Decreasing sequences of $\sigma$-fields and a measure change for Brownian motion

Abstract

Let $(\mathscr{F}_t)_{t \geq 0}$ be the filtration of a Brownian motion $(B(t))_{t \geq 0}$ on $(\Omega,\mathscr{F},P)$. An example is given of a measure $Q \sim P$ (in the sense of absolute continuity) for which $(\mathscr{F}_t)_{t \geq 0}$ is not the filtration of any Brownian motion on $(\Omega,\mathscr{F},Q)$. This settles a 15-year-old question.