2268 CHAPTER 65. STOCHASTIC INTEGRATION

The existence of the integral follows from Lemma 65.14.2. From the assumption ofweak continuity, supt∈[0,T ] |X (t)| ≤ C (ω) for a.e.ω . For the first part of the argument,assume C does not depend on ω off a set of measure zero. Let

M (t)≡∫ t

0ZdW

Let {ek} be an orthonormal basis for H and let Pn be the orthogonal projection ontospan(e1, · · · ,en). For each ei

limk→∞

∣∣∣(X (s)−X lk (s) ,ei

)∣∣∣= 0

and so, by weak continuity,

limk→∞

Pn

(X (s)−X l

k (s))= 0 for a.e.ω

Then

limk→∞

∫Ω

∫ T

0

∣∣∣Pn

(X (s)−X l

k (s))∣∣∣2 ∥Z (s)∥2

L2dsdP = 0

because you can apply the dominated convergence theorem with respect to the measure∥Z (s)∥2

L2dsdP.

Therefore,

limk→∞

P

([sup

t∈[0,T ]

∣∣∣∣∫ t

0R((

Z (s)◦ J−1)∗Pn∆k (s))◦ JdW (s)

∣∣∣∣≥ ε/2

])= 0 (65.14.24)

Here is why. By the Burkholder Davis Gundy theorem, Theorem 63.4.4 and Corollary65.11.1 which describes the quadratic variation of the stochastic integral,

∫Ω

(sup

t∈[0,T ]

∣∣∣∣∫ t

0R((

Z (s)◦ J−1)∗Pn∆k (s))

dW (s)∣∣∣∣)

dP

≤C∫

Ω

(∫ T

0

∣∣∣Pn

(X (s)−X l

k (s))∣∣∣2 ||Z (s)||2L2

ds)1/2

dP

Consider the following two probabilities.

P

([sup

t∈[0,T ]

∣∣∣∣∫ t

0R((

Z (s)◦ J−1)∗ ( I−Pn)X (s))◦ JdW (s)

∣∣∣∣≥ ε/2

])(65.14.25)

P

([sup

t∈[0,T ]

∣∣∣∣∫ t

0R((

Z (s)◦ J−1)∗ ( I−Pn)X lk (s)

)◦ JdW (s)

∣∣∣∣≥ ε/2

])(65.14.26)

By Corollary 63.4.5 which depends on the Burkholder Davis Gundy inequality andCorollary 65.11.1 which describes the quadratic variation of the stochastic integral, given