Precise Asymptotic Formulae for the First Hitting Times of Bessel Processes

Abstract

We study the first hitting time to $b$ of a Bessel process with index $\nu$ starting from $a$, which is denoted by $\tau_{a,b}^{(\nu)}$, in the case when $0<b<a$. When $\nu>1$ and $\nu-1/2$ is not an integer, we obtain that $\mathbf P(t<\tau_{a,b}^{(\nu)}<\infty)$ is asymptotically equal to $\kappa_1^{(\nu)}t^{-\nu}+\kappa_2^{(\nu)} t^{-\nu-1}$ as $t\to\infty$ for some explicit constants $\kappa_1^{(\nu)}$ and $\kappa_2^{(\nu)}$. The constant $\kappa_1^{(\nu)}$ is known and the aim is to get $\kappa_2^{(\nu)}$. Combining our result with the known facts, we obtain the precise asymptotic formula for every index $\nu$.