8.2 The Crucial Lemma 139Speaking mathematically, this means that — even whenSis a martingale,
as is the case in the example of the suicide strategy — the condition (H·S)t≥
−1 a.s., for allt≥0, and the final outcome (H·S)∞do not determine the
processH·Suniquely. In particular there is no hope to derive from (8.4) and
the a.s. convergence of the sequence of random variables ((Hn·S)∞)∞n=0,a
convergence property of the sequence of processes (Hn·S).
The idea to remedy the situation is to notice the following fact: the suicide
strategy is a silly investment and obviously there are better trading strategies,
e.g., not to gamble at all. By discarding such “silly investments”, we hopefully
will be able to improve the situation.
Here is the way to formalise the idea of discarding “silly investments”:
denote byDthe set of all random variableshsuch that there is a random
variablef≥hand a sequence (Hn)∞n=0of admissible trading strategies satis-
fying (8.4), and such that (Hn·S)∞converges a.s. tof.Wecallf 0 a maximal
element ofDif the conditionsh≥f 0 andh∈Dimply thath=f 0.
For example, in the context of the random walkS=(St)∞t=0on which we
constructed the above “suicide strategy”,h≡−1isanelementofD, but not
a maximal element. A maximal element dominatinghis, for example,f 0 ≡0.
More generally, it is not hard to prove under the assumptions of Lemma
8.2.3 that, for a givenf=(H·S)∞≥−1, whereHis an admissible integrand,
there is a maximal elementf 0 ∈Ddominatingf(see Lemma 9.4.4).
The point of the above concept is that, in the proof of Lemma 8.2.3, we may
assume without loss of generality thatfis a maximal element ofD. Under this
additional assumption it is indeed possible to derive from the a.s. convergence
of the sequence of random variables ((Hn·S)∞)∞n=0some information on the
convergence of the sequence of processes (Hn·S)∞n=0.
As the proof of this result is another nice illustration of our general ap-
proach of replacing “martingale arguments” by “economically motivated argu-
ments” relying on the assumption(NFLVR), we sketch the argument. Again
we observe that the proof does not make use of the local boundedness ofS.
Lemma 8.2.5 (9.4.6).LetSbe anRd-valued semi-martingale and letfbe
a maximal element ofD.Let(Hn)∞n=1be a sequence of admissible integrands
as in Lemma 8.2.3.
Then the sequence of random variables
Fn,m=sup
0 ≤t<∞|(Hn·S)t−(Hm·S)t| (8.7)tends to zero in probability, asn, m→∞.
Proof.Suppose to the contrary that there isα>0, and sequences (nk,mk)∞k=1
tending to infinity s.t.P[sup 0 ≤t((Hnk·S)t−(Hmk·S)t)>α]≥α,foreach
k∈N.