The Mathematics of Arbitrage

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.