Hitting distributions of geometric Brownian motion

T. Byczkowski¹ ; M. Ryznar¹

¹ Institute of Mathematics and Informatics Wrocław University of Technology Wybrzeże Wyspiańskiego 27 50-370 Wrocław, Poland

Studia Mathematica, Tome 173 (2006) no. 1, pp. 19-38

Résumé

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.

Détail

DOI : 10.4064/sm173-1-2

Keywords: tau first hitting time point geometric brownian motion exp drift geq starting here brownian motion starting provide integral formula density function stopped exponential functional tau int tau determine its asymptotic behaviour infinity although basically rely methods developed cite bgs present paper covers arbitrary drifts geq provides significant unification extension results above mentioned paper corollary provide integral formula asymptotic behaviour infinity poisson kernel half spaces brownian motion drift real hyperbolic spaces arbitrary dimension