A decomposition of the Brownian path
type: Item
instance of: scholarly article
title: A decomposition of the Brownian path
zbMATH Open document ID: 0615.60075
DOI: 10.1016/0167-7152(87)90061-7
author: Steven Shreve
author: Ioannis Karatzas
published in: Statistics \& Probability Letters
publication date: 1987-01-01T00:00:00Z
review text: The Brownian path \(\{\omega\) (s),s\(\leq t\}\) is transformed into the path \(\{\) \({\hat \omega}\)(s),s\(\leq t\}\) such that the resulting process is again Brownian on [0,t] and \[ ({\hat \omega}(t),\gamma_ t({\hat \omega}),M(t,{\hat \omega},\sigma_ t({\hat \omega}))=(\omega (t),\gamma_ t(\omega),L(t,\omega),\Gamma_+(t,\omega)) \] on \(\{\omega (t)<0\}\), where L(t,\(\omega)\) is local time of the origin, \(\gamma_ t(\omega)=\sup \{s\leq t:\omega (s)=0\}\), \(\Gamma_+(t,\omega)=meas\{s\in [0,t]:\omega\) (s)\(\geq 0\}\), M(t,\({\hat \omega}\))\(=\max \{{\hat \omega}(s)\); \(s\leq t\},\) \(\sigma_ t({\hat \omega})\) the (a.s. unique) location where M(t,\({\hat \omega}\)) is achieved. Several consequences are considered.
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 60J55
zbMATH DE Number: 3996802
zbMATH Keywords: Brownian motion and bridge
zbMATH Keywords: occupation time
zbMATH Keywords: conditioning
zbMATH Keywords: path decomposition
zbMATH Keywords: local time
reviewed by: R. Mikulevicius
MaRDI profile type: Publication
full work available at URL: 0167-7152(87)90061-7
OpenAlex ID: W2084140702
cites work: Q3959169
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cites work: Path Decomposition and Continuity of Local Time for One-Dimensional Diffusions, I
description: scientific article; zbMATH DE number 3996802