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

Abstract

Sharpening the main result of the preceding paper, it is shown that if, $B_t,0 \leq t < \infty$ is a standard Brownian motion on $(\Omega,\mathscr{F},P)$, then for any $\varepsilon > 0$ there is a probability measure $Q$ with $(1 - \varepsilon)P \leq Q \leq (1= \varepsilon)P$ such that the filtration of B cannot be generated by any Brownian motion on $(\Omega,\mathscr{F},Q)$.