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

Definition:

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.

Theorem:

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).

Proof:

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].