A remark on Tsirelson's stochastic differential equation

  • M. Émery
  • W. Schachermayer
Questions de Filtrations
Part of the Lecture Notes in Mathematics book series (LNM, volume 1709)


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).


Brownian Motion Stochastic Differential Equation Strong Solution Innovation Problem Natural Filtration 
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© Springer-Verlag 1999

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  • M. Émery
  • W. Schachermayer

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