73.4. THE MAIN ESTIMATE 2469

Next it is necessary to drop the assumption that ⟨BX (a) ,X (a)⟩ ∈ L∞ (Ω). Note that X ln

is right continuous and BX ln progressively measurable. Thus,⟨

BX ln (t) ,X

ln (t)

⟩= ∑

i

⟨BX l

n (t) ,ei

⟩2

where {ei} is the set defined in Lemma 76.2.1 each in V . Thus⟨BX l

n,Xln⟩

is also progres-sively measurable and right continuous, and one can define the stopping time

σnq ≡ inf

{t :⟨

BX ln (t) ,X

ln (t)

⟩> q}, (73.3.5)

the first hitting time of an open set. Also, for each ω, there are only finitely many valuesfor⟨BX l

n (t) ,Xln (t)

⟩and so σn

q = ∞ for all q large enough.From localization of the stochastic integral,⟨

B∫ t∧σn

q

a∧σnq

ZdW,X (a)

⟩=

⟨B∫ t

aX[0,σn

q]ZdW,X (a)

⟩=

∫ t

a

(X[0,σn

q]Z ◦ J−1

)∗BX (a)◦ JdW

=∫ t∧σn

q

a∧σnq

(Z ◦ J−1)∗BX (a)◦ JdW

Then it follows that, using the stopping time,

mn−1

∑j=0

⟨B∫ tn

j+1∧t∧σnq

tnj∧t∧σn

q

ZdW,X(tn

j)⟩

=∫ t∧σn

q

0

(Z ◦ J−1)∗BX l

n ◦ JdW

where X ln is the step function

X ln (t) =

mn−1

∑k=0

X (tnk )X[tn

k ,tnk+1)

(t) .

Thus the given sum equals the local martingale∫ t

0

(Z ◦ J−1)∗BX l

n ◦ JdW.

Note that the sum 73.3.3 does not depend on J or on U1 so the same must be true ofwhat it equals although it does not look that way. The question of convergence as n→∞ isconsidered later.

What follows is the main estimate and discrete formulas.

73.4 The Main EstimateThe argument will be based on a formula which follows in the next lemma.