65.11. THE QUADRATIC VARIATION OF THE STOCHASTIC INTEGRAL 2257

For the particular ω of interest,

=∫ t∧τn

aX[a,τm]ΦdW

and this equals

=∫ t

aX[a,τn]X[a,τm]ΦdW =

∫ t

aX[a,τn]ΦdW

for all ω, in particular for the given ω . Therefore, for the particular ω of interest,∫ t

aX[a,τn]ΦdW =

∫ t

aX[a,τm]ΦdW

Thus the limit exists because for all n large enough, the integral is eventually constant.Then

∫ ta ΦdW is Ft adapted because for U an open set in H,(∫ t

aΦdW

)−1

(U) = ∪∞n=1

((∫ t

aX[a,τn]ΦdW

)−1

(U)∩ [τn > t]

)∈Ft .

The next lemma says that even when∫ t

a Φ(s)dW (s) is only a local martingale relativeto a suitable localizing sequence, it is still the case that∫ t∧σ

aΦdW =

∫ t

aX[a,σ ]ΦdW.

Lemma 65.10.5 Let Φ be progressively measurable and suppose there exists the localizingsequence described above. Then if σ is a stopping time,∫ t∧σ

aΦdW (s) =

∫ t

aX[a,σ ]ΦdW (s)

Proof: Let {τn} be the localizing sequence described above for which, when the localmartingale is stopped, it results in a martingale, (satisfying 1 and 2 on Page 2255). Thenby definition,∫ t∧σ

aΦdW (s) ≡ lim

n→∞

∫ t∧τn∧σ

aΦdW (s)

= limn→∞

∫ t∧τn

aX[a,σ ]ΦdW (s) =

∫ t

aX[a,σ ]ΦdW (s)

since t ∧ τn = t for all n large enough.

65.11 The Quadratic Variation Of The Stochastic Inte-gral

An important corollary of Lemma 65.9.1 concerns the quadratic variation of∫ t

a ΦdW . It isconvenient here to use the notation

∫ ta ΦdW ≡Φ ·W (t) . Recall this is a local submartingale

[Φ ·W ] such that∥Φ ·W (t)∥2

H = [Φ ·W ] (t)+N (t)