262 13 The Banach Space of Workable Contingent Claims in Arbitrage Theory
Proof.PutL=(H^1 −H^2 ), whereH^1 andH^2 are both maximal, and so
thatg=(L·S)∞.SinceLis acceptable and (L·S)∞≥−‖g−‖∞we find
by proposition 13.2.13 thatLis admissible. For a well-chosen elementQ∈
Me, the processL·Sis a uniformly integrable martingale and henceLis
maximal.
Corollary 13.3.5.Suppose thatSis a locally bounded semi-martingale that
satisfies the (NFLVR) property. IfV andWare maximal admissible strate-
gies, if((V−W)·S)∞ is uniformly bounded from below, thenV−W is
admissible and maximal.
Corollary 13.3.6.Suppose thatSis a locally bounded semi-martingale that
satisfies the (NFLVR) property. Bounded contingent claims inGare charac-
terised as
G∞=G∩L∞=Kmax∩L∞
={(H·S)∞|H·Sis bounded}.Remark 13.3.7.The vector spaceG∞should not be mixed up with the cone
K∩L∞. As shown in Chap. 9 and [DS 94a], the contingent claim−1canbe
inKbut by the no-arbitrage property, the contingent claim +1 cannot be in
K. The vector spaceG∞was used in the study of the convex setM(S), see
Chap. 9, [AS 94] and [J 92].
Definition 13.3.8 (Notation).We define the following norm on the spaceG:
‖g‖=inf{a|g=(H^1 ·S)∞−(H^2 ·S)∞,
H^1 ,H^2 a-admissible and maximal}
.
The norm on the spaceGis quite natural and is suggested by its definition.
It is easy to verify that‖.‖is indeed a norm. We will investigate the relation
of this norm to other norms, e.g.L∞andL^1 -norms.
Proposition 13.3.9.Suppose that S is a locally bounded semi-martingale
that satisfies the (NFLVR) property. Ifg=(H·S)∞whereH is workable
then for every stopping timeT,gT=(H·S)T∈Gand‖gT‖≤‖g‖.
Proof. Follows immediately from the definition and the proof of Corol-
lary 13.2.6 above.
Proposition 13.3.10.Suppose thatSis a locally bounded semi-martingale
that satisfies the (NFLVR) property. Ifg∈G∞then, as shown above,g∈
K‖g−‖∞and−g∈K‖g+‖∞.Hence
‖g‖≤min(
‖g+‖∞,‖g−‖∞)
≤max(
‖g+‖∞,‖g−‖∞)
=‖g‖∞.