160 9 Fundamental Theorem of Asset Pricing
We close this section by quoting a result due to ́Emery and Ansel and
Stricker. The result states that under suitable conditions the stochastic inte-
gral of a local martingale is again a local martingale. A counter-example due
to [E 80] shows that in general a stochastic integral of a local martingale need
not be a local martingale. From Theorem 9.2.2 it follows that ifMis a local
martingale with respect to a measureP,thenH·Mis a local martingale if
and only if it is a special semi-martingale, i.e. if it is locally integrable. The
next theorem gives us a criterion that is related to admissibility ofH.
Theorem 9.2.9.IfMis a local martingale and if His an admissible in-
tegrand forM,thenH·M is a local martingale. ConsequentlyH·M is
a super-martingale.
Proof.We refer to [E 80] and [AS 94, Corollaire 3.5]. It is an easy consequence
of Fatou’s lemma that ifH·Mis a local martingale uniformly bounded from
below, then it is a super-martingale.
9.3 No Free Lunchwith Vanishing Risk........................
The main result of this section states that for a semi-martingale S, un-
der the condition of no free lunch with vanishing risk(NFLVR), the limit
(H·S)∞= limt→∞(H·S)texists and is finite whenever the integrandHis
admissible. To get a motivation for this result, consider the case where we
already know that there is an equivalent local martingale measureQ.Inthis
case, by Theorem 9.2.9, the stochastic integralH·Sis aQ-local martingale
ifHis admissible. This implies that it is a super-martingale and the classical
convergence theorem shows that the limit (H·S)∞= limt→∞(H·S)t,exists
and is finite almost everywhere. But of course we do not know yet that there
is an equivalent martingale measureQand the art of the game is to derive
the convergence result simply from the property(NFLVR). We start with two
preparatory results.
Proposition 9.3.1.IfSis a semi-martingale with the property (NFLVR),
then the set
{(H·S)∞|His 1 -admissible and of bounded support}is bounded inL^0.
Proof.H1-admissible means thatHisS-integrable and (H·S)t≥−1. Being
of bounded support means thatHis 0 outside [[0,T]] w h e r eTis a positive real
number. The limit (H·S)∞= limt→∞(H·S)texists without difficulty because
(H·S)tbecomes eventually constant. Suppose that the set{(H·S)∞|His
1-admissible and of bounded support}is not bounded inL^0. This implies the
existence of a sequenceHnof 1-admissible integrands of bounded support and