The Mathematics of Arbitrage

9.7 Simple Integrands 199

P[Tm<1]≤P

[

sup

0 ≤t≤ 1

|Wtm|>m

]

+

∑

m≥ 1

2 −(n+m)

≤

√

2

π

1

m

e−

m 22

+2−m

and hence

∑

P[Tm<1]<∞. This implies, via the Borel-Cantelli lemma,
that for almost allω∈Ω,Tm(ω) becomes eventually 1.
The processZmis now defined as

Zmt :=

{

Ytm+αm(Wtm+m^2 t)fort≤Tm,

YTmm+αm(WTmm+m^2 Tm)forTm≤t≤ 1.

The sequenceαmwill be chosen later, but will satisfy 0<αm≤1.
The processZmis clearly bounded by 1 + (m+m^2 )αm≤1+m+m^2.
Finally we define

St:=

{ 1

2 Z

1

t for 0≤t≤^1 ,

Sm− 1 +2−mZtm−(m−1) form− 1 ≤t≤m.

The processSis c`adl`ag and|S|≤

∑

m≥ 12

−m(1 +m+m (^2) )α
m≤24. It is
a semi-martingale with decompositionS=M+A,whereAis given by the
recurrence relations
Am−1+t−Am− 1 =

{

2 −mαmm^2 t fort≤Tm,

2 −mαmm^2 Tm forTm≤t≤ 1.

The martingaleMis uniformly bounded on each interval [[0,m]].
With respect to its natural filtration, augmented with the zero sets,Sis
a special semi-martingale and the filtration satisfies the usual conditions. The
last statement is not trivial to verify but it follows from the same property of
the filtration of the Brownian motion.

Lemma 9.7.9.For each sequence(αm)m≥ 1 in]0,1], the processSfails the
equivalent (local) martingale property.

Proof.Consider the sequence (Hm)m≥ 1 defined as

Hm=α−m^1 m−^22 m (^1) (]m− 1 ,m]\Q)×Ω.
EachHmis a deterministic process, hence predictable. The process (Hm·S)is
uniformly bounded from below by−1and((Hm·S)∞)m≥ 1 equalsm^12 WTmm+
Tm≥Tm−m^1.
BecauseTm=1formbig enough we see that (Hm·S)∞tends to 1
formtending to∞. This clearly violates(NFLVR). Because of the Main
Theorem 9.1.1 we see thatScannot have an equivalent martingale measure.