2276 CHAPTER 66. THE INTEGRAL∫ t

0 (Y,dM)H

One also sees that E(∥Y∥2 N̂ (t)

)= 0.

Now it follows from Corollary 63.3.3 that

[(Ms−Mr)] = [Ms]− [Mr] = [M]s− [M]r

Hence

[(Y,(Ms−Mr))] (t)≤ ∥Y∥2 [Ms−Mr] (t) = ∥Y∥2 ([M]s (t)− [M]r (t))

as claimed.The last claim is easy. Let τ p be a localizing sequence for which Mτ p is a martingale.

Then ∫ t∧τ p

0(Y,dM) ≡

m−1

∑i=0

(Yi,M (t ∧ ti+1∧ τ p)−M (t ∧ ti∧ τ p))H

=m−1

∑i=0

(Yi,Mτ p (t ∧ ti+1)−Mτ p (t ∧ ti))H

a finite sum of martingales.Note that this is just a definition and did not use the above localization lemma. In

particular, τ p is not restricted to having only the partition points as values.Next one needs to generalize past the elementary functions.Continue writing M in place of Mτ p in what follows. Consider an elementary function

Y ≡mn−1

∑k=0

YkX(tk,tk+1] (t)

where YkM∗ ∈ L2 (Ω). Consider

∫ t

0(Y,dM)≡

mn−1

∑k=0

(Yk,M (t ∧ tk+1)−M (t ∧ tk)) (66.0.3)

Then it is routine to verify that

E

(mn−1

∑k=0

(Yk,M (t ∧ tk+1)−M (t ∧ tk))H

)2

=mn−1

∑k=0

E((Yk,M (t ∧ tk+1)−M (t ∧ tk))

2H

)(66.0.4)

This is because the mixed terms all vanish. This follows from the following reasoning. Lett j < tk

E((Yk,M (t ∧ tk+1)−M (t ∧ tk))H

(Yj,M

(t ∧ t j+1

)−M

(t ∧ t j+1

))H

)= E

(E((Yk,∆kM (t))H (Yj,∆ jM (t))H |Ftk

))