2278 CHAPTER 66. THE INTEGRAL∫ t

0 (Y,dM)H

Definition 66.0.7 Let G denote those functions Y which are adapted and have the propertythat for each p,

limn→∞

E(∫ T

0∥Y −Y n∥2

H d [M]τ p

)= 0

for some sequence Y n of elementary functions for which ∥Y n (t)∥M∗ ∈ L2 (Ω) for each t.Here d [M]τ p signifies the Lebesgue Stieltjes measure determined by the increasing functiont→ [Mτ p ] (t). Let Mτ p be an L2 martingale. Recall that τ p is just a localizing sequence forthe local martingale M.

It is not known whether this increasing function is absolutely continuous.

Definition 66.0.8 Let Y ∈ G . Then∫ t

0(Y,dMτ p)≡ lim

n→∞

∫ t

0(Y n,dMτ p) in L2 (Ω)

For example, suppose Y is a bounded continuous process having values in H. Then youcould look at the left step functions

Y n (t)≡mn−1

∑i=0

Y (ti)X[ti,ti+1) (t)

The Y n would converge to Y pointwise on [0,T ] for each ω and these Y n are bounded. Infact, in this case, these converge uniformly to Y on [0,T ]. Thus this is an example of thesituation in the above definition. In this case, the integrand would be bounded by C forsome C and

E(∫ T

0Cd [M]τ p

)= E

([M]τ p (T )

)= E

(∥Mτ p (T )∥2

)< ∞

by assumption. Hence, by the dominated convergence theorem,

limn→∞

E(∫ T

0∥Y −Y n∥2

H d [M]τ p

)= 0.

What if [M]τ p were bounded and absolutely continuous with respect to Lebesgue mea-sure? This could be the case if you had τ p a stopping time of the form

τ p = inf{t : [M] (t)> p}

Then if Y ∈ L2 ([0,T ]×Ω,H) , and progressively measurable there are left step functionswhich converge to Y in L2 ([0,T ]×Ω,H) . Say d [Mτ p ] = k (t,ω)dm where k is bounded.Then

E(∫ T

0∥Y −Y n∥2

H d [M]τ p

)= E

(∫ T

0∥Y −Y n∥2

H kdt)→ 0