\(\varepsilon\)-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.


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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[6] BERKELEY, CALIFORNIA 94729 E-MAIL: feldman@math.berkeley.edu
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