Hitting distributions of geometric Brownian motion

@article{T2006,
abstract = {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 $A(τ) = ∫_0^τ X²(t)dt$ 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.},
author = {T. Byczkowski, M. Ryznar},
journal = {Studia Mathematica},
keywords = {hyperbolic spaces; stable processes; Poisson kernel},
language = {eng},
number = {1},
pages = {19-38},
title = {Hitting distributions of geometric Brownian motion},
url = {http://eudml.org/doc/285235},
volume = {173},
year = {2006},
}

TY - JOUR
AU - T. Byczkowski
AU - M. Ryznar
TI - Hitting distributions of geometric Brownian motion
JO - Studia Mathematica
PY - 2006
VL - 173
IS - 1
SP - 19
EP - 38
AB - 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 $A(τ) = ∫_0^τ X²(t)dt$ 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.
LA - eng
KW - hyperbolic spaces; stable processes; Poisson kernel
UR - http://eudml.org/doc/285235
ER -