140 8 Arbitrage Theory in Continuous Time: an Overview
Define the stopping timesτkasτk=inf{t|(Hnk·S)t−(Hmk·S)t≥α},so that we haveP[τk<∞]≥α.
DefineLkasLk=Hnk (^1) ]] 0,τk]]+Hmk (^1) ]]τk,∞[[. Clearly the processLkis
predictable andLk·S≥−1.
Translating the formal definition into prose: the trading strategyLkcon-
sists of following the trading strategyHnkup to timeτk, and then switching
toHmk. The idea is thatLkproduces a sensibly better final result (Lk·S)∞
than either (Hnk·S)∞or (Hmk·S)∞, which will finally lead to a contradiction
to the maximality assumption onf.
Why isLk“sensibly better” thanHnkorHmk?Forlargek, the ran-
dom variables (Hnk·S)∞as well as (Hmk·S)∞will both be close tof
in probability; for the sake of the argument, assume that both are in fact
equal tof(keeping in mind that the difference is “small with respect to
convergence in probability”). A moment’s reflection reveals that this im-
plies that the random variable (Lk·S)∞ equalsf plus the random vari-
able ((Hnk·S)τk−(Hmk·S)τk) (^1) {τk<∞}. The latter random variable is non-
negative and with probabilityαgreater than or equal toα; this means that
this difference betweenfand (Lk·S)∞is not “small with respect to conver-
gence in probability”; this is, what we had in mind when saying thatLkis a
“sensible” improvement as compared toHnkorHmk.
Modulo some technicalities, which are worked out in Lemma 9.4.6 below,
this gives the desired contradiction to the maximality assumption onf,thus
finishing the (sketch of the) proof of Lemma 8.2.5.
Lemma 8.2.5 is our first step towards a proof of Lemma 8.2.3: it gives
some information on the convergence of the sequence of processes (Hn·S)∞n=0
in terms of the maximal functions defined in (8.7). But the assertion that
these maximal functions tend to zero in probability is still much weaker than
the convergence of (Hn·S)∞n=0with respect to the semi-martingale topology,
which we finally need in order to be able to apply M ́emin’s theorem. There is
still a long way to go!
But it is time to finish this “guided tour” towards a proof of Theorem 8.2.2
and to advise the interested reader to find the remaining part of the proof in
Chap. 9 below. We hope that we have succeeded to give some motivation for
the proof and for the “economically motivated” arguments underlying it.
8.3 Sigma-martingales and the Non-locally Bounded Case
Case
To finish this chapter we return to the basic assumption in Sect. 8.2 that the
processSislocally bounded. What happens if we drop this — technically very
convenient — assumption?