The Mathematics of Arbitrage

180 9 Fundamental Theorem of Asset Pricing

(R ̃k·S)t=(Rk·S)t

=(Rk·A)t+(Rk·M)t

≥max

(

(Lik·A)t,(Ljk·A)t

)

+(Rk·M)t

≥max

(

(Lik·A)t,(Ljk·A)t

)

+max

(

(Lik·M)t,(Ljk·M)t

)

−δk

≥max

(

(Lik·S)t,(Ljk·S)t

)

−δk

≥− 1 −δk.

At timeτkthe jump ∆(R ̃k·S)iseither∆(Lik·S)or∆(Ljk·S)and
hence (Rk·S)τk≥− 1 −δkbecause the left limit of (R ̃k·S)atτkis at least
max((Lik·S)τk−,(Ljk·S)τk−)−δk.

The integrands (1 +δk)−^1 R ̃kare 1-admissible. We will use them to con-
struct a contradiction to the maximal property off 0 = limk→∞(Ljk·S)∞=
limm→∞(Hm·S)∞.
(
R ̃k
1+δk

·S−Lik·S

)

∞

=

1

1+δk

(

(R ̃k−Lik)·S

)

∞−

δk

1+δk

(Lik·S)∞

=

(

1

1+δk

)

(

(R ̃k−Lik)·A

)

∞

+

1

1+δk

(

(R ̃k−Lik)·M

)

∞−

δk

1+δk

(Lik·S)∞.

This first term is estimated from below

Q

[(

(R ̃k−Lik)·A

)

∞>

α

2

]

>

α

2

and

(

(R ̃k−Lik)·A

)

∞≥^0.

The second term is estimated from above

Q

[(

(R ̃k−Lik)·M

)

∞≤−δk

]

<δkand

(

(R ̃k−Lik)·M

)

∞→^0.

The third term tends to zero sinceδk→0. From Lemma 9.8.1 we know
that there are convex combinationsVk∈conv{R ̃k,R ̃k+1,...}such that (Vk·
S)∞will converge to a functiong. Because (Lik·S)∞→f 0 and because

Q

[

((R ̃k−Lik)·S)∞>α 2 −δk

]

>α 2 −δkwe deduce from Lemma 9.8.6 that

Q[g>f 0 ]>0. Alsog≥f 0 , a contradiction to the construction off 0.

Final part of the proof of Theorem 9.4.2.From Lemmas 9.4.12 and 9.4.11
we deduce the existence of 1-admissible integrandsLk∈conv{Hk,Hk+1,...}
such thatLk·MandLk·Aboth converge in the semi-martingale topology. The
sequence (Lk·S)k≥ 1 is therefore convergent in the semi-martingale topology.
M ́emin’s theorem (see [M 80]) now implies the existence of a predictable pro-
cessLsuch thatLk·S→L·Sin the semi-martingale topology. In particular
Lis 1-admissible and the final value satisfies