The Mathematics of Arbitrage

14.4 The GeneralRd-valued Case 295

Theorem 14.4.1.Under the assumption (NFLVR) the coneCis weak-star-
closed inL∞(Ω,F,P). Hence there is a probability measureQ 1 ∼Psuch
that
EQ 1 [f]≤ 0 , forf∈C.

Remark 14.4.2.In the case, whenSis bounded,Q 1 is already a martingale
measure forS,andwhenSis locally bounded,Q 1 is a local martingale mea-
sure forS(compare Theorem 9.1.1 and Corollary 9.1.2).
To take care of the non-locally bounded case we have to take care of the
“big jumps” ofS. We shall distinguish between the jumps ofSoccurring at
accessible stopping times and those occurring at totally inaccessible stopping
times.
We start with an easy lemma which will allow us to change the measure
Q 1 countably many times without loosing the equivalence toP.

Lemma 14.4.3.Let(Qn)∞n=1be a sequence of probability measures on the
probability space(Ω,F,P)such that eachQn is equivalent toP.Suppose
further that the sequence of strictly positive numbers(εn)n≥ 1 is such that

(1)‖Qn−Qn+1‖<εn+1,
(2)ifQn[A]≤εn+1 2 nthenP[A]≤ 2 −n.

Then the sequence(Qn)n≥ 1 converges with respect to the total variation
norm to a probability measureQ, which is equivalent toP.

Proof of Lemma 14.4.3.Clearly the second assumption implies thatεn+1≤
2 −nand hence the sequence (Qn)n≥ 1 converges in variation norm to a prob-
ability measureQ. We have to show thatQ∼P.Foreachnwe letqn+1be
defined as the Radon-Nikod ́ym derivative ofQn+1with respect toQn. Clearly
for eachn≥1wethenhave

∫

| 1 −qn+1|dQn≤εn+1and hence the Markov
inequality implies thatQn[| 1 −qn+1|≥ 2 −n]≤ 2 nεn+1.Thehypothesison
the sequence (εn)n≥ 2 then implies thatP[| 1 −qn+1|≥ 2 −n]≤ 2 −n.Fromthe
Borel-Cantelli lemma it also follows that a.s. the series

∑

n≥ 2 |^1 −qn|converges
and hence the product

∏

n≥ 2 qnconverges to a functionqa.s. different from

0. Clearlyq=ddQQ 1 which shows thatQ∼Q 1 ∼P.
   

We are now ready to take the crucial step in the proof of the main theorem. To
make life easier we make the simplifying assumption thatSdoes not jump at
predictable times. In the Proof of the Main Theorem 14.1.1 below we finally
shallalsodealwiththecaseofthepredictablejumps.

Proposition 14.4.4.Let S =(St)t∈R+ be an Rd-valued semi-martingale
which isquasi-left-continuous, i.e., such that, for every predictable stopping
timeTwe haveST=ST−almost surely.
Suppose, as in Theorem 14.4.1 above, thatQ 1 ∼Pis a probability measure
verifying
EQ 1 [f]≤ 0 , forf∈C.