30.2. KOLMOGOROV ČENTSOV CONTINUITY THEOREM 811

Then there exists a stochastic process Y such that for a.e. ω, t → Y (t)(ω) is Hölder con-tinuous with exponent γ < β

αand for each t, P([∥X (t)−Y (t)∥> 0]) = 0. (Y is a version

of X .)

Proof: Let rmj denote j

( T2m

)where j ∈ {0,1, · · · ,2m} . Also let Dm =

{rm

j

}2m

j=1and

D = ∪∞m=1Dm. Consider the set,

[∥X (t)−X (s)∥> δ ]

By 30.7,

P([∥X (t)−X (s)∥> δ ])δα ≤

∫[||X(t)−X(s)||>δ ]

∥X (t)−X (s)∥α dP

≤ C |t− s|1+β . (30.8)

Letting t = rkj+1, s = rk

j ,and δ = 2−γk where γ ∈(

0, β

α

), this yields

P([∥∥∥X

(rk

j+1

)−X

(rk

j

)∥∥∥> 2−γk])≤C2αγk

(T 2−k

)1+β

=CT 1+β 2k(αγ−(1+β ))

There are 2k of these differences so letting Nk = ∪2k

j=1

[∥∥∥X(

rkj+1

)−X

(rk

j

)∥∥∥> 2−γk]

itfollows

P(Nk)≤C2αγk(

T 2−k)1+β

2k =C2k(αγ−β )T 1+β .

Since γ < β/α, ∑∞k=1 P(Nk) ≤ CT 1+β

∑∞k=1 2k(αγ−β ) < ∞ and so by the Borel Cantelli

lemma, Lemma 26.1.2, there exists a set of measure zero N, such that if ω /∈ N, then ω

is in only finitely many Nk. In other words, for ω /∈ N, there exists M (ω) such that ifk ≥M (ω) , then for each j,∣∣∣∣∣∣X (rk

j+1

)(ω)−X

(rk

j

)(ω)∣∣∣∣∣∣≤ 2−γk. (30.9)

It follows from Lemma 30.2.1 that t → X (t)(ω) is Holder continuous on D with Holderexponent γ. Note the constant is a measurable function of ω, depending on the number ofmeasurable Nk which contain ω .

By Lemma 30.1.3, one can define Y (t)(ω) to be the unique function which extendsd → X (d)(ω) off D for ω /∈ N and let Y (t)(ω) = 0 if ω ∈ N. Thus by Lemma 30.1.3t → Y (t)(ω) is Holder continuous. Also, ω → Y (t)(ω) is measurable because it is thepointwise limit of measurable functions

Y (t)(ω) = limd→t

X (d)(ω)XNC (ω) . (30.10)

It remains to verify the claim that Y (t)(ω) = X (t)(ω) a.e.

X[∥Y (t)−X(t)∥>ε]∩NC (ω)≤ lim infd→t

X[∥X(d)−X(t)∥>ε]∩NC (ω)