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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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Pitman, J. and Yor, M. , Random Brownian scaling identities and splicing of Bessel processes, Ann. Probab. , 26 (4) (1998), 1683–1702. MR 2000m :60097 Yor M. Random

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Itô measure for Bessel processes with dimension d = 2(1 − α ), 0 < α < 1, to appear in Studia Sci. Math. Hungar. , 45(1) (2008). Getoor, R. K. , The Brownian escape process

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A closed-form expression is obtained for the conditional probability distribution of ∫t 0 R 2 s ds given R t, where (R s, s ≧ 0) is a Bessel process of dimension δ > 0 started from 0, in terms of parabolic cylinder functions. This is done by inverting the following Laplace transform also known as the generalized Lévy’s stochastic area formula: [exp(λ220tRs2ds)|Rt=a]=(λtsinh(λt))δ/2exp(a22t(λtcoth(λt)1)). We also examine the joint distribution of (R 2 t, ∫t 0 R 2 s ds.

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PITMAN, J. and YOR, M., Bessel processes and infinitely divisible laws, Stochastic integrals (Proc. Sympos., Univ. Durham, Durham, 1980), ed. by D. Williams, Lecture Notes in Math., 851, Springer, Berlin, 1981. MR 82j :60149

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311 367 PITMAN, J. and YOR, M., Bessel processes and infinitely divisible laws, Stochastic integrals (Proc. Sympos., Univ. Durham, Durham, 1980), ed. by D. Williams, Lecture

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-dimensional Brownian motion and the three-dimensional Bessel process, Advances in Appl. Probability 7(3) (1975), 511–526. MR 51 #11677 Pitman J. W. One-dimensional Brownian motion and the

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