2468 CHAPTER 73. THE HARD ITO FORMULA, IMPLICIT CASE

=mn−1

∑k=0

⟨(Zn

k )∗BX (a) ,

(W(t ∧ tn

k+1)−W (t ∧ tn

k ))⟩

U ′1,U1

=mn−1

∑k=0

(Znk )∗BX (a)

(W(t ∧ tn

k+1)−W (t ∧ tn

k ))

=∫ t

aZ∗nBX (a)dW

Note that the restriction of (Zn)∗BX (a) is in

L (U1,R)0 ⊆L2

(JQ1/2U,R

).

Recall also that the space on the left is dense in the one on the right. Now let {gi} be anorthonormal basis for Q1/2U, so that {Jgi} is an orthonormal basis for JQ1/2U. Then

∞

∑i=1

∣∣∣((Zn)∗BX (a)−

(Z ◦ J−1)∗BX (a)

)(Jgi)

∣∣∣2=

∞

∑i=1

∣∣⟨BX (a) ,(Zn−Z ◦ J−1)(Jgi)

⟩∣∣2≤ ⟨BX (a) ,X (a)⟩

∞

∑i=1

⟨B(Zn−Z ◦ J−1)(Jgi) ,

(Zn−Z ◦ J−1)(Jgi)

⟩

≤ ⟨BX (a) ,X (a)⟩∥B∥∞

∑i=1

∥∥(Zn−Z ◦ J−1)(Jgi)∥∥2

W

= ⟨BX (a) ,X (a)⟩∥B∥∥∥Zn−Z ◦ J−1∥∥2

L2(JQ1/2U,W)

When integrated over [a,b]×Ω, it is given that this converges to 0, if it is assumed that⟨BX (a) ,X (a)⟩ ∈ L∞ (Ω) , which is assumed for now.

It follows that, with this assumption,

Z∗nBX (a)→(Z ◦ J−1)∗BX (a)

in L2([a,b]×Ω,L2

(JQ1/2U,R

)). Writing this differently, it says

Z∗nBX (a)→((

Z ◦ J−1)∗BX (a)◦ J)◦ J−1 in L2

([a,b]×Ω,L2

(JQ1/2U,R

))It follows from the definition of the integral that the Ito integrals converge. Therefore,⟨

B∫ t

aZdW,X (a)

⟩=∫ t

a

(Z ◦ J−1)∗BX (a)◦ JdW

The term on the right is a martingale because the one on the left is.