2184 CHAPTER 64. WIENER PROCESSES

Corollary 64.1.4 Let {ξ i}∞

i=1 be independent random variables each normal with mean 0and variance 1. Then

W (t,ω)≡∞

∑i=1

(X[0,t],gi

)L2 ξ i (ω)

is a real Wiener process. Furthermore, the distribution of W (t)−W (s) is the same as thedistribution of W (t− s) and W is Holder continuous with exponent γ for any γ < 1/2. Alsofor each α > 1,

E(|W (t)−W (s)|α

)≤Cα |t− s|α/2 E

(|W (1)|α

)64.2 Nowhere Differentiability of Wiener Processes

If W (t) is a Wiener process, it turns out that t →W (t,ω) is nowhere differentiable fora.e. ω. This fact is based on the independence of the increments and the fact that theseincrements are normally distributed.

First note that W (t)−W (s) has the same distribution as (t− s)1/2 W (1) . This is be-cause they have the same characteristic function. Next it follows that because of the inde-pendence of the increments and what was just noted that,

P(∩5

r=1 [|W (t + rδ )−W (t +(r−1)δ )| ≤ Kδ ])

=5

∏r=1

P([|W (t + rδ )−W (t +(r−1)δ )| ≤ Kδ ])

=5

∏r=1

P([∣∣∣δ 1/2W (1)

∣∣∣≤ Kδ

])=

(1√2π

∫ K√

δ

−K√

δ

e−12 t2

dt

)5

≤ Cδ5/2. (64.2.4)

With this observation, here is the proof which follows [120] and according to this referenceis due to Payley, Wiener and Zygmund and the proof is like one given by Dvoretsky, Erdösand Kakutani.

Theorem 64.2.1 Let W (t) be a Wiener process. Then there exists a set of measure 0, Nsuch that for all ω /∈ N,

t→W (t,ω)

is nowhere differentiable.

Proof: Let [0,a] be an interval. If for some ω, t→W (t,ω) is differentiable at some s,then for some n, p > 0, ∣∣∣∣W (t,ω)−W (s,ω)

t− s

∣∣∣∣≤ p

whenever |t− s|< 5a2−n ≡ 5δ n. Define Cnp by{ω : for some s ∈ [0,a),

∣∣∣∣W (t,ω)−W (s,ω)

t− s

∣∣∣∣≤ p if |t− s| ≤ 5δ n

}. (64.2.5)