Sojourn times for the Brownian motion
type: Item
instance of: scholarly article
title: Sojourn times for the Brownian motion
zbMATH Open document ID: 0916.60073
DOI: 10.1155/S1048953398000203
published in: Journal of Applied Mathematics and Stochastic Analysis
publication date: 1999-07-19T00:00:00Z
full work available at URL: https://eudml.org/doc/48175
review text: Let \(\omega(\alpha)\) be a sojourn time of a standard Brownian motion spent in the set \((\alpha,\infty)\), \(\alpha>0\) in the time interval \((0,1)\), and \(\omega^*(\alpha)\) be the corresponding sojourn time of the reflecting Brownian motion \(\xi(t)=| w(t)|\). Denote by \(\tau(\alpha)\) the Brownian local time at level \(\alpha\). The author uses a combinatorial approach to determine the moment of \(\tau(\alpha)\), \(\omega(\alpha)\) and \(\omega^*(\alpha)\). For instance, he proves that \[ M_r(\alpha) =E\bigl(\omega (\alpha)\bigr)^r=m_{2r} (\alpha)/(2^rr!) \] and \[ M^*_r(\alpha)= E\bigl(\omega^* (\alpha)\bigr)^r =(r-1)!2^{-(r-1)}\sum^m_{k=1}m_{ 2r}\bigl((2k-1)\alpha\bigr)/ \bigl \{ (r-k) !(r+k-1)!\bigr\}, \] where \(m_r(\alpha)=E(\tau(\alpha))^r\), \(r=1,2,\dots\). These relations enable one to find an explicit formula for the distribution function of \(\omega^*(\alpha)\) via Laplace-Stieltjes transform.
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60J55
zbMATH DE Number: 1221901
zbMATH Keywords: reflecting Brownian motion
zbMATH Keywords: local time
zbMATH Keywords: moments
zbMATH Keywords: sojourn time
author: Lajos Takács
MaRDI profile type: Publication
OpenAlex ID: W2073781308
description: scientific article; zbMATH DE number 1221901