Hitting distributions of geometric Brownian motion

Tom 173 / 2006

T. Byczkowski, M. Ryznar Studia Mathematica 173(2006), 19-38 MSC: Primary 60J65; Secondary 60J60. DOI: 10.4064/sm173-1-2

Streszczenie

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 $E B^2(t) = 2t$. 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 covers the case of arbitrary drifts $\mu \geq 0$ 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.

Autorzy

  • T. ByczkowskiInstitute of Mathematics and Informatics
    Wrocław University of Technology
    Wybrzeże Wyspiańskiego 27
    50-370 Wrocław, Poland
    e-mail
  • M. RyznarInstitute of Mathematics and Informatics
    Wrocław University of Technology
    Wybrzeże Wyspiańskiego 27
    50-370 Wrocław, Poland

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