Periodica Mathematica Hungarica

, Volume 62, Issue 1, pp 75–101

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

Article

## 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.

## Key words and phrases

reflecting Brownian motion Bessel process hitting time linear boundary time reversal time inversion Brownian bridge Bessel bridge

60J65 60J60

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