222 11 Change of Num ́eraire
Me(P)={
Q
∣
∣
∣
∣
Qis equivalent toP
and the processSis aQ-local martingale}
M(P)=
{
Q
∣
∣
∣
∣
Qis absolutely continuous with respect toP
and the processSis aQ-local martingale}
.
We identify absolutely continuous measures with their Radon-Nikod ́ym deriva-
tives. It is clear that the setMe(P)isL^1 -dense inM(P).
11.3 Duality Relation
In this section we extend the duality formula of Delbaen [D 92] and Chap. 9
to the case of unbounded functions. We denote byC◦the polar of the coneC,
i.e.
C◦={
f|f∈L^1 (P)andforeachh∈Cwe haveEP[fh]≤ 0}
.
Theorem 11.3.1.IfS is a locally bounded semi-martingale that satisfies
(NFLVR) with respect to general admissible integrands then
M(P)=C◦∩{Q|Qprobability measure,Q P}.Proof.IfQ∈M(P) then for each admissible integrandHwe have, by the
Ansel-Stricker theorem, [AS 94], thatH·Sis aQ-local martingale and hence
it is a super-martingale. ThereforeEQ[f]≤0foreachf∈K.Thesame
inequality pertains for elements ofC.
Conversely ifQis a probability measure inC◦thenSwill be aQ-local
martingale. Indeed takeTnan increasing sequence of stopping times,Tn↗∞,
such that eachSTnis bounded. For eachs<tand eachA∈Fswe have that
(^1) A(STtn−SsTn)isinCand hence we haveEQ[ (^1) A(StTn−STsn)]≤0. Replacing
(^1) Aby− (^1) Agives thatEQ[ (^1) A(StTn−STsn)] = 0. These equalities show thatS
is aQ-local martingale.
Corollary 11.3.2.Suppose that the locally bounded semi-martingaleSsatis-
fies the (NFLVR) property with respect to general admissible integrands. The
setM(P)is then closed inL^1 (P).
We remark that this is essentially a consequence of the local boundedness
ofS. It is easy to give counter-examples in the general case.
Theorem 11.3.3.If the locally bounded semi-martingale S satisfies the
(NFLVR) property with respect to general admissible integrands, then for
bounded elementsfinL∞we have that