2248 CHAPTER 65. STOCHASTIC INTEGRATION
Proof: First note that(∫ t
aΦdW,
∫ t
aΨdW
)H=
14
[∣∣∣∣∫ t
a(Φ+Ψ)dW
∣∣∣∣2H−∣∣∣∣∫ t
a(Φ−Ψ)dW
∣∣∣∣2H
]
and so from the above theorem,
E((∫ t
aΦdW,
∫ t
aΨdW
)H
)=
= E
(14
[∣∣∣∣∫ t
a(Φ+Ψ)dW
∣∣∣∣2H−∣∣∣∣∫ t
a(Φ−Ψ)dW
∣∣∣∣2H
])14
E(∫ t
a||Φ+Ψ||2
L2(Q1/2U,H) ds)+
14
E(∫ t
a||Φ−Ψ||2
L2(Q1/2U,H) ds)
= E(∫ t
a
14
[||Φ+Ψ||2
L2(Q1/2U,H) + ||Φ−Ψ||2L2(Q1/2U,H)
]ds)
= E(∫ t
a(Φ,Ψ)L2(Q1/2U,H) ds
).
Now consider the last claim. First suppose L = lXA where A ∈Fa, and l ∈L (H,H).Also suppose Φ is an elementary function
Φ =n
∑i=0
ψ iX(si,si+1]
Then
L∫ t
aΦdW = lXA
n
∑i=0
ψ i (W (t ∧ si+1)−W (t ∧ si))
=n
∑i=0
lXAψ i (W (t ∧ si+1)−W (t ∧ si))
Thus 65.5.13 also holds for L a simple function which is Fa measurable. For generalL ∈ L∞ (Ω,L (H,H)) , approximating with a sequence of such simple functions Ln yields
L∫ t
aΦdW = lim
n→∞Ln
∫ t
aΦdW = lim
n→∞
∫ t
aLnΦdW =
∫ t
aLΦdW
because LnΦ→ LΦ in L2([a,T ]×Ω;L2
(Q1/2U,H
)). Now what about general Φ? Let
{Φn} be elementary functions converging to Φ ◦ J−1 in L2([a,T ]×Ω;L2
(JQ1/2U,H
)).
Then by definition of the integral,
L∫ t
aΦdW = lim
n→∞L∫ t
aΦndW = lim
n→∞
∫ t
aLΦndW =
∫ t
aLΦdW