2488 CHAPTER 73. THE HARD ITO FORMULA, IMPLICIT CASE

where a is the limit in probability of the term

qk−1

∑j=1

⟨B(∆X (t j)−∆M (t j)) ,∆X (t j)−∆M (t j)

⟩(73.7.27)

Let Pn be the projection onto span(e1, · · · ,en) where {ek} is an orthonormal basis for Wwith each ek ∈V . Then using

BX(t j+1

)−BX (t j)−

(BM

(t j+1

)−BM (t j)

)=∫ t j+1

t j

Y (s)ds

the troublesome term of 73.7.27 above is of the form

qk−1

∑j=1

∫ t j+1

t j

⟨Y (s) ,∆X (t j)−∆M (t j)

⟩ds

=qk−1

∑j=1

∫ t j+1

t j

⟨Y (s) ,∆X (t j)−Pn∆M (t j)

⟩ds

+qk−1

∑j=1

∫ t j+1

t j

⟨Y (s) ,−(I−Pn)∆M (t j)

⟩ds

which equals

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 (73.7.28)

+qk−1

∑j=1

⟨B(∆X (t j)−∆M (t j)) ,−( I−Pn)

(M(t j+1

)−M (t j)

)⟩(73.7.29)

The reason for the Pn is to get Pn(M(t j+1

)−M (t j)

)in V . The sum in 73.7.29 is dominated

by (qk−1

∑j=1

⟨B(∆X (t j)−∆M (t j)) ,(∆X (t j)−∆M (t j))

⟩)1/2

·

(qk−1

∑j=1

∣∣⟨B( I−Pn)∆M (t j) ,( I−Pn)∆M (t j)⟩∣∣2)1/2

(73.7.30)

Now it is known from the above that ∑qk−1j=1

⟨B(∆X (t j)−∆M (t j)) ,(∆X (t j)−∆M (t j))

⟩converges in probability to a ≥ 0. If you take the expectation of the square of the otherfactor, it is no larger than

∥B∥E

(qk−1

∑j=1

∥∥( I−Pn)∆M (t j)∥∥2

W

)