On certain almost Brownian filtrations

Abstract

A consequence of Vershik's results on discrete-time filtrations is the existence, in continuous time, of filtrations F = ( F t ) t ⩾ 0 which are “Brownian after zero” (that is, for each ɛ > 0 , F ɛ = ( F ɛ + t ) t ⩾ 0 is generated by F ɛ and some F ɛ -Brownian motion), but not generated by F 0 and any Brownian motion. Among the filtrations that are Brownian after zero, how are the truly Brownian ones characterized? An answer is given by the self-coupling criterion (ii) of Theorem 1. This criterion is always satisfied when F is immersible into the filtration of an infinite-dimensional Brownian motion.