904 CHAPTER 33. QUADRATIC VARIATION AND STOCHASTIC INTEGRATION

and so Q(t)− [M] (t) equals a martingale. Thus from Lemma 32.2.1, Q(t)− [M] (t) = 0.This proves the first part of the following important result.

Theorem 33.5.1 Let H be a Hilbert space and suppose (M,Ft) , t ∈ [0,T ] is a uni-formly bounded continuous martingale with values in H. Also let

{tnk

}mnk=1 be a sequence of

partitions 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 ∧ tn

k+1)−M (t ∧ tn

k )∥∥2

H

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

Proof: It only remains to show the claim about the case where M is a local martingale.Suppose M is only a continuous local martingale. By Proposition 32.3.3 there exists anincreasing localizing sequence {τk} such that Mτk is a uniformly bounded martingale. Then

P(∪∞k=1 [τk = ∞]) = 1

As above, let

Qn (t)≡mn−1

∑k=0

∥∥M(t ∧ tn

k+1)−M (t ∧ tn

k )∥∥2

H

where there are mn points in Pn where as before, Pn ⊆ Pn+1 for all n.Let η ,ε > 0 be given. Then there exists k large enough that P([τk = ∞]) > 1−η/2.

This is because the sets [τk = ∞] increase to Ω other than a set of measure zero. Then forthis k, [∣∣Qτk

n − [M]τk (t)∣∣> ε

]∩ [τk = ∞] = [|Qn− [M] (t)|> ε]∩ [τk = ∞]

Thus

P([|Qn− [M] (t)|> ε]) ≤ P([|Qn− [M] (t)|> ε]∩ [τk = ∞])

+P([τk < ∞])

≤ P([∣∣Qτk

n − [M]τk (t)∣∣> ε

])+η/2

The convergence in probability of Qτkn (t) to [M]τk (t) follows from the convergence in

L2 (Ω) shown earlier for bounded martingales, and so if n is large enough, the right sideof the above inequality is less than η/2+η/2 = η . Since η was arbitrary, this provesconvergence in probability. ■

33.6 The Case of f ∈ L2 (Ω× [0,T ] ;L2 (U,H))

At this point I will discontinue the general treatment in terms of arbitrary martingales andsuppose that [M] (t) = F (t) a continuous increasing function which depends only on t andnot on ω . In fact, the most interest is centered on the Wiener process in which [M] (t) = atfor a> 0 and this is the case considered here. Of course a does not matter so we will simplyassume [M] (t) = t.