Daniell Kolmogorov Extension / Consistency Theorem Proof, Example

The Kolmogorov Continuity Theorem is a theorem which states that a suitably 'consistent' collection of finite-dimensional distributions will define a stochastic process. The theorem was discovered by Percy John Daniell, the British Mathematician and was credited to the Soviet mathematician Andrey Nikolaevich Kolmogorov, hence the name the Daniell-Kolmogorov Existence Theorem. It is also called as the Daniell-Kolmogorov extension theorem or the Kolmogorov consistency theorem. This theorem provides existence results for nice probability measures on path (function) spaces.

Daniell Kolmogorov Extension Theorem or Kolmogorov Consistency Theorem


The Daniell-Kolmogorov extension or consistency theorem is one of the first deep theorems of the stochastic processes theory. It provides existence results for nice probability measures on function spaces.


Ε (|| Xt - Xs ||α) ≤ c |t-s|^1+z, Let α, ε, c >0. If a d-dimensional process (Xt)t ∈ [0,1] defined on a probability space (Ω, ƒ, P) satisfies for s,t ∈ [0,1], then there exists a modification of the process (Xt)t∈[0,1] that is a continuous process and whose paths are γ-Hölder for every γ ∈ [0, ε/α).


We make the proof for d=1 and let the reader extend it as an exercise to the case d ≥ 2. For n ∈ N, we denote Dn = {k/2n, k=0,...,2n} and D = Un ∈ N Dn. Let γ ∈ [0, ε/α). From Chebychev’s inequality: P (max1 ≤ k ≤ 2n |Xk/2n} - Xk-1/2n| ≥ 2-γ n) = P (U1 ≤ k ≤ 2n |Xk/2n} - Xk-1/2n| ≥ 2-γ n) ≤ Σk=12n P |Xk/2n} - Xk-1/2n| ≥ 2-γ n) ≤ Σk=12n E|Xk/2n} - Xk-1/2n| ≥ 2-γ n) / 2-γ α n ≤ c 2-n(ε - γ α) Therefore, since γ α < ε, we deduce Σk=n+∞Σ P (max1 ≤ k ≤ 2n |Xk/2n} - Xk-1/2n| ≥ 2-γ n) < +∞. From the Borel-Cantelli lemma, we can thus find a set Ω* ∈ ƒ such that P (Ω*) = 1 and such that for ω ∈ Ω*, there exists N(ω) such that for n ≥ N(ω), max1 ≤ k ≤ 2n |Xk/2n} - Xk-1/2n| ≥ 2-γ n. In particular, there exists an almost surely finite random variable C such that for every n ≥ 0, max1 ≤ k ≤ 2n |Xk/2n} - Xk-1/2n| ≥ C 2-γ n. We now claim that the paths of the restricted process X / Ω* are consequently γ-Hölder on D. Indeed, let s,t ∈ D, t ≠ s. We can find n ≥ 0 such that 1 / 2n+1 ≤ |s-t| ≤ 1/2n. We now pick an increasing and stationary sequence (sk)k ≥ n converging toward s, such that sk ∈ Dk and |sk+1 - sk| = 2-(k+1) or 0. In the same way, we can find an analogue sequence (tk)k ≥ n that converges toward t and such that sn and tn are neighbors in Dn. We have then: Xt - Xs = Σi=n+∞ (Xsi+1 - Xsi) + (Xsn - Xtn) + Σi=n+∞Σ (Xti - Xti+1), where the above sums are actually finite. Therefore, |Xt - Xs| ≤ C 2{-γ n + 2 Σk=n+∞ C 2-γ(k+1) ≤ 2C Σk=n+∞ 2-γk ≤ 2C / 1-2 2-γ n Hence the paths of X/Ω* are γ-Hölder on the set D. For ω ∈ Ω*, let t → X't (ω) be the unique continuous function that agrees with t → Xt (ω) on D. For ω ∉ Ω*, we set X't (ω) = 0. The process (X't)t ∈ [0,1] is the desired modification of (Xt)t ∈ [0,1].

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