A simplification of Tsirelson’s argument on the non-Brownian character of the Walsh processes. (Une simplification de l’argument de Tsirelson sur le caractère non-brownien des processus de Walsh.) (French) Zbl 0947.60052

de Meyer, Bernard

Azéma, Jacques (ed.) et al., Séminaire de probabilités XXXIII. Berlin: Springer. Lect. Notes Math. 1709, 217-220 (1999).

The short note gives a new proof of the fact that a Walsh process is not adapted to any filtration generated by a Brownian motion. The proof generalizes to spider-type martingales (martingale araignée) in the sense that a martingale with respect to a Brownian filtration on a cob-web like structure (toile) is necessarily constant. The notions of martingale araignée and toile were introduced by M. T. Barlow, M. Émery, F. B. Knight, S. Song and M. Yor [in: Séminaire de probabilités XXXII. Lect. Notes Math. 1686, 264-305 (1998; Zbl 0914.60064)].
For the entire collection see [Zbl 0924.00016].

Reviewer: René L.Schilling (Nottingham)

MSC:

60G44	Martingales with continuous parameter
60J65	Brownian motion

Keywords:

Walsh process; Brownian filtration; spider martingale

Citations:

Zbl 0914.60064