Letters, Elsevier, vol. 83(5), pages 1404-1410.
    6. Casse, Jérôme & Marckert, Jean-François, 2016. "Processes iterated ad libitum," Stochastic Processes and their Applications, Elsevier, vol. 126(11), pages 3353-3376.
    7. Bertoin, Jean, 1996. "Iterated Brownian motion and stable() subordinator," Statistics & Probability Letters, Elsevier, vol. 27(2), pages 111-114, April.
    8. Endre Csáki & Antónia Földes, 2020. "Random Walks on Comb-Type Subsets of $$\mathbb {Z}^2$$ Z 2," Journal of Theoretical Probability, Springer, vol. 33(4), pages 2233-2257, December.
    9. Nicolas Curien & Takis Konstantopoulos, 2014. "Iterating Brownian Motions, Ad Libitum," Journal of Theoretical Probability, Springer, vol. 27(2), pages 433-448, June.
    10. Neuenschwander, Daniel, 1998. "Law of the iterated logarithm for Lévy's area process composed with Brownian motion," Statistics & Probability Letters, Elsevier, vol. 40(4), pages 371-377, November.
    11. Csáki, Endre & Földes, Antónia, 1997. "On the logarithmic average of iterated processes," Statistics & Probability Letters, Elsevier, vol. 33(4), pages 347-358, May.

    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:11:y:1998:i:2:d:10.1023_a:1022679721638. 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.