73.3. PRELIMINARY RESULTS 2467
= E
(⟨BX(tn
j),
(E∫ tn
j+1∧σ
tnj∧σ
ZdW |Ftnj
)⟩)
= E(⟨
BX(tn
j),E(∫ tn
j+1∧σ
0ZdW −
∫ tnj∧σ
0ZdW |Ftn
j
)⟩)= E
(⟨BX(tn
j),0⟩)
= 0
and so this is a martingale by Lemma 63.1.1. I want to write the formula in 73.3.3 as astochastic integral. First note that W has values in U1.
Consider one of the terms of the sum more simply as⟨B∫ b
aZdW,X (a)
⟩, a = tn
k ∧ t, b = tnk+1∧ t.
Then from the definition of the integral, let Zn be a sequence of elementary functions con-verging to Z ◦ J−1 in L2
([a,b]×Ω,L2
(JQ1/2U,W
))and∥∥∥∥∫ t
aZdW −
∫ t
aZndW
∥∥∥∥L2(Ω,W )
→ 0
Using a maximal inequality and the fact that the two integrals are martingales along with theBorel Cantelli lemma, there exists a set of measure 0 N such that for ω /∈N, the convergenceof a suitable subsequence of these integrals, still denoted by n, is uniform for t ∈ [a,b]. Itfollows that for such ω,⟨
B∫ t
aZdW,X (a)
⟩= lim
n→∞
⟨B∫ t
aZndW,X (a)
⟩. (73.3.4)
Say
Zn (u) =mn−1
∑k=0
Znk X[tn
k ,tnk+1)
(u)
where Znk has finitely many values in L (U1,W )0 , the restrictions of maps in L (U1,W ) to
JQ1/2U, and the tnk refer to a partition of [a,b]. Then the product on the right in 73.3.4 is of
the formmn−1
∑k=0
⟨BZn
k(W(t ∧ tn
k+1)−W (t ∧ tn
k )),X (a)
⟩W ′,W
Note that it makes sense because Znk is the restriction to J
(Q1/2U
)of a map from U1 to W
and so BZnk is a map from U1 to W ′. Then the Wiener process has values in U1 so when you
apply BZnk to W
(t ∧ tn
k+1
)−W
(t ∧ tn
k
), you get something in W ′ and so the duality pairing
is between W ′ and W as shown. Also, Znk
(W(t ∧ tn
k+1
)−W
(t ∧ tn
k
))gives something in W
because the Wiener process has values in U1 and Znk acts on these things to give something
in W . Thus the above equals
=mn−1
∑k=0
⟨BX (a) ,Zn
k(W(t ∧ tn
k+1)−W (t ∧ tn
k ))⟩
W ′,W