2570 CHAPTER 76. IMPLICIT STOCHASTIC EQUATIONS

for a constant C independent of n. Therefore,

limn→∞

E

(∣∣∣∣∣∫ T

0 X[0,τ p](Φn ◦ J−1

)∗Bun ◦ JdW

−∫ T

0 X[0,τ p](Φ◦ J−1

)∗Bu◦ JdW

∣∣∣∣∣)

= 0

Hence

P(An∩ [τ p = ∞])≤ 1ε

E

(∣∣∣∣∣∫ T

0 X[0,τ p](Φn ◦ J−1

)∗Bun ◦ JdW

−∫ T

0 X[0,τ p](Φ◦ J−1

)∗Bu◦ JdW

∣∣∣∣∣)

and solimn→∞

P(An∩ [τ p = ∞]) = 0

Then

P(An) =∞

∑p=1

P(An∩ ([τ p = ∞]\ [τ p−1 < ∞]))

and so from the dominated convergence theorem,

limn→∞

P(An) =∞

∑p=1

limn→∞

P(An∩ ([τ p = ∞]\ [τ p−1 < ∞])) = ∑p

0 = 0.

There was nothing special about T. The same argument holds for all t and so M (t) men-tioned above has been identified as

∫ t0(Φ◦ J−1

)∗Bu◦ JdW. Then from 76.4.51

limn→∞

E

(sup

t∈[0,T ]

∣∣∣∣∫ t

0

(Φ◦ J−1)∗Bu◦ JdW −

∫ t

0

(Φn ◦ J−1)∗Bun ◦ JdW

∣∣∣∣)

= 0

It follows from the usual Borel Cantelli argument that there is a set of measure zero and afurther subsequence such that off this set, all the above convergences happen and also∫ t

0

(Φn ◦ J−1)∗Bun ◦ JdW →

∫ t

0

(Φ◦ J−1)∗Bu◦ JdW

uniformly on [0,T ].The rest of the argument is identical. This yields the following theorem.

Theorem 76.4.9 Suppose V ≡ Lp ([0,T ]×Ω,V ) where p ≥ 2,with the σ algebra of pro-gressively measurable sets and Vω = Lp ([0,T ] ,V ).

Φ ∈ L2([0,T ]×Ω,L2

(Q1/2U,W

)),

f ∈ V ′ ≡ Lp′ ([0,T ]×Ω,V ′)

and both are progressively measurable. Suppose that

λB+A(ω) : Vω → V ′ω , λB+A : V → V ′