2204 CHAPTER 64. WIENER PROCESSES
because ψk (t j)−ψk(t j−1
)is normally distributed having variance t j− t j−1 and mean 0.
Now letting N→ ∞, this implies
E
(exp
m
∑j=1
i(h j,W (t j)−W
(t j−1
))U
)
=m
∏j=1
exp
(−1
2(t j− t j−1
) ∞
∑k=1
λ k (h j,ek)2U
)
=m
∏j=1
exp(−1
2(t j− t j−1
)(Qh,h)U
)=
m
∏j=1
exp(i(h j,W (t j)−W
(t j−1
))U
)(64.5.25)
because of the fact shown above that (h,W (t)−W (s)) is normally distributed with mean 0and variance (t− s)(Qh,h). By Theorem 59.13.3 on Page 1898, this shows the incrementsare independent.
Next consider the continuity assertion. Recall
W (t) =∞
∑k=1
√λ kekψk (t)
where ψk is a real Wiener process. Therefore, letting 2m > 2,m ∈ N and using 64.1.3 forψk and Jensen’s inequality along with Lemma 64.3.1,
E(|W (t)−W (s)|2m
)= E
∣∣∣∣∣ ∞
∑k=1
√λ kek (ψk (t)−ψk (s))
∣∣∣∣∣2m
= E
((∞
∑k=1
λ k |ψk (t)−ψk (s)|2
)m)
≤ E
( ∞
∑k=1
λ k
)m−1∞
∑k=1
λ k |ψk (t)−ψk (s)|2m
≤ Cm
∞
∑k=1
λ kE(|ψk (t)−ψk (s)|
2m)
(64.5.26)
≤ Cm |t− s|m (64.5.27)
By the Kolmogorov Čentsov Theorem, Theorem 62.2.2, it follows that off a set of measure0, t→W (t,ω) is Holder continuous with exponent γ such that
γ <m−1
2m.
Finally, from 64.5.24 taking r = 1,
E (exp i(h,W (t)−W (s))U ) = exp(−1
2(t− s)(Qh,h)
)