2242 CHAPTER 65. STOCHASTIC INTEGRATION

It follows that Q1/21 (U1)⊆ J

(Q1/2 (U)

).

One can also show that W (t) ≡ ∑Lk=1 ψk (t)Jgk where the ψk (t) are the real Wiener

processes described earlier and {gk} is an orthonormal basis for Q1/2 (U), is a Q1 Wienerprocess. To see this, recall the above definition of a Wiener process in terms of HilbertSchmidt operators, the convergence happening in U1 in this case. Then by independence ofthe ψ j,

E

((h,

L

∑k=1

ψk (t− s)Jgk

)(l,

L

∑j=1

ψ j (t− s)Jg j

))

= E

(∑k(h,Jgk)(l,Jg j)ψk (t− s)ψ j (t− s)

)

= ∑k(h,Jgk)(l,Jgk)E

(ψ

2k (t− s)

)= (t− s)∑

k(h,Jgk)(l,Jgk)

= (t− s)∑k(J∗h,gk)Q1/2(U) (J

∗l,gk)Q1/2(U) = (t− s)(J∗h,J∗l)Q1/2(U)

= (t− s)(JJ∗h, l)U1≡ (t− s)(Q1h, l)U1

65.5 The General IntegralIt is time to generalize the integral. The following diagram illustrates the ingredients of thenext lemma.

W (t) ∈U1J← Q1/2U Φ→ H

U↓ Q1/2

U1 ⊇ JQ1/2U J←1−1

Q1/2U

Φn ↘ ↓ Φ

H

Lemma 65.5.1 Let Φ ∈ L2([a,T ]×Ω;L2

(Q1/2U,H

))and suppose also that Φ is pro-

gressively measurable with respect to the usual filtration associated with the Wiener pro-cess

W (t) =L

∑k=1

ψk (t)Jgk

which has values in U1 for U1 a separable real Hilbert space such that J : Q1/2U → U1is Hilbert Schmidt and one to one, {gk} an orthonormal basis in Q1/2U. Then lettingJ−1 : JQ1/2U → Q1/2U be the map described in Definition 65.4.1, it follows that

Φ◦ J−1 ∈ L2([a,T ]×Ω;L2

(JQ1/2U,H

)).

Also there exists a sequence of elementary functions {Φn} having values in L (U1,H)0which converges to Φ◦ J−1 in L2

([a,T ]×Ω;L2

(JQ1/2U,H

)).