# zbMATH — the first resource for mathematics

Precise asymptotic formulae for the first hitting times of Bessel processes. (English) Zbl 1419.60037
Summary: 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 $$\nu$$.
##### MSC:
 60G40 Stopping times; optimal stopping problems; gambling theory 60J60 Diffusion processes
##### Keywords:
Bessel process; hitting time; tail probability
Full Text: