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

$d{X}_t=a[t,(X_s,s\le t)]dt+d{W}_t,X_0=0,$

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 ${ℱ}^W$ of the Brownian motion. This gives a weak solution, but since the process $X$ is not ${ℱ}_{\mathrm{\infty }}^W$-measurable, not a strong solution.

Tsirelson's Drift

Let

• ${ℱ}_t^W=\sigma (W_s:0\le s\le t)$ and $\{{ℱ}_t^W{\}}_{t\in {ℝ}_{+}}$ be the natural Brownian filtration that satisfies the usual conditions,
• $t_0=1$ and $(t_n{)}_{n\in -\mathbb{N}}$ be a descending sequence $t_0>t_{-1}>t_{-2}>\dots ,$ such that $\underset{n\to -\mathrm{\infty }}{\lim }{t}_n=0$,
• $\mathrm{\Delta }{X}_{t_n}=X_{t_n}-X_{t_{n-1}}$ and $\mathrm{\Delta }{t}_n=t_n-t_{n-1}$,
• $\{x\}=x-\lfloor x\rfloor$ be the decimal part.

Tsirelson now defined the following drift

$a[t,(X_s,s\le t)]=\sum _{n\in -\mathbb{N}}\left\{\frac{\mathrm{\Delta }{X}_{t_n}}{\mathrm{\Delta }{t}_n}\right\}{1}_{(t_n,t_{n+1}]}(t).$

Let the expression

${\eta }_n={\xi }_n+\{{\eta }_{n-1}\}$

be the abbreviation for

$\frac{\mathrm{\Delta }{X}_{t_{n+1}}}{\mathrm{\Delta }{t}_{n+1}}=\frac{\mathrm{\Delta }{W}_{t_{n+1}}}{\mathrm{\Delta }{t}_{n+1}}+\left\{\frac{\mathrm{\Delta }{X}_{t_n}}{\mathrm{\Delta }{t}_n}\right\}.$

Theorem

According to a theorem by Tsirelson and Yor:

1) The natural filtration of $X$ has the following decomposition

${ℱ}_t^X={ℱ}_t^W\vee \sigma (\{{\eta }_{n-1}\}),\mathrm{\forall }t\ge 0,\mathrm{\forall }{t}_n\le t$

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

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

Literature

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