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Anticipation cancelled by a Girsanov transformation: A paradox on Wiener space. (English) Zbl 0796.60082

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.

MSC:

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