65.1. INTEGRALS OF ELEMENTARY PROCESSES 2229

Definition 65.1.2 Let Φ(t)∈L (U,H) be constant on each interval, (tm, tm+1] determinedby a partition of [a,T ] , 0 ≤ a = t0 < t1 · · · < tn = T. Then Φ(t) is said to be elementary ifalso Φ(tm) is Ftm measurable and Φ(tm) equals a sum of the form

Φ(tm)(ω) =m

∑j=1

Φ jXA j

where Φ j ∈L (U,H), A j ∈Ftm . What does the measurability assertion mean? It meansthat if O is an open (Borel) set in the topological space L (U,H), Φ(tm)

−1 (O) ∈ Ftm .Thus an elementary function is of the form

Φ(t) =n−1

∑k=0

Φ(tk)X(tk,tk+1] (t) .

Then for Φ elementary, the stochastic integral is defined by

∫ t

aΦ(s)dW (s)≡

n−1

∑k=0

Φ(tk)(W (t ∧ tk+1)−W (t ∧ tk)) .

It is also sometimes denoted by Φ ·W (t) .

The above definition is the same as saying that for t ∈ (tm, tm+1],∫ t

aΦ(s)dW (s) =

m−1

∑k=0

Φ(tk)(W (tk+1)−W (tk))

+Φ(tm)(W (t)−W (tm)) . (65.1.3)

The following lemma will be useful.

Lemma 65.1.3 Let f ,g∈ L2 (Ω;H) and suppose g is G measurable and f is F measurablewhere F ⊇ G . Then

E (( f ,g)H |G ) = (E ( f |G ) ,g)H a.e.

Similarly if Φ is G measurable as a map into L (U,H) with∫Ω

||Φ||2 dP < ∞

and f is F measurable as a map into U such that f ∈ L2 (Ω;H) , then

E (Φ f |G ) = ΦE ( f |G ) .

Proof: Let A∈ G . Let {gn} be a sequence of simple functions, measurable with respectto G ,

gn (ω)≡mn

∑k=1

ankXEn

k(ω)