12.4 The Existence of an Absolutely Continuous Local Martingale Measure 249
It follows that
L ̃∞> 0 PF-a.s..
It is chosen in such a way thatLS ̃ is aPF-local martingale and therefore the
setK ̃ 1 constructed with the 1-admissible, with respect toPF, integrands, is
bounded inL^0 (PF).
In particular this also excludes the possibility of immediate arbitrage for
Swith respect toPF.
Step2:Ssatisfies(NA)with respect toPF(and with respect to general
admissible integrands).
Since by step 1 immediate arbitrage is excluded, the violation of the(NA)
property would, by Lemma 12.3.1, give us a predictable integrandHsuch that
forPF the integrand is of finite support, isS-integrable and 1-admissible.
When the support ofHiscontainedin]]σ 1 ,σ 2 ]] it gives an outcome at leastε
on the set{σ 1 <∞}. All this, of course, with respect toPF.
The rest of the proof is devoted to the transformation of this phenomenon
to a situation valid forP.
Without loss of generality we may suppose that for the measurePwe
haveσ 1 ≤σ 2 ≤T, we replace, e.g., the stopping timeσ 2 by max(σ 1 ,σ 2 )and
then we replaceσ 1 andσ 2 by, respectively, min(T,σ 1 )andmin(T,σ 2 ). All
these substitutions have no effect when seen under the measurePF.Since
PF[{σ 1 <σ 2 <∞}]>0, we certainly have thatP[{σ 1 <σ 2 <T}]>0.
Roughly speaking we will now use the strategyHto construct arbitrage
on the setF and we use the processL^1 to construct a sure win on the set
Fc,asontheinterval[[0,T[[, the processL^1 −1equalsK·Sfor a well-chosen
integrandK. When we add the two integrands,HandK, we should obtain
an integrand that gives arbitrage on Ω with respect toPand this will provide
the desired contradiction.
Let the sequence of stopping timesτnbe defined as
τn=inf{
t∣
∣
∣
∣Lt≤1
n}
.
We have thatτn↗TforPandτn↗∞for the measurePF.Sincewe
have thatLτn >0 a.s., we also have thatUτn>0 a.s.. It follows that on
theσ-algebraFτnthe two measures,PandPFare equivalent. We can there-
fore conclude that for eachnthe integrandH (^1) [[ 0,τn]]as well as the integrand
K (^1) [[ 0,τn]]isS-integrable and 1-admissible forP. The last integrand still has
to be renormalised.
In fact on the setFitself, the lower bound−1 for the processK·Sis too
low since it will be compensated at most byε. We therefore transformKin
such a way that it will stay above 2 εbut will nevertheless give outcomes that
are very big on the setFc. Let us define
K ̃=K (^1) {σ 1 <T}ε
2
Lσ 1 ,
K ̃n=K ̃ (^1) [[ 0,τn]],