Tsirelson's stochastic differential equation (also Tsirelson's drift or Tsirelson's equation) is a stochastic differential equation which has a weak solution but no strong solution. It is therefore a counter-example and named after its discoverer Boris Tsirelson. Tsirelson's equation is of the form
where is the one-dimensional Brownian motion. Tsirelson chose the drift to be a bounded measurable function that depends on the past times of but is independent of the natural filtration of the Brownian motion. This gives a weak solution, but since the process is not -measurable, not a strong solution.
Tsirelson's Drift Let and be the natural Brownian filtration that satisfies the usual conditions, and be a descending sequence such that , and , be the decimal part.
Tsirelson now defined the following drift
Let the expression
be the abbreviation for
Theorem According to a theorem by Tsirelson and Yor:
1) The natural filtration of has the following decomposition
2) For each the are uniformly distributed on and independent of resp. .
3) is the -trivial σ-algebra, i.e. all events have probability or .
See also Tanaka equation
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