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\(\varepsilon\)-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.)
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