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Square-root boundaries for Bessel processes and the hitting times of radial Ornstein-Uhlenbeck processes

Hamana Yuji

Opuscula Mathematica

2023

Vol. 43, no. 2

145--172

This article deals with the first hitting times of a Bessel process to a square-root boundary. We obtain the explicit form of the distribution function of the hitting time by means of zeros of the confluent hypergeometric function with respect to the first parameter. In deducing the distribution function, the time that a radial Ornstein-Uhlenbeck process reaches a certain point is very useful and plays an important role. We also give its distribution function in the case that the starting point is closer to the origin than the arrival site.

Asymptotic expansions for the first hitting times of bessel processes

Hamana Yuji, Kaikura Ryo, Shinozaki Kosuke

Opuscula Mathematica

2021

Vol. 41, no. 4

509--537

We study a precise asymptotic behavior of the tail probability of the first hitting time of the Bessel process. We deduce the order of the third term and decide the explicit form of its coefficient.

Supremum distribution of Bessel process of drifting Brownian motion

Pyć A., Żak T., Serafin G.

Probability and Mathematical Statistics

2015

Vol. 35, Fasc. 2

201--222

Let us assume that (B(1)t, B(2)t, B(3)t + μt) is a three-dimensional Brownian motion with drift μ, starting at the origin. Then Xt = ∥(B(1)t, B(2)t, B(3)t + μt)∥, its distance from the starting point, is a diffusion with many applications. We investigate the supremum of (Xt), give an infinite-series formula for its distribution function and an exact estimate of the density of this distribution in terms of elementary functions.