104 6 The Dalang-Morton-Willinger Theorem
Proof.The proof is by induction onT.ForT= 1 the statement is Stricker’s
lemma, Theorem 6.4.2 (i). The inductive hypothesis reads:
K 2 =
{T
∑
t=2(Ht,∆St)∣
∣
∣
∣
∣
(Ht)Tt=2predictable andRd-valued}
(6.14)
is closed inL^0 (Ω,FT,P). The basic ingredient of the proof is the reduction of
the integrandsHto a canonical form. However, in this multiperiod setting we
have to be more careful, since the elementsf∈Kcan, a priori, be represented
in many different ways.
LetP:L^0 (Ω,F 0 ,P;Rd)→L^0 (Ω,F 0 ,P;Rd) be the projection on theF 0 -
predictable range of ∆S 1 as in Lemma 6.2.1 and let
H 1 ={H|HisRd-valuedF 0 -measurable andPH=H}. (6.15)In other words, the elements ofH 1 are in canonical form for ∆S 1 .LetI^1
be the linear mappingI^1 :H 1 →L^0 (Ω,FT,P),I^1 (H 1 )=(H 1 ,∆S 1 ). As
in Sect. 6.4 above,I^1 is continuous and injective. LetF 1 ⊂H 1 be defined
byF 1 =(I^1 )−^1 (K 2 ∩I^1 (H 1 )). ClearlyF 1 is a closed subspace ofH 1 since
K 2 is closed by hypothesis. AlsoF 1 is stable in the sense of Lemma 6.2.1.
This means that there is a projection-valuedF 0 -measurable map, calledP 0 :
L^0 (Ω,F 0 ,P;Rd)→L^0 (Ω,F 0 ,P;Rd), so thatf∈F 1 if and only ifP 0 f=f
a.s.. Now we take
E 1 ={H 1 ∈H 1 |P 0 H 1 =0}
={
H 1 ∈L^0 (Ω,F 0 ,P;Rd)∣
∣P(Id−P 0 )H 1 =H 1 }.The elementsH 1 inE 1 are in canonical form and the integrals (H 1 ,∆S 1 )
cannot be obtained by stochastic integrals on (St)Tt=2(see (6.14)). We have
that
K=
{ T
∑
t=1(Ht,∆St)∣
∣
∣
∣
∣
(Ht)Tt=1isRd-valued, predictable andH 1 ∈E 1}
.
Moreover the decomposition of elementsf∈Kintof=(H 1 ,∆S 1 )+f 2 where
H 1 ∈E 1 andf 2 ∈K 2 is unique.
Let nowfn=(H 1 n,∆S 1 )+f 2 nbe a sequence inKwithH 1 n∈E 1 ,f 2 n∈K 2
so thatfn→falmost surely. We have to show thatf∈K. We will show
that (H 1 n)∞n=1 is bounded a.s. and the selection principle will do the rest.
LetA={lim sup|Hn 1 |=∞}. By Proposition 6.3.4 we have that there is a
F 0 -measurably parameterised subsequence (τn)∞n=1so that:|H 1 τn|→∞onA
andHτ 1 n →H 1 onAcfor someH 1 ∈E 1. We will show thatP[A]=0.If
this were not the case we could apply Proposition 6.3.3 and suppose that we
have H
1 τn
|H 1 τn|→ψ^1 a.s. on the setA,whereψ^1 =ψ^11 Ais someF^0 -measurable
function supported byAwhere it takes values in the unit sphere ofRd. Clearly