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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
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    References listed on IDEAS

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    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.
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    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.

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