64.6. WIENER PROCESSES, ANOTHER APPROACH 2211
=1√
2π |h|
∫∞
−∞
eu∫
∞
ue− 1
2y2
|h|2 dydu =1√2π
∫∞
−∞
∫∞
u/|h|eue−
12 v2
dvdu
=1√2π
∫∞
−∞
∫ |h|v−∞
eue−12 v2
dudv =1√2π
∫∞
−∞
e−12 v2
e|h|vdv
=1√2π
√2√
πe|h|2/2 = e
12 |h|
2< ∞
If h = 0,W (h) would be 0 because by the construction, E(
W (0)2)= (0,0)H = 0. Then
∫Ω
eW (h)dP =∫
Ω
e0dP = 1
Consider the last claim. It is enough to assume p is an integer.∣∣∣∣∣ n
∑k=0
W (h)k
k!− eW (h)
∣∣∣∣∣ =
∣∣∣∣∣ ∞
∑k=n+1
W (h)k
k!
∣∣∣∣∣= ∣∣∣W (h)n+1∣∣∣ ∣∣∣∣∣ ∞
∑k=0
W (h)k
(n+1+ k)!
∣∣∣∣∣=
∣∣∣W (h)n+1∣∣∣ ∣∣∣∣∣ ∞
∑k=0
W (h)k
k!k!
(n+1+ k)!
∣∣∣∣∣≤
∣∣∣W (h)n+1∣∣∣ 1(n+1)!
∣∣∣∣∣ ∞
∑k=0
W (h)k
k!
∣∣∣∣∣=∣∣∣∣∣W (h)n+1
(n+1)!
∣∣∣∣∣eW (h)
This converges to 0 for each ω because it says nothing more than that the nth term of aconvergent sequence converges to 0.
∫Ω
(∣∣∣∣∣W (h)n+1
(n+1)!
∣∣∣∣∣eW (h)
)2p
dP =∫
Ω
(W (h)n+1
(n+1)!
)2p(eW (h)
)2pdP
=
(1
(n+1)!
)2p 1√2π |h|
∫R
e− 1
2x2
|h|2 e2pxx2p(n+1)dx
=
(1
(n+1)!
)2p 1√2π |h|
e2p|h|2∫R
e− 1
2|h|2(x−2p|h|2)
2
x2p(n+1)dx
≤(
1(n+1)!
)2p 22p(n+1)√
2π |h|e2p|h|2
∫R
e− 1
2|h|2(x−2p|h|2)
2 (x−2p |h|2
)2p(n+1)dx
+
(1
(n+1)!
)2p 22p(n+1)√
2π |h|e2p|h|2
∫R
e− 1
2|h|2(x−2p|h|2)
2 (2p |h|2
)2p(n+1)dx