13.6 The SpaceGunder a Num ́eraire Change 275is constructed with thed-dimensional processS,thespaceG
(S
V,
1
V)
is the
space of workable contingent claims constructed with the (d+ 1)-dimensional
process
(S
V,
1
V)
.
Theorem 13.6.2.Suppose thatSis a locally bounded semi-martingale that
satisfies the (NFLVR) property. Suppose thatV is a strictly positive process
of the formV=H·S+1where1+(H·S)∞is strictly positive and where
(H·S)∞is maximal admissible. Suppose that the processV^1 is locally bounded.
The mapping
φ: G(S)−→ G(SV,V^1 )
g −→ Vg∞
defines an isometry betweenG(S)=G(S,1)andG(VS,V^1 ).
Proof.SupposeV =H·S+1 where 1+(H·S)∞is strictly positive and
where (H·S)∞is maximal admissible. Take an admissible, with respect to
the processS,strategyK. The processKV·Sis the outcome of the strategy
K′=(K,(K·S)−−KS−), see also the Remark 13.2.10 above and Chap. 11.
From Theorem 13.2.5 above it follows that there is an elementQ∈Mesuch
thatEQ[(K·S)∞]=0andsuchthatEQ[V∞]=1.ThemeasureQ ̃defined
asdQ ̃=V∞dQis therefore an element ofM
(S
V,
1
V)
such thatEQ ̃[KV∞·S]=0.It follows that the contingent claimV^1 ∞−1 is maximal and admissible for the
process
(S
V,
1
V)
and hence the contingent claim KV∞·Sis workable. It followsthat the mappingφmapsKmax, and hence alsoG(S), intoG
(S
V,
1
V)
.
If we apply the num ́eraireV^1 to the system(S
V,
1
V)
we find the (d+1)-
dimensional process (S, V). However, becauseV is given by a stochastic in-
tegral with respect toS,wehavethatG(S, V)=G(S). It follows that the
mapping that associates with each elementk∈G
(S
V,
1
V)
, the elementkV∞
mapsG
(S
V,
1
V)
intoG(S). The mappingφis clearly bijective.
LetG={Q∈M(S)|EQ[V∞]=1}andG ̃={Q ̃∈M(VS,V^1 )|EQ ̃[V^1 ∞]=1}.Since both sets are dense in, respectively,M(S)andM
(S
V,
1
V)
,itisclear
that for every elementg∈G(S),
2 ‖g‖=sup{EQ[|g|]|Q∈M(S)}
=sup{EQ[|g|]|Q∈G}=sup{
EQ ̃
[
|g|
V∞]∣∣
∣
∣
Q ̃∈G ̃
}
=sup{
EQ ̃
[
|g|
V∞]∣∣
∣
∣
Q ̃∈M
(
S
V
,
1
V
)}
=2
∥
∥
∥
∥
g
V∞∥
∥
∥
∥.
This shows thatφis also an isometry.