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

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

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

Tsirelson's Drift

Let

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

Tsirelson now defined the following drift

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

Let the expression

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

be the abbreviation for

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

Theorem

According to a theorem by Tsirelson and Yor:

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

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

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

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

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.

References

1. Tsirel'son, Boris S. (1975). "An Example of a Stochastic Differential Equation Having No Strong Solution". Theory of Probability & Its Applications. 20 (2): 427–430. doi:10.1137/1120049.
2. Rogers, L. C. G.; Williams, David (2000). Diffusions, Markov Processes and Martingales: Volume 2, Itô Calculus. United Kingdom: Cambridge University Press. p. 156.
3. Yano, Kouji; Yor, Marc (2010). "Around Tsirelson's equation, or: The evolution process may not explain everything". Probability Surveys. 12: 1–12. arXiv: 0906.3442 . doi:10.1214/15-PS256.