The Mathematics of Arbitrage

9.4 Proof of the Main Theorem 177

Proof.Letε>0andtakec 0 as in Lemma 9.4.9 i.e.

Q

[(∑

λiKci·M

)∗

>ε

]

<ε

for all (λ 1 ...λn) convex combination and allc ≥c 0. By enlargingc 0 we
also may suppose that supn‖(Kcn·S)∗‖L^2 (Q) ≤ ε 3 (see the proof of the
Lemma 9.4.9). Corollary 9.2.4 now implies that for allnand every stopping
timeσ

‖∆(Kcn·M)σ‖L (^2) (Q)≤ε.
Take nowhpredictable and bounded by 1, takec≥c 0 ,λ 1 ...λnaconvex
combination. Defineσas
σ=inf

{

t

∣

∣

∣

∣

∣

∣

∣

∣

∣

(n

∑

i=1

λi(Kci·M)t

)∣

∣

∣

∣

∣

>ε

}

.

The following estimate holds:

sup

t≤σ

∣

∣

∣

∣

∣

(n

∑

i=1

λiKic

)

·M

∣ ∣ ∣ ∣ ∣ t

≤ε+

∑

λi

∣

∣∆(Kci·M)σ

∣

∣.

TheL^2 -norm of the left hand side is therefore smaller than 2εand we have
anL^2 -martingale. This implies that the martingale

(

h

∑

λiKci

)

(^1) [[ 0,σ]]·Mis
inL^2 and its norm is smaller than 2ε. Hence
Q

[((

h

∑

λiKci

)

·M

)∗

>

√

ε

]

≤Q

[((

h

∑

λiKci

)

(^1) [[ 0,σ]]·M

)∗

>

√

ε

]

+Q[σ<∞]

≤

4 ε^2

ε

+ε=5ε. 

Lemma 9.4.11.There is a sequence of convex combinationsLn∈conv{Hk,k≥
n}such that(Ln·M)converges in the semi-martingale topology.

Proof.We use the notation introduced before Lemma 9.4.9. Forε=^1 nwe
apply Lemma 9.4.10 to findcnsuch that

D

((m

∑

i=1

λiKcin

)

·M

)

≤

1

n

for all convex weightsλ 1 ...λm.

For eachnand eachkwe have (Hk (^1) [[ 0,Tckn]]·M)∗≤cn+|∆(Hk·M)Tckn|and
an application of Corollary 9.2.4 yields that eachHk (^1) [[ 0,Tckn]]·Mis anL^2 (Q)-
martingale with boundcn+3‖q‖L (^2) (Q). A standard diagonalisation argument
shows the existence of convex weightsλk 0 ,λk 1 ,...,λkNk, such that