Recognising Whether a Filtration is Brownian: a Case Study

Abstract

A filtration on a probability space is said to be Brownian when it is generated by some Brownian motion started from 0. Recognising whether a given filtration F = (F _t ) _t≥0 is Brownian may be a difficult problem; but when F is Brownian after zero, a necessary and sufficient condition for F to be Brownian is available, namely, the self-coupling property (ii) of Theorem 1 of [4]. (‘Brownian after zero’ means that for each ɛ > 0, the shifted filtration F ^ɛ = (F _ɛ+t ) _t≥0 is generated by its initial σ-field F _ɛ and by some F ^ɛ-Brownian motion.) In all concrete examples where this self-coupling criterion has been used to establish Brownianity, another, more constructive proof was also available. The situation presented below is different. We are interested in a certain process, introduced in 1991 by Beneš, Karatzas and Rishel; the natural filtration of this process turns out to be also generated by some Brownian motion, but we have not been able to exhibit such a generating Brownian motion; the general, non constructive criterion is the only proof we know that this filtration is indeed Brownian.