2296 CHAPTER 67. THE EASY ITO FORMULA

Claim: limn→∞ X[0,τn] = 1.Proof of claim: From maximal estimates as in the construction of the stochastic integral

and the Borel Cantelli lemma, it follows that there exists a subsequence still denoted by nand a set of measure zero N such that for ω /∈ N1,∫ t

0ΦndW →

∫ t

0ΦdW

uniformly on [0,T ] . Also one can show that off a set of measure zero, there is a subsequencestill called n such that

∫ t0 φ n (s)ds→

∫ t0 φ (s)ds uniformly on [0,T ] . Here is why.

E(∣∣∣∣∫ t

0φ n (s)ds−

∫ t

0φ (s)ds

∣∣∣∣)≤ ∫Ω

∫ T

0|φ n−φ |dtdP

which is given to converge to 0. Thus

P(

maxt∈[0,T ]

∣∣∣∣∫ t

0φ n (s)ds−

∫ t

0φ (s)ds

∣∣∣∣> λ

)≤ P

(∫ T

0|φ n (s)−φ (s)|ds > λ

)

≤ 1λ

∫[∫ T

0 |φn(s)−φ(s)|ds>λ ]

∫ T

0|φ n (s)−φ (s)|dsdP

≤ 1λ

∫Ω

∫ T

0|φ n (s)−φ (s)|dsdP

Thus

P(

maxt∈[0,T ]

∣∣∣∣∫ t

0φ n (s)ds−

∫ t

0φ (s)ds

∣∣∣∣> 2−k)≤ 2k

∫Ω

∫ T

0|φ n (s)−φ (s)|dsdP

If n > nk, the right side is less than 2−k. Use φ nk. Then there exists a set of measure zero

N2 such that for ω /∈ N2, ∣∣∣∣∫ t

0φ n (s)ds−

∫ t

0φ (s)ds

∣∣∣∣→ 0

uniformly. Hence, you can take a couple of subsequences and assert that there exists asubsequence still called n and a set of measure zero N such that Xn (t)→ X (t) uniformlyon [0,T ] for each ω /∈ N. Since |X (t,ω)| < M, it follows that for each ω /∈ N, when n islarge enough, τn = ∞ and this proves the claim.

From the claim, it follows that X[0,τn]Φn→ Φ◦ J−1 in L2([0,T ]×Ω;L2

(Q1/2U,H

))and X[0,τn]φ n → φ in L1 ([0,T ]×Ω;H ) . Thus you can replace Φn in the above withX[0,τn]Φn and φ n with X[0,τn]φ n. Thus there exists a subsequence, still called n and aset of measure zero N such that for ω /∈ N,∫ t

0X[0,τn]ΦndW →

∫ t

0ΦdW