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 ${\mathcal {F}}^{W}$ 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 $\{{\mathcal {F}}_{t}^{W}\}_{t\in \mathbb {R} _{+}}$ 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 $\lim _{n\to -\infty }t_{n}=0$,
• $\Delta X_{t_{n}}=X_{t_{n}}-X_{t_{n-1}}$ and $\Delta t_{n}=t_{n}-t_{n-1}$,
• $\{x\}=x-\lfloor x\rfloor$ be the decimal part.

Tsirelson now defined the following drift

$a[t,(X_{s},s\leq t)]=\sum \limits _{n\in -\mathbb {N} }{\bigg \{}{\frac {\Delta X_{t_{n}}}{\Delta t_{n}}}{\bigg \}}1_{(t_{n},t_{n+1}]}(t).$

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 $n\in -\mathbb {N}$ 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 $1$.

Literature

• Rogers, L. C. G.; Williams, David (2000). Diffusions, Markov Processes and Martingales: Volume 2, Itô Calculus. United Kingdom: Cambridge University Press. pp. 155–156.