The Mathematics of Arbitrage

194 9 Fundamental Theorem of Asset Pricing

Since|K ̃n|≤1 we still have

E

[(

(K ̃n·Mn)TNnn+1

) 2 ]

≤c+5.

On the other hand

(K ̃n·S)∞=(K ̃n·S)TNnn+1

=(K ̃n·M)TNnn+1+

∑Nn

k=0

|fkn||AnTkn+1−AnTkn|

≥(K ̃n·M)TNnn+1+

∑Nn

k=0

(fkn)

(

AnTkn+1−AnTkn

)

≥(K ̃n·M)TNnn+1+(Kn·S)∞−(Kn·M)TNnn+1.

Because the sequences (K ̃n·M)TNnn+1and (Kn·M)TNnn+1are bounded in

L^2 and the sequence (Kn·S)+∞is unbounded inL^0 , the sequence (K ̃n·S)∞is
necessarily unbounded inL^0. On the other hand sup 0 ≤t(K ̃n·S)−t is a bounded
sequence inL^0. Indeed fort=Tknwe have

(K ̃n·S)−Tkn≤(K ̃n·M)Tkn≤sup

0 ≤t

|(K ̃n·M)t|.

And forTkn≤t≤Tkn+1we find:

(K ̃n·S)−t ≤(K ̃n·S)−Tn

k

+|fkn||St−STkn|≤2+sup

0 ≤t

|(K ̃n·M)t|

and hence

(^33)
3
3 sup 0 ≤t(
K ̃n·S)−t
(^33)
3
3
2

≤2+

(^33)
3
3 sup 0 ≤t(
K ̃n·M)t
(^33)
3
3
2
≤2+2‖(K ̃n·M)t‖ 2
≤2+2

√

c+ 5 (by Doob’s maximum inequality).

This proves the boundedness inL^0. From Lemma 9.7.3 it now follows that
(K ̃n·S)+t is bounded inL^0. This contradicts the choice of the sequenceK ̃n.

The following example shows that the requirement thatS is locally
bounded cannot be dropped. The same notation will also be used in a later
example.

Example 9.7.5.We suppose that on a probability space (Ω,F,P) following
sequences of variables are defined: a sequence (γn)n≥ 1 of Gaussian normalised
N(0,1) variables, a sequence (φn)n≥ 1 of random variables with distribution
P[φn=1]=2−nandP[φn=0]=1− 2 −n. All these variables are supposed