2140 CHAPTER 63. THE QUADRATIC VARIATION OF A MARTINGALE

Then for each ω, the integrand converges as l→ ∞ to

∑k, j

fkg j((M (t ∧ tk+1)−M (t ∧ tk)) ,

(N(t ∧ t j+1

)−N (t ∧ t j)

))But also you can do a sloppy estimate which will allow the use of the dominated conver-gence theorem.∥∥∥∥∥∑k, j fkg j (Mτ l (t ∧ tk+1)−Mτ l (t ∧ tk)) ,

(Nτ l(t ∧ t j+1

)−Nτ l (t ∧ t j)

)∥∥∥∥∥≤∑

k, j| fk|∣∣g j∣∣4M∗N∗ ∈ L1 (Ω)

by assumption. Thus the left side of 63.5.16 converges as l→ ∞ to∫Ω

∑k, j

fkg j((M (t ∧ tk+1)−M (t ∧ tk)) ,

(N(t ∧ t j+1

)−N (t ∧ t j)

))dP

=∫

Ω

(∫ t

0f dM,

∫ t

0gdN

)U

dP

Note for each ω, the inside integral in 63.5.13 is just a Stieltjes integral taken withrespect to the increasing integrating function [M].

Of course, with this estimate it is obvious how to extend the integral to a larger class offunctions.

Definition 63.5.3 Let ν (ω) denote the Radon measure representing the functional

Λ(ω)(g)≡∫ T

0gd [M] (t)(ω)

(t→ [M] (t)(ω) is a continuous increasing function and ν (ω) is the measure representingthe Stieltjes integral, one for each ω .) Then let GM denote functions f (s,ω) which are thelimit of such elementary functions in the space L2

(Ω;L2 ([0,T ] ,ν (·))

), the norm of such

functions being

|| f ||2G ≡∫

Ω

∫ T

0f (s)2 d [M] (s)dP

For f ∈ G just defined, ∫ t

0f dM ≡ lim

n→∞

∫ t

0fndM

where { fn} is a sequence of elementary functions converging to f in

L2 (Ω;L2 ([0,T ] ,ν (·))

).

Now here is an interesting lemma.