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

where is the one-dimensional Brownian motion. Tsirelson chose the drift to be a bounded measurable function that depends on the past times of but is independent of the natural filtration of the Brownian motion. This gives a weak solution, but since the process is not -measurable, not a strong solution.

Tsirelson's Drift

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Let

  •   and   be the natural Brownian filtration that satisfies the usual conditions,
  •   and   be a descending sequence   such that  ,
  •   and  ,
  •   be the decimal part.

Tsirelson now defined the following drift

 

Let the expression

 

be the abbreviation for

 

Theorem

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According to a theorem by Tsirelson and Yor:

1) The natural filtration of   has the following decomposition

 

2) For each   the   are uniformly distributed on   and independent of   resp.  .

3)   is the  -trivial σ-algebra, i.e. all events have probability   or  .[2][3]

Literature

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  • 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

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  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.