Decreasing sequences of \(\sigma\)-fields and a measure change for Brownian motion. I. (English) Zbl 0870.60078

The authors give a negative answer to the following question [originally raised by D. W. Stroock and M. Yor, Ann. Sci. Éc. Norm. Super., IV. Sér. 13, 95-164 (1980; Zbl 0447.60034)]. Let \((\Omega,{\mathfrak F},\mathbb{P},\{B(t)\}_{t\geq 0},\{{\mathfrak F}_t\}_{t\geq 0})\) be a real Brownian motion and \(\mathbb{Q}\) any equivalent probability measure w.r.t. \(\mathbb{P}\). Does there always exist a Brownian motion \((\Omega,{\mathfrak F},\mathbb{Q},\{B'(t)\}_{t\geq 0},\{{\mathfrak F}_t'\}_{t\geq 0})\) such that \({\mathfrak F}_t'={\mathfrak F}_t\)? (If only \({\mathfrak F}_t'\subset{\mathfrak F}_t\) were required, the above question would have an affirmative answer in Girsanov’s theorem.)
The construction of the counterexample relies essentially on the following theorem on reverse filtrations \(\{{\mathfrak F}_n\}_{n\in\mathbb{N}}\), i.e., decreasing sequences of sub-\(\sigma\)-fields \(\{{\mathfrak F}_n\}_{n\in\mathbb{N}}\): There exists a measure \(m\) equivalent to the Bernoulli product measure \(\lambda\) on \(\{0,1\}^{\mathbb{N}}\) such that \((\{0,1\}^{\mathbb{N}},\{{\mathfrak F}_n\}_{n\in\mathbb{N}},m)\) admits no standard extension. Here, a standard extension is another triple \((\widetilde{\mathcal H},\{\widetilde{\mathfrak F}_n\}_{n\in\mathbb{N}},\widetilde m)\) such that (a) \(\widetilde{\mathfrak F}_n=\sigma(\widetilde X_k, k\geq n)\) is generated by independent random variables \(\{\widetilde X_n\}_{n\in\mathbb{N}}\) (standardness), (b) there is a measure preserving map \(\pi:\widetilde{\mathcal H}\to\{0,1\}^{\mathbb{N}}\) such that \(\pi^{-1}({\mathfrak F}_n)\subset\widetilde{\mathfrak F}_n\) and (c) \(\widetilde{\mathbb{E}}(X\circ\pi\mid \widetilde{\mathfrak F}_n)= \mathbb{E}(X\mid{\mathfrak F}_n)\circ\pi\) for all bounded real-valued random variables \(X\) on \(\{0,1\}^{\mathbb{N}}\). The measure \(m\) is explicitly constructed on the coordinate functions \(\{X_n\}_{n\in\mathbb{N}}\) of \(\{0,1\}^{\mathbb{N}}\) (taken as random variables) by means of its conditional probabilities, \(m(X_n=1\mid X_k,\;k\geq n+1)=(1\pm\delta_n)/2\), \(\delta_n\in (0,1)\). In order to prove standardness, A. M. Vershik’s criterion for standardness [Sov. Math., Dokl. 11, 1007-1011 (1970); translation from Dokl. Akad. Nauk SSSR 193, 748-751 (1970; Zbl 0238.28011)] is being used.
In the situation of the above question, the counterexample for \(\mathbb{Q}\) is now given by the following construction: Choose a strictly decreasing sequence of positive numbers \(t_n\to 0\) and put \(X_n:=\text{\textbf{1}}_{(0,\infty)}(B(t_n)-B(t_{n+1}))\) and \(\pi:\Omega\ni\omega\to (X_1(\omega),X_2(\omega),\dots)\). The measure \(\mathbb{Q}\) is given by \({d\mathbb{Q}\over d\mathbb{P}}:= {dm\over d\lambda}\circ\pi\) and it can be seen that \((\Omega, \{\sigma(B(s), s\leq t_n)\}_{n\in\mathbb{N}}, \mathbb{Q})\) and \(\pi\) extend \((\{0,1\}^{\mathbb{N}}, \{\sigma(X_k, k\geq n)\}_{n\in\mathbb{N}}, m)\). By the above theorem, this extension cannot be standard, i.e. \(\sigma(B_s, s\leq t_n)\) cannot be generated by independent random variables, in particular not by another Brownian motion. [For part II see below].


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