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Brownian filtrations are not stable under equivalent time-changes. (English) Zbl 0949.60087

Azéma, Jacques (ed.) et al., Séminaire de probabilités XXXIII. Berlin: Springer. Lect. Notes Math. 1709, 267-276 (1999).
The aim of this article is to construct, on the Wiener space \((W, {\mathcal F}_t, {\mathcal F}_{\infty}, \lambda)\), a family of stopping times \(T_t\), \(t\geq 0\), such that a.s. \(t\rightarrow T_t\) is null at zero, differentiable, and with derivative uniformly as close as 1 as possible, and such that the time-changed filtration \({\mathcal F}_{T_t}\) is not cosy (hence not generated by any Brownian motion). This shows roughly that Brownian filtrations are not necessarily stable under (non-deterministic) time-changes. The construction of this counterexample is deeply connected with the so-called innovation phenomenon, which was first observed by A. M. Vershik [St. Petersbg. Math. J. 6, No. 4, 705-761 (1995); translation from Algebra Anal. 6, No. 4, 1-68 (1994; Zbl 0853.28009)] in the framework of decreasing filtrations in discrete time.
For the entire collection see [Zbl 0924.00016].

MSC:

60J65 Brownian motion

Citations:

Zbl 0853.28009
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