Hitting distributions of geometric Brownian motion

Abstract

Letbe the first hitting time of the point 1 by the geometric Brownian motion X(t) = xexp(B(t) −2µt) with drift µ > 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(�) = R � 0 X 2 (t)dt and determine its asymptotic behaviour at infinity. Although we basically rely on methods developed in (BGS), the present paper also covers the case of arbitrary drifts µ > 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.