2348 CHAPTER 68. A DIFFERENT KIND OF STOCHASTIC INTEGRATION

constant term is 0 and consider

E⟨DF, f X[0,t]

⟩≡ E

⟨∑k

Dk (F)hk, f X[0,t]

⟩

Since hk ∈ H = L2 (0,∞;U) , so is hkX[0,t]. Thus the above reduces to

= ∑k

E⟨Dk (F)hk, f X[0,t]

⟩= ∑

kE(∫

∞

0Dk (F (W (h1) , · · ·W (hn)))hkX[0,t] f dt

)Since F is just a polynomial and W is linear and X q

[0,t] = X[0,t], this equals

∑k

E(∫

∞

0Dk(F(W(X[0,t]h1

), · · ·W

(X[0,t]hn

)))hkX[0,t] f dt

)Let Ft = F

(W(X[0,t]h1

), · · ·W

(X[0,t]hn

))and so the above is nothing more than

E⟨DF, f X[0,t]

⟩= E ⟨DFt , f ⟩

and since f ∈ D(δ ) ,∣∣E ⟨DF, f X[0,t]⟩∣∣= |E ⟨DFt , f ⟩| ≤C ( f )∥Ft∥L2(Ω)

Also

∥Ft∥2L2(Ω) =

∫Ω

F(W(X[0,t]h1

), · · ·W

(X[0,t]hn

))2 dP

=∫

Ω

X[0,t]F (W (h1) , · · ·W (hn))2 dP

≤∫

Ω

F (W (h1) , · · ·W (hn))2 dP

Thus for such F which have zero constant term,∣∣E ⟨DF, f X[0,t]⟩∣∣≤C∥F∥L2(Ω)

Now what if F is a constant a? In this case, DF = Da = 0∣∣E ⟨Da, f X[0,t]⟩∣∣= 0≤ ∥a∥L2(Ω)

It follows that X[0,t] f ∈ D(δ ) whenever f is.Note how it was essential in this argument to have F be a polynomial or perhaps more

generally an analytic function. However, in the definition of the Skorokhod integral, onemust test with functions F which are smooth and have polynomial growth. In particular,this would include functions which are infinitely differentiable with compact support, noneof which have valid power series.