2442 CHAPTER 72. THE HARD ITO FORMULA

Proof: We have

P([∥ fn− f∥Lp([0,T ],E) > λ

])≤ 1

λ

∫Ω

∥ fn− f∥Lp([0,T ],E) dP

≤ 1λ∥ fn− f∥Lp([0,T ]×Ω,E)

Hence there exists a subsequence nk such that

P([∥∥ fnk − f

∥∥Lp([0,T ],E) > 2−k

])≤ 2−k

Then by the Borel Cantelli lemma, it follows that there exists a set of measure zero N suchthat for all k large enough and ω /∈ N,∥∥ fnk − f

∥∥Lp([0,T ],E) ≤ 2−k

Because of this lemma, it can also be assumed that for a.e. ω pointwise convergenceis obtained on [0,T ] as well as convergence in Lp ([0,T ]). This kind of assumption will betacitly made whenever convenient in the context of the above lemma.

Also recall the diagram for the definition of the integral.

U↓ Q1/2

U1 ⊇ JQ1/2U J←1−1

Q1/2U

Φn ↘ ↓ Φ

H

The idea was to get∫ t

0 ΦdW where Φ ∈ L2([0,T ]×Ω;L2

(Q1/2U,H

)). Here W (t) was a

cylindrical Wiener process. This meant that it was a Q1 Wiener process on U1 for Q1 = JJ∗

and J was a Hilbert Schmidt operator mapping Q1/2U to U1.

72.3 The SituationNow consider the following situation.

Situation 72.3.1 Let X satisfy the following.

X (t) = X0 +∫ t

0Y (s)ds+

∫ t

0Z (s)dW (s) , (72.3.2)

X0 ∈ L2 (Ω;H) and is F0 measurable, where Z is L2(Q1/2U,H

)progressively measurable

and ∫ T

0

∫Ω

||Z (s)||2L2(Q1/2U,H) dPdt < ∞

so that the stochastic integral makes sense. Also X has a measurable representative X̄which has values in V . (For a.e.t, X̄ (t) = X (t) for P a.e. ω). This representative satisfies

X̄ ∈ L2 ([0,T ]×Ω,B ([0,T ]×F ,H))∩Lp ([0,T ]×Ω,B ([0,T ])×F ,V )