The Mathematics of Arbitrage

176 9 Fundamental Theorem of Asset Pricing

(K·S)t=

∑n

i=1

λi (^1) {t>Tci}

(

(Hi·S)min(t,τ,σ)−(Hi·S)min(Tci,τ,σ)

)

≥

∑n

i=1

λi (^1) {t>Tci}(− 1 −N)

≥−(N+1)

∑n

i=1

λi (^1) {t>Tci}
≥−(N+1)Ft
whereF is the processF=
∑n
i=1λi^1 ]]Tci,∞[[.Fis an increasing adapted left
continuous process, it is therefore predictable. By constructionEQ[F∞]≤δ^2
and thereforeQ[F∞>δ]≤δ. This implies that the stopping timeν, defined
asν=inf{t|Ft>δ},satisfiesQ[ν<∞]<δ<α 4.
This implies thatK′=K (^1) [[ 0,ν]],satisfies
‖(K′·S)∗‖L (^2) (Q)≤δ
and
Q[(K′·M)∗>α]>α−Q[τ<∞]−Q(ν<∞)≥
α
2
as well as
(K′·S)≥−(N+1)δ.
The integrandLδ= K
′
(N+1)δtherefore is 1-admissible and
‖(Lδ·S)∗‖L^2 (Q)≤

(

1

N+1

)

.

FurthermoreQ

[

(Lδ·M)∗>(N+1)α δ

]

α 2.
Forδtending to zero this produces a contradiction to Lemma 9.4.8.

The following lemma relates, in theL^0 -topology, the maximal function of
a local martingale with the maximal function of a stochastic integral for an
integrand that is bounded by 1. The proof uses the fact that the sequence
(Hn·M)n≥ 1 is a sequence of localL^2 -martingales with uniformL^2 -control of
the jumps.

Lemma 9.4.10.With the same notation as in Lemma 9.4.9, for allε> 0
there isc 0 > 0 such that for allhpredictable |h|≤ 1 , all convex weights
(λ 1 ...λn)and allc≥c 0

Q

{[(

h

∑n

i=1

λiKic

)

·M

]∗

>ε

}

<ε.

In particularD

(∑

λiKci·M

)∗

< 2 εwhereDis the quasi-norm introduced
in Sect. 9.2 and inducing the semi-martingale topology.