13.3 The Banach Space Generated by Maximal Contingent Claims 265Proof.As in the previous result, for a contingent claimg=(H^1 ·S)∞−(H^2 ·
S)∞whereH^1 andH^2 are maximal admissible, let us put:
β=sup{EQ[|g|]|Q∈M}
≤sup{
EQ
[
g+]
|Q∈M
}
+sup{
EQ
[
g−]
|Q∈M
}
=2‖g‖.From Chap. 11 it follows that there is a maximal strategyK, such that|g|≤
β+(K·S)∞. This inequality shows that
β+((K·S)∞)≥(H^1 ·S)∞−(H^2 ·S)∞
β+((K·S)∞)≥(H^2 ·S)∞−(H^1 ·S)∞.As in previous result we obtain thatK−H^1 +H^2 andK−H^2 +H^1 areβ-
admissible and maximal. Since 2(H^1 −H^2 )=(K−H^2 +H^1 )−(K−H^1 +H^2 ),
we obtain the inequality‖ 2 g‖≤β.
Corollary 13.3.18.Suppose thatSis a locally bounded semi-martingale that
satisfies the (NFLVR) property. Ifg∈G, then there is a sequence of elements
Qn∈Mesuch that
(1)EQn[g+]→sup{EQ[g+]|Q∈M},
(2)EQn[g−]→sup{EQ[g−]|Q∈M},
(3)EQn[|g|]→sup{EQ[|g|]|Q∈M}.
Proof.It suffices to take a sequence that satisfies the third line.
Remark and Example 13.3.19.For a contingent claimf∈Kmaxwe do
not necessarily have that
‖f‖=inf{a|f∈Ka}.Indeed take a processSsuch that there is only one risk neutral measureQ.In
this case the norm on the spaceGis (half) theL^1 (Q)-norm.Asiswell-known
the market is complete (see e.g. Chap. 9) andG=
{
f|f∈L^1 (Q),EQ[f]=0}
.
It follows thatKmaxa =
{
f|f∈L^1 (Q),EQ[f]=0,f≥−a}
.Thisconemay
contain contingent claims with‖f−‖∞=aand with arbitrary smallL^1 (Q)-
norm.
This example also shows that the spaceG, which in this example is a hy-
perplane inL^1 , can be isomorphic to anL^1 -space. It also shows that the
coneKmaxis not necessarily closed. Indeed the coneKmaxcontains all con-
tingent claimsf∈L∞with the propertyEQ[f] = 0. This set is dense in
G=
{
f|f∈L^1 (Q),EQ[f]=0}
. However, we have the following result.
Proposition 13.3.20.Suppose thatSis a locally bounded semi-martingale
that satisfies the (NFLVR) property. The conesKmaxa are closed in the spaceG.
Proof.Take a sequencefninKmaxa and tending toffor the norm ofG.Since
clearlyf≥−a, the contingent claimfis the outcome of an admissible and
by Corollary 13.2.19, also of a maximal strategy.