2172 CHAPTER 63. THE QUADRATIC VARIATION OF A MARTINGALE

Lemma 63.8.4 Let {X (t)} be a real continuous martingale adapted to the filtration Ft

for t ∈ [a,b] some interval such that for all t ∈ [a,b] ,E(

X (t)2)< ∞. Suppose also that{

X (t)2− t}

is a martingale. Then for λ real,

E(

eiλX(b))= E

(eiλX(a)

)e−(b−a) λ2

2

Proof: Let λ ∈ [−p, p] where for most of the proof, p is fixed but arbitrary. Let{

tnk

}2n

k=0be uniform partitions such that tn

k − tnk−1 = δ n ≡ (b−a)/2n. Now for ε > 0 define a stop-

ping time τε,n to be the first time, t such that there exist s1,s2 ∈ [a, t] with |s1− s2| < δ nbut

|X (s1)−X (s2)|= ε.

If no such time exists, then τε,n ≡ b.Then τε,n really is a stopping time because from continuity of X (t) and denoting by

r,r1 elements of Q, then

[τε,n > t] =∞⋃

m=1

⋂0≤r1,r2≤t,|r1−r2|≤δ n

[|X (r1)−X (r2)| ≤ ε− 1

m

]∈Ft

because to be in [τε,n > t] it means that by t the absolute value of the differences mustalways be less than ε. Hence [τε,n ≤ t] = Ω\ [τε,n > t] ∈Ft .

Now consider [τε,n = b] for various n. By continuity, it follows that for each ω ∈Ω,

τε,n (ω) = b

for all n large enough. Thus/0 = ∩∞

n=1 [τε,n < b] ,

the sets in the intersection decreasing. Thus there exists n(ε) such that

P([

τε,n(ε) < b])

< ε. (63.8.42)

Denote τε,n(ε) as τε for short and it will always be assumed that n(ε) is at least this largeand that limε→0+ n(ε) = ∞. In addition to this, n(ε) will also be large enough that

1− λ2

2δ n(ε) > 0

for all λ ∈ [−p, p] . To save on notation, t j will take the place of tnj . Then consider the

stopping times τε ∧ t j for j = 0,1, · · · ,2n(ε).Let y j ≡ X (τε ∧ t j)−X

(τε ∧ t j−1

), it follows from the definition of the stopping time

that ∣∣y j∣∣≤ ε (63.8.43)

because both τε ∧ t j and τε ∧ t j−1 are less than τε and closer together than δ n(ε) and so if∣∣y j∣∣> ε, then τε ≤ t j, t j−1 and so y j would need to equal 0.