2310 CHAPTER 67. THE EASY ITO FORMULA

and so by the Itô isometry, ∣∣∣∣ft1 − ft2∣∣∣∣

L2(Ω×[0,t1];Rn)= 0.

Letting N ∈ N, it follows that

M (t) = E (M (0))+∫ t

0fN (s, ·)T dW

for all t ≤ N. Let g = fN for t ∈ [0,N] . Then asside from a set of measure zero, this is welldefined and for all t ≥ 0

M (t) = E (M (0))+∫ t

0g(s, ·)T dW

Surely this is an incredible theorem. Note that it implies all the martingales adapted toGt which are in L2 for each t must be continuous a.e. and are obtained from an Ito integral.Also, any such martingale satisfies M (0) = E (M (0)) . Isn’t that amazing? Also note thatthis featured Rn as where W has its values and n was arbitrary. One could have n = 1 ifdesired.

The above theorems can also be obtained from another approach. It involves showingthat random variables of the form

φ (W(t1) , · · · ,W(tk))

are dense in L2 (Ω,GT ). This theorem is interesting for its own sake and it involves inter-esting results discussed earlier. Recall the Doob Dynkin lemma, Lemma 59.3.6 on Page1868 which is listed here.

Lemma 67.9.6 Suppose X,Y1,Y2, · · · ,Yk are random vectors, X having values in Rn andY j having values in Rp j and

X,Y j ∈ L1 (Ω) .

Suppose X is σ (Y1, · · · ,Yk) measurable. Thus

{X−1 (E) : E Borel

}⊆

{(Y1, · · · ,Yk)

−1 (F) : F is Borel ink

∏j=1Rp j

}

Then there exists a Borel function, g :∏kj=1Rp j → Rn such that

X = g(Y) .

Recall also the submartingale convergence theorem.

Theorem 67.9.7 (submartingale convergence theorem) Let

{(Xi,Si)}∞

i=1

be a submartingale with K ≡ supE (|Xn|)<∞. Then there exists a random variable X , suchthat E (|X |)≤ K and

limn→∞

Xn (ω) = X (ω) a.e.