2262 CHAPTER 65. STOCHASTIC INTEGRATION
It follows one can consider ∫ T
aLΦdW.
Assume to begin with that Φ∈L2([a,T ]×Ω;L2
(Q1/2U,H
)). Next recall the situation
in which the definition of the integral is considered.
U↓ Q1/2
U1 ⊇ JQ1/2U J←1−1
Q1/2U
Φn ↘ ↓ Φ
H
Letting {Φn} be an approximating sequence of elementary functions satisfying
E(∫ T
a
∣∣∣∣Φn−Φ◦ J−1∣∣∣∣2L2(JQ1/2U,H) dt
)→ 0,
it is also the case that
E(∫ T
a
∣∣∣∣LΦn−LΦ◦ J−1∣∣∣∣2L2(JQ1/2U,H1)
dt)→ 0
By the definition of the integral, for each t∫ t
aLΦdW = lim
n→∞
∫ t
aLΦndW = lim
n→∞L∫ t
aΦndW
= L limn→∞
∫ t
aΦndW = L
∫ t
aΦdW
The second equality is obvious for elementary functions.Now consider the case where Φ is only stochastically square integrable so that all is
known is that
P([∫ T
a||Φ||2
L2(Q1/2U,H) dt < ∞
])= 1.
Then define τn as above
τn ≡ inf{
t :∫ t
a||Φ||2
L2(Q1/2U,H) dt ≥ n}
This sequence of stopping times works for LΦ also. Recall there were two conditions thesequence of stopping times needed to satisfy. The first is obvious. Here is why the secondholds.∫ T
a
∣∣∣∣X[a,τn]LΦ∣∣∣∣2
L2(Q1/2U,H1)dt ≤ ||L||2
∫ T
a
∣∣∣∣X[a,τn]Φ∣∣∣∣2
L2(Q1/2U,H) dt
= ||L||2∫
τn
a||Φ||2
L2(Q1/2U,H) dt ≤ ||L||2 n