6
The Dalang-Morton-Willinger Theorem
6.1 Statement of the Theorem
In Chap. 2 we only dealt with finite probability spaces. This was mainly done
because of technical difficulties. As soon as the probability space (Ω,FT,P)
is no longer finite, the corresponding function spaces such asL^1 (Ω,FT,P)or
L∞(Ω,FT,P) are infinite dimensional and we have to fall back on functional
analysis. In this chapter we will present a proof of the Fundamental Theorem
of Asset Pricing, Theorem 2.2.7, in the case of general (Ω,FT,P), but still in
finite discrete time. Since discounting does not present any difficulty, we will
suppose that thed-dimensional price processShas already been discounted as
in Sect. 2.1. Also the notion of the classHof trading strategies does not present
any difficulties and we may adopt Definition 2.1.4 verbatim also for general
(Ω,FT,P) as long as we are working in finite discrete time. In this setting
we can state the following beautiful version of the Fundamental Theorem of
Asset Pricing, due to Dalang, Morton and Willinger [DMW 90].
Theorem 6.1.1.Let(Ω,FT,P)be a probability space and let(St)Tt=0be an
Rd-valued stochastic process adapted to the discrete time filtration(Ft)Tt=0.
Suppose further that the no-arbitrage (NA) condition holds:
K∩L^0 +(Ω,FT,P)={ 0 }, (6.1)where
K={T
∑
t=1(Ht,∆St)∣
∣
∣
∣
∣
H∈H
}
Then there exists an equivalent probability measureQ,Q∼Pso that(i) St∈L^1 (Ω,FT,P),t=0,...,T,
(ii) (St)Tt=0is aQ-martingale, i.e.,E[St|Ft− 1 ]=St− 1 ,fort=1,...,T,
(iii)ddQPis bounded, i.e.ddQP∈L∞(Ω,FT,P).