Let τ be the first hitting time of the point 1 by the geometric Brownian motion X(t) = x exp(B(t) - 2μt) with drift μ ≥ 0 starting from x > 1. Here B(t) is the Brownian motion starting from 0 with EB²(t) = 2t. We provide an integral formula for the density function of the stopped exponential functional and determine its asymptotic behaviour at infinity. Although we basically rely on methods developed in [BGS], the present paper covers the case of arbitrary drifts μ ≥ 0 and provides a significant unification and extension of the results of the above-mentioned paper. As a corollary we provide an integral formula and give the asymptotic behaviour at infinity of the Poisson kernel for half-spaces for Brownian motion with drift in real hyperbolic spaces of arbitrary dimension.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm173-1-2,
author = {T. Byczkowski and M. Ryznar},
title = {Hitting distributions of geometric Brownian motion},
journal = {Studia Mathematica},
volume = {173},
year = {2006},
pages = {19-38},
zbl = {1088.60085},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm173-1-2}
}
T. Byczkowski; M. Ryznar. Hitting distributions of geometric Brownian motion. Studia Mathematica, Tome 173 (2006) pp. 19-38. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm173-1-2/