2290 CHAPTER 67. THE EASY ITO FORMULA
Lemma 67.2.1 Suppose η j are real random variables E(
η2j
)< ∞, such that ηk is mea-
surable with respect to G j for all j > k where {Gk} is increasing. Then
E
[m−1
∑k=0
ηk−m−1
∑k=0
E (ηk|Gk)
]2 (67.2.2)
= E
(m−1
∑k=0
η2k−E (ηk|Gk)
2
)
Proof: First consider a mixed term i < k.
E ((η i−E (η i|Gi))(ηk−E (ηk|Gk)))
This equals
E (η iηk)−E (η iE (ηk|Gk))−E (ηkE (η i|Gi))+E (E (η i|Gi)E (ηk|Gk))
= E (η iηk)−E (E (η iηk|Gk))−E (ηkE (η i|Gi))+E (E (ηkE (η i|Gi) |Gk))
= E (η iηk)−E (E (η iηk|Gk))−E (ηkE (η i|Gi))+E (ηkE (η i|Gi))
= E (η iηk)−E (η iηk)−E (ηkE (η i|Gi))+E (ηkE (η i|Gi)) = 0
Thus 67.2.2 equalsm−1
∑k=0
E((ηk−E (ηk|Gk))
2)
which equals
m−1
∑k=0
E(η
2k)−2E (ηkE (ηk|Gk))+E
(E (ηk|Gk)
2)
=m−1
∑k=0
E(η
2k)−2E (E (ηkE (ηk|Gk)) |Gk)+E
(E (ηk|Gk)
2)
=m−1
∑k=0
E(η
2k)−2E (E (ηk|Gk)E (ηk|Gk))+E
(E (ηk|Gk)
2)
=m−1
∑k=0
E(η
2k)−2E
(E (ηk|Gk)
2)+E
(E (ηk|Gk)
2)
=m−1
∑k=0
E(η
2k)−E
(E (ηk|Gk)
2)