The Mathematics of Arbitrage

14.5 Duality Results and Maximal Elements 313

S)∗≤w. It follows that random variables of the form (^1) A

(

φ·Sit−φ·Sis

)

or

− (^1) A

(

φ·Sti−φ·Ssi

)

,wheres<t,A∈Fsand (Si)i=1...dare the coordinates
ofS, are results ofw-admissible integrands. Thereforeφ·Sis aQ-martingale
andQ∈Meσ.

Corollary 14.5.21.Ifw≥ 1 is a feasible weight function then the set

{Q|Q∈Meσ,EQ[w]<∞}

is dense inMeσfor the variation norm.

Proof.If the set would not be dense then by the Hahn-Banach theorem, there
existsQ 0 ∈Meσand a bounded functiongsuch that

EQ 0 [g]>sup{EQ[g]|Q∈Meσ,EQ[w]<∞}=α.

This, together with Theorem 14.5.9, would then imply

α 0 =inf{α| there isHadmissible andg≤α+(H·S)∞}

=sup

Q∈Meσ

EQ[g]

> sup

Q∈Meσ

EQ[w]<∞

EQ[g]

=inf{α| there isHw-admissible andg≤α+(H·S)∞}.

But aw-admissible integrandHsuch that (H·S)∞+α≥gis already admis-
sible, proving that the strict inequality cannot hold. Indeed the processH·S
is aQ-super-martingale for each elementQ∈Meσsuch thatEQ[w]<∞.
Therefore the processH·Sis bounded below by−α−‖g‖∞and this means
thatHis admissible.

Remark 14.5.22.An interesting question is, whether by taking the supremum
in Theorem 14.5.9, we have, for general unbounded functionsg,torestrictto
those elementsQ∈Meσsuch that for the feasible weight functionwwe have
EQ[w]<∞. More precisely is there a contingent claimg≥−wsuch that

sup

Q∈Meσ

EQ[g]> sup

Q∈Meσ

EQ[w]<∞

EQ[g].

An inspection of the proof of the above theorem shows that we used theQ-
integrability of the feasible weight functionwin order to conclude that the
w-admissible integrandHdefined aQ-super-martingaleH·S.
The next example, however, shows that it might happen that, for some
sigma-martingale measure,H·Sis a super-martingale, while for other sigma-
martingale measures, it fails to be so.