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  • 1 Mathematical Department, Åbo Akademi University, Fänriksgatan 3 B, FIN-20500 Åbo, Finland
  • 2 Laboratoire de Probabilités et Modèles aléatoires, Université Pierre et Marie Curie, 4, Place Jussieu, Case 188, F-75252 Paris Cedex 05, France
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Abstract  

Firstly, we compute the distribution function for the hitting time of a linear time-dependent boundary ta + 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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