63.6. ANOTHER LIMIT FOR QUADRATIC VARIATION 2149

Theorem 63.6.4 Let H be a Hilbert space and suppose (M,Ft) , t ∈ [0,T ] is a uniformlybounded continuous martingale with values in H. Also let

{tnk

}mnk=1 be a sequence of parti-

tions satisfying

limn→∞

max{∣∣tn

i − tni+1∣∣ , i = 0, · · · ,mn

}= 0, {tn

k }mnk=1 ⊆

{tn+1k

}mn+1k=1 .

Then

[M] (t) = limn→∞

mn−1

∑k=0

∣∣M (t ∧ tnk+1)−M (t ∧ tn

k )∣∣2H

the limit taking place in L2 (Ω). In case M is just a continuous local martingale, the abovelimit happens in probability.

Proof: First suppose M is uniformly bounded.

mn−1

∑k=0

∣∣M (t ∧ tnk+1)−M (t ∧ tn

k )∣∣2H

=mn−1

∑k=0

∣∣M (t ∧ tnk+1)∣∣2−|M (t ∧ tn

k )|2−2

mn−1

∑k=0

(M (t ∧ tn

k ) ,M(t ∧ tn

k+1)−M (t ∧ tn

k ))

= |M (t)|2H −2mn−1

∑k=0

(M (t ∧ tn

k ) ,M(t ∧ tn

k+1)−M (t ∧ tn

k ))

= |M (t)|2H −2mn−1

∑k=0

RM (t ∧ tnk )(M(t ∧ tn

k+1)−M (t ∧ tn

k ))

= |M (t)|2H −2mn−1

∑k=0

RM (tnk )(M(t ∧ tn

k+1)−M (t ∧ tn

k ))

Then by Lemma 63.6.3, the right side converges to

|M (t)|2H −2∫ t

0(RM)dM

Therefore, in L2 (Ω) ,

limn→∞

mn−1

∑k=0

∣∣M (t ∧ tnk+1)−M (t ∧ tn

k )∣∣2H +2

∫ t

0(RM)dM = |M (t)|2H

That term on the left involving the limit is increasing and equal to 0 when t = 0. Therefore,it must equal [M] (t).

Next suppose M is only a continuous local martingale. By Proposition 63.2.2 thereexists an increasing localizing sequence {τk} such that Mτk is a uniformly bounded mar-tingale. Then

P(∪∞k=1 [τk = ∞]) = 1