2312 CHAPTER 67. THE EASY ITO FORMULA

Now consider the martingale, {E (gM|Gm)}∞

m=1. where here

gM (ω)≡

 g(ω) if g(ω) ∈ [−M,M]M if g(ω)> M−M if g(ω)<−M

and M is chosen large enough that

||g−gM||L2(Ω) < ε/4. (67.9.20)

Now the terms of this martingale are uniformly bounded by M because

|E (gM|Gm)| ≤ E (|gM| |Gm)≤ E (M|Gm) = M.

It follows the martingale is certainly bounded in L1 and so the martingale convergencetheorem stated above can be applied, and so there exists f measurable in σ (Gm,m < ∞)such that limm→∞ E (gM|Gm)(ω) = f (ω) a.e. Also | f (ω)| ≤M a.e. Since all functions arebounded, it follows that this convergence is also in L2 (Ω).

Now letting A ∈ σ (Gm,m < ∞) , it follows from the dominated convergence theoremthat ∫

Af dP = lim

m→∞

∫A

E (gM|Gm)dP =∫

AgMdP

Now GT = σ (W(tk) , tk ≤ T ) = σ (Gm,m≥ 1) and so the above equation implies that f =gM a.e.

By the Doob Dynkin lemma listed above, there exists a Borel measurable h : Rnm→ Rsuch that

E (gM|Gm) = h(Wt1 , · · · ,Wtm) a.e.Of course h is not in C∞

c (Rnm) . Let m be large enough that

||gM−E (gM|Gm)||L2 = || f −E (gM|Gm)||L2 <ε

4. (67.9.21)

Let λ (Wt1 ,··· ,Wtm)be the distribution measure of the random vector (Wt1 , · · · ,Wtm) . Thus

λ (Wt1 ,··· ,Wtm)is a Radon measure and so there exists φ ∈Cc (Rnm) such that(∫

Ω

|E (gM|Gm)−φ (Wt1 , · · · ,Wtm)|2 dP

)1/2

=

(∫Ω

|h(Wt1 , · · · ,Wtm)−φ (Wt1 , · · · ,Wtm)|2 dP

)1/2

=

(∫Rnm|h(x1, · · · ,xm)−φ (x1, · · · ,xm)|2 dλ (Wt1 ,··· ,Wtm)

)1/2

< ε/4.

By convolving with a mollifier, one can assume that φ ∈ C∞c (Rnm) also. It follows from

67.9.20 and 67.9.21 that

||g−φ (Wt1 , · · · ,Wtm)||L2

≤ ||g−gM||L2 + ||gM−E (gM|Gm)||L2

+ ||E (gM|Gm)−φ (Wt1 , · · · ,Wtm)||L2

≤ 3(

ε

4

)< ε