The Mathematics of Arbitrage

9.4 Proof of the Main Theorem 175

the technique. Let us introduce the following sequence of stopping times (cis
supposed to be>0).
Tcn=inf{t||(Hn·M)t|≥c}. The local martingales (Hn·M) will be

stopped atTcn, causing an errorKcn·MwhereKcn=Hn (^1) ]]Tcn,∞[[.
Lemma 9.4.9.For al lε> 0 ,thereisc 0 > 0 such that for arbitraryn, for all
convex weights(λ 1 ,...,λn)and allc≥c 0 , we have

Q

[(n

∑

i=1

λiKic·M

)∗

>ε

]

<ε.

Proof.Suppose on the contrary that there isα>0 such that for allc 0 there
are convex weights (λ 1 ,...,λn)andc≥c 0 , such that

Q

[(n

∑

i=1

λiKci·M

)∗

>α

]

>α.

From this we will deduce the existence of a sequence of 1-admissible inte-

grandsLnsuch that supn‖(Ln·S)∗‖L (^2) (Q)is bounded and such that (Ln·M)∗
is unbounded inL^0 (Q). This will contradict Lemma 9.4.8.
LetN be large enough so thatQ[q>N]< α 4 (rememberq =supn
supt|(Hn·S)t|). This is easy sinceqis finite a.s.. If we defineτas the stopping
time
τ=inf{t| for somen≥1:|(Hn·S)t|>N}
we trivially haveQ[τ<∞]<α 4. From Lemma 9.4.8, applied withλ=sup
‖(Hn·S)∗‖L^2 (Q), we deduce that limc→∞supnQ[Tcn<∞]≤limc→∞supnQ
[(Hn·M)∗≥c]=0.For0<δ<α 4 ,letc 1 be chosen so that for allnand all
c≥c 1 we haveQ[Tcn<∞]<δ^2 .Foreachnwe have
‖(Kcn·S)∗‖L^2 (Q)≤‖2(Hn·S)∗ (^1) {Tcn<∞}‖L^2 (Q)
≤ 2 ‖q‖L (^2) (Q)Q[Tcn<∞]
1
(^2).
If follows that there isc 2 so that for allnand allc≥c 2
‖(Kcn·S)∗‖L (^2) (Q)≤δ.
Forc≥max(c 1 ,c 2 )takeλ 1 ...λna convex combination that guarantees
Q

[(∑

n

i=1λiK

i

c·M

)∗

>α

]

>αand letσ=inf

{

t|

∣

∣

(∑n

i=1λiK

i

c·M

)

t

∣

∣≥α

}

.

PutK=

(∑n

i=1λiK

i

c

)

(^1) [[ 0,min(τ,σ)]].
ClearlyQ[(K·M)∗ ≥α]>α−Q[τ<∞]=^34 α and the inequality
(K·S)∗≤
∑n
i=1λi(K
i
c·S)
∗implies‖(K·S)∗‖
L^2 (Q)≤δ. Let us now investigate
whetherKis admissible.