168 9 Fundamental Theorem of Asset Pricing
by the larger outcome 0. Replacinggby a maximal element is in this sense a
“best try”.
Of course, this is only a very simple example and the reader may construct
examples where even more pathological phenomena occur. But the present
example shows in a convincing way, that the convergence of the final outcome
gndoes not imply any kind of convergence of the corresponding integrandsHn.
The difficulties arising from the above introduced “suicide strategies”Hn
were already addressed in [HP 81].
We finish this remark by giving an example of a process (St)t≥ 0 such
thatK =K 0 ∩L∞is notσ(L∞,L^1 )-closed. This underlines again the im-
portance of considering the coneC 0 of elements dominated by elements of
K 0 , a phenomenon already encountered in the Kreps-Yan theorem (see [S 94,
Theorem 3.1]). The example is in discrete time. We consider a sequenceYn
of independent variables taking 3 possible values{a, b, c}. The probability is
defined asP[Yn=a]=^12 ;P[Yn=b]=^12 − 4 −n;P[Yn=c]=4−n.We
again use the sequence of Rademacher functions defined this time asrn=1
ifYn=a,andrn=−1ifYn=borc.LetTbe defined as the firstnso that
Yn=c. It is clear thatP[there isnsuch thatYn=c]≤^13. We define the
processSasSm=
∑min(m,T)
n=1 rn. More precisely we take the sum of the first
mRademacher functions but we stop the process atT. The original measure
is clearly a martingale measure forS. Let us now defineBnas the set{T>n}
and letHnbe the doubling strategy starting at timen. From the definition
ofTit follows that the final outcomegn=(Hn·S)∞=− (^1) Bn. The sequence
gntends weak-star tog=− (^1) {T=∞}. This random variableg, however, is
not in the setK. Suppose on the contrary thatHis a predictable integrand
such that (H·S)∞=− (^1) {T=∞}.Ontheset{T≤n− 1 }we can without
disturbing the final outcome, replaceH 1 ,...,Hnby 0. This new integrand is
still denoted byH.Letnownbe the first integer such thatHnis not iden-
tically 0. On the set{T=n}the productHnrnis also the final outcome.
Since this set is disjoint from the set{T=∞}we find thatHn=0ontheset
{T=n}.ThevariableHnisFn− 1 -measurable and by independence ofFn− 1
andYnwe therefore haveHn=0ontheset{T>n− 1 }. This contradicts
the assumption onn.
For the rest of the proof of Theorem 9.4.2 we will denote byf 0 a maximal
element ofD,(fn)n≥ 1 is a sequence of elements, obtained asfn=(Hn·S)∞,
whereHnare 1-admissible strategiesHn, and the sequencefnconverges to
f 0 almost surely. Remark that if we can prove thatf 0 ∈K 0 , we finish the
proof of Theorem 9.4.2.
Lemma 9.4.6.With the notation introduced above we have that the random
variables
Fn,m=
(
(Hn−Hm)·S)∗
=sup
t∈R+|(Hn·S)t−(Hm·S)t|tend to zero in probability asn, m→∞.