9.1 Introduction 151strictly positive with positive probability. The economic interpretation is that
by betting on the processSand without bearing any risk, it should not be
possible to make something out of nothing. If one wants to make this intu-
itive idea precise, several problems arise. First of all one has to restrict the
choice of the integrandsH to make sure that (H·S)∞exists. Besides the
qualitative restrictions coming from the theory of stochastic integration, one
has to avoid problems coming from so-called doubling strategies. This was
already noted in the papers [HK 79] and [HP 81]. To explain this remark let
us consider the classical doubling strategy. We draw a coin and when heads
comes out the player is paid 2 times his bet. If tails comes up, the player loses
his bet. The strategy is well-known: the player doubles his bet until the first
time he wins. If he starts with 1e, his final gain ( = last pay out−total sum
of the preceding bets) is almost surely 1e. He has an almost sure win. The
probability that heads will eventually show up is indeed one, even if the coin is
not fair. However, his accumulated losses are not bounded below. Everybody,
especially the casino boss, knows that this is a very risky way of winning 1e.
This type of strategy has to be ruled out: there should be a lower bound on
the player’s loss. The described doubling strategy is known for centuries and
in French it is still referred to as “la martingale”.
One possible way to avoid these difficulties is to restrict oneself to simple
predictable integrands. These are defined as linear combinations of buy and
hold strategies. Mathematically sucha buy and hold strategy is described as
an integrand of the formH=f (^1) ]]T 1 ,T 2 ]],whereT 1 ≤T 2 are finite stopping
times andfisFT 1 , measurable. The advantage of using such integrands is that
they have a clear interpretation: when timeT 1 (ω) comes up, buyf(ω) units of
the financial asset, keep them until timeT 2 (ω) and sell. A linear combination
of such integrands is called a simple integrand. An elementary integrand is
a linear combination of buy and hold strategies with stopping times that are
deterministic. This terminology agrees with standard terminology of stochas-
tic integration (see [P 90, DM 80, CMS 80]). Even if the processSis not a semi-
martingale the stochastic integral (H·S)forH=f (^1) ]]T 1 ,T 2 ]]can be defined as
the process (H·S)t=f·(Smin(t,T 2 )−Smin(t,T 1 )). Also the definition of the limit
(H·S)∞= limt→∞(H·S)t=f·(ST 2 −ST 1 ) poses no problem. The net profit of
the strategy is precisely (H·S)∞. The use of stopping times is interpreted as
the use of signals coming from available, observable information. This explains
why in financial theories the filtration and the derived concepts such as pre-
dictable processes, are important. It is clear that the use of simple integrands
rules out the introduction of doubling strategies. This led [HK 79, K 81, HP 81]
to define no-arbitrage and no free lunch in terms of simple integrands and
to obtain theorems relating these notions to the existence of an equiva-
lent martingale measure. In various directions these results were extended
in [DH 86, Str 90, DMW 90, AS 93, MB 91, L 92, D 92, S 94, K 93].
To relate our work to earlier results, let us summarise the present state of
the art. The case when the time set is finite is completely settled in [DMW 90]
and the use of simple or even elementary integrands is no restriction at all