On the sample path behavior of the first passage time process of a Brownian motion with drift (Q913362)

Let \(\{\) W(t), \(t\geq 0\}\) be a Wiener process and consider the Brownian motion \(X(t)=\mu t+\sigma W(t)\) (t\(\geq 0\), \(\mu >0\), \(\sigma\neq 0)\) as well as the first passage time process \(M(t)=\inf \{s\geq 0:\) X(s)\(\geq t\}\) (t\(\geq 0)\). The authors investigate the properties of the increments \[ \Delta (T)=\sup_{0\leq t\leq T-a(T)}(M(t+a(T))-M(T)) \] where \(0<a(T)\leq T\) 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).