2264 CHAPTER 65. STOCHASTIC INTEGRATION
This expression is a function of ω .I want to write this in the form of a stochastic integral. To begin with, consider one of
the terms. For simplicity of notation, consider(∫ b
aZ (u)dW,X (a)
)H
where Z ∈ L2([a,b]×Ω,L2
(Q1/2U,H
))and X (a) ∈ L2 (Ω,H). Also assume the func-
tion of ω, |X (a)|H , is bounded. There is an Ito integral involved in the above. LetZn be a sequence of elementary functions defined on [a,b] which converges to Z ◦ J−1
in L2([a,b]×Ω,L2
(JQ1/2U,H
)). Then by the definition of the integral,∥∥∥∥∫ t
aZ (u)dW −
∫ t
aZn (u)dW
∥∥∥∥L2(Ω,H)
→ 0
Also, by the use of a maximal inequality and the fact that the two integrals above aremartingales, there is a subsequence, still called n and a set of measure zero N such that forω /∈ N, the convergence ∫ t
aZn (u)dW (ω)→
∫ t
aZ (u)dW (ω)
is uniform for t ∈ [a,b]. Therefore, for such ω,(∫ t
aZ (u)dW,X (a)
)H= lim
n→∞
(∫ t
aZn (u)dW,X (a)
)H
Say Zn (u) = ∑mn−1k=0 Zn
k X[tnk ,t
nk+1)
(u) where Znk has finitely many values in L (U1,H)0 , the
restrictions of L (U1,H) to JQ1/2U . Then the inner product in the above formula on theright is of the form
mn−1
∑k=0
(Zn
k(W(t ∧ tn
k+1)−W (t ∧ tn
k )),X (a)
)H
=mn−1
∑k=0
((W(t ∧ tn
k+1)−W (t ∧ tn
k )),(Zn
k )∗X (a)
)U1
=mn−1
∑k=0
R((Zn
k )∗X (a)
)(W(t ∧ tn
k+1)−W (t ∧ tn
k ))
≡∫ t
aR (Z∗nX (a))dW
where R is the Riesz map from U1 to U ′1. Note that R (Z∗nX (a)) has values in
L (U1,R)0 ⊆L2
(JQ1/2U,R
).