Hitting distributions of geometric Brownian motion

• Autorzy: T. Byczkowski , M. Ryznar
• Języki publikacji: 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.

Kategorie tematyczne

• 60J60: Diffusion processes
• 60J65: Brownian motion

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2006
• Tom: 173
• Numer: 1
• Strony: 19-38

Daty

wydano

2006

Twórcy

autor

T. Byczkowski

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

autor

M. Ryznar

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

Identyfikatory

DOI

10.4064/sm173-1-2