2176 CHAPTER 63. THE QUADRATIC VARIATION OF A MARTINGALE

in the inequality, 63.8.48 and conclude∣∣∣∣E (eiλX(b))

eλ22 b−E

(eiλX(a)e

λ22 a)∣∣∣∣

≤ O(1)((b−a)−

(E(

X (b)2)−E

(X (a)2

)))= 0.

Therefore,

E(

eiλX(b))= E

(eiλX(a)

)e−

λ22 (b−a)

This proves the lemma because p was arbitrary.Now from this lemma, it is not hard to establish Levy’s theorem.

Theorem 63.8.5 Let {X (t)} be a real continuous martingale adapted to the filtration

Ft for t ∈ [0,a] some interval such that for all t ∈ [0,a] ,E(

X (t)2)< ∞. Suppose also

that{

X (t)2− t}

is a martingale. Then for s < t,X (t)− X (s) is normally distributedwith mean 0 and variance t − s. Also if 0 ≤ t0 < t1 < · · · < tm ≤ b, then the increments{

X (t j)−X(t j−1

)}are independent.

Proof: Let the t j be as described above and consider the interval [tm−1, tm] in place of[a,b] in Lemma 63.8.4. Also let λ k for k = 1,2, · · · ,m be given. For t ∈ [tm−1, tm] , andλ m ̸= 0,

Zλ m (t) =1

λ m

m−1

∑j=1

λ j(X (t j)−X

(t j−1

))+(X (t)−X (tm−1))

Then it is clear that{

Zλ m (t)}

is a martingale on [tm−1, tm] . What is possibly less clear is that{Zλ m (t)

2− t}

is also a martingale. Note that Zλ m (t) = X (t)+Y where Y is measurable inFtm−1 . Therefore, for s < t,s ∈ [tm−1, tm] ,

E(

Zλ m (t)2− t|Fs

)= E

(X (t)2 +2X (t)Y +Y 2− t|Fs

)

= X (s)2− s+2E (X (t)Y |Fs)+Y 2

= X (s)2− s+2Y X (s)+Y 2 = Zλ m (s)2− s

and so Lemma 63.8.4 can be applied to conclude

E(

eiλZλm (tm))= E

(eiλZλm (tm−1)

)e−

λ22 (tm−tm−1).

Now letting λ = λ m,

E(

ei∑mj=1 λ j(X(t j)−X(t j−1))

)= E

(ei∑

m−1j=1 λ j(X(t j)−X(t j−1))

)e−

λ2m2 (tm−tm−1).