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.
A closed-form expression is obtained for the conditional probability distribution of ∫t0R2sds given Rt, where (Rs, 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: . We also examine the joint distribution of (R2t, ∫t0R2sds.
PITMAN, J. and YOR, M., Besselprocesses 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