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. Tsirelson's equation is of the form
d X t = a [ t , ( X s , s ≤ t ) ] d t + d W t , X 0 = 0 , {\displaystyle dX_{t}=a[t,(X_{s},s\leq t)]dt+dW_{t},\quad X_{0}=0,}
where W t {\displaystyle W_{t}} is the one-dimensional Brownian motion. Tsirelson chose the drift a {\displaystyle a} to be a bounded measurable function that depends on the past times of X {\displaystyle X} but is independent of the natural filtration F W {\displaystyle {\mathcal {F}}^{W}} of the Brownian motion. This gives a weak solution, but since the process X {\displaystyle X} is not F ∞ W {\displaystyle {\mathcal {F}}_{\infty }^{W}} -measurable, not a strong solution.
Tsirelson's Drift Let
F t W = σ ( W s : 0 ≤ s ≤ t ) {\displaystyle {\mathcal {F}}_{t}^{W}=\sigma (W_{s}:0\leq s\leq t)} and { F t W } t ∈ R + {\displaystyle \{{\mathcal {F}}_{t}^{W}\}_{t\in \mathbb {R} _{+}}} be the natural Brownian filtration that satisfies the usual conditions,
t 0 = 1 {\displaystyle t_{0}=1} and ( t n ) n ∈ − N {\displaystyle (t_{n})_{n\in -\mathbb {N} }} be a descending sequence t 0 > t − 1 > t − 2 > … , {\displaystyle t_{0}>t_{-1}>t_{-2}>\dots ,} such that lim n → − ∞ t n = 0 {\displaystyle \lim _{n\to -\infty }t_{n}=0} ,
Δ X t n = X t n − X t n − 1 {\displaystyle \Delta X_{t_{n}}=X_{t_{n}}-X_{t_{n-1}}} and Δ t n = t n − t n − 1 {\displaystyle \Delta t_{n}=t_{n}-t_{n-1}} ,
{ x } = x − ⌊ x ⌋ {\displaystyle \{x\}=x-\lfloor x\rfloor } be the decimal part. Tsirelson now defined the following drift
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