86 6 The Dalang-Morton-Willinger Theorem
The proof of the theorem is not a trivial extension of the case of finite Ω
(we remark, however, that, in the cased= 1, the verification of the theorem
is indeed almost a triviality). Besides the original proof (see [DMW 90]) based
on a measurable selection theorem, many authors have tried to give other and
perhaps easier proofs. Below we repeat some of these proofs.
We first prove the theorem forT= 1. This seemingly easy case contains
already the essential difficulties of the proof. We shall develop some abstract
notions, which will be convenient for the proof, but which shall also turn out
to be of independent interest: the concept of the predictable range of a process
and the concept of a random subsequence (fτk)∞k=1of a sequence (fn)∞n=1of
random variables. We also give a different proof, due to C. Rogers [R 94],
based on the idea of utility maximisation.
To pass from the caseT= 1 to the general caseT≥1 we shall again
offer two alternative proofs. The first will follow the original argument of
[DMW 90] by considering, for eacht=1,...,T, the one-step process (St− 1 ,St)
to which the previous result applies. The equivalent martingale measureQ
for (S 0 ,...,ST) is then obtained by concatenating the densities of the corre-
sponding one step measures, i.e., by multiplicatively composing them (com-
pare Sect. 2.3 above). The second proof will proceed by directly generalising
the arguments for the caseT= 1 to the general case.
We elaborate on these proofs because each of them gives different — and
important — insights into the problem: the argument relying on concatenation
in particular makes clear that the process (S 0 ,...,ST) is free of arbitrage iff
each of the one step processes (St− 1 ,St) is so (see also Theorem 2.3.2 and
Lemma 5.1.5 above); the direct argument shows the closedness of the coneC
of super-replicable claims inL^0 (Ω,FT,P), a result which will play a central
role in the further development of the theory.
The proofs we present below are a mixture of the proofs in [S 92], [D 92],
[R 94], [St 97] and [KS 01].
The proof of the Dalang-Morton-Willinger Theorem for the one-period
caseS=(S 0 ,S 1 ) uses several ingredients that will be explained in the follow-
ing sections. The first problem we have to solve is the problem of redundancy.
Thedgiven assets may have redundancy in the sense that some of them are lin-
ear combinations of others. This linear dependence, given the informationF 0 ,
may, however, depend onω∈Ω. We have to find a way to describe this math-
ematically. We will do this in Sect. 6.2 where we will introduce the notion of
“the predictable range”. It is a coordinate-free approach to describe in anF 0 -
measurable way, the conditional linear (in-)dependence betweenS^11 ,...,Sd 1.
The second ingredient we need is the selection principle.
6.2 The PredictableRange...................................
Let us fix twoσ-algebrasF 0 ⊂F 1 on Ω and a probability measurePonF 1.
For anF 1 -measurable mappingX:Ω→Rdwe will try to find the smallest