Hitting distributions of geometric Brownian motion

Abstract

Let $\tau$ be the first hitting time of the point $1$ by the geometric Brownian motion $X(t)= x \exp(B(t)-2\mu t)$ with drift $\mu \geq 0$ starting from $x>1$. Here $B(t)$ is the Brownian motion starting from $0$ with $E B^2(t) = 2t$. We provide an integ