13.2 Maximal Admissible Contingent Claims 255We suppose from now on that the processSis a fixedd-dimensional locally
bounded semi-martingale and that it satisfies the property(NFLVR).Theset
of local martingale measures is therefore, according to the previous theorems
not empty. In Sect. 13.7, we will also make use of finitely additive measures.
So we letba(Ω,F,P) be the Banach space of all finitely additive measures
that are absolutely continuous with respect toP, i.e.ba(Ω,F,P) is the dual of
L∞(Ω,F,P). We will use Roman lettersP,Q,Q^0 ,...forσ-additive measures
and Greek letters for elements ofbawhich are not necessarilyσ-additive. We
say that a finitely additive measureμis absolutely continuous with respect to
the probability measurePifP[A] = 0 impliesμ[A] = 0 for any setA∈F.
Let us put:
Me={
Q
∣
∣
∣
∣
Qis equivalent toP
and the processSis aQ-local martingale}
M=
{
Q
∣
∣
∣
∣
Qis absolutely continuous with respect toP
and the processSis aQ-local martingale}
Mba={
μ∣
∣
∣
∣
μis inba(Ω,F∞,P)
and for every elementh∈C: Eμ[h]≤ 0}
We identify, as usual, absolutely continuous measures with their Radon-
Nikod ́ym derivatives. It is clear that, under the hypothesis(NFLVR),theset
Me(P)isdenseinM(P) for the norm ofL^1 (Ω,F,P). This density together
with Fatou’s lemma imply that for random variablesgthat are bounded below
we have the equality
sup{EQ[g]|Q∈Me}=sup{EQ[g]|Q∈M}.We will use this equality freely.
As shown in Remark 9.5.10 the setMeis weak-star-dense, i.e. for the
topologyσ(ba, L∞), in the setMba.
The first two sets are sets ofσ-additive measures, the third set is a set of
finitely additive measures. ClearlyMe⊂M⊂Mbaand sinceSis locally
bounded the setMis closed inL^1 (Ω,F,P). If needed we will add the process
Sin parenthesis, e.g.Me(S), to make clear that we are dealing with a set of
local martingale measures for the processS.
13.2 Maximal Admissible Contingent Claims
We now give the definition of a maximal admissible contingent claim and its
relation to the existence of an equivalent martingale measure. As mentioned
above we always suppose thatS is ad-dimensional locally bounded semi-
martingale that satisfies the(NFLVR)property.
Definition 13.2.1.IfUis a non-empty subset ofL^0 , then we say that a con-
tingent claimf∈Uis maximal inU, if the propertiesg≥fa.s. andg∈U
imply thatg=fa.s..