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On hitting times of affine boundaries by reflecting Brownian motion and Bessel processes

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Abstract

Firstly, we compute the distribution function for the hitting time of a linear time-dependent boundary t ↦ a + bt, a ≥ 0, b ∈ ℝ, by a reflecting Brownian motion. The main tool hereby is Doob’s formula which gives the probability that Brownian motion started inside a wedge does not hit this wedge. Other key ingredients are the time inversion property of Brownian motion and the time reversal property of diffusion bridges. Secondly, this methodology can also be applied for the three-dimensional Bessel process. Thirdly, we consider Bessel bridges from 0 to 0 with dimension parameter δ > 0 and show that the probability that such a Bessel bridge crosses an affine boundary is equal to the probability that this Bessel bridge stays below some fixed value.

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Authors and Affiliations

  1. Mathematical Department, Åbo Akademi University, Fänriksgatan 3 B, FIN-20500, Åbo, Finland

    Paavo Salminen

  2. Laboratoire de Probabilités et Modèles aléatoires, Université Pierre et Marie Curie, 4, Place Jussieu, Case 188, F-75252, Paris Cedex 05, France

    Marc Yor

Authors
  1. Paavo Salminen
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  2. Marc Yor
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Correspondence to Paavo Salminen.

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Dedicated to Endre Csáki and Pál Révész on the occasion of their 75th birthdays

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Salminen, P., Yor, M. On hitting times of affine boundaries by reflecting Brownian motion and Bessel processes. Period Math Hung 62, 75–101 (2011). https://doi.org/10.1007/s10998-011-5075-2

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  • DOI: https://doi.org/10.1007/s10998-011-5075-2

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