67.9. SOME REPRESENTATION THEOREMS 2307

Theorem 67.9.4 Let F ∈ L2 (Ω,Gt ,P) . Then there exists a unique Gt adapted

f ∈ L2 (Ω× [0, t] ;Rn)

such that F = E (F)+∫ t

0 f(s,ω)T dW.

Proof: By Lemma 67.9.3, the span of functions of the form

exp(∫ t

0hT dW− 1

2

∫ t

0h ·hdt

)where h is a vector valued deterministic step function of the sort described in this lemma,are dense in L2 (Ω,Gt ,P). Given F ∈ L2 (Ω,Gt ,P) , {Gk}∞

k=1 be functions in the subspace oflinear combinations of the above functions which converge to F in L2 (Ω,Gt ,P). For eachof these functions there exists fk an adapted step function such that

Gk = E (Gk)+∫ t

0fk (s,ω)T dW.

Then from the Itô isometry, and the observation that E (Gk−Gl)2→ 0 as k, l→ ∞ by the

above definition of Gk in which the Gk converge to F in L2 (Ω) ,

0 = limk,l→∞

E((Gk−Gl)

2)

= limk,l→∞

E

((E (Gk)+

∫ t

0fk (s,ω)T dW−

(E (Gl)+

∫ t

0fl (s,ω)T dW

))2)

= limk,l→∞

{E (Gk−Gl)

2 +2E (Gk−Gl)∫

Ω

∫ t

0(fk− fl)

T dWdP

+∫

Ω

(∫ t

0(fk− fl)

T dW)2

dP

}

= limk,l→∞

{E (Gk−Gl)

2+∫

Ω

(∫ t

0(fk− fl)

T dW)2

dP

}=

limk,l→∞

∫Ω

(∫ t

0(fk− fl)

T dW)2

dP = limk,l→∞

||fk− fl ||L2(Ω×[0,T ];Rn) (67.9.19)

Going from the third to the fourth equations, is justified because∫Ω

∫ t

0(fk− fl)

T dWdP = 0

thanks to the fact that the Ito integral is a martingale which equals 0 at t = 0.This shows {fk}∞

k=1 is a Cauchy sequence in L2 (Ω× [0, t] ;Rn,P) , where P denotesthe progressively measurable sets. It follows there exists a subsequence and

f ∈ L2 (Ω× [0, t] ;Rn)