A remark on Tsirelson's stochastic differential equation

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Abstract

Tsirelson's stochastic differential equation is called “celebrated and mysterious” by Rogers and Williams [16]. This note aims at making it a little more celebrated and a little less mysterious.

Using a deterministic time-change, we translate the study of Tsirelson's equation into the study of “eternal” Brownian motion on the circle. This allows us to show that the filtration generated by any solution of Tsirelson's equation is also generated by some Brownian motion (which, however, cannot be the Brownian motion driving the equation, because the equation has no strong solution).