316 14 The FTAP for Unbounded Stochastic Processes
Lemma 14.5.24.Ifg∈Kw, then there is a maximal elementh∈Kwsuch
thath≥g.
Proof.It is sufficient to show that every increasing sequence inKwhas an
upper bound inKw.Solethn,h 1 =g, be an increasing sequence inKw.For
eachntakeHn,w-admissible so thathn=(Hn·S)∞. As in the previous
proof we then find, as an application of Theorem 14.5.13, that there isH^0 ,
w-admissible such that (H^0 ·S)∞≥limnhn. This concludes the proof of the
lemma.
Proof of Theorem 14.5.12.IfEQ[w]<∞thenH·Sis aQ-super-martingale
and hence (2) and (3) are equivalent. Also it is clear that (2) implies (1).
Indeed ifgis the result of aw-admissible integrand thenEQ[g]≤0foreach
Q∈Meσsuch that alsoEQ[w]<∞. It follows thathis necessarily maximal.
The only remaining part is that (1) implies (2). Since alwaysEQ[h]≤ 0
forQ∈Meσsuch that alsoEQ[w]<∞, we obtain already that for measures
Qsatisfying these assumptions,h+isQ-integrable. So fix such a measure
Q. Now letw 1 =h++w. Clearlyw 1 is a feasible weight function. We will
work with the setKw 1. The problem is, however, that we do not (yet) know
thathis still maximal in the bigger coneKw 1. From the construction ofw 1
it follows that, for elementsQ∈Meσ,wehaveEQ[w 1 ]<∞if and only if
EQ[w]<∞.Nowletg≥hbe the result of aw 1 -admissible integrand. Hence
g=(K·S)∞whereKisw 1 -admissible. Since (K·S)∞≥g≥h≥−wand
sinceKisw 1 -admissible we have thatKis alreadyw-admissible. (Remember
thatEQ[w 1 ]<∞if and only ifEQ[w]<∞) From the maximality ofhin
Kwit then follows thatg=h, i.e.his maximal inKw 1 .Thiscanthenbe
translated into (
h
w 1
+L∞+
)
∩Cw∞ 1 ={ 0 }.Using Yan’s separation theorem (Theorem 5.2.2 above) in the same way
as in the proof of Theorem 14.5.9 above, we find a measureQ 1 such that
EQ 1 [w 1 ]<∞,Q 1 ∈MeσandEQ 1 [h]≥0.
The following theorem generalises a result due to Ansel-Stricker and Jacka,
[AS 94] and [J 92].
Theorem 14.5.25.Letwbe a feasible weight function and letf≥−w.The
following assertions are equivalent
(1)there is a measureQ∈Meσsuch thatEQ[w]<∞and such that
EQ[f]= sup
R∈Meσ
ER[w]<∞ER[f]<∞(2)f can be hedged, i.e. there isα∈R,Q∈Meσsuch thatEQ[w]<∞,
aw-admissible integrandH such thatH·Sis aQ-uniformly integrable
martingale, and such thatf=α+(H·S)∞.