The Mathematics of Arbitrage

14.5 Duality Results and Maximal Elements 305

The proof of the main theorem is complete now.

For later use, let us resume in the subsequent proposition what we have
shown in the above proof.

Proposition 14.4.5.Denote byMesthe set of probability measuresQequiv-
alent toPsuch that, for admissible integrands, the processH·S becomes
a super-martingale. More precisely

Mes={Q|Q∼Pand for eachf∈C:EQ[f]≤ 0 }.

IfSsatisfies (NFLVR), then

Meσ={Q|Sis aQsigma-martingale},

is dense inMes.

Theorem 14.4.6.The setMeσis a convex set.

Proof.LetQ 1 ,Q 2 ∈Meσand letφ 1 ,φ 2 be strictly positive real-valuedS-
integrable predictable processes, such that fori=1,2,φi·Sis anH^1 (Qi)-
martingale. Take nowφ= min(φ 1 ,φ 2 ). Since 0<φ≤φ 1 ,φ·Sis still an
H^1 (Q 1 )-martingale. Similarlyφ·Sis still anH^1 (Q 2 )-martingale. From this

it follows thatφ·Sis anH^1

(

Q 1 +Q 2

2

)

-martingale. 

14.5 Duality Results and Maximal Elements

In this section we suppose without further notice thatSis anRd-valued semi-
martingale that satisfies the(NFLVR)property, so that the set

Meσ={Q|Q∼PandSis aQsigma-martingale}

is non-empty. We remark that when the price processSis locally bounded
then the setMeσcoincides with the setMe(S) as introduced in Chap. 9, i.e.
the set of all equivalent local martingale measures for the processS.
In the case of locally bounded processes we showed the following duality
equality, (see [D 92, Theorem 6.1] for the case of continuous bounded processes,
El Karoui-Quenez [EQ 95] for theL^2 case, Theorem 9.5.8 for the case of
bounded functions and Theorem 11.3.4 for the case of positive functions). The
duality argument was used by El Karoui-Quenez [EQ 95]. For a non-negative
random variablegwe have:

sup

Q∈Meσ

EQ[g]=inf{α|there isHadmissible andg≤α+(H·S)∞}.

Using this equality we were able to derive a characterisation of maximal ele-
ments, see Corollary 11.4.6.