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Zbl 0835.60073
Attal, S.; Burdzy, K.; Émery, M.; Hu, Y.
On Brownian filtrations and transformations. (Sur quelques filtrations et transformations browniennes.) (French)
[A] Azéma, J. (ed.) et al., Séminaire de probabilités XXIX. Berlin: Springer-Verlag. Lect. Notes Math. 1613, 56-69 (1995). ISBN 3-540-60219-4/pbk

Let $(\Omega, {\cal F}, P)$ be a Wiener space, carrying a one-dimensional Brownian motion, with filtration $({\cal F}_t)_{t > 0}$. The Goswami-Rao filtration is the filtration $({\cal G}_t)_{t > 0}$ such that ${\cal G}_t$ consists of events in ${\cal F}_t$ which are invariant (up to null sets) by the map $\omega \mapsto - \omega$ on the Wiener space. The main result states that the Goswami-Rao filtration is a Brownian filtration, namely that it is generated by a Brownian motion. The proof consists in exhibiting a Brownian motion which generates the filtration, and three different such Brownian motions are constructed. The construction of these Brownian motions suggests the study of more general Brownian transformations, which are endomorphisms of the Wiener space, preserving the Wiener measure. In particular, it is proved that there exist Brownian transformations which are adapted and representable in the sense of the first author [Ann. Inst. Henri Poincaré, Probab. Stat. 31, No. 3, 467-484 (1995; Zbl 0827.60004)], and which can have arbitrary period.
[Ph.Biane (Paris)]

MSC 2000:: *60J65 Brownian motion

Keywords: predictable representation; Brownian motion; Goswami-Rao filtration

Citations: Zbl 0827.60004

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