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Higher-Order Derivative of Self-Intersection Local Time for Fractional Brownian Motion

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  • Qian Yu

    (Nanjing University of Aeronautics and Astronautics
    School of Statistics, East China Normal University)

Abstract

We consider the existence and Hölder continuity conditions for the k-th-order derivatives of self-intersection local time for d-dimensional fractional Brownian motion, where $$k=(k_1,k_2,\ldots , k_d)$$ k = ( k 1 , k 2 , … , k d ) . Moreover, we show a limit theorem for the critical case with $$H=\frac{2}{3}$$ H = 2 3 and $$d=1$$ d = 1 , which was conjectured by Jung and Markowsky [7].

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:34:y:2021:i:4:d:10.1007_s10959-021-01093-6
    DOI: 10.1007/s10959-021-01093-6
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    References listed on IDEAS

    as
    1. Nualart, David & Xu, Fangjun, 2019. "Asymptotic behavior for an additive functional of two independent self-similar Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 3981-4008.
    2. Markowsky, Greg, 2008. "Renormalization and convergence in law for the derivative of intersection local time in," Stochastic Processes and their Applications, Elsevier, vol. 118(9), pages 1552-1585, September.
    3. 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.
    4. Song, Jian & Xu, Fangjun & Yu, Qian, 2019. "Limit theorems for functionals of two independent Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4791-4836.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
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