  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 $E^0 B^2(t) = 2t$. 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 $\mu
\geq 0$ 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.
