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Multidimensional hitting time results for Brownian bridges with moving hyperplanar boundaries

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  • Atkinson, Michael P.
  • Singham, Dashi I.

Abstract

We calculate several hitting time probabilities for a correlated multidimensional Brownian bridge process, where the boundaries are hyperplanes that move linearly with time. We compute the probability that a Brownian bridge will cross a moving hyperplane if the endpoints of the bridge lie on the same side of the hyperplane at the starting and ending times, and we derive the distribution of the hitting time if the endpoints lie on opposite sides of the moving hyperplane. Our third result calculates the probability that this process remains between two parallel hyperplanes, and we extend this result in the independent case to a hyperrectangle with moving faces. To derive these quantities, we rotate the coordinate axes to transform the problem into a one-dimensional calculation.

Suggested Citation

  • Atkinson, Michael P. & Singham, Dashi I., 2015. "Multidimensional hitting time results for Brownian bridges with moving hyperplanar boundaries," Statistics & Probability Letters, Elsevier, vol. 100(C), pages 85-92.
    • Handle: RePEc:eee:stapro:v:100:y:2015:i:c:p:85-92
    DOI: 10.1016/j.spl.2015.02.006
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    References listed on IDEAS

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    1. Abundo, Mario, 2002. "Some conditional crossing results of Brownian motion over a piecewise-linear boundary," Statistics & Probability Letters, Elsevier, vol. 58(2), pages 131-145, June.
    2. Yin, Chuancun, 1999. "The joint distribution of the hitting time and place to a sphere or spherical shell for Brownian motion with drift," Statistics & Probability Letters, Elsevier, vol. 42(4), pages 367-373, May.
    3. Metzler, Adam, 2010. "On the first passage problem for correlated Brownian motion," Statistics & Probability Letters, Elsevier, vol. 80(5-6), pages 277-284, March.
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    Cited by:

    1. Stefan Ankirchner & Christophette Blanchet-Scalliet & Nabil Kazi-Tani, 2019. "The De Vylder-Goovaerts conjecture holds true within the diffusion limit," Post-Print hal-01887402, HAL.
    2. Stefan Ankirchner & Christophette Blanchet-Scalliet & Nabil Kazi-Tani, 2018. "The De Vylder-Goovaerts conjecture holds true within the diffusion limit," Working Papers hal-01887402, HAL.

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