65.10. THE STOCHASTIC INTEGRAL AS A LOCAL MARTINGALE 2255

Lemma 65.9.1 Let Φ be progressively measurable and in

L2([a,T ]×Ω;L2

(Q1/2U,H

))Let W (t) be a cylindrical Wiener process as described above. Then for τ a stopping time,X[a,τ]Φ is progressively measurable, in K, and

∫ t∧τ

aΦdW =

∫ t

aX[a,τ]ΦdW.

65.10 The Stochastic Integral As A Local MartingaleWith Lemma 65.9.1, it becomes possible to define the stochastic integral on functionswhich are only stochastically square integrable.

Definition 65.10.1 Φ is stochastically square integrable in L2(Q1/2U,H

)if Φ is progres-

sively measurable and

P([∫ T

a∥Φ(s)∥2

L2(Q1/2U,H) ds < ∞

])= 1

Thus equivalently, there exists N such that P(N) = 0 and for ω /∈ N,∫ T

a∥Φ(s,ω)∥2

L2(Q1/2U,H) ds < ∞.

Lemma 65.10.2 Suppose Φ is L2(Q1/2U,H

)progressively measurable and

P([∫ T

a∥Φ∥2

L2(Q1/2U,H) ds < ∞

])= 1.

Define

τn (ω)≡ inf{

t ∈ [a,T ] :∫ t

a∥Φ∥2

L2(Q1/2U,H) ds≥ n},

By convention, let inf /0 = ∞. Then τn is a stopping time. Furthermore, τn has the followingproperties.

1. {τn} is an increasing sequence and for ω outside a set of measure zero N, for everyt ∈ [a,T ] there exists n such that τn (ω) > t. (It is a localizing sequence of stoppingtimes.)

2. For each n, X[a,τn]Φ is progressively measurable and

E(∫ T

a

∥∥X[a,τn]Φ∥∥2

L2(Q1/2U,H) dt)< ∞