138 8 Arbitrage Theory in Continuous Time: an Overview
One easily verifies (arguing either “mathematically” or “economically”)
that the processK·Ssatisfies
(K·S)t≥−M a.s., for all t,whereMis the uniform lower bound forH·S,and
lim
t→∞
(K·S)t=∞ a.s. onA.HenceKdescribes a trading scheme, where the investor can lose at most
a fixed amount of money, while, with strictly positive probability, she ulti-
mately becomes infinitely rich. Intuitively speaking, this is “something like
an arbitrage”, and it is an easy task to formally deduce from these proper-
ties ofKa “free lunch with vanishing risk”: for example, it suffices to de-
fineKn=n^1 K (^1) ]] 0,τn∧Tn]], for a sequence of (deterministic) times (Tn)∞n=0,to
letfn=(Kn·S)∞ =(Kn·S)τn∧Tn and to definegn=fn∧(γ−β) (^1) B
whereB=
⋂∞
n=0{τn≤Tn}.If(Tn)∞
n=1tends to infinity sufficiently fast, we
haveP[B]>0, and one readily verifies that (gn)∞n=1converges uniformly to
(γ−β) (^1) B.
Summing up, we have shown that(NFLVR)implies that, forβ<γ,the
processH·Salmost surely upcrosses the interval ]β, γ[ only finitely many
times. Whence (H·S)tconverges almost surely to a random variable (H·S)∞
with values inR∪{∞}.Thefactthat(H·S)∞is a.s. finitely valued follows from
another application (similar to but simpler than the above) of the assumption
of(NFLVR), which we leave to the reader.
We now start to sketch the main arguments underlying the proof of Lemma
8.2.3. The strategy is to obtain from assumption (8.4) and from suitable mod-
ifications of the original sequence (Hn)∞n=0more information on the conver-
gence of the sequence ofprocesses(Hn·S)∞n=0. Eventually we shall be able
to reduce the problem to the case where (Hn·S)∞n=0converges in the semi-
martingale topology; at this stage M ́emin’s theorem [M 80] will give us the
desired limiting trading strategyH.
So, what can we deduce from assumption (8.4) and the a.s. convergence
of (fn)∞n=0=((Hn·S)∞)∞n=0for the convergence of the sequence of processes
(Hn·S)∞n=0? The unpleasant answer is: a priori, we cannot deduce anything.
To see this, recall the “suicide” strategyHwhich we have discussed in the
context of inequality (8.3) above: it designs an admissible way to lose onee.
Speaking mathematically, the corresponding stochastic integralH·Sstarts
at (H·S) 0 =0,satisfies(H·S)t≥−1 almost surely, for allt≥0, as well as
(H·S)∞=−1. But clearly this is not the only admissible way to lose onee
and there are many other trading strategiesKon the processShaving the
same properties. Taking up again the example discussed after (8.3), a trivial
example is, to first wait without playing for a fixed number of games of the
roulette, and to start the suicide strategy only after this waiting period; of
course, this is a different way of losing onee.