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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 \(\nu\).
60G40 Stopping times; optimal stopping problems; gambling theory
60J60 Diffusion processes
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