2054 CHAPTER 62. STOCHASTIC PROCESSES

=CT 1+β 2k(αγ−(1+β ))

There are 2k of these differences and so letting

Nk = ∪2k

j=1

[∣∣∣∣∣∣X (rkj+1

)−X

(rk

j

)∣∣∣∣∣∣> 2−γk]

it follows

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 59.1.2, 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, thereexists M (ω) such that if k ≥M (ω) , then for each j,∣∣∣∣∣∣X (rk

j+1

)(ω)−X

(rk

j

)(ω)∣∣∣∣∣∣≤ 2−γk. (62.2.7)

It follows from Lemma 62.2.1 that t → X (t)(ω) is Holder continuous on D with Holderexponent γ. Note the constant is a measurable function of ω, depending on how manymeasurable Nk which contain ω .

By Lemma 62.1.3, 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 62.1.3t → 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) . (62.2.8)

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)

because if ω ∈ N both sides are 0 and if ω ∈ NC then the above limit in 62.2.8 holds andso if ||Y (t)(ω)−X (t)(ω)|| > ε, the same is true of ||X (d)(ω)−X (t)(ω)|| whenever dis close enough to t and so by Fatou’s lemma,

P([||Y (t)−X (t)||> ε]) =∫

X[||Y (t)−X(t)||>ε]∩NC (ω)dP

≤∫

lim infd→t

X[||X(d)−X(t)||>ε] (ω)dP

≤ lim infd→t

∫X[||X(d)−X(t)||>ε] (ω)dP

≤ lim infd→t

P([||X (d)−X (t)||α > ε

α])

≤ lim infd→t

ε−α

∫[||X(d)−X(t)||α>εα ]

||X (d)−X (t)||α dP

≤ lim infd→t

Cεα|d− t|1+β = 0.