Tsirelson's stochastic differential equation

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

${\displaystyle dX_{t}=a[t,(X_{s},s\leq t)]dt+dW_{t},\quad X_{0}=0,}$

where ${\displaystyle W_{t}}$ is the one-dimensional Brownian motion. Tsirelson chose the drift ${\displaystyle a}$ to be a bounded measurable function that depends on the past times of ${\displaystyle X}$ but is independent of the natural filtration ${\displaystyle {\mathcal {F}}^{W}}$ of the Brownian motion. This gives a weak solution, but since the process ${\displaystyle X}$ is not ${\displaystyle {\mathcal {F}}_{\infty }^{W}}$-measurable, not a strong solution.