68.4. THE SKOROKHOD INTEGRAL 2349

How does the Skorokhod integral relate to the Ito integral? What about elementaryfunctions and so forth? Let 0 = t0 < t1 < · · ·< tn = T. Consider

n−1

∑k=0

FkX(tk,tk+1)

As shown above, this is one of the things in D(δ ) .

δ

(X(0,t)

n−1

∑k=0

FkX(tk,tk+1)

)= δ

(n−1

∑k=0

FkX[tk∧t,t∧tk+1]

)

=n−1

∑k=0

FkW(X[tk∧t,t∧tk+1]

)−⟨

DFk,X[tk∧t,t∧tk+1]

⟩=

n−1

∑k=0

Fk

(W(X(0,t∧tk+1)

)−W

(X(0,t∧tk)

))−⟨

DFk,X[tk∧t,t∧tk+1]

⟩In terms of the Wiener process, this is of the form

=n−1

∑k=0

Fk (W (t ∧ tk+1)−W (t ∧ tk))−⟨

DFk,X[0,t∧tk+1]−X[0,t∧tk]

⟩H

What ifFk = Fk

(W(X[0,tk]h1

), · · ·W

(X[0,tk]hn

))?

Let Ft ≡ σ(W(X[0,t]h

): h ∈ H

). Then this is clearly a filtration. If Fk is as just described,

then Fk is Ftk adapted.⟨DFk,X[0,t∧tk+1]−X[0,t∧tk]

⟩=∫

∞

0∑s

Ds (Fk)X(0,tk)hsX(t∧tk,t∧tk+1) = 0

because the intervals are disjoint. In this case, the troublesome term at the end vanishesand you are left with

n−1

∑k=0

Fk (W (t ∧ tk+1)−W (t ∧ tk)) (68.4.25)

which is similar to the usual definition for the Ito integral.What if F ∈ L2 (Ω× [0,T ]) and is progressively measurable. Does it have a Skorokhod

integral, and if so, is it the same as the Ito integral? Recall the following useful lemma. Itis Lemma 65.3.1 on Page 2233.

Lemma 68.4.7 Let Φ : [0,T ]×Ω→ E, be B ([0,T ])×F measurable and suppose

Φ ∈ K ≡ Lp ([0,T ]×Ω;E) , p≥ 1

Then there exists a sequence of nested partitions, Pk ⊆Pk+1,

Pk ≡{

tk0 , · · · , tk

mk

}