13.2 Maximal Admissible Contingent Claims 259
Proof.This is a rephrasing of the Theorem 13.2.14 since by Theorem 13.2.5,
the condition on the existence of an equivalent risk neutral measure is equiv-
alent with the maximality property.
The previous Corollary 13.2.15 will begeneralised to sequences (see Corol-
lary 13.2.18 below). We first prove the following Proposition.
Proposition 13.2.16.Suppose thatSis a locally bounded semi-martingale
that satisfies the (NFLVR) property. If(fn)n≥ 1 is a sequence inKmax 1 ,such
that
(1)The sequencefn→fin probability
(2)for allnwe havef−fn≥−δnwhereδnis a sequence of strictly positive
numbers tending to zero,
thenfis inKmaxtoo, i.e. it is maximal admissible.
Proof.Ifgis a maximal contingent claim such thatg≥f,thenwehave
g−fn≥−δn. Since eachfnis maximal we find thatg−fnis acceptable and
henceδn-admissible by Proposition 13.2.13. Sinceδntends to zero, we find
that the(NFLVR)property implies thatg−fntends to zero in probability.
This means thatg=fand hencefis maximal.
Corollary 13.2.17.IfS is a locally bounded semi-martingale that satisfies
the (NFLVR) property, if(an)n≥ 1 is a sequence of strictly positive real num-
bers such that ∞
∑
n=1
an<∞,
if for eachn,Hnis anan-admissible maximal strategy, then we have that the
series
f=
∑∞
n=1
(Hn·S)∞
converges in probability to a maximal contingent claim.
Proof.Lethn=(Hn·S)∞, the partial sumsfN =
∑N
n=1hnare outcomes
of
∑∞
n=1an-admissible strategies.For an arbitrary elementQ∈M
ewe have
that
EQ[(hn+an)]≤an.
It follows that the series of positive functions
∑∞
n=1(hn+an)convergesin
L^1 (Q) and hence the series
∑∞
n=1hnalso converges inL
(^1) (Q). The series
f=
∑∞
n=1hn= limfntherefore also converges to a contingent claimfinP.
From the Proposition 13.2.16, we now deduce thatfis maximal.