80 5 The Kreps-Yan Theorem
prove versions of the theorem, where the closure with respect to the weak-
star topology is replaced by the closure with respect to some finer topology
(ideally the topology of uniform convergence, which allows for an obvious and
convincing economic interpretation).
To do so, let us contemplate once more, where the above encountered diffi-
culties related to the weak-star topology originated from: they are essentially
caused by our restriction to consider onlysimple, admissible trading strategies.
These nice and simple objects can be defined without any limiting procedure,
but we should not forget, that — except for the case of finite discrete time —
they are only auxiliary gimmicks, playing the same role as step functions in in-
tegration theory. The concrete examples of trading strategies (e.g., replicating
a European call option) encountered in Chap. 4 for the case of the Bachelier
and the Black-Scholes model led us already out of this class: of course, they
are not simple trading strategies. This is similar to the situation in classi-
cal integration theory, where the most basic examples, such as polynomials,
trigonometric functions etc, of course fail to be step functions.
Hence we have to pass to a suitable class of more general trading strategies
then just the simple, admissible ones. Among other pleasant and important
features, this will have the following effects on the corresponding setsCand
K: these sets will turn out to be “closer to their closures” (ideally they will
already be closed in the relevant topology, see Theorem 6.9.2 and 8.2.2 below),
than the above considered setsCsimpleandKsimple. The reason is that the
passage from simple to more general integrandsinvolves already a limiting
procedure.
We shall take up the theme of developing a no-arbitrage theory based on
general (i.e., not necessarily simple) integrands in Chap. 8 below. But before
doing so we shall investigate in Chap. 6 in more detail, the situation of a
finite time index setT={ 0 , 1 ,...,T}, where — as opposed to Chap. 2 — we
drop the assumption that (Ω,F,P) is finite. This is the theme of the Dalang-
Morton-Willinger theorem. In this setting the simple integrands are already
the general concept. It turns out that the assumption of(NA)(without any
strengthening of “no free lunch” type) implies already that the coneCsimpleis
closed w.r.t. the relevant topologieswhich will allow us to directly apply the
Kreps-Yan theorem.
In order to prove the Dalang-Morton-Willinger theorem in full generality
it will be convenient to state a slightly more general version of the Kreps-Yan
theorem. While in Theorem 5.2.2 above we considered the duality between
L∞andL^1 only, we now state the theorem for the duality betweenLpand
Lq, for arbitrary 1≤p≤∞and^1 p+^1 q= 1. We observe that this is still a
more restricted degree of generality then the one considered by D. Kreps in
his original paper [K 81], who worked with an abstract dual pair〈E, E′〉of
vector lattices.
We use the occasion to give a slightly different proof than in Theorem
5.2.2 above, replacing the exhaustion argument by a somewhat more direct