Hitting distributions of geometric Brownian motion

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dc.creatorByczkowski, T.
dc.creatorRyznar, M.
dc.date2005-03-03
dc.date.accessioned2026-07-07T05:17:38Z
dc.date.available2026-07-07T05:17:38Z
dc.descriptionLet $τ$ be the first hitting time of the point 1 by the geometric Brownian motion $X(t)= x \exp(B(t)-2μt)$ with drift $μ\geq 0$ 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 $A(τ)=\int_0^τX^2(t) dt$ 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 $μ\geq 0$ 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.
dc.description18 pages
dc.identifierhttps://arxiv.org/abs/math/0503060
dc.identifierhttp://arxiv.org/abs/math/0503060
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/74376
dc.subjectProbability
dc.subject60J65; 60J60
dc.titleHitting distributions of geometric Brownian motion
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