884 CHAPTER 33. QUADRATIC VARIATION AND STOCHASTIC INTEGRATION
because it is a finite sum of continuous random variables. In the argument to get 33.2, I willwrite M rather than Mτ l . Then it is understood that you can let l→ ∞ to obtain the desiredformula for a local martingale. Thus, in what follows, M is a bounded martingale.
E
(∥∥∥∥∫ t
0f dM
∥∥∥∥2
U
)
= E
(mn−1
∑k=0
fk (M (t ∧ tk+1)−M (t ∧ tk)) ,mn−1
∑j=0
f j(M(t ∧ t j+1
)−M (t ∧ t j)
))
Let M (t ∧ tk+1)−M (t ∧ tk) = ∆Mk. Thus ∆Mk is Ftk+1 measurable. Consider a mixed termin the above in which j < k
E ( fk∆Mk, f j∆M j) = E(E(
fk∆Mk, f j∆M j|Ftk
))= E
(fk, f j∆M jE
(∆Mk|Ftk
))= 0
because E(M (t ∧ tk+1)−M (t ∧ tk) |Ftk
)= M (t ∧ tk+1∧ tk)−M (t ∧ tk) . Thus
E
(∥∥∥∥∫ t
0f dM
∥∥∥∥2
U
)=
mn−1
∑k=0
E ( fk∆Mk, fk∆Mk)
=mn−1
∑k=0
E(E(( fk∆Mk, fk∆Mk) |Ftk
))(33.4)
=mn−1
∑k=0
E(
f 2k E((∆Mk,∆Mk) |Ftk
))(33.5)
now
E((M (t ∧ tk+1) ,M (t ∧ tk)) |Ftk
)=
(E(M (t ∧ tk+1) |Ftk
),M (t ∧ tk)
)= ∥M (t ∧ tk)∥2
and so
E((∆Mk,∆Mk) |Ftk
)= E
((M (t ∧ tk+1)−M (t ∧ tk) ,M (t ∧ tk+1)−M (t ∧ tk)
)|Ftk
)=
E(∥M (t ∧ tk+1)∥2 |Ftk
)+E
(∥M (t ∧ tk)∥2 |Ftk
)−2E
((M (t ∧ tk+1) ,M (t ∧ tk)) |Ftk
)= E
(∥M (t ∧ tk+1)∥2 |Ftk
)−∥M (t ∧ tk)∥2
Therefore, the right side of 33.5 is
mn−1
∑k=0
E(
f 2k ∥M (t ∧ tk+1)∥2
)−E
(f 2k ∥M (t ∧ tk)∥2
).
Now recall that ∥M (t)∥2 = [M] (t)+N (t) where N (t) is a martingale. It then reduces to
mn−1
∑k=0
E(
f 2k ([M (t ∧ tk+1)]+N (t ∧ tk+1))
)−E
(f 2k ([M] (t ∧ tk)+N (t ∧ tk))
)