On the local time of the Brownian motion
type: Item
instance of: scholarly article
title: On the local time of the Brownian motion
zbMATH Open document ID: 0845.60082
author: Lajos Takács
published in: The Annals of Applied Probability
publication date: 1996-09-16T00:00:00Z
review text: Let \(W(t)\), \(t \in [0,1]\), be a reflected Brownian motion, started at zero, and denote by \(\tau(\alpha)\) its local time at level \(\alpha \geq 0\). An explicit formula is derived for the density and the moments of the distribution of the random variable \(\tau(\alpha)\). The proof of these formulas uses a simple random walk approximation for the Brownian motion. One can compute explicitly the binomial moments for the local times of the simple random walk, and obtain the binomial moments for the limit distribution. From this one can compute the moments, and then the Laplace transform, and then by inversion, the distribution function and the density of the distribution.
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60J55
zbMATH DE Number: 854863
zbMATH Keywords: local times
zbMATH Keywords: random walk
zbMATH Keywords: reflected Brownian motion
zbMATH Keywords: Laplace transform
reviewed by: Philippe Biane
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1177004703
OpenAlex ID: W2094671679
DOI: 10.1214/AOAP/1177004703
description: scientific article; zbMATH DE number 854863