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 $\mu \ge 0$ starting from $x>1$. Here $B(t)$ is the Brownian motion starting from 0 with $E^0{B}^2(t)=2t$. We provide an integral formula for the density function of the stopped exponential functional $A(\tau )={\int }_0^{\tau }{X}^2(t)dt$ and determine its asymptotic behaviour at infinity. Although we basically rely on methods developed in \cite{BGS}, the present paper also covers the case of arbitrary drifts $\mu \ge 0$ and provides a significant unification and extension of results of the above-mentioned paper. As a corollary we provide an integral formula and give asymptotic behaviour at infinity of the Poisson kernel for half-spaces for Brownian motion with drift in real hyperbolic spaces of arbitrary dimension.

18 pages