2200 CHAPTER 64. WIENER PROCESSES

and in general,

E((h,(W (t)−W (s)))2m

U

)=Cm (Qh,h)m (t− s)m .

Now it follows from Minkowsky’s inequality applied to the two integrals ∑∞i=1 and

∫Ω

that

[E(|W (t)−W (s)|2m

)]1/m=

[E

((∞

∑k=1

(ek,W (t)−W (s))2

)m)]1/m

≤∞

∑k=1

[E((ek,W (t)−W (s))2m

)]1/m

=∞

∑k=1

[Cm (Qek,ek)m (t− s)m]

1/m

= C1/mm |t− s|

(∞

∑k=1

(Qek,ek)

)≡C′m |t− s| .

Hence there exists a constant Cm such that

E(|W (t)−W (s)|2m

)≤Cm |t− s|m

By the Kolmogorov Čentsov Theorem, Theorem 62.2.2, it follows that off a set ofmeasure 0, t→W (t,ω) is Holder continuous with exponent γ such that

γ <m−1

2m, m > 2.

Finally, from 64.5.19 with r = 1,

E (exp i(h,W (t)−W (s))U ) = exp(−1

2(t− s)(Qh,h)

)which is the same as E (exp i(h,W (t− s))U ) due to the fact W (0) = 0.

The above has shown that W (t) satisfies the conditions of Lemma 64.4.2 and so it is amartingale with respect to the filtration given there. What is its quadratic variation?

E(||W (t)||2

)=

∞

∑k=1

E ((W (t) ,ek)(W (t) ,ek)) =∞

∑k=1

(Qek,ek) t = trace(Q) t

Is it the case that [W ] (t) = trace(Q) t? Let the filtration be as in Lemma 64.4.2 and letA ∈Fs. Then using the result of that lemma,∫

A

(||W (t)||2− t trace(Q) |Fs

)dP

=∫

A

(||W (t)−W (s)||2 +2(W (t) ,W (s))−||W (s)||2

−(t− s) traceQ− traceQs|Fs)dP