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. Tsirelson's equation is of the form
 * $$dX_t = a[t,(X_s, s\leq t)]dt + dW_t, \quad X_0=0,$$

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

Tsirelson's Drift
Let
 * $$\mathcal{F}_t^{W}=\sigma(W_s : 0 \leq s \leq t)$$ and $$\{\mathcal{F}_t^{W}\} _{t\in \R_+}$$ be the natural Brownian filtration that satisfies the usual conditions,
 * $$t_0=1$$ and $$(t_n)_{n\in-\N}$$ be a descending sequence $$t_0>t_{-1}>t_{-2} >\dots,$$ such that $$\lim_{n\to -\infty } t_n=0$$,
 * $$\Delta X_{t_n}=X_{t_n}-X_{t_{n-1}}$$ and $$\Delta t_n=t_n-t_{n-1}$$,
 * $$\{x\}=x-\lfloor x \rfloor$$ be the decimal part.

Tsirelson now defined the following drift
 * $$a[t,(X_s, s\leq t)]=\sum\limits_{n\in -\N}\bigg\{\frac{\Delta X_{t_n}}{\Delta t_n}\bigg\}1_{(t_n,t_{n+1}]}(t).$$

Let the expression
 * $$\eta_n=\xi_n+\{\eta_{n-1}\}$$

be the abbreviation for
 * $$\frac{\Delta X_{t_{n+1}}}{\Delta t_{n+1}}=\frac{\Delta W_{t_{n+1}}}{\Delta t_{ n+1}}+\bigg\{\frac{\Delta X_{t_n}}{\Delta t_n}\bigg\}.$$

Theorem
According to a theorem by Tsirelson and Yor:

1) The natural filtration of $$X$$ has the following decomposition
 * $$\mathcal{F}_t^{X}=\mathcal{F}_t^{W} \vee \sigma\big(\{\eta_{n-1}\}\big),\quad \forall t\geq 0, \quad \forall t_n\leq t$$

2) For each $$n\in -\N$$ the $$\{\eta_n\}$$ are uniformly distributed on $$[0,1)$$ and independent of $$(W_t)_{t\geq 0}$$ resp. $$\mathcal{F}_{\infty}^{W}$$.

3) $$\mathcal{F}_{0+}^{X}$$ is the $$P$$-trivial σ-algebra, i.e. all events have probability $$0$$ or $$1$$.