228 11 Change of Num ́eraire
Remark 11.4.5.We conjecture that the assumption thatV−^1 is locally bounded
can be removed.†
Proof. (1)and(2)are equivalent:SinceSsatisfies the(NFLVR)property
with respect to general admissible integrands, there is an equivalent local
martingale measure,QforS. Because the stochastic integralH·Sis bounded
below, the theorem of Ansel-Stricker, see [AS 94], implies that it, and hence
alsoV, is a local martingale. Since the final valueV∞ofVis strictly positive,
the result in Dellacherie-Meyer [DM 80, Theorem 17, p. 85] implies that the
processV is bounded away from zero a.s.. We can now apply Theorem 11.4.2
to see that (1) and (2) are already equivalent.
(1)implies(4): In caseV−^1 is locally bounded we have thatS ̃is also locally
bounded. It has the(NA)property and the productVS ̃is a local martingale.
Therefore the process has the(NFLVR)property and by Theorem 11.2.3 and
Theorem 11.2.9 it has an equivalent local martingale measure.
(1)and/or(2)imply(3): Now we apply the statement that (1) implies (4)
on the processV′=^12 (1 +V). This process is defined usingH 2 instead ofH.It
has the advantage thatV^1 ′is bounded. LetQ ̃be an equivalent local martingale
measure for (VS′,V^1 ′). Since the last coordinateX=V^1 ′ is bounded and is a
Q ̃-local martingale it is a strictly positive bounded martingale, starting at
- When we define the probability measureQbydQ=X∞dQ ̃,weobtain
thatS=VS′V′is aQ-local martingale andV′is aQ-uniformly integrable
martingale. This implies thatH·Sis aQ-uniformly integrable martingale.
The proof that (1) and/or (2) implies (3) is complete.
(3)implies(1): IfH·Sis aQ-uniformly integrable martingale for some
Q∈Me(P)then(H·S)∞is necessarily maximal. Indeed if say (K·S)∞≥
(H·S)∞for some admissibleK, then by taking expectations with respect
toQ, applying the super-martingale property ofK·Sand the martingale
property of (H·S)wesee
0=EQ[(H·S)∞]≤EQ[(K·S)∞]≤ 0.It follows thatEQ[(K·S)∞]=0and(H·S)∞=(K·S)∞. This completes
the proof that (3) implies (1).
(4)implies(2): Since the existence of an equivalent local martingale mea-
sure implies the(NA)property with respect to general admissible integrands,
this is trivial.
Corollary 11.4.6.If the locally bounded semi-martingaleSsatisfies (NFLVR)
with respect to general admissible integrands then for an admissible integrand
Hthe following are equivalent:
†Note added in this reprint: The hypothesis of local boundedness is not needed
since the processVcan be used as a martingale measure density. If (4) is satisfied
then even the existence of an equivalent sigma-martingale measure implies the
(NA)propertry for (S ̃).