134 8 Arbitrage Theory in Continuous Time: an Overview
from a good old friend from functional analysis, the theorem of Krein-Smulian
as worked out in Proposition 5.2.4 above.
Once we have reduced the problem to the case of sequences (Hn)∞n=0
we may apply another good friend from functional analysis, the theorem
of Banach-Steinhaus (also called principle of uniform boundedness): ifase-
quence(gn)∞n=0in a dual Banach spaceX∗is weak-star convergent, the norms
(‖gn‖)∞n=0remain bounded. This result implies (see Chap. 9 below) that we
may reduce to the case where the sequence (Hn)∞n=0admits a uniform admis-
sibility boundMsuch thatHn·S≥−M, for alln∈N.
Putting together these results from general functional analysis, it will suf-
fice to prove the following result to complete the proof of Theorem 8.2.2.
Crucial Lemma 8.2.3.Under the hypotheses of Theorem 8.2.2, let(Hn)∞n=0
be a sequence of admissible integrands such that
(Hn·S)t≥− 1 , a.s., fort≥ 0 andn∈N. (8.4)Assume also thatfn=(Hn·S)∞converges almost surely tof. Then there
is an admissible integrandHsuch that
(H·S)∞≥f. (8.5)The admissible strategyHcan be chosen in such a way that(H·S)∞is a
maximal element in the setK.
To convince ourselves that Lemma 8.2.3 indeed implies Theorem 8.2.2, we
still have to justify one more reduction step which is contained in the state-
ment of Lemma 8.2.3: we may reduce to the case, when (fn)∞n=0converges
almost surely. This is done by an elementary lemma in the spirit of Kom-
los’ theorem (Lemma 9.8.1, compare also Lemma 6.6.1 above). In its simplest
form it states the following: Let (fn)∞n=0be an arbitrary sequence of random
variables uniformly bounded from below. Then we may find convex combina-
tionshn∈conv{fn,fn+1,...}converging almost surely to anR∪{+∞}-valued
random variablef. For more refined variations on this theme see Chap. 15.
Note that the passage to convex combinations does not cost anything in the
present context, where our aim is to find a limit to a given sequence in a locally
convex vector space; hence the above argument allows us to reduce to the case
where we may assume, in addition to (8.4), that (fn)∞n=0=((Hn·S)∞)∞n=0
converges almost surely to a functionf:Ω→R∪{+∞}. Using the assumption
(NFLVR)we can quickly show in the present context thatf must be a.s.
finitely valued.
Summing up, Lemma 8.2.3 is a statement about the possibility of passing
to a limitH, for a given sequence (Hn)∞n=0of admissible integrands. The
crucial hypothesis is the uniform one-sided boundedness (8.4); apart from this
strong assumption, we only have an information on the a.s. convergence ofthe
terminal values((Hn·S)∞)∞n=0, but we do not have any a priori information
on the convergence ofthe processes((Hn·S)t≥ 0 )∞n=0.