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

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 $P$ -trivial σ-algebra, i.e. all events have probability $0$ or $1$ .[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

Tsirel'son, Boris S. (1975). "An example of a stochastic differential equation that has no strong solution". Teor. veroyatnost. I primes. 20 (2): 427–430. doi:10.1137/1120049.
Rogers, L. C. G.; Williams, David (2000). Diffusions, Markov Processes and Martingales: Volume 2, Itô Calculus. United Kingdom: Cambridge University Press. p. 156.
Yano, Kouji; Yor, Marc (2010). "Around Tsirelson's equation, or: The evolution process may not explain everything". Probab. Surveys. 12: 1–12. arXiv:0906.3442. doi:10.1214/15-PS256.

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[1] Tsirel'son, Boris S. (1975). "An example of a stochastic differential equation that has no strong solution". Teor. veroyatnost. I primes. 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". Probab. Surveys. 12: 1–12. arXiv:0906.3442. doi:10.1214/15-PS256.