$\epsilon$-close measures producing nonisomorphic filtrations

Abstract

A consequence of the preceding two papers is this. Let $\{\mathscr{A}_t: 0 \leq t < \infty\}$ be the filtration of a stochastic process on $(\Omega, \mathscr{A},P)$. Under a mild assumption on the process, there exist, for any $\varepsilon > 0$, uncountably many probability measures $Q_\alpha$ with $(1 - \varepsilon) P \leq Q_\alpha \leq (1+ \varepsilon)P$ so that no two of the filtrations $(\Omega, (\mathscr{A}_t)_{o \leq t}, Q_\alpha)$ and $(\Omega (\mathscr{A}_t)_{o\leq t}, Q_\beta), \alpha \not= \beta$, can be generated by equivalent stochastic processes.