64.3. WIENER PROCESSES IN SEPARABLE BANACH SPACE 2185

Thus ∪n,p∈NCnp contains the set of ω such that t →W (t,ω) is differentiable for somes ∈ [0,a).

Now define uniform partitions of [0,a),{

tnk

}2n

k=0 such that∣∣tnk − tn

k−1∣∣= a2−n ≡ δ n

LetDnp ≡ ∪2n−1

i=0

(∩5

r=1 [|W (tni + rδ n)−W (tn

i +(r−1)δ n)| ≤ 10pδ n])

If ω ∈Cnp, then for some s∈ [0,a), the condition of 64.2.5 holds. Suppose k is the numbersuch that s ∈ [tn

k−1, tnk ). Then for r ∈ {1,2,3,4,5} ,∣∣W (

tnk−1 + rδ n,ω

)−W

(tnk−1 +(r−1)δ n,ω

)∣∣≤∣∣W (

tnk−1 + rδ n,ω

)−W (s,ω)

∣∣+ ∣∣W (s,ω)−W(tnk−1 +(r−1)δ n,ω

)∣∣≤ 5pδ n +5pδ n = 10pδ n

Thus Cnp ⊆ Dnp. Now from 64.2.4,

P(Dnp)≤ 2nCδ5/2n =Ca5/22n (2−n)5/2

=C(√

a)5 2−

32 n (64.2.6)

LetCp = ∪∞

n=1∩∞k=n Ckp ⊆ ∪∞

n=1∩∞k=n Dkp.

It was just shown in 64.2.6 that P(∩∞

k=nDkp)= 0 and so Cp has measure 0. Thus

∪∞p=1Cp, the set of points, ω where t →W (t,ω) could have a derivative has measure 0.

Taking the union of the exceptional sets corresponding to intervals [0,n) for n ∈ N, thisproves the theorem.

This theorem on nowhere differentiability is very important because it shows it is doubt-ful one can define an integral

∫f (s)dW (s) by simply fixing ω and then doing some sort

of Stieltjes integral in time. The reason for this is that the nowhere differentiability of Wimplies it is also not of bounded variation on any interval since if it were, it would equal thedifference of two increasing functions and would therefore have a derivative at a.e. point.

I have presented the theorem on nowhere differentiability for one dimensional Wienerprocesses but the same proof holds with minor modifications if you have defined the Wienerprocess in Rn or you could simply consider the components and apply the above result.

64.3 Wiener Processes In Separable Banach SpaceHere is an important lemma on which the existence of Wiener processes will be based.

Lemma 64.3.1 There exists a sequence of real Wiener processes, {ψk (t)}∞

k=1 which havethe following properties. Let t0 < t1 < · · ·< tn be an arbitrary sequence. Then the randomvariables {

ψk (tq)−ψk(tq−1

): (q,k) ∈ (1,2, · · · ,n)× (k1, · · · ,km)

}(64.3.7)

are independent. Also each ψk is Holder continuous with exponent γ for any γ < 1/2 andfor each m ∈ N there exists a constant Cm independent of k such that∫

Ω

|ψk (t)−ψk (s)|2m dP≤Cm |t− s|m (64.3.8)