Anticipation cancelled by a Girsanov transformation: A paradox on Wiener space
type: Item
instance of: scholarly article
title: Anticipation cancelled by a Girsanov transformation: A paradox on Wiener space
zbMATH Open document ID: 0796.60082
author: Hans Föllmer
author: P. Imkeller
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1994-09-20T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77470
review text: A Wiener process defined as the coordinate process \(X\) on the Wiener space \((\Omega,{\mathcal F},P)\), \(\Omega=C([0,1])\), remains a semimartingale w.r.t. \(({\mathcal G}_ t)\), the canonical filtration \(({\mathcal F}_ t)\) enlarged by the terminal value \(X_ 1\), and it has the semimartingale decomposition \[ X_ t=W_ t+\int^ t_ 0{X_ 1-X_ s\over 1- s}ds,\qquad 0\leq t\leq 1. \] The elimination of the drift by means of a Girsanov transformation leads to a new probability measure \(Q\) on each \(\sigma\)-field \({\mathcal G}_ t\), \(0\leq t<1\), which turns \((X_ s,{\mathcal G}_ s)\) into a Wiener process up to each time point \(t<1\). But any such \(Q\) on \(C([0,1])\) should coincide with the Wiener measure \(P\), which contradicts the fact that the Girsanov density at time \(t\) is not identically to 1. The authors study this apparent paradoxon and show that the probability measure determined by Girsanov transformation does not live on \((\Omega,{\mathcal F})\) but it can be constructed on \(\Omega\times R^ 1\) and is given by \(\bar Q=P\otimes{\mathcal N}(0,1)\). In the second part of the paper the authors consider the more general case of an enlargement of \(({\mathcal F}_ t)\) by a random variable \(G\) under which \(X\) remains a semimartingale.
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 60H05
zbMATH DE Number: 566214
zbMATH Keywords: enlargement of filtration
zbMATH Keywords: Wiener process
zbMATH Keywords: semimartingale
zbMATH Keywords: Wiener measure
zbMATH Keywords: Girsanov transformation
reviewed by: Rainer Buckdahn
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 566214