264 13 The Banach Space of Workable Contingent Claims in Arbitrage Theory
Proof.Take a sequence of real numbers such thatan↘‖g‖.Foreachnwe
takeHnandKnmaximal andan-admissible such thatg=(Hn·S)∞−(Kn·
S)∞. From the Theorem 13.2.3 cited above we deduce that there are convex
combinationsVn∈conv{Hn,Hn+1,...}andWn∈conv{Kn,Kn+1,...}such
that (Vn·S)∞→hand (Wn·S)∞→k. Clearlyg=h−k,h≥−‖g‖
andk≥−‖g‖. However, at this stage we cannot assert thathand/orkare
maximal. Theorem 13.2.3 above, however, allows us to find a maximal strategy
Rsuch that (R·S)∞≥h≥−‖g‖.ThestrategyR−H^1 +K^1 is acceptable
and satisfies
((R−H^1 +K^1 )·S)∞=(R·S)∞−g≥h−g=k≥−‖g‖.From the Proposition 13.2.13 above it follows thatU =R−H^1 +K^1 is
‖g‖-admissible and maximal. By definition ofU andRwe have thatg=
(R·S)∞−(U·S)∞.
Corollary 13.3.14.With the notation of the above Theorem 13.3.13:(R·
S)∞+‖g‖≥g+and(U·S)∞+‖g‖≥g−. Hence we find
sup{
EQ
[
g+]
|Q∈M
}
≤‖g‖
sup{
EQ
[
g−]
|Q∈M
}
≤‖g‖.Theorem 13.3.15.Suppose thatSis a locally bounded semi-martingale that
satisfies the (NFLVR) property. Ifg∈Gthen
‖g‖=sup{
EQ
[
g+]
|Q∈M
}
=sup{
EQ
[
g+]
|Q∈Me}
=sup{
EQ
[
g−]
|Q∈M
}
=sup{
EQ
[
g−]
|Q∈Me}
.
Proof.Putβ=sup{EQ[g+]|Q∈M}, where the random variablegis de-
composed asg=(H^1 ·S)∞−(H^2 ·S)∞withH^1 andH^2 maximal. From
Corollary 11.3.5 to Theorem 11.3.4, we recall that there is a maximal strategy
K^1 such thatg+≤β+(K^1 ·S)∞, implying thatK^1 isβ-admissible. The
strategyK^2 =K^1 −H^1 +H^2 is alsoβ-admissible and by Proposition 13.2.13
therefore maximal. SinceK^1 −K^2 =H^1 −H^2 we obtain that‖g‖≤β.Since
the opposite inequality is already shown in Corollary 13.3.14, we therefore
proved the theorem.
Remark 13.3.16.IfQis a martingale measure for the process (H^1 −H^2 )·S,
then of courseEQ[g+]=EQ[g−]. But not all elements in the setMare
martingale measures for this process and hence the equality of the suprema
does not immediately follow from martingale considerations.
Theorem 13.3.17.Suppose thatSis a locally bounded semi-martingale that
satisfies the (NFLVR) property. The norm of the spaceGis also given by the
formula
2 ‖g‖=sup{EQ[|g|]|Q∈M}=sup{EQ[|g|]|Q∈Me}.