On the extension of an identity in law between the Brownian bridge and the variance of the Brownian motion
type: Item
instance of: scholarly article
title: On the extension of an identity in law between the Brownian bridge and the variance of the Brownian motion
zbMATH Open document ID: 0802.60073
DOI: 10.1016/0304-4149(94)90135-X
author: Ching-Sung Chou
published in: Stochastic Processes and their Applications
publication date: 1994-04-10T00:00:00Z
review text: M. Yor established in 1991 that the laws of \(\int^ 1_ 0 [B_ s- s^{-1} \int^ s_ 0 B_ u du]^ 2 ds\) and of \(\int^ 1_ 0 \widetilde{B}_ s ds\) were equal, \(B\) being standard real Brownian motion and \(\widetilde{B}\) standard real Brownian bridge, and he calculated the Laplace transform of \(\int^ 1_ 0 B_ s B_ s'ds\) (resp. \(\int^ 1_ 0 \widetilde{B}_ s \widetilde{B}_ s' ds\)), \(B'\) (resp. \(\widetilde{B}'\)) being i.i.d. with \(B\) (resp. \(\widetilde{B}\)). The author presents an extension of these results to the case where \(B'\) (resp. \(\widetilde {B}'\)) is replaced by \(\rho B+ \sqrt{1-\rho^ 2} B'\) (resp. \(\rho \widetilde{B}+ \sqrt{1- \rho^ 2} \widetilde{B}'\)), \(\rho\) being a constant. His calculation uses Mercer's \(L^ 2\)-decomposition of \(B_ s\) (resp. \(\widetilde{B}_ s\)) and diagonalization of the kernel \(s\Lambda s'\) (resp. \(s\Lambda s'- ss'\)).
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 519768
zbMATH Keywords: Mercer's decomposition
zbMATH Keywords: real Brownian motion
zbMATH Keywords: real Brownian bridge
reviewed by: Jacques Franchi
MaRDI profile type: Publication
cites work: Fubini's theorem for double Wiener integrals and the variance of the Brownian path
cites work: An explanation of the Ciesielski-Taylor theorem
full work available at URL: 0304-4149(94)90135-x
OpenAlex ID: W2008181897
description: scientific article; zbMATH DE number 519768