The Mathematics of Arbitrage

9.4 Proof of the Main Theorem 179

values in{+1,− 1 },α >0 and two increasing sequences (ik,jk)k≥ 1 such that
Q[φk>α]>αwhere

φk=

∫

[0,∞[

hkud

(

(Lik−Ljk)·A

)

u

=

∫

[0,∞[

hku(Liuk−Ljuk)dAu

=

∫

[0,∞[

|Liuk−Ljuk||dAu|.

We now define the integrandRkas

Rk=

(

Ljk+^12 (1 +hk)(Lik−Ljk)

)

=^12

(

Lik+Ljk+hk(Lik−Ljk)

)

.

The idea is simple ifhk= 1 i.e. if (Lik−Ljk)·dA≥0wetakeLik,if
hk=−1 i.e if (Lik−Ljk)dA≤0wetakeLjk.InsomesenseRktakes the best
of both. The processes (Rk−Lik)and(Rk−Ljk)·Adefine positive measures
and are therefore increasing. Indeed

(Rk−Lik)·A=

((

Ljk−Lik

)

+^12

(

1+hk

)(

Lik−Ljk

))

·A

=^12

((

hk− 1

)(

Lik−Ljk

))

·Aand

(Rk−Ljk)·A=^12

((

hk− 1

)(

Lik−Ljk

))

·A.

Both measures are positive by the construction ofhk.Also

φk=

(

(Rk−Lik)·A

)

∞+

(

(Rk−Ljk)·A

)

∞.

We may therefore suppose thatQ

[(

(Rk−Lik)·A

)

∞>

α

2

]

α 2 (if neces-
sary we interchangeikandjkand take subsequences to keep them increas-
ing). Because (Rk−Lik)·M =^12 ((hk−1)(Lik−Ljk)·M) and because
(Lik−Ljk)·Mtend to zero in the semi-martingale topology on [0,∞[we
deduce that the maximal functions ((Rk−Lik)·M)∗tend to zero in proba-
bility. The same holds for ((Rk−Ljk)·M)∗.Letnow(δk)k≥ 1 be a sequence
of strictly positive numbers tending to 0. By taking subsequences and by
the above observation we may suppose thatQ[((Rk−Lik)·M)∗>δk or
((Rk−Ljk)·M)∗>δk]<δkholds for allk. This implies that the stopping
timeτkdefined asτk=inf{t|(Rk·M)t≤max((Lik·M)t,(Ljk·M)t)−δk}

satisfiesQ[τk <∞]<δk. Define nowR ̃k =Rk (^1) [[ 0,τk]]. We claim that the
integrandsR ̃kare (1 +δk)-admissible!
Fort<τkwe have