The Mathematics of Arbitrage

14.5 Duality Results and Maximal Elements 309

by the inequalityH·S≥−EQ[w|Ft]whereEQ[w]<∞.Comparethe
formulation of Theorems 15.B and 15.C. For the convenience of the reader let
us rephrase the results of Chap. 15 in the present setting.

Theorem 14.5.13 (Theorem 15.D).LetQbe a probability measure, equiv-
alent toP.LetMbe an Rd-valuedQ-local martingale andw ≥ 1 a Q-
integrable function.
Given a sequence(Hn)n≥ 1 ofM-integrableRd-valued predictable processes
such that
(Hn·M)t≥−EQ[w|Ft], for alln, t ,

then there are convex combinations

Kn∈conv{Hn,Hn+1,...},

and there is a super-martingale(Vt)t∈R+,V 0 ≤ 0 , such that

lims↘t

s∈Q+

lim

n→∞

(Kn·M)s=Vt, fort∈R+,a.s.,

and anM-integrable predictable processH^0 such that

((H^0 ·M)t−Vt)t∈R+ is increasing.

In addition,H^0 ·Mis a local martingale and a super-martingale.

Corollary 14.5.14 (Corollary 15.4.11).LetSbe a semi-martingale taking
values inRdsuch thatMeσ(S)=∅andw≥ 1 a weight function such that there
is someQ∈Meσ(S)withEQ[w]<∞.
Then the convex cone

{g|there is a(1,w)-admissible integrandHsuch thatg≤(H·S)∞}

is closed inL^0 (Ω)with respect to the topology of convergence in measure.

Letw≥1 be such that there isQ∈Meσ,withEQ[w]<∞.Theset

Kw={(H·S)∞|Hisw-admissible}

is a cone in the space of measurable functionsL^0. As in Chap. 9 we need
the cone of all elements that are dominated by outcomes ofw-admissible
integrands:

Cw^0 ={g|g≤(H·S)∞,whereHisw-admissible}.

IfH isw-admissible andEQ[w]<∞for someQ∈Meσ, then it follows
from the results in [AS 94] that the processH·Sis aQ-super-martingale.
Therefore the limit (H·S)∞exists andEQ[(H·S)∞]≤0. It also follows
that for elementsg ∈C^0 w,wehavethat−∞ ≤EQ[g]≤0. We will use