13.3 The Banach Space Generated by Maximal Contingent Claims 263The following lemma is an easy exercise in integration theory and imme-
diately gives the relation with theL^1 -norm.
Proposition 13.3.11.Iff∈L^1 (Ω,F,Q)for some probability measureQ,if
EQ[f]=0,iff=g−h,wherebothEQ[g]≤ 0 andEQ[h]≤ 0 ,then
‖f‖L (^1) (Q)=2EQ[f+]=2max(EQ[f+],EQ[f−])
≤2max
(
‖g−‖∞,‖h−‖∞)
.
Proof.The first line is obvious and shows that the obvious decomposition
f=(f+−E[f+])−(f−−E[f−]) is best possible. So let us concentrate on
the last line. Iff=g−hthen we have the following inequalities:
f+‖g−‖∞−‖h−‖∞=g+‖g−‖∞−(h+‖h−‖∞)
(
f+‖g−‖∞−‖h−‖∞)+
≤g+‖g−‖∞
(
f+‖g−‖∞−‖h−‖∞)−
≤h+‖h−‖∞.These inequalities together withEQ[g]≤0andEQ[h]≤0, imply that
‖f+‖g−‖∞−‖h−‖∞‖L^1 (Q)≤‖g−‖∞+‖h−‖∞.It is now easy to see that
‖f‖L (^1) (Q)≤‖g−‖∞+‖h−‖∞+
∣
∣‖g−‖∞−‖h−‖∞∣
∣
≤2max(
‖g−‖∞,‖h−‖∞)
.
Corollary 13.3.12.Suppose thatSis a locally bounded semi-martingale that
satisfies the (NFLVR) property. Ifg∈Gthen
2 ‖g‖≥sup{
‖g‖L^1 (Q)|Q∈M}
Proof.Takeg=(H^1 ·S)∞−(H^2 ·S)∞∈GwhereH^1 andH^2 are both
maximal anda-admissible. For everyQ∈Mwe have thatEQ[(H^1 ·S)∞]≤ 0
andEQ[(H^2 ·S)∞]≤0. The lemma shows that
‖g‖L (^1) (Q)≤2max
(
‖(H^1 ·S)∞‖L (^1) (Q),‖(H^2 ·S)∞‖L (^1) (Q)
)
≤ 2 a.By taking the infimum over all decompositions and by taking the supremum
over all elements inMwe find the desired inequality.
The next theorem shows that in some sense there is an optimal decompo-
sition. The proof relies on Theorem 13.2.3 above and on the technical Lemma
9.8.1.
Theorem 13.3.13.Suppose thatSis a locally bounded semi-martingale that
satisfies the (NFLVR) property. Ifg∈Gthen there exist two‖g‖-admissible
maximal strategiesRandUsuch thatg=(R·S)∞−(U·S)∞.