2516 CHAPTER 74. A MORE ATTRACTIVE VERSION

+2∫ t

0⟨BX ,dM⟩+

[R−1BM,M

](t)−a

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)

⟩(74.6.23)

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 74.6.23 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 (74.6.24)

+qk−1

∑j=1

⟨B(∆X (t j)−∆M (t j)) ,−( I−Pn)

(M(t j+1

)−M (t j)

)⟩(74.6.25)

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

)−M (t j)

)in V . The sum in 74.6.25 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

(74.6.26)

Now it is known from the above that

qk−1

∑j=1

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

⟩