Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Artykuły Online First

Streszczenie

We investigate the first hitting times of radial Ornstein–Uhlenbeck processes in the case when the hitting site is closer to the origin than the starting point. The Laplace transform of the first hitting time is represented by the ratio of confluent hypergeometric functions of the second kind, so-called Tricomi functions. We apply the Heaviside expansion theorem and give an explicit form of the distribution functions by means of the zeros of Tricomi functions with respect to the first variable. Moreover, by using the asymptotics of Whittaker functions, which can be written in terms of Tricomi functions, and their derivatives with respect to the first parameter, the asymptotic behavior of tail probabilities and formulas for the probability density functions are also derived from the distribution function.