2682 CHAPTER 79. INCLUDING STOCHASTIC INTEGRALS

4. One can also generalize to the case where Φ is only progressively measurable andinstead of being in L2

([0,T ]×Ω;L2

(Q1/2U,H

)), you have only that

P(∫ T

0∥Φ(t)∥2

L2dt < ∞

)= 1

This is done by using an appropriate sequence of stopping times called a localizingsequence. More generally a local martingale is a stochastic process M (t) adapted tothe filtration for which there is a locallizing sequence of stopping times {τn} suchthat limn→∞ τn = ∞ and Mτn is a martingale. Local martingales will occur in theestimates which are encountered in what follows.

5. Denoting by M (t) the stochastic integral, M (t) =∫ t

0 ΦdW, the quadratic variation isgiven by

[M] (t) =∫ t

0∥Φ∥2

L2ds

6. We will also need a part of the Burkholder Davis Gundy inequality [77], Theorem63.4.4 which in terms of this stochastic integral is of the form

∫Ω

M∗dP≤CE

((∫ T

0∥Φ∥2

L2ds)1/2

), C some constant

where M (t) is the above stochastic integral and

M∗ ≡ sup{∥M (t)∥H : t ∈ [0,T ]}

Now let Φ ∈ L2([0,T ]×Ω;L2

(Q1/2U,W

)). Let an orthonormal basis for Q1/2U be

{gi} and an orthonormal basis for W be { fi}. Then { fi⊗gi} is an orthonormal basis forL2(Q1/2U,W

). Hence,

Φ = ∑i

∑j

Φi j fi⊗g j

where fi⊗ g j (y) ≡ (g j,y)Q1/2U fi. Let E be a separable real Hilbert space which is densein V. Then without loss of generality, one can assume that the orthonormal basis for W areall vectors in E. Thus the orthogonal projection of Φ onto the closed subspace

span({ fi⊗gi} , i, j ≤ n)

given by

Φn ≡n

∑i=1

n

∑j=1

Φi j fi⊗g j

Then Φn ∈ L2([0,T ]×Ω;L2

(Q1/2U,E

))and also

limn→∞∥Φn−Φ∥L2([0,T ]×Ω;L2(Q1/2U,W)) = 0