2232 CHAPTER 65. STOCHASTIC INTEGRATION

Then∫ t

a Φ(s)dW is a continuous square integrable H valued martingale with respectto the σ algebras of 65.1.2 on [0,T ] and

E

(∣∣∣∣∫ t

aΦ(s)dW

∣∣∣∣2H

)=∫ t

aE(||Φ◦ J||2L2(U0,H)

)ds

Proof: Start with the left side. Denote by ∆kW (t)≡W (t ∧ tk+1)−W (t ∧ tk) . Then

E

(∣∣∣∣∫ t

aΦ(s)dW

∣∣∣∣2H

)= E

∣∣∣∣∣n−1

∑k=0

Φ(tk)∆kW (t)

∣∣∣∣∣2

H

 .

Consider a mixed term for j < k. Using Lemma 65.1.3 and the fact that W (t) is a martin-gale,

E((Φ(tk)∆kW (t) ,Φ(t j)∆ jW (t))H

)= E

(E((Φ(tk)∆kW (t) ,Φ(t j)∆ jW (t))H

)|Ftk

)= E

((Φ(t j)∆ jW (t) ,E

(Φ(tk)∆kW (t) |Ftk

)))= E

((Φ(t j)∆ jW (t) ,Φ(tk)E

(∆kW (t) |Ftk

)))= E ((Φ(t j)∆ jW (t) ,Φ(tk)0)) = 0.

Therefore, from Lemma 65.1.4, and letting{

f j}

be an orthonormal basis for H, it followsthat since the mixed terms disappeared,

E

(∣∣∣∣∫ t

aΦ(s)dW

∣∣∣∣2H

)=

n−1

∑k=0

E ((Φ(tk)∆kW (t) ,Φ(tk)∆kW (t)))

=n−1

∑k=0

E

(∞

∑j=1

(Φ(tk)∆kW (t) , f j)2

)=

n−1

∑k=0

∞

∑j=1

E((Φ(tk)∆kW (t) , f j)

2)

=n−1

∑k=0

∞

∑j=1

(t ∧ tk+1− t ∧ tk)E(∣∣∣∣J∗Φ(tk)

∗ f j∣∣∣∣2

U0

)=

n−1

∑k=0

(t ∧ tk+1− t ∧ tk)E(∣∣∣∣J∗Φ(tk)

∗∣∣∣∣2L2(H,U0)

)=

n−1

∑k=0

(t ∧ tk+1− t ∧ tk)E(||Φ(tk)J||2L2(U0,H)

)=∫ t

aE(||Φ◦ J||2L2(U0,H)

)ds

It is obvious that∫ t

a Φ(s)dW is a continuous square integrable martingale from the defini-tion, because it is just a finite sum of such things.

Of course this is a version of the Ito isometry. The presence of the J is troublesomebut it is hidden in the definition of W on the left side of the conclusion of the proposition.In finite dimensions one could just let J = I and this fussy detail would not be there tocause confusion. The next task is to generalize the above integral to a more general classof functions and obtain a process which is not explicitly dependent on J.