65.1. INTEGRALS OF ELEMENTARY PROCESSES 2231

where {gk} is an orthonormal basis for U0. Let f ∈ H. Then considering one of the termsof the integral defined above,

E((Φ(tk)(W (t ∧ tk+1)−W (t ∧ tk)) , f )2

)= E

(((W (t ∧ tk+1)−W (t ∧ tk)) ,Φ(tk)

∗ f)2)

= (t ∧ tk+1− t ∧ tk)E(∣∣∣∣J∗Φ(tk)

∗ f∣∣∣∣2

U0

).

Proof: For simplicity, write ∆Wk (t) for W (t ∧ tk+1)−W (t ∧ tk) and ∆k (t)= (t ∧ tk+1)−(t ∧ tk) . If Φ(tk) were a constant, then the result would follow right away from the fact thatW (t) is a Wiener process. Therefore, suppose for disjoint Ei,

Φ(tk)(ω) =m

∑i=1

ΦiXEi (ω)

where Φi ∈L (U,H) and Ei ∈Ftk . Then, since the Ei are disjoint,

E((Φ(tk)(W (t ∧ tk+1)−W (t ∧ tk)) , f )2

)=

m

∑i=1

E(((∆kW (t)) ,Φ∗i f XEi)

2)=

m

∑i=1

∫Ω

(XEi ((∆kW (t)) ,Φ∗i f )2

)dP

Each Ei is Ftk measurable. By Lemma 64.4.2, and the properties of the Wiener process,this equals

m

∑i=1

P(Ei)∫

Ω

(((∆kW (t)) ,Φ∗i f )2

)dP =

m

∑i=1

P(Ei)∆kt (QΦ∗i f ,Φ∗i f )U

where Q = JJ∗. Then the above reduces to

(t ∧ tk+1− t ∧ tk)E(∣∣∣∣J∗Φ(tk)

∗ f∣∣∣∣2

U0

).

Now here is a major result on the integral of elementary functions. The last assertion inthe following proposition is called the Ito isometry.

Proposition 65.1.5 Let Φ(t) be an elementary process as defined in Definition 65.1.2 andlet W (t) be a Wiener process.

W (t) =∞

∑k=1

ψk (t)Jgk

where J : U0→U is Hilbert Schmidt and the ψk are real independent Wiener processes asdescribed above.

U0{gk}

J→ UW (t)

Φ→ H