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Zbl 0796.60082
Föllmer, Hans; Imkeller, Peter
Anticipation cancelled by a Girsanov transformation: A paradox on Wiener space. (English)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 29, No.4, 569-586 (1993). ISSN 0246-0203

A Wiener process defined as the coordinate process $X$ on the Wiener space $(\Omega,{\cal F},P)$, $\Omega=C([0,1])$, remains a semimartingale w.r.t. $({\cal G}\sb t)$, the canonical filtration $({\cal F}\sb t)$ enlarged by the terminal value $X\sb 1$, and it has the semimartingale decomposition $$X\sb t=W\sb t+\int\sp t\sb 0{X\sb 1-X\sb s\over 1- s}ds,\qquad 0\le t\le 1.$$ The elimination of the drift by means of a Girsanov transformation leads to a new probability measure $Q$ on each $\sigma$-field ${\cal G}\sb t$, $0\le t<1$, which turns $(X\sb s,{\cal G}\sb s)$ 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 $(\Omega,{\cal F})$ but it can be constructed on $\Omega\times R\sp 1$ and is given by $\bar Q=P\otimes{\cal N}(0,1)$. In the second part of the paper the authors consider the more general case of an enlargement of $({\cal F}\sb t)$ by a random variable $G$ under which $X$ remains a semimartingale.
[R.Buckdahn (Berlin)]

MSC 2000:: *60J65 Brownian motion
60J45 Probabilistic potential theory
60G44 Martingales with continuous parameter
60H05 Stochastic integrals

Keywords: enlargement of filtration; Wiener process; semimartingale; Wiener measure; Girsanov transformation

Cited in: Zbl 0988.60042

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