2216 CHAPTER 64. WIENER PROCESSES

and (W ( f1) ,W ( f2) , · · · ,W ( fn)) is normally distributed with mean 0. Next define F =σ (W (h) : h ∈ H) .

Consider the special example where H = L2 (0,∞;R) , real valued functions which aresquare integrable with respect to Lebesgue measure. Note that for each t ∈ [0,∞),X[0,t) ∈H. Let

W (t)≡W(X(0,t)

)Then from definition, if t1 < t2 < · · ·< tm, the increments {W (ti)−W (ti−1)} are indepen-dent. This is because, due to the linearity of W, each of these equals

W(X(0,ti)−X(0,ti−1)

)=W

(X(ti−1,ti)

)and from Corollary 64.6.1, the random vector

(W(X(t1,t2)

), · · · ,W

(X(tm,tm−1)

))is nor-

mally distributed with covariance equal to a diagonal matrix. Also

E(

W (t)2)= E

(W(X(0,t)

)2)=∫

∞

0X 2

(0,t)ds = t.

More generally,

W (t)−W (s) =W(X(0,t)

)−W

(X(0,s)

)=W

(X(s,t)

)W (t− s) =W

(X(0,t−s)

)so both W (t)−W (s) and W (t− s) are normally distrubuted with mean 0 and variance t−s.What about the Holder continuity? The characteristic function of W (t)−W (s) is

E(

eiλ (W (t−s)))= e

12 λ

2|t−s|

Consider a few derivatives of the right side with respect to λ and then let λ = 0. This willyield E ((W (t)−W (s))n) for n = 1,2,3,4.

0, |s− t| ,0,3 |s− t|2

You see the pattern. By induction, you can show that E((W (t)−W (s))2m

)=Cm |t− s|m.

By the Kolmogorov Centsov theorem, Theorem 62.2.3,

E(

sup0≤s<t≤T

∥W (t)−W (s)∥(t− s)γ

)≤Cm

whenever γ < β/α = m−12m . Thus the above is true whenever γ < 1/2. It follows that there

exists a set of measure zero off which t →W (t) is Holder continuous with exponent γ <1/2.

Thus this gives a construction of the real Wiener process. Now consider the normalfiltration

Fs ≡ ∩t>sσ (W (u)−W (r) : 0≤ r < u≤ t)

By Lemma 64.4.2, {W (t)} is a martingale with respect to this filtration, because of theindependence of the increments.