2324 CHAPTER 68. A DIFFERENT KIND OF STOCHASTIC INTEGRATION

Also for a ∈ Λ, define

Ha (x)≡∞

∏j=1

Hai (x)

This is well defined because H0 (x) = 1 and all but finitely many terms of this infiniteproduct are therefore equal to 1. Now let {ei} be an orthonormal basis for H. For a ∈ Λ,

Φa ≡√

a!∞

∏i=1

Hai (W (ei)) ∈ L2 (Ω)

Suppose a,b ∈ Λ.∫Ω

ΦaΦbdP =√

a!√

b!∫

Ω

∞

∏i=1

Hai (W (ei))Hbi (W (ei))dP

Now recall from Corollary 64.6.1 the random variables {W (ei)} are independent. There-fore, the above equals

√a!√

b!∞

∏i=1

∫Ω

Hai (W (ei))Hbi (W (ei))dP =

{1 if a = b0 if a ̸= b

Thus {Φa : a ∈ Λ} is an orthonormal set in L2 (Ω).

Lemma 68.2.5 If sk → h, then for n ∈ N, there is a subsequence, still called sk for whichW (sk)

n→W (h)n in L2 (Ω).

Proof: If sk→ h, does W (sk)n→W (h)n in L2 (Ω) for some subsequence? First of all,

∥W (h)−W (sk)∥2L2(Ω) = |sk−h|2H → 0

and so there is a subsequence, still called k such that W (sk)(ω)→W (h)(ω) for a.e. ω .Consider ∫

Ω

|W (h)n−W (sk)n|2 dP (68.2.8)

Does this converge to 0? The integrand is bounded by 2(

W (h)2n +W (sk)2n). Since

W (h) ,W (sk) are symmetric,∫Ω

(2(

W (h)2n +W (sk)2n))2

dP≤ 8∫

Ω

(W (h)4n +W (sk)

4n)

dP

= 16∫

Ω∩[W (h)≥0]e4nW (h)dP+16

∫Ω∩[W (sk)≥0]

e4nW (sk)dP

≤ 16∫

Ω

e4nW (h)dP+16∫

Ω

e4nW (sk)dP

≤ 16e12 |4nh|+16e

12 |4nsk|