274 13 The Banach Space of Workable Contingent Claims in Arbitrage Theory
For eachnthe set{Q|Q∈M;EQ[fn]=0}is a norm dense and (for the
norm topology) aGδ-set inM.SinceMis a complete space for theL^1 -norm,
we may apply Baire’s category theorem. Therefore the intersection over alln,
{Q|Q∈M; for alln:EQ[fn]=0}is still a denseGδ-set ofM. Because,
by corollary 13.2.18, the set{Q|Q∈Me; for alln:EQ[fn]=0}is non-
empty, an easy argument using convex combinations yields that{Q|Q∈
Me;EQ[f] = 0 for allf∈V}is dense inM.
Corollary 13.5.4.IfSis a continuousd-dimensional semi-martingale with
the (NFLVR) property, if there is a continuous local martingaleWsuch that
〈W, S〉=0butd〈W, W〉is not singular tod〈S, S〉,thenGis not a separable
space.
Proof.This follows from the previous corollary and from Theorem 13.5.1
above.
13.6 The SpaceG under a Num ́eraire Change
If we change the num ́eraire, e.g. we change from one reference currency to
another, what will happen with the spaceG? Referring to Chap. 11 and espe-
cially the proofs of Theorem 11.4.2 and 11.4.4 therein, we expect that there
is an obvious transformation which should be the mathematical translation
of the change of currency. More precisely we want the contingent claims of
Gto be multiplied with the exchange ratio between the two currencies. This
section will give some precise information on this problem.
We start with the investigation of how the set of equivalent martingale
measures is changed.
Suppose thatVis a strictly positive process of the formV=H·S+1 where
1+(H·S)∞is strictly positive and where (H·S)∞is maximal admissible.
Suppose also that the processV^1 is locally bounded. This hypothesis allows
us to use, without restriction, the theory developed so far. With each element
RofM(S) we asssociate the measureR ̃defined bydR ̃=V∞dR.Ofcourse
this measure is not a probability measure since we do not necessarily have
thatER[V∞] = 1. But from Theorem 13.5.2 above it follows, however, that
the setG={Q∈M(S)|EQ[V∞]=1}is a denseGδ-set ofM(S). Likewise
the setG ̃=
{
Q ̃∈M
(S
V,
1
V)∣∣
∣EQ ̃
[ 1
V∞]
=1
}
is a denseGδ-set ofM(S
V,
1
V)
.
The following theorem is obvious.
Theorem 13.6.1.Suppose thatSis a locally bounded semi-martingale that
satisfies the (NFLVR) property. With the above notations, the relationdR ̃=
V∞dR, defines a bijection between the setsGandG ̃.
In the following theorem we make use of the notation introduced in The-
orem 13.3.2. The spaceG(S) is the space of workable contingent claims that