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Zbl 0699.60018
Deheuvels, Paul; Steinebach, Josef
On the sample path behavior of the first passage time process of a Brownian motion with drift. (English)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 26, No.1, 145-179 (1990). ISSN 0246-0203

Let $\{$ W(t), $t\ge 0\}$ be a Wiener process and consider the Brownian motion $X(t)=\mu t+\sigma W(t)$ (t$\ge 0$, $\mu >0$, $\sigma\ne 0)$ as well as the first passage time process $M(t)=\inf \{s\ge 0:$ X(s)$\ge t\}$ (t$\ge 0)$. The authors investigate the properties of the increments $$ \Delta (T)=\sup\sb{0\le t\le T-a(T)}(M(t+a(T))-M(T)) $$ where $0<a(T)\le T$ is a function satisfying some regularity conditions. The main results give a complete enough strong law for the lim sup and lim inf behaviour of $\Delta$ (T).
[P.Revesz]

MSC 2000:: *60F15 Strong limit theorems
60G17 Sample path properties
60G50 Sums of independent random variables
60J65 Brownian motion

Keywords: Brownian motion with drift; strong limit theorems; first passage time process

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