2256 CHAPTER 65. STOCHASTIC INTEGRATION
Proof: It follows from Proposition 62.7.5 that τn is a stopping time because it is thefirst hitting time of a closed set by an adapted continuous process.
It remains to verify the two claims. There exists a set of measure 0, N such that forω /∈ N ∫ T
a∥Φ∥2
L2(Q1/2U,H) dt < ∞
Therefore, for such ω, there exists n large enough that∫ t
a∥Φ∥2
L2(Q1/2U,H) ds < n
and so τn (ω)≥ t. Now consider the second claim.
E(∫ T
a
∥∥X[a,τn]Φ∥∥2
L2(Q1/2U,H) dt)
= E(∫
τn(ω)∧T
a∥Φ∥2
L2(Q1/2U,H) dt)≤ E (n) = n.
With this lemma, it is possible to give the following definition.
Definition 65.10.3 Suppose Φ is L2(Q1/2U,H
)progressively measurable and
P([∫ T
a∥Φ∥2
L2(Q1/2U,H) ds < ∞
])= 1. (65.10.18)
More generally, suppose there exists a localizing sequence of stopping times τn having thetwo properties of Lemma 65.10.2. Then for all ω not in the exceptional set N.∫ t
aΦdW ≡ lim
n→∞
∫ t
aX[a,τn]ΦdW
Lemma 65.10.4 The above definition is well defined. For all ω not in a set of measurezero, ∫ t
aΦdW (ω)≡ lim
n→∞
∫ t
aX[a,τn]ΦdW (ω)
the function on the right being constant for all n large enough for a given ω . The randomvariable
∫ ta ΦdW is also Ft adapted.
Proof: Let {τn} be a sequence of stopping times as described in 1 and 2 of Lemma65.10.2. Such a sequence exists by Lemma 65.10.2. It makes sense to define the randomvariable ∫ t
aX[a,τn]ΦdW
Now what if both τm and τn are at least as large as t for some ω? Do the two randomvariables coincide at that value of ω? Say m > n so that τm (ω)≥ τn (ω)> t. For the givenω, ∫ t
aX[a,τm]ΦdW =
∫ t∧τm
aX[a,τm]ΦdW