The Mathematics of Arbitrage

13.4 Some Results on the Topology ofG 271

with the usual Wiener measurePand where we take the uniform distribution
mon [− 1 ,+1] as the second factor. The measure is thereforeP×m.
The set of equivalent local martingales measures can also be characterised.
Since Brownian motion has only one local martingale measure we see that for
eachQ∈Meand for eacht<1wehavethatQ=Pon theσ-algebraFt=
Ht. Therefore alsoQ=PonH 1. From the existence theorem of conditional
distributions, or the desintegration theorem of measures, we then learn thatQ
is necessarily of the formQ[dω×dx]=P[dω]μω[dx], whereμis a probability
kernelμ:Ω×B[− 1 ,+1]→[0,1], measurable forH 1 .InorderforQto be a local
martingale measureμshould satisfy

∫

[− 1 ,+1]xμω(dx) = 0 for almost allω.In
order to be equivalent toP×m, a.s. the measureμωshould be equivalent to
m. This can easily be seen by using the density ofQwith respect toP×m.
IfHis a predictable strategy then it is clear that it is predictable with
respect to the filtration of the Brownian motion. A strategyHis therefore
S-integrable if and only if

∫ 1

0 H

2
t dt <∞a.s.. It follows that a necessary
condition for a predictable processHto be 1-admissible isH·W≥−1. We
can change the value ofHat time 1 without perturbing the integralH·W.
In order to obtain a characterisation of 1-admissible integrands forS,weonly
need a condition onH 1 in order to have, in addition, that (H·S) 1 ≥−1. The
outcome at time 1 is (H·S) 1 =(H·W) 1 +H 1 gand this is almost surely bigger
than−1 if and only if|H 1 |≤1+(H·W) 1 almost surely. If we are looking for
1-admissible maximal contingent claims the condition onHbecomes

(1)H·Wis a uniformly integrable martingale forPandf=(H·W) 1 ≥− 1
(2)|H 1 |≤1+f.

From this it follows that a random variablekis inGif and only if it is of the
form
k=f 1 −f 2 +g(h 1 −h 2 )

where

(1)f 1 ,f 2 ,h 1 ,h 2 areH 1 -measurable;
(2)f 1 ,f 2 ≥−afor some positive real numbera;
(3)EP[f 1 ]=EP[f 2 ]=0;
(4)|h 1 |≤a+f 1 and|h 2 |≤a+f 2.

If we want to find a better description we observe that iffisH 1 -measurable,
integrable and positive then we can takef 1 =f 2 =f−EP[f] and hence the
condition onh 1 andh 2 becomes|h 1 |,|h 2 |≤f. It follows that the spaceGis
the space of all functionskof the form

f+gh

where

EP[f] = 0 and whereh, fare bothH 1 -measurable and integrable.