310 14 The FTAP for Unbounded Stochastic Processes
this result frequently. We also remark that ifHisw-admissible and if (H·
S)∞≥−wthenHis already (1,w)-admissible. Indeed because of the super-
martingale property ofH·Swe have that (at least for thoseQ∈Meσsuch
thatEQ[w]<∞):
(H·S)t≥EQ[(H·S)∞|Ft]≥EQ[−w|Ft].By Lemma 14.5.4 this means thatHis (1,w)-admissible.
Theorem 14.5.15.Ifw≥ 1 and if there is someQ∈Meσsuch thatEQ[w]<
∞,then
Cw∞=
{
h|h∈L∞andhw∈Cw^0}
is weak-star-closed inL∞(Q).
Proof.This is just a reformulation of Corollary 14.5.14 cited above.
We now prove the duality result stated in Theorem 14.5.9. The proof is
broken up into several lemmata. As we will work with functionsw≥1that
are not necessarily feasible weight functions we will make use of a larger class
of equivalent measures namely:
Mes,w={Q∼P|EQ[w]<∞and for eachh∈Cw:EQ[h]≤ 0 }.The reader can check thatMes,wis the set of equivalent probability measures
so thatwis integrable and with the property that for aw-admissible integrand
H, the processH·Sis a super-martingale. When we work with admissible
integrands, i.e. withwidentically equal to 1, then we simply drop, as in
Proposition 14.4.5, the subscriptw.
Lemma 14.5.16.Ifw≥ 1 has a finite expectation for at least one element
Q∈Meσ,ifgis a random variable such thatg≥−wthen:
sup
Q∈Mes,w
EQ[w]<∞EQ[g]≤inf{α| there isHw-admissible andg≤α+(H·S)∞}.Proof.The proof follows the same lines as the proof of Theorem 11.3.4. If
w≥1andQ∈Mes,wthen as observed above, the processH·Sis aQ-
super-martingale for eachHthat isw-admissible. Therefore the inequality
g≤α+(H·S)∞implies thatEQ[g]≤α.
Remark 14.5.17.If, under the same hypothesis of the Theorem 14.5.9 above,
supQ∈Meσ;EQ[w]<∞EQ[g]=∞, then also inf{α|there isHw-admissible
andg≤α+(H·S)∞}=∞. This simply means that no matter how big
the constantais taken, there is now-admissible integrandHsuch thatg≤
a+(H·S)∞.