9.5 The Set of Representing Measures 185set we will use is precisely the setCintroduced in Sections 9.2, 9.3 and 9.4.
In Sect. 9.4, Theorem 9.4.2, it is proved thatCis weak-star-closed inL∞.
In the case of processes which are not necessarily continuous,Cis the exact
substitute for the setW^0 −L∞+, so useful in the continuous case. The polar
C◦of the coneCis by definition
C◦={g|g∈L^1 ,E[gh]≤0 for allhinC}.Theorem 9.5.7.
M(P)={Q|Q∈L^1 ,Q[Ω] = 1andQ∈C◦}.Proof.IfQis inM(P)thenforH admissible we know by Theorem 9.2.9
thatH·Sis aQ-super-martingale. ThereforeEQ[h]≤0foreveryhinC.
Conversely letQbe inL^1 , of norm 1 andQ∈C◦.Theset−L∞+ is a subset
ofCand hence every element ofC◦is inL^1 +. ThereforeQis a probability
measure. IfTis a stopping time andSTis bounded then the random variables
α(SuT−StT) (^1) Aforu≥t,αreal andAinFt,areinCand henceQis a local
martingale measure forS.
The following theorem is the precise form of the duality equality stated
above. We will prove it for bounded functions, referring to [AS 94] for the case
of measurable functions with bounded negative parts.
Theorem 9.5.8.For everyfinL∞we have
sup
Q∈Me(P)
EQ[f]= sup
Q∈M(P)
EQ[f]
=inf{x| there ish∈Cwithx+h≥f}
=inf{x| there ish∈Cwithx+h=f}.
Proof.From the definition ofCit follows thatx+h≥fforhinCif and only
if there ishinCwithf=x+h. The second equality is therefore obvious.
From the preceding theorem it follows that
sup
Q∈M(P)
EQ[f]≤ inf{x| there ish∈Cwithx+h≥f}.
Ifz<inf{x| there ish∈Cwithx+h≥f}thenf−zis not an element of
the weak-star-closed coneC. By the Hahn-Banach theorem there is a signed
measureQ∈L^1 ,EQ[h]≤0 for allhinCandEQ[f−z]>0. The preceding
theorem shows thatQcan normalise asQ[Ω] = 1 and then it is inM(P). It
follows thatz<EQ[f]≤supR∈M(P)ER[f]. This shows that
sup
Q∈M(P)
EQ[f]≥ inf{x| there ish∈Cwithx+h=f}.
Remark 9.5.9.The infimum is a minimum sinceC is weak-star and hence
norm closed.