On certain probabilities equivalent to Wiener measure, d’après Dubins, Feldman, Smorodinsky and Tsirelson. (English) Zbl 0947.60079

Schachermayer, W.

Azéma, Jacques (ed.) et al., Séminaire de probabilités XXXIII. Berlin: Springer. Lect. Notes Math. 1709, 221-239 (1999).

The author gives an alternative proof of a result originally due to L. Dubins, J. Feldman, M. Smorodinsky and B. Tsirelson [Ann. Probab. 24, No. 2, 882-904 (1996; Zbl 0870.60078)], see also J. Feldman and B. Tsirelson [ibid., 905-911 (1996; Zbl 0870.60079)]: there exists a probability measure \(\mathbb{Q}\) on a standard Wiener space \((\Omega,{\mathfrak A}, \mathbb{P},\{{\mathfrak F}_t\})\) that is equivalent to the Wiener measure \(\mathbb{P}\), but any \(\mathbb{Q}\)-Brownian motion generates a strictly smaller filtration than the canonical filtration \(\{{\mathfrak F}_t\}\). The proof of Schachermayer is actually a careful redaction of the original proof by Dubins et al. and includes the strengthening due to Feldman and Tsirelson. The methods of proofs are essentially the same where, on the technical side, the notion of standard extension of a process (or substandard process) is replaced by the notion of generating parametrization in the sense of M. Smorodinsky [Isr. J. Math. 107, 327-331 (1998; Zbl 0921.60074)]. The presentation is self-contained, worked out in detail and seems to be more easily accessible than the original papers. For yet another version of the proof of this result see the paper reviewed below.
For the entire collection see [Zbl 0924.00016].

Reviewer: René L.Schilling (Nottingham)

MSC:

60J65	Brownian motion
28C20	Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
60G07	General theory of stochastic processes

Keywords:

Brownian motion; Brownian filtration; parametrization of process

Citations:

Zbl 0947.60080; Zbl 0870.60078; Zbl 0870.60079; Zbl 0921.60074