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.
[1] 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
Let Tsirelson now defined the following drift Let the expression be the abbreviation for According to a theorem by Tsirelson and Yor: 1) The natural filtration of
-trivial σ-algebra, i.e. all events have probability