2350 CHAPTER 68. A DIFFERENT KIND OF STOCHASTIC INTEGRATION
such that the step functions given by
Φrk (t) ≡
mk
∑j=1
Φ
(tk
j
)X[tk
j−1,tkj )(t)
Φlk (t) ≡
mk
∑j=1
Φ
(tk
j−1
)X[tk
j−1,tkj )(t)
both converge to Φ in K as k→ ∞ and
limk→∞
max{∣∣∣tk
j − tkj+1
∣∣∣ : j ∈ {0, · · · ,mk}}= 0.
Also, each Φ
(tk
j
),Φ(
tkj−1
)is in Lp (Ω;E). One can also assume that Φ(0) = 0. The mesh
points{
tkj
}mk
j=0can be chosen to miss a given set of measure zero. In addition to this, we
can assume that ∣∣∣tkj − tk
j−1
∣∣∣= 2−nk
except for the case where j = 1 or j =mnk when this is so, you could have∣∣∣tk
j − tkj−1
∣∣∣< 2−nk .
Theorem 68.4.8 Let F ∈ L2 (Ω× [0,T ]) and is progressively measurable. Then it has aSkorokhod integral which coincides with the Ito integral.
Proof: From Lemma 68.4.7, there is a sequence of left step functions denoted here as{F l
k
}∞
k=1 which converges to F in L2 (Ω× [0,T ]) where F lk
(tk
j
)= F
(tk
j
). We can take a
subsequence if necessary and assume∥∥∥F lk −F
∥∥∥L2([0,T ]×Ω)
< 2−k
Here the{
tkj
}are mesh points corresponding to the kth partition described above. Thus
each F lk
(tk
j
)is in L2 (Ω). By Lemma 68.4.3 there exists a random variable Gl
k
(tk
j
)which
is a polynomial function of some W (h) for h ∈ L2(
0, tkj
)which can approximate F l
k
(tk
j
)as closely as desired in L2 (Ω). Then choosing these sufficiently close, it can be assumedthat the step functions
Glk ≡
mk−1
∑j=0
Glk
(tk
j
)X(
tkj ,t
kj+1
)also converge in L2 (Ω× [0,T ]) to F . Of course, each of these last step functions are inD(δ ).
The idea is to show that δ(Gl
k
)is Cauchy in L2 (Ω) as k→∞ and then use the fact that,
since δ is an adjoint, it must be a closed operator. This will show that F ∈ L2 (Ω× [0,T ]) ,considered as a subspace of L2
(Ω;L2 (0,∞,R)
), is in D(δ ) and δ (F) is equal to the above