The Mathematics of Arbitrage

15.5 Application 355

is increasing.
By applying Theorem 15.D to theQ 0 -local martingaleS— and by passing
to convex combinations, if necessary — the process

W ̃t= lim

ss∈↘Qt

+

lim

n→∞

(Hn·S)s

is well-defined and we may find a predictableS-integrable processHsuch that
H·S−W ̃is increasing; asW−W ̃is increasing too, we obtain in particular
thatH·S−Wis increasing.
AsH·S≥Wwe deduce from [AS 94] that, for eachQ∈Me(S),H·Sis
aQ-local martingale and aQ-super-martingale. By the maximality condition
ofW we must haveH·S =W thus finishing the proof of the Optional
Decomposition Theorem 15.5.1.
We still have to prove the claim. This essentially follows from Corol-
lary 15.4.11.
Let us defineL∞w to be the space of all measurable functionsgsuch that
g
w is essentially bounded. This space is the dual of the spaceL

1
w−^1 (Q^0 )of
functionsgsuch thatEQ 0 [w|g|]<∞. By the Banach-Dieudonn ́e theorem or
the Krein-Smulian theorem (see Chap. 9 for a similar application), it follows
from Corollary 15.4.11 that the set

B={h||εh|≤wandεh∈C 1 , 2 wfor someε> 0 },

is a weak-star-closed convex cone inL∞w (the setC 1 , 2 wwasdefinedinDefi-
nition 15.4.8 above). Now as easily seen, if the claim were not true, then the
said functionfis not inB.SinceB−L∞w+⊂Bwe have by Yan’s separation
theorem ([Y 80]), that there is a strictly positive functionh∈L^1 w− 1 such that
EQ 0 [hf]>0andsuchthatEQ 0 [hg]≤0 for allg∈B. If we normalisehso
thatEQ 0 [h] = 1 we obtain an equivalent probability measureQ,dQ=hdQ 0
such thatEQ[f]>0. But sinceSis dominated by the weight functionw,we
have that the measureQis an equivalent martingale measure for the pro-
cessS. The processW is therefore a local super-martingale underQ.But
the densityhis such thatEQ[w]<∞and therefore the processuWv,being
dominated by 2w, is a genuine super-martingale underQ. However, this is
a contradiction to the inequalityEQ[f]>0. This ends the proof of the claim
and the proof of the Optional Decomposition Theorem.

Remark 15.5.6.LetusstressoutthatwehaveprovedabovethatinTheo-
rem 15.5.1 for each processWwithW−V increasing,W aQ-local super-
martingale for eachQ∈Me(S)andWbeing maximal with respect to this
property in the sense of Lemma 15.5.4, we obtain the semi-martingale repre-
sentationW=H·S.

Remark 15.5.7.Referring to the notation of the proof of the optional decom-
position theorem and the claim made in it, the fact that the coneBis weak-
star-closed inL∞w yields a duality equality as well as the characterisation of