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

[math]\displaystyle{ dX_t = a[t,(X_s, s\leq t)]dt + dW_t, \quad X_0=0, }[/math]

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

Tsirelson's Drift

Let

• [math]\displaystyle{ \mathcal{F}_t^{W}=\sigma(W_s : 0 \leq s \leq t) }[/math] and [math]\displaystyle{ \{\mathcal{F}_t^{W}\} _{t\in \R_+} }[/math] be the natural Brownian filtration that satisfies the usual conditions,
• [math]\displaystyle{ t_0=1 }[/math] and [math]\displaystyle{ (t_n)_{n\in-\N} }[/math] be a descending sequence [math]\displaystyle{ t_0\gt t_{-1}\gt t_{-2} \gt \dots, }[/math] such that [math]\displaystyle{ \lim_{n\to -\infty } t_n=0 }[/math],
• [math]\displaystyle{ \Delta X_{t_n}=X_{t_n}-X_{t_{n-1}} }[/math] and [math]\displaystyle{ \Delta t_n=t_n-t_{n-1} }[/math],
• [math]\displaystyle{ \{x\}=x-\lfloor x \rfloor }[/math] be the decimal part.

Tsirelson now defined the following drift

[math]\displaystyle{ 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). }[/math]

Let the expression

[math]\displaystyle{ \eta_n=\xi_n+\{\eta_{n-1}\} }[/math]

be the abbreviation for

[math]\displaystyle{ \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\}. }[/math]

Theorem

According to a theorem by Tsirelson and Yor:

1) The natural filtration of [math]\displaystyle{ X }[/math] has the following decomposition

[math]\displaystyle{ \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 }[/math]

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

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

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.