73.7. THE ITO FORMULA 2489

= ∥B∥E

(qk−1

∑j=1

∥∥∥∥( I−Pn)∫ t j+1

t j

Z (s)dW (s)∥∥∥∥2

W

)

= ∥B∥qk−1

∑j=1

E

(∥∥∥∥∫ t j+1

t j

( I−Pn)Z (s)dW (s)∥∥∥∥2)

= ∥B∥qk−1

∑j=1

E(∫ t j+1

t j

∥( I−Pn)Z (s)∥2L2(Q1/2U,W) ds

)≤ ∥B∥E

(∫ T

0∥( I−Pn)Z (s)∥2

L2(Q1/2U,H) ds)

Now letting {gi} be an orthonormal basis for Q1/2U,

= ∥B∥∫

Ω

∫ T

0

∞

∑i=1∥( I−Pn)Z (s)(gi)∥2

W dsdP (73.7.31)

The integrand ∑∞i=1 ∥( I−Pn)Z (s)(gi)∥2

W converges to 0. Also, it is dominated by

∞

∑i=1∥Z (s)(gi)∥2

W ≡ ∥Z∥2L2(Q1/2U,W)

which is given to be in L1 ([0,T ]×Ω) . Therefore, from the dominated convergence theo-rem, the expression in 73.7.31 converges to 0 as n→ ∞.

Thus the expression in 73.7.30 is of the form fkgnk where fk converges in probabilityto a1/2 as k→ ∞ and gnk converges in probability to 0 as n→ ∞ independently of k. Nowthis implies fkgnk converges in probability to 0. Here is why.

P([| fkgnk|> ε]) ≤ P(2δ | fk|> ε)+P(2Cδ |gnk|> ε)

≤ P(

2δ

∣∣∣ fk−a1/2∣∣∣+2δ

∣∣∣a1/2∣∣∣> ε

)+P(2Cδ |gnk|> ε)

where δ | fk|+Cδ |gkn|> | fkgnk| and limδ→0 Cδ =∞. Pick δ small enough that ε−2δa1/2 >ε/2. Then this is dominated by

≤ P(

2δ

∣∣∣ fk−a1/2∣∣∣> ε/2

)+P(2Cδ |gnk|> ε)

Fix n large enough that the second term is less than η for all k. Now taking k large enough,the above is less than η . It follows the expression in 73.7.30 and consequently in 73.7.29converges to 0 in probability.

Now consider the other term 73.7.28 using the n just determined. This term is of theform

qk−1

∑j=1

∫ t j+1

t j

⟨Y (s) ,X

(t j+1

)−X (t j)−Pn

(M(t j+1

)−M (t j)

)⟩ds =