## $$\varepsilon$$-close measures producing nonisomorphic filtrations.(English)Zbl 0879.60077

Summary: A consequence of the preceding two papers [see the author, L. Dubins, M. Smorodinsky and B. Tsirelson, ibid. 24, No. 2, 882-904 (1996; Zbl 0870.60078) and the author and B. Tsirelson, ibid. 24, No. 2, 905-911 (1996; Zbl 0870.60079)] is this. Let $$\{{\mathcal A}_t:0\leq t<\infty\}$$ be the filtration of a stochastic process on $$(\Omega,{\mathcal A},P)$$. 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 $$(\Omega, ({\mathcal A}_t)_{0\leq t}, Q_\alpha)$$ and $$(\Omega, ({\mathcal A}_t)_{0\leq t}, Q_\beta)$$, $$\alpha\neq\beta$$, can be generated be equivalent stochastic processes.

### MSC:

 60J65 Brownian motion 60G07 General theory of stochastic processes 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)

### Citations:

Zbl 0870.60078; Zbl 0870.60079
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### References:

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