11.2 Basic Theorems 221Proof.The necessity is clear. IfQis an equivalent local martingale mea-
sure, then the Radon-Nikod ́ym derivative ddQP defines a strictly positiveP-
martingaleLsuch thatLSis aP-local martingale. Also the processSneces-
sary satisfies the(NA)property with respect to general admissible integrands.
The converse is less obvious. We recall from Corollary 9.3.9, that it is
sufficient to prove thatSsatisfies(NA)with respect to general admissible
integrands and that the setK 1 is bounded inL^0 .IfLis a strictly positive
local martingale, then the sequence of stopping times defined as
Tn=inf{t|Lt≥n}satisfiesP[Tn=∞]→1andLTnis a uniformly integrable martingale. These
properties follow from the fact thatLis a super-martingale and the fact that
the jumps ofLare necessarily integrable. Also we may and do suppose that
L 0 =1.Foreachnthe measureQndefined byddQPn=LTnis a local martingale
measure for the stopped processSTn. It follows that the setK 1 is bounded
when restricted to the event{Tn=∞}. BecauseP[Tn=∞]→1, this implies
thatK 1 is bounded inL^0.
The theorem yields the following result, see [DS 94a] and Chap. 10 for a dif-
ferent approach and for related results. For details on continuous martingales
and Bessel processes we refer to Revuz-Yor [RY 91].
Corollary 11.2.10.IfRis the Bessel(3) process, stopped at time 1 and with
its natural filtration thenRallows arbitrage with respect to general admissible
integrands.
Proof.The processL=R^1 is a local martingale and from stochastic calculus
it follows that it is the only local martingaleXsuch thatX 0 =1andsuch
thatXRis a local martingale. If nowQwere a local martingale measure
forR, then the martingaleXdefined asEP[ddQP |Ft] satisfies thatXRis
a local martingale and henceX=L.SinceLis only a local martingale and
not a true martingale we arrive at a contradiction. It follows thatRdoes not
have an equivalent local martingale measure. Since it satisfies the second part
of the preceding theorem, it cannot satisfy the(NA)property with respect to
general admissible integrands.
Remark 11.2.11.The elementL 1 −1 is not maximal in the setK 1 constructed
with the processL. To see this recall thatE[L 1 ]<1andthatL 1 −E[L 1 ]
is by the predictable representation property ofL, the result of a uniformly
integrable martingale of the formK·L. It is clear that (K·L) 1 =(L 1 −E[L 1 ])>
L 1 −1.
If a locally bounded semi-martingaleSsatisfies the(NFLVR)property
with respect to general admissible integrands, then the following two non-
empty sets will play a role in the theory: