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On a theorem of Tsirelson with respect to Brownian and non-Brownian filtrations. (Autour d’un théorème de Tsirelson sur des filtrations browniennes et non browniennes.) (French) Zbl 0914.60064
Azéma, Jacques (ed.) et al., Séminaire de probabilités XXXII. Berlin: Springer. Lect. Notes Math. 1686, 264-305 (1998).
The paper is motivated by recent advances made by Tsirelson and co-authors on Brownian filtrations, i.e. natural filtrations of Brownian motions. One of the most striking result in that field proved by Tsirelson is that the filtration generated by a so-called Walsh process is not Brownian. This provides in particular explicit examples of non-Brownian filtrations which nonetheless enjoy the predictable representation property with respect to a Brownian motion (then of course the natural filtration of the latter is smaller than the initial filtration). Roughly, the paper not only recasts Tsirelson’s ideas in the framework of the general theory of processes, but it also introduces and studies new concepts like the so-called spider-martingales, and presents interesting applications to honest times (a random variable $$L$$ valued in $$[0,\infty]$$ is called honest for a filtration $$({\mathcal F}_t)$$ if for each $$t\geq 0$$, there is an $${\mathcal F}_t$$-measurable variable $$\ell_t$$ such that $$L=\ell_t$$ on the event $$\{L\leq t\})$$. In particular, it is shown that if $$({\mathcal F}_t)$$ is a Brownian filtration and $$L$$ a honest time, then $${\mathcal F}_{L+}={\mathcal F}_L\vee\sigma(A)$$ where $$A$$ is a single event.
For the entire collection see [Zbl 0893.00035].
Reviewer: J.Bertoin (Paris)

##### MSC:
 60J65 Brownian motion 60G48 Generalizations of martingales
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