224 11 Change of Num ́eraire
Theorem 11.3.4.Suppose that the locally bounded martingaleSsatisfies the
(NFLVR) property with respect to general admissible integrands. Iff≥ 0 ,or
more generally iffis bounded below by a constant, then
sup
Q∈Me(P)EQ[f]= sup
Q∈M(P)EQ[f]=inf{x|∃h∈Kx+h≥f}
and when the expression is finite
=min{x|∃h∈Kx+h≥f}.Proof.We supp ose that f ≥0. The first equality follows again from the
density ofMe(P)inthesetM(P) and Fatou’s lemma. The left hand side
is smaller than the right hand side exactly as in the proof of the previ-
ous theorem. We remark that this already implies that we have equality as
soon as supQ∈Me(P)EQ[f]=∞.Letnowzbe a real number such that
z>supQ∈Me(P)EQ[f]. For all natural numbers we therefore have that
z>supQ∈Me(P)EQ[f∧n]. The theorem for bounded functions now implies
the existence ofhn∈Kand 0≤xn<zsuch thatf∧n≤xn+hn.Wemay
extract subsequences and suppose that the bounded sequencexnconverges
to a real numberx≤z. The functionshnare bigger than−xnand therefore
the result of anxnand hence az-admissible strategyHn. The sequence of
functionshnis inKz, a bounded convex set ofL^0 (P). Using Lemma 9.8.1
we may take convex combinations ofhnthat converge almost everywhere to
a functionh. We still have thath+x≥f. The properties ofKzlisted above
(see Theorem 11.2.6 (2)), imply that there is an elementg∈Kzsuch that
g≥h. This element clearly satisfiesx+g≥fand hence we obtain
z≥inf{x|∃h∈Kx+h≥f}.We therefore see that
sup
Q∈Me(P)EQ[f]= sup
Q∈M(P)EQ[f]=inf{x|∃h∈Kx+h≥f}.To see that the infimum is a minimum we take a sequencexntending to the
infimum and a corresponding sequence of outcomeshn. We can apply the
same reasoning to see that the infimum is attained.
Corollary 11.3.5.Suppose that the locally bounded semi-martingaleSsat-
isfies the (NFLVR) property with respect to general admissible integrands. If
f≥ 0 and ifx=supQ∈Me(P)EQ[f]<∞, then there is a maximal element
g∈Ksuch thatf≤x+g.
Proof.This follows from the proof of the theorem.