2258 CHAPTER 65. STOCHASTIC INTEGRATION

where N is a local martingale. Recall the quadratic variation is unique so that if it acts likethe quadratic variation, then it is the quadratic variation. Recall also why this was so. Ifyou have a local martingale equal to the difference of increasing adapted processes whichequals 0 when t = 0, then the local martingale was equal to 0. Of course you can substitutea for 0.

Corollary 65.11.1 Suppose Φ is L2(Q1/2U,H

)progressively measurable and has the lo-

calizing sequence with the two properties in Lemma 65.10.2. Then the quadratic variation,[Φ ·W ] is given by the formula

[Φ ·W ] (t) =∫ t

a||Φ(s)||2

L2(Q1/2U,H) ds

Proof: By the above discussion,∫ t

a ΦdW is a local martingale. Let {τn} be a local-izing sequence for which the stopped local martingale is a martingale and ΦX[a,τn] is inL2([a,T ]×Ω,L2

(Q1/2U,H

)). Also let σ be a stopping time with two values no larger

than T . Then from Lemma 65.10.5,

E

(∣∣∣∣∫ τn∧σ

aΦdW

∣∣∣∣2H−∫

τn∧σ

a||Φ(s)||2

L2(Q1/2U,H) ds

)

E

(∣∣∣∣∫ T∧τn∧σ

aΦdW

∣∣∣∣2H−∫ T∧τn∧σ

a||Φ(s)||2

L2(Q1/2U,H) ds

)

= E

(∣∣∣∣∫ T

aX[a,τn]X[0,σ ]ΦdW

∣∣∣∣2H−∫ T

aX[a,τn]X[0,σ ] ||Φ(s)||2

L2(Q1/2U,H) ds

)

= E(∫ T

a

∥∥X[a,τn]X[0,σ ]Φ∥∥2

L2dt)−E

(∫ T

a

∣∣∣∣X[a,τn]X[0,σ ]Φ(s)∣∣∣∣2

L2ds)= 0

thanks to the Ito isometry. There is also no change in letting σ = t. You still get 0. Itfollows from Lemma 63.1.1, the lemma about recognizing a martingale when you see one,that

t→∣∣∣∣∫ t∧τn

aΦdW

∣∣∣∣2H−∫ t∧τn

a||Φ(s)||2

L2(Q1/2U,H) ds

is a martingale. Therefore,∣∣∣∣∫ t

aΦdW

∣∣∣∣2H−∫ t

a||Φ(s)||2

L2(Q1/2U,H) ds

is a local martingale and so, by uniqueness of the quadratic variation,

[Φ ·W ] (t) =∫ t

a||Φ(s)||2

L2(Q1/2U,H) ds

Here is an interesting little lemma which seems to be true.