164 9 Fundamental Theorem of Asset Pricing
hn≥gn.Ifhn=(Ln·S)∞then‖h−n‖∞≤n^1 and henceLnisn^1 -admissible
by Proposition 9.3.6 and the property(NA)ofS. Lemma 9.8.1 allows us to
replacehnbyfn∈conv{hn,hn+1,...}such thatfnconverges tof 0 :Ω→
[0,∞] in probability. LetHnbe the corresponding convex combination of the
integrands (Lk)k≥n. ObviouslyHnis still^1 n-admissible andfn−tends to 0 in
L∞.Fornlarge enough we have‖gn−g 0 ‖∞≤α 2 and henceP[hn>0]≥
P[gn>α 2 ]>α 2. Lemma 9.8.1 now shows thatP[f 0 >0]>0.
The following corollary relates the condition(NFLVR)with the condition
(d) in [D 92] (which in turn is just reformulating the concept of(NFLBR)to
be defined in Sect. 9.6 below).
Corollary 9.3.8.The semi-martingaleSsatisfies the condition (NFLVR) if
and only if for a sequence(gn)n≥ 1 inK 0 , the condition‖gn−‖∞→ 0 implies
thatgntends to 0 in probability.
Proof.We first observe that the condition stated in the corollary implies(NA).
The corollary is now a direct consequence of the Proposition 9.3.7 and the
Lemma 9.8.1.
Corollary 9.3.9.Under the assumption (NA), the semi-martingaleSsatis-
fies the condition (NFLVR) if and only if the set
{(H·S)∞|H 1 -admissible and of bounded support}is bounded inL^0.
Proof.From the proof of Proposition 9.3.2, it follows that the set{sup 0 ≤t(H·
S)t|H1-admissible}is also bounded inL^0. If the sequence (gn)n≥ 1 inK 0 ,
satisfies‖g−n‖∞→0, then by the(NA)property and Proposition 9.3.6,gn=
(Hn·S)∞whereHnisεn-admissible withεn=‖g−n‖∞. The sequenceε^1 ngn
has to be bounded which is only possible whengntends to 0 in probability.
The conclusion now follows from the preceding corollary.
9.4 Proof of the Main Theorem
In this section we prove the main theorem of the paper. The proof follows
the following plan: prove that the setC, introduced in Sect. 9.2, is weak-star-
closed inL∞and apply the separation theorem of Kreps and Yan (see [S 94]),
which in turn is a consequence of the Hahn-Banach theorem. We use similar
arguments as in [D 92] and [S 94]. The technicalities are, however, different
and more complicated.
Definition 9.4.1 (compare [MB 91] and [S 94], Definition 3.4).Asub-
setDofL^0 isFatou closedif for every sequence(fn)n≥ 1 uniformly bounded
from below and such thatfn→falmost surely, we havef∈D.