View source for 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]].<ref>{{cite journal|first=Boris S.|last=Tsirel'son |title= An Example of a Stochastic Differential Equation Having No Strong Solution|journal= Theory of Probability & Its Applications|volume=20 |number=2 |date=1975 |pages=427–430 |doi=10.1137/1120049}}</ref> Tsirelson's equation is of the form
:<math>dX_t = a[t,(X_s, s\leq t)]dt + dW_t, \quad X_0=0,</math>
where <math>W_t</math> is the one-dimensional [[Brownian motion]]. Tsirelson chose the drift <math>a</math> to be a bounded [[measurable function]] that depends on the past times of <math>X</math> but is independent of the [[natural filtration]] <math >\mathcal{F}^W</math> of the Brownian motion. This gives a weak solution, but since the process <math>X</math> is not <math>\mathcal{F}_{\infty}^W</math>-measurable, not a strong solution.

== Tsirelson's Drift ==
Let
*<math>\mathcal{F}_t^{W}=\sigma(W_s : 0 \leq s \leq t)</math> and <math>\{\mathcal{F}_t^{W}\} _{t\in \R_+}</math> be the natural Brownian filtration that satisfies the usual conditions,
*<math>t_0=1</math> and <math>(t_n)_{n\in-\N}</math> be a descending sequence <math>t_0>t_{-1}>t_{-2} >\dots,</math> such that <math>\lim_{n\to -\infty } t_n=0</math>,
* <math>\Delta X_{t_n}=X_{t_n}-X_{t_{n-1}}</math> and <math>\Delta t_n=t_n-t_{n-1}</math>,
* <math>\{x\}=x-\lfloor x \rfloor</math> be the [[decimal place|decimal part]].

Tsirelson now defined the following drift
:<math>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>\eta_n=\xi_n+\{\eta_{n-1}\}</math>
be the abbreviation for
:<math>\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 [[Marc Yor|Yor]]:

1) The natural filtration of <math>X</math> has the following decomposition
: <math>\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>n\in -\N</math> the <math>\{\eta_n\}</math> are [[Continuous uniform distribution|uniformly distributed]] on <math>[0,1)</math> and independent of <math>(W_t)_{t\geq 0}</math> resp. <math>\mathcal{F}_{\infty}^{W}</math>.

3) <math>\mathcal{F}_{0+}^{X}</math> is the <math>P</math>-[[trivial σ-algebra]], i.e. all events have probability <math>0</math> or <math>1</math>.<ref>{{cite book|first1=L. C. G.|last1=Rogers|first2=David|last2=Williams |publisher=Cambridge University Press |title=Diffusions, Markov Processes and Martingales: Volume 2, Itô Calculus |place=United Kingdom |date=2000 |pages=156}}</ref><ref>{{cite journal|first1=Kouji|last1=Yano|first2=Marc|last2=Yor |title=Around Tsirelson's equation, or: The evolution process may not explain everything |journal=  Probability Surveys|volume=12 |date=2010 |pages=1–12 |doi=10.1214/15-PS256|arxiv=0906.3442 }}</ref>

== Literature ==
* {{cite book|first1=L. C. G.|last1=Rogers|first2=David|last2=Williams |publisher=Cambridge University Press |title=Diffusions, Markov Processes and Martingales: Volume 2, Itô Calculus |place=United Kingdom |date=2000 |pages=155–156}}

== References ==
<references />

[[Category:Stochastic differential equations]]
[[Category:Stochastic processes]]