IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v23y2010i4d10.1007_s10959-009-0221-y.html
   My bibliography  Save this article

Regularity of Intersection Local Times of Fractional Brownian Motions

Author

Listed:
  • Dongsheng Wu

    (University of Alabama in Huntsville)

  • Yimin Xiao

    (Michigan State University)

Abstract

Let $B^{\alpha_{i}}$ be an (N i ,d)-fractional Brownian motion with Hurst index α i (i=1,2), and let $B^{\alpha_{1}}$ and $B^{\alpha_{2}}$ be independent. We prove that, if $\frac{N_{1}}{\alpha_{1}}+\frac{N_{2}}{\alpha_{2}}>d$ , then the intersection local times of $B^{\alpha_{1}}$ and $B^{\alpha_{2}}$ exist, and have a continuous version. We also establish Hölder conditions for the intersection local times and determine the Hausdorff and packing dimensions of the sets of intersection times and intersection points. One of the main motivations of this paper is from the results of Nualart and Ortiz-Latorre (J. Theor. Probab. 20:759–767, 2007), where the existence of the intersection local times of two independent (1,d)-fractional Brownian motions with the same Hurst index was studied by using a different method. Our results show that anisotropy brings subtle differences into the analytic properties of the intersection local times as well as rich geometric structures into the sets of intersection times and intersection points.

Suggested Citation

  • Dongsheng Wu & Yimin Xiao, 2010. "Regularity of Intersection Local Times of Fractional Brownian Motions," Journal of Theoretical Probability, Springer, vol. 23(4), pages 972-1001, December.
    • Handle: RePEc:spr:jotpro:v:23:y:2010:i:4:d:10.1007_s10959-009-0221-y
    DOI: 10.1007/s10959-009-0221-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-009-0221-y
    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/s10959-009-0221-y?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. David Nualart & Salvador Ortiz-Latorre, 2007. "Intersection Local Time for Two Independent Fractional Brownian Motions," Journal of Theoretical Probability, Springer, vol. 20(4), pages 759-767, December.
    2. Rosen, Jay, 1987. "The intersection local time of fractional Brownian motion in the plane," Journal of Multivariate Analysis, Elsevier, vol. 23(1), pages 37-46, October.
    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. Jingjun Guo & Yaozhong Hu & Yanping Xiao, 2019. "Higher-Order Derivative of Intersection Local Time for Two Independent Fractional Brownian Motions," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1190-1201, September.
    2. Junna Bi & Fangjun Xu, 2016. "A first-order limit law for functionals of two independent fractional Brownian motions in the critical case," Journal of Theoretical Probability, Springer, vol. 29(3), pages 941-957, September.
    3. Paul Jung & Greg Markowsky, 2015. "Hölder Continuity and Occupation-Time Formulas for fBm Self-Intersection Local Time and Its Derivative," Journal of Theoretical Probability, Springer, vol. 28(1), pages 299-312, March.

    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. Bardet, Jean-Marc, 2002. "Bivariate occupation measure dimension of multidimensional processes," Stochastic Processes and their Applications, Elsevier, vol. 99(2), pages 323-348, June.
    2. Jingjun Guo & Yaozhong Hu & Yanping Xiao, 2019. "Higher-Order Derivative of Intersection Local Time for Two Independent Fractional Brownian Motions," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1190-1201, September.
    3. David Nualart & Salvador Ortiz-Latorre, 2007. "Intersection Local Time for Two Independent Fractional Brownian Motions," Journal of Theoretical Probability, Springer, vol. 20(4), pages 759-767, December.
    4. Paul Jung & Greg Markowsky, 2015. "Hölder Continuity and Occupation-Time Formulas for fBm Self-Intersection Local Time and Its Derivative," Journal of Theoretical Probability, Springer, vol. 28(1), pages 299-312, March.
    5. Chen, Zhenlong & Wang, Jun & Wu, Dongsheng, 2020. "On intersections of independent space–time anisotropic Gaussian fields," Statistics & Probability Letters, Elsevier, vol. 166(C).
    6. Yan, Litan & Shen, Guangjun, 2010. "On the collision local time of sub-fractional Brownian motions," Statistics & Probability Letters, Elsevier, vol. 80(5-6), pages 296-308, March.
    7. Jaramillo, Arturo & Nualart, David, 2017. "Asymptotic properties of the derivative of self-intersection local time of fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 127(2), pages 669-700.
    8. Qian Yu, 2021. "Higher-Order Derivative of Self-Intersection Local Time for Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 34(4), pages 1749-1774, December.
    9. Jung, Paul & Markowsky, Greg, 2014. "On the Tanaka formula for the derivative of self-intersection local time of fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 124(11), pages 3846-3868.
    10. Chen, Chao & Yan, Litan, 2011. "Remarks on the intersection local time of fractional Brownian motions," Statistics & Probability Letters, Elsevier, vol. 81(8), pages 1003-1012, August.
    11. Junna Bi & Fangjun Xu, 2016. "A first-order limit law for functionals of two independent fractional Brownian motions in the critical case," Journal of Theoretical Probability, Springer, vol. 29(3), pages 941-957, September.

    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:23:y:2010:i:4:d:10.1007_s10959-009-0221-y. 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.