{
  "id": "math/0503060",
  "version": "v1",
  "published": "2005-03-03T14:46:50.000Z",
  "updated": "2005-03-03T14:46:50.000Z",
  "title": "Hitting distributions of geometric Brownian motion",
  "authors": [
    "T. Byczkowski",
    "M. Ryznar"
  ],
  "comment": "18 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "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 $E^0 B^2(t) = 2t$. 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 also covers the case of arbitrary drifts $\\mu \\geq 0$ 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-03-03T14:46:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J65",
      "60J60"
    ],
    "keywords": [
      "geometric brownian motion",
      "hitting distributions",
      "asymptotic behaviour",
      "integral formula",
      "real hyperbolic spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......3060B"
    }
  }
}