2466 CHAPTER 73. THE HARD ITO FORMULA, IMPLICIT CASE
Lemma 73.3.1 In the above situation, there exists a set of measure zero N ⊆Ω and a densesubset of [0,T ] , D such that for ω /∈ N, BX (t,ω) = B(X (t,ω)) for all t ∈ D.
Theorem 73.3.2 Let Z be progressively measurable and in
L2([0,T ]×Ω,L2
(Q1/2U,W
)).
Also suppose X is progressively measurable and in L2 ([0,T ]×Ω,W ). Let{
tnj
}mn
j=0be a
sequence of partitions of the sort in Lemma 73.1.1 such that if
Xn (t)≡mn−1
∑j=0
X(tn
j)X[tn
j ,tnj+1)
(t)≡ X ln (t)
then Xn→ X in Lp ([0,T ]×Ω,W ) . Also, it can be assumed that none of these mesh pointsare in the exceptional set off which BX (t) = B(X (t)). (Thus it will make no differencewhether we write BX (t) or B(X (t)) in what follows for all t one of these mesh points.)Then the expression
mn−1
∑j=0
⟨B∫ tn
j+1∧t
tnj∧t
ZdW,X(tn
j)⟩
=mn−1
∑j=0
⟨BX(tn
j),∫ tn
j+1∧t
tnj∧t
ZdW
⟩(73.3.3)
is a local martingale which can be written in the form∫ t
0
(Z ◦ J−1)∗BX l
n ◦ JdW
where
X ln (t) =
mn−1
∑k=0
X (tnk )X[tn
k ,tnk+1)
(t)
Proof: First suppose that⟨BX(tnk
),X(tnk
)⟩∈ L∞ (Ω). Then⟨
BX(tn
j),∫ tn
j+1∧t
tnj∧t
ZdW
⟩
is in L1 (Ω) for each t since both entries are in L2 (Ω). Why is this a martingale?
E
(⟨BX(tn
j),∫ tn
j+1∧t
tnj∧t
ZdW
⟩)= E
(E
(⟨BX(tn
j),∫ tn
j+1∧t
tnj∧t
ZdW
⟩|Ftn
j
))
= E
(⟨BX(tn
j),E
(∫ tnj+1∧t
tnj∧t
ZdW |Ftnj
)⟩)= E
(⟨BX(tn
j),0⟩)
= 0
because the stochastic integral is a martingale. Now let σ be a bounded stopping time.
E
(⟨BX(tn
j),∫ tn
j+1∧σ
tnj∧σ
ZdW
⟩)= E
(E
(⟨BX(tn
j),∫ tn
j+1∧σ
tnj∧σ
ZdW
⟩|Ftn
j
))