130 8 Arbitrage Theory in Continuous Time: an Overview
doubles his bet until the first time he wins. If he starts with 1e, his final gain
(last payout minus the total sum of the preceding losses) is 1ealmost surely.
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 a 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.
Here is the definition of the class of integrands which turns out to be
appropriate for our purposes.
Definition 8.1.1.Fix anRd-valued stochastic processS=(St)t≥ 0 as defined
in Chap. 5, which we now alsoassume to be a semi-martingale.AnRd-valued
predictable process H =(Ht)t≥ 0 is called an admissible integrandfor the
semi-martingaleS,if
(i) HisS-integrable, i.e., the stochastic integralH·S=((H·S)t)t≥ 0 is well-
defined in the sense of stochastic integration theory for semi-martingales,
(ii)there is a constantMsuch that
(H·S)t≥−M, a.s., for allt≥ 0.Let us comment on this definition: we place ourselves into the “th ́eorie
g ́en ́ erale” of integration with respect to semi-martingales: here we are on safe
grounds as the theory, developed in particular by P.-A. Meyer and his school,
tells us precisely what it means that a predictable processHisS-integrable
(see Chap. 7 above). But in order to be able to apply this theory we have to
make sure thatSis a semi-martingale: this is precisely the class of processes
allowing for a satisfactory integration theory, as we know from the theorem
of Bichteler and Dellacherie ([B 81], [DM 80]; see also [P 90]).
How natural is the assumption thatSis a semi-martingale from an eco-
nomic point of view? In fact, it fits very nicely into the present no-arbitrage
framework: it is shown in Theorem 9.7.2 below that, for a locally bounded,
cadlag processS, the assumption, that the closure ofCsimplewith respect to
the norm topology ofL∞(P) intersectsL∞(P)+only in{ 0 }, implies already
thatSis a semi-martingale. The semi-martingale property therefore is im-
plied by a very mild strengthening of the no-arbitrage condition for simple,
admissible integrands. Loosely speaking, the message of this theorem is that
a no-arbitrage theory for a stochastic processSmodelling a financial market,
only makes sense if we start with the assumption thatSis a semi-martingale.
For example, this rules out fractional Brownian motion (except for Brownian
motion itself, of course). There is no reasonable no-arbitrage theory for these
processes in the present setting of frictionless trading in continuous time. How-
ever, if one introduces transaction costs, then for fractional Brownian motion
the picture changes completely and the arbitrage opportunities disappear (see
[G 05]).