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

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

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

Let

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

Tsirelson now defined the following drift

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

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

be the abbreviation for

${\displaystyle {\frac {\Delta X_{t_{n+1}}}{\Delta t_{n+1}}}={\frac {\Delta W_{t_{n+1}}}{\Delta t_{n+1}}}+{\bigg \{}{\frac {\Delta X_{t_{n}}}{\Delta t_{n}}}{\bigg \}}.}${\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 \}}.}

According to a theorem by Tsirelson and Yor:

1) The natural filtration of ${\displaystyle X}${\displaystyle X} has the following decomposition

${\displaystyle {\mathcal {F}}_{t}^{X}={\mathcal {F}}_{t}^{W}\vee \sigma {\big (}\{\eta _{n-1}\}{\big )},\quad \forall t\geq 0,\quad \forall t_{n}\leq t}${\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}

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

3) ${\displaystyle {\mathcal {F}}_{0+}^{X}}${\displaystyle {\mathcal {F}}_{0+}^{X}} is the ${\displaystyle P}${\displaystyle P}-trivial σ-algebra, i.e. all events have probability ${\displaystyle 0}${\displaystyle 0} or ${\displaystyle 1}${\displaystyle 1}.^{[2] [3]}

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

• Stochastic differential equations
• Stochastic processes