65.5. THE GENERAL INTEGRAL 2247
mn
∑j=0
∞
∑k=1
(ψk(t ∧ tn
j+1)−ψk
(t ∧ tn
j))((
f nj −Φ
(tn
j)◦ J−1)(Jgk)
)The first expression does not depend on J or U1. I need only argue that the second expres-sion converges to 0 as n→ ∞. The infinite sum converges in L2 (Ω;H) and also, as in theabove diversion, the independence of the ψk implies that
E
∣∣∣∣∣ mn
∑j=0
∞
∑k=1
(ψk(t ∧ tn
j+1)−ψk
(t ∧ tn
j))((
f nj −Φ
(tn
j)◦ J−1)(Jgk)
)∣∣∣∣∣2
H
=
mn
∑j=0
(t ∧ tn
j+1− t ∧ tnj) ∞
∑k=1
E(∣∣( f n
j −Φ(tn
j)◦ J−1)(Jgk)
∣∣2H
)
=mn
∑j=0
(t ∧ tn
j+1− t ∧ tnj)
E(∣∣∣∣ f n
j −Φ(tn
j)◦ J−1∣∣∣∣2
L2(JQ1/2U,H)
)=
∫ t
aE(∣∣∣∣Φn−Φ
n ◦ J−1∣∣∣∣2L2(JQ1/2U,H)
)ds
which is given to converge to 0 since both converge to Φ◦J−1. Consequently, the stochasticintegral defined above does not depend on J or U1.
It is interesting to note that in the above definition, the approximate problems do appearto depend on J and U1 but the limiting stochastic process does not. Since it is the case thatthe stochastic integral is independent of U1 and J, it can only be dependent on Q1/2U andU, and so we refer to W (t) as a cylindrical process on U. By Lemma 65.4.3 you can takeU1 = U and so you can consider the finite sums defining the Wiener process to be in Uitself. From the proof of this lemma, you can even have J being the identity on the spanof the first n vectors in the orthonormal basis for Q1/2U . The case where Q is trace classfollows in the next section. In this case, W is an actual Q Wiener process on U .
The following corollary follows right away from the above theorem.
Corollary 65.5.4 Let Φ,Ψ ∈ L2([a,T ]×Ω;L2
(Q1/2U,H
))and suppose they are both
progressively measurable. Then
E((∫ t
aΦdW,
∫ t
aΨdW
)H
)= E
(∫ t
a(Φ,Ψ)L2(Q1/2U,H) ds
)Also if L is in L∞ (Ω,L (H,H)) and is Fa measurable, then
L∫ t
aΦdW =
∫ t
aLΦdW (65.5.13)
and
E((
L∫ t
aΦdW,
∫ t
aΨdW
)H
)= E
(∫ t
a(LΦ,Ψ)L2(Q1/2U,H) ds
). (65.5.14)