5.2 No Free Lunch 79(i) Is it possible, in general, to replace the net (gα)α∈Iabove by a sequence
(gn)∞n=0?
(ii) Can we choose the net (gα)α∈I (or, hopefully, the sequence (gn)∞n=0)
such that (gα)α∈Iremains bounded inL∞(P) (or at least such that the
negative parts ((gα)−)α∈Iremain bounded)?
Note that this latter issue is crucial from an economic point of view, as
it pertains to the question whether the approximation offby (gα)α∈I
can be donerespecting a finite credit line.
(iii) Is it really necessary to allow for the “throwing away of money”, i.e., to
pass fromKsimpletoCsimple?
It turns out that questions (i) and (ii) are intimately related and, in gen-
eral, the answer to these questions is no. In fact, the study of the pathologies
of the operation of taking the weak-star-closure is an old theme of functional
analysis. On the very last pages of S. Banach’s original book ([B 32]) the fol-
lowing example is given: there is a separable Banach spaceXsuch that, for
every given fixed numbern≥1(sayn= 147), there is a convex coneCin
the dual spaceX∗, such thatCC(1)C(2)...C(n)=C(n+1)=C,
whereC(k)denotes the sequential weak-star-closure ofC(k−1), i.e., the limits
of weak-star convergentsequences(xi)∞i=0, withxi∈C(k−1),andCdenotes
the weak-star-closure ofC. In other words, by taking the limits of weak-star
convergentsequencesinCwe do not obtain the weak-star-closure ofCimme-
diately, but we have to repeat this operation preciselyntimes, when finally
this process stabilises to arrive at the weak-star-closureC.
In Banach’s book this construction is done forX=c 0 andX∗=^1 while
our present context isX=L^1 (P)andX∗=L∞(P). Adapting the ideas
from Banach’s book, it is possible to construct a semi-martingaleSsuch that
the corresponding convex coneCsimplehas the following property: taking the
weak-star sequential closure (Csimple)(1), the resulting set intersectsL∞+(P)
only in{ 0 }; but doing the operation twice, we obtain the weak-star-closure
C(2)=CandCintersectsL∞+(P) in a non-trivial way (see Example 9.7.8
below). Hence we cannot — in general — reduce to sequences (gn)∞n=0in the
definition of(NFL). The construction of this example uses a process with
jumps; for continuous processes the situation is, in fact, nicer, and in this
case it is possible to give positive answers to questions (i) and (ii) above (see
[Str 90], [D 92], and Chap. 9 below).
Regarding question (iii), the dividing line again is the continuity of the
processS(see [Str 90] and [D 92] for positive results for continuous processes,
and [S 94] and [S 04a] for counter-examples).
Summing up the above discussion: the theorem of Kreps and Yan is a
beautiful and mathematically precise extension of the fundamental theorem of
asset pricing 2.2.7 to a general framework of stochastic processes in continuous
time. However, in general, the concept of passing to the weak-star-closure does
not allow for a clear-cut economic interpretation. It is therefore desirable to