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• # Artykuł - szczegóły

## Studia Mathematica

2006 | 173 | 1 | 19-38

## Hitting distributions of geometric Brownian motion

EN

### Abstrakty

EN
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.

19-38

wydano
2006

### Twórcy

autor
• Institute of Mathematics and Informatics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
autor
• Institute of Mathematics and Informatics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland