2118 CHAPTER 63. THE QUADRATIC VARIATION OF A MARTINGALE

Then from Fatou’s lemma,

E

(∑k≥0

(ξ k,(M (τk+1∧ t)−M (τk ∧ t)))

)2≤

lim infq→∞

E

( q

∑k=0

(ξ k,(M (τk+1∧ t)−M (τk ∧ t)))

)2

≤ C2E(||M (t)||2

)Now here is an interesting lemma which will be used to prove uniqueness in the main

result.

Lemma 63.1.5 Let Ft be a normal filtration and let A(t) ,B(t) be adapted to Ft , continu-ous, and increasing with A(0) = B(0) = 0 and suppose A(t)−B(t) is a martingale. ThenA(t)−B(t) = 0 for all t.

Proof: I shall show A(l) = B(l) where l is arbitrary. Let M (t) be the name of themartingale. Define a stopping time

τ ≡ inf{t > 0 : |M (t)|>C}∧ l∧ inf{t > 0 : A(t)>C}∧ inf{t > 0 : B(t)>C}

where inf( /0)≡ ∞ and denote the stopped martingale

Mτ (t)≡M (t ∧ τ) .

Then I claim this is also a martingale with respect to the filtration Ft because by Doob’soptional sampling theorem for martingales, if s < t,

E (Mτ (t) |Fs)≡ E (M (τ ∧ t) |Fs) = M (τ ∧ t ∧ s) = M (τ ∧ s) = Mτ (s)

Note the bounded stopping time is τ∧t and the other one is σ = s in this theorem. Then Mτ

is a continuous martingale which is also uniformly bounded. It equals Aτ −Bτ . The stop-ping time ensures Aτ and Bτ are uniformly bounded by C. Thus all of |Mτ (t)| ,Bτ (t) ,Aτ (t)are bounded by C on [0, l] . Now let Pn≡{tk}n

k=1 be a uniform partition of [0, l] and letMτ (Pn) denote

Mτ (Pn)≡max{|Mτ (ti+1)−Mτ (ti)|}ni=1 .

Then

E(

Mτ (l)2)= E

(n−1

∑k=0

Mτ (tk+1)−Mτ (tk)

)2

Now consider a mixed term in the sum where j < k.

E((Mτ (tk+1)−Mτ (tk))

(Mτ(t j+1

)−Mτ (t j)

))