Kenneth Kuttler

2344 CHAPTER 68. A DIFFERENT KIND OF STOCHASTIC INTEGRATION

Since δ is an adjoint map, it is clearly linear. Hence, if h is arbitrary, h ̸= 0 of course,

δ (Fh) = ∥h∥δ

(F(

W (h)∥h∥

))= ∥h∥F

W (h)∥h∥

−∥h∥n

∑i=1

Di (F)

(hi,

h∥h∥

)= FW (h)−

∑i=1

Di (F)(hi,h)H = FW (h)−⟨DF,h⟩ (68.4.22)

Note how this looks just like integration by parts. More generally,

∑j=1

Fjh j

∑j=1

δ (Fjh j) =m

∑j=1

FjW (h j)−⟨DFj,h j

⟩Are functions like ∑

mj=1 Fjh j where Fj is a polynomial in variables of the form W (h)

dense in Lp (Ω,H)? It was shown earlier in Lemma 68.4.3 that polynomial functions F inthe W (h) are dense in Lp (Ω) for any p. Let s(ω) = ∑

nk=1 hkXEk be a simple function.

Then XEk is clearly in Lp (Ω) and so there exists Fk a polynomial in the W (h) which isas close as desired to XEk in Lp. Hence ∑

nk=1 hkFk is close to s in Lp (Ω,H) and so since

these simple functions are dense, it follows that these kinds of functions are indeed dense inLp (Ω,H), this for any p > 1. The above discussion is summarized in the following lemma.

Lemma 68.4.5 Functions of the form ∑nk=1 Fkhk where Fk is a polynomial in the W (h)

(Fj ∈P) are dense in Lp (Ω,H) for any p > 1. Also each function of this form is in Dδ

and

∑j=1

Fjh j

∑j=1

δ (Fjh j) =m

∑j=1

FjW (h j)−⟨DFj,h j

⟩What does D do to δ (Fh)? It is shown above that δ (Fh) = FW (h)− ⟨DF,h⟩ . Say

F = F (W (h1) , · · · ,W (hn)) . Then when you do D to δ (Fh), you would get

Fh+n

∑k=1

Dk (F)W (h)hk−n

∑k=1

∑j=1

D j (Dk (F))h j (hk,h)

In other words,Fh+W (h)D(F)−D⟨DF,h⟩

Recall that DG is well defined. This means that we can replace {h1, · · · ,hn,h} with anorthonormal basis

{e1, · · · ,ep,h

}as in

G(W (h1) , · · · ,W (hn) ,W (h)) = Ĝ(W (e1) , · · · ,W (ep) ,W (h))

where we assume ∥h∥= 1 for simplicity. Thus the above equals

D(δ (F)) = D(δ(F̂))

= F̂h+W (h)D(F̂)−D

⟨DF̂ ,h

⟩Now consider E

(δ (Fh)2

)= E ⟨D(δ (Fh)) ,Fh⟩ . Thus the following must be considered.

E⟨F̂h+W (h)D

(F̂)−D

⟨DF̂ ,h

⟩, F̂h

⟩(68.4.23)

2344 CHAPTER 68. A DIFFERENT KIND OF STOCHASTIC INTEGRATIONSince 6 is an adjoint map, it is clearly linear. Hence, if / is arbitrary, h 4 0 of course,5(Fh) = wear)" Fa MLO) (a)= n)-YDUF ) (hi, h) yy = FW (h) — (DF,h) (68.4.22)Note how this looks just like integration by parts. More generally,(Zn) = Y 5 (Fh) “he W (hj) — (DFj,hj)3Are functions like vei Fh; where F; is a polynomial in variables of the form W (h)dense in L? (Q,H)? It was shown earlier in Lemma 68.4.3 that polynomial functions F inthe W (h) are dense in L? (Q) for any p. Let s(@) = Yy_, hy Ze, be a simple function.Then 2g, is clearly in L? (Q) and so there exists F, a polynomial in the W (h) which isas close as desired to 2p, in L’. Hence Yy_, hy Fy is close to s in L? (Q,H) and so sincethese simple functions are dense, it follows that these kinds of functions are indeed dense inL? (Q,H), this for any p > 1. The above discussion is summarized in the following lemma.Lemma 68.4.5 Functions of the form Yi_, Fihy where F, is a polynomial in the W (h)(F; © F) are dense in L? (Q,H) for any p > 1. Also each function of this form is in Déand_(Zn) = 8h) = EWU) (DF)j=lWhat does D do to 6(Fh)? It is shown above that 6 (Fh) = FW (h) — (DF,h). SayF =F (W(h,),--- ,W (Ay)). Then when you do D to 6 (FA), you would getFh+ ¥. Dy (F)W(h)he— YY Dj (Dy hj (hx, h)k= k=1 j=In other words,Fh+W (h)D(F)—D(DF,h)Recall that DG is well defined. This means that we can replace {h1,--- ,4,,h} with anorthonormal basis {e1,-+- ,ep,/} as inG(W (h1),--+ ,W (In) ,W (h)) = G(W (e1) ++ ,W (ep) ,W (h))where we assume ||/|| = 1 for simplicity. Thus the above equalsD(6(F)) =D(6(F)) = Fh+W (h)D(F) —D(DF,h)Now consider E (5 (F n)’) = E(D(6(Fh)), Fh). Thus the following must be considered.E(Fh+W (h)D(F) —D(DF,h) ,Fh) (68.4.23)