## 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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