On hitting times of affine boundaries by reflecting Brownian motion and Bessel processes
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.
Key words and phrasesreflecting Brownian motion Bessel process hitting time linear boundary time reversal time inversion Brownian bridge Bessel bridge
Mathematics subject classification numbers60J65 60J60
Unable to display preview. Download preview PDF.
- M. Abramowitz and I. Stegun, Mathematical functions, Dover publications, Inc., New York, 1970.Google Scholar
- L. Breiman, First exit times from a square root boundary, Proc. 5th Berkeley Symp. Math. Stat. Probab. Vol. 2, University of California, Berkeley, 1967, 9–16.Google Scholar
- W. Feller, An introduction to probability theory and its applications, Vol. II., Second edition, John Wiley & Sons Inc., New York, 1971.Google Scholar
- J. Pitman and M. Yor, The law of the maximum of a Bessel bridge, Electron. J. Probab., 4 (1999), 35 pp. (electronic).Google Scholar
- P. Salminen and M. Yor, On some probabilistic occurrences of the Poisson summation formula, in preparation.Google Scholar