2146 CHAPTER 63. THE QUADRATIC VARIATION OF A MARTINGALE

where Nk is a martingale which equals 0 for t ≤ tk. The above equals

E(∫ t

0∥ f∥2 d [M]

)≡ E

(∫ t

0∥ f∥2 dν

)the integral inside being the ordinary Lebesgue Stieltjes integral for the step function whereν is the measure determined by the positive linear functional

Λg =∫ T

0gd [M]

where the integral on the right is the ordinary Stieltjes integral. Thus, the following in-equality is obtained.

E

(∣∣∣∣∫ t

0f dM

∣∣∣∣2)≤ E

(∫ t

0∥ f∥2 d [M] ,

)(63.6.24)

Now what would it take for

E

(∣∣∣∣∫ t

0f dM

∣∣∣∣2)

(63.6.25)

to be well defined? A convenient condition would be to insist that each ∥ fk∥M∗ is inL2 (Ω) where

M∗ (ω)≡ supt∈[0,T ]

|M (t)(ω)|H

Is this condition also sufficient for the above integral 63.6.25 to be finite? From theabove, that integral equals

E

(m−1

∑k=0∥ fk∥2 |M (t ∧ tk+1)−M (t ∧ tk)|2

)

≤ E

(4

m−1

∑k=0∥ fk∥2 (M∗)2

)Thus the condition that for each k,∥ fk∥M∗ ∈ L2 (Ω) is sufficient for all of the above toconsist of real numbers and be well defined.

Definition 63.6.2 A function f is called an elementary function if it is a step function of theform given in 63.6.22 where each fk is Ftk measurable and for each k,∥ fk∥M∗ ∈ L2 (Ω).Define GM to be the collection of functions f having values in H ′ which have the propertythat there exists a sequence of elementary functions { fn} with fn→ f in the space

L2 (Ω;L2 ([0,T ] ,ν)

)Then picking such an approximating sequence,∫ t

0f dM ≡ lim

n→∞

∫ t

0fndM

the convergence happening in L2 (Ω).