250 12 Absolutely Continuous Local Martingale Measures
H ̃=H (^1) {σ 1 <T},
H ̃n=H ̃ (^1) [[ 0,τ
n]].
From the preceding considerations it follows that the integrandsH ̃nare
all 1-admissible forPand that the integrandsK ̃nare 2 ε-admissible forP.The
outcomes (K ̃n·S)τntend to∞onFc∩{σ 1 <T}, and the outcomes (H ̃n·S)τn
become larger thanεon the setF∩{σ 1 <T}. When we add them we see
that on the set{σ 1 <T}we have
lim inf
n→∞
((H ̃+K ̃)·S)τn= lim inf
n→∞
((H ̃n+K ̃n)·S)τn≥
ε
2
.
Define now the stopping timeμas
μ=τnifnis the first number such that ((H ̃n+K ̃n)·S)τn≥
ε
4.
The stopping timeμis finite on the set{σ 1 <τ}. The integrandJ=(H ̃+
K ̃) (^1) ]] 0,μ]]is nowS-integrable and is certainly 1+ε
2 admissible. By the definition
of the stopping timeμwe have that (J·S)μ≥ε 41 {σ 1 <T}, producing arbitrage.
Since the processSsatisfied the(NA)property, we arrived at a contradiction.
Step 2 is therefore completed and this ends the proof of the theorem.
Acknowledgement
The results of this paper were presented at the seminar at Tokyo University
in summer 1993. We thank Professor Kotani and Professor Kusuoka for the
invitation and for discussions on this topic. We also thank W. Brannath for
discussions on the current proof of Theorem 12.3.7, and an anonymous referee
for valuable suggestions.