Tsirelson's stochastic differential equation explained

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,} 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 $\backslash mathcal^W$ of the Brownian motion. This gives a weak solution, but since the process X is not l{F} ^{W} _{inf ty} -measurable, not a strong solution.

Tsirelson's Drift

Let

l{F} ^{W} _{t} = \sigma(W _{s} : 0 \leq s \leq t) and \{ l{F} ^{W} _{t} \} ^{t\in \R _{+}} be the natural Brownian filtration that satisfies the usual conditions,t _{0=1} and (t _{n)} _{n\in-\N} be a descending sequence t _{0>t} _{-1} >t _{-2} > ... , such that \lim _{n\to} 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} \ { (\Delta X _{t _{n}})/(\Delta t _{n}) \ } 1 _{(t _{n,t} _{n+1} ]} (t). Let the expressionη _{n=\xi} _{n+\{η} _{n-1} \} be the abbreviation for(\Delta X _{t _{n+1}})/(= \Delta t _{n+1}) (\Delta W _{t _{n+1}})/(+ \ { \Delta t) (\Delta X _{t _{n}})/(\Delta t _{n}) \ }.

Theorem

According to a theorem by Tsirelson and Yor:

1) The natural filtration of

X has the following decompositionl{F} ^{X} _{t} = l{F} ^{W} _{t} \vee \sigma ( \{ η _{n-1} \} ) , \forall t\geq 0 , \forall t _{n\leq} t

2) For each

n\in - \N the \{ η _{n\}} are uniformly distributed on [ 0 , 1 ) and independent of (W _{t)} _{t\geq} resp. l{F} ^{W} _{inf ty} .

3)

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

Literature

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

References

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