8.2 The Crucial Lemma 133up, which will cause the huddle — including your originale— to disappear.
Translating this story into the language of stochastic integration, we have a
martingaleS(in fact, a random walk) and an admissible trading strategyH
such that (H·S) 0 = 0 while (H·S)∞=−1 so that we have a strict inequality
in (8.3).
A continuous analogue of the suicide strategy is given by the processSt=
exp
(
Wt−^12 t)
,whereWis a standard Brownian motion. This process starts at
1 and moves up and down in ]0,∞[ according to a fair game (it is a martingale).
But, asttends to infinity,Sttends to 0 almost surely. The reader can see that
the processScan assume quite high values but eventually the player loses the
initial betS 0 =1.
We now discuss the difficult implication(NFLVR)⇒(EMM)of Theorem
8.2.1. It is reduced to the subsequent theorem which may be viewed as the
“abstract” version of Theorem 8.2.1:
Theorem 8.2.2 (Theorem 9.4.2).In the setting of Theorem 8.2.1 assume
that (ii) holds true, i.e., thatSsatisfies (NFLVR).
Then the coneC⊆L∞(P)is weak-star-closed.
The fact that Theorem 8.2.2 implies Theorem 8.2.1 follows immediately
from the Kreps-Yan Theorem 5.2.2, i.e., we find a probability measureQ∼P
such thatSis a localQ-martingale. Theorem 8.2.2 tells us that we don’t have
to bother about passing to the weak-star-closure ofCany more, as assumption
(ii) of Theorem 8.2.1 implies thatCalready is weak-star-closed.Inotherwords,
our program of choosing the“right”class of admissible integrands has been
successful: the passage to the limit which was necessary in the context of the
Kreps-Yan theorem, i.e., the passage fromCsimpleto its weak-star-closure, is
already taken care of by the passages to the limit in the stochastic integration
theory from simple to general admissible integrands.
In fact, Theorem 8.2.2 tells us that — under the assumption of(NFLVR)
—Cequals precisely the weak-star-closure ofCsimple.ThefactthatCsimpleis
weak-star dense inCfollows from Chap. 7 where the general theory of stochas-
tic integration is based on the idea of approximating a general integrand by
simple integrands.
By rephrasing Theorem 8.2.1 in the form of Theorem 8.2.2, we did not
come closer to a proof yet. But we see more clearly, what the heart of the
matteris:foranet(Hα)α∈Iof admissible integrands,fα=(Hα·S)∞and
gα≤fαsuch that (gα)α∈Iweak-star converges inL∞(P)tosomef,wehave
to show that we can find an admissible integrandHsuch thatf≤(H·S)∞.
This will prove Theorem 8.2.2 and therefore 8.2.1. Loosely speaking, we have
to be able to pass from a net (Hα)α∈Iof admissible trading strategies to a
limiting admissible trading strategyH.
The first good news on our way to prove this result is that in the present
context we may reduce from the case of a general net (Hα)α∈Ito the case of
a sequence (Hn)∞n=0and therefore to a sequencefn=(Hn·S)∞. This follows