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The first exit time of fractional Brownian motion from an unbounded domain

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  • Zhou, Yinbing
  • Lu, Dawei

Abstract

Consider a fractional Brownian motions starting at the interior point x0,h‖x0‖+2K∈Rd+1 with the constant K>1, for some fixed x0∈Rd, of an unbounded domain D=x,y∈Rd+1:y>h‖x‖, The function h is a nondecreasing, lower semicontinuous, and convex function on [0,∞) with h(0) being finite. Here we take h−1x=Axαlogxβwith a positive constant A for x>K. It is evident that h−1(x) exhibits monotonic behavior for sufficiently large values of x. Let τD denote the first time that the fractional Brownian motion exits from D. In most cases, we give the asymptotically equivalent estimate of logPτD>t. The proof methods are based on the earlier works of Li, Shi, Lifshits, and Aurzada.

Suggested Citation

  • Zhou, Yinbing & Lu, Dawei, 2025. "The first exit time of fractional Brownian motion from an unbounded domain," Statistics & Probability Letters, Elsevier, vol. 218(C).
    • Handle: RePEc:eee:stapro:v:218:y:2025:i:c:s0167715224002888
    DOI: 10.1016/j.spl.2024.110319
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    References listed on IDEAS

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    1. Jumarie, Guy, 2005. "Merton's model of optimal portfolio in a Black-Scholes Market driven by a fractional Brownian motion with short-range dependence," Insurance: Mathematics and Economics, Elsevier, vol. 37(3), pages 585-598, December.
    2. Alayed, Haneen & DeBlassie, Dante, 2021. "Brownian motion with a horizontal Bessel drift in a parabolic-type domain," Stochastic Processes and their Applications, Elsevier, vol. 140(C), pages 183-215.
    3. Dawei Lu & Lixin Song, 2011. "The First Exit Time of a Brownian Motion from the Minimum and Maximum Parabolic Domains," Journal of Theoretical Probability, Springer, vol. 24(4), pages 1028-1043, December.
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