The Mathematics of Arbitrage

9.7 Simple Integrands 201

min((Lj·S)m,1)−min((Lj·S)m− 1 ,1)≤(Lj·S)m−(Lj·S)m− 1.

The processLjis bounded in intervals [0,m] and becauseSis also uniformly
bounded with only one jump in each interval [k, k+ 1], the semi-martingale
Lj·Sis locally bounded, therefore special and decomposed asLj·S=Lj·
M+Lj·A. The local martingale part is a square integrable martingale and
hence:
EP[(Lj·M)m−(Lj·M)m− 1 ]=0.

This yields the following estimates:

EP[min((Lj·S)m,1)−min((Lj·S)m− 1 ,1)]

≤EP[(Lj·S)m−(Lj·S)m− 1 ]

≤EP[(Lj·A)m−(Lj·A)m− 1 ]

≤EP

[∫

]m− 1 ,m]

Ljuαmm^2 du

]

≤ 2 m+1m^2 αm.

This implies that

EP[min((Lj·S)m 0 ,1)]

≥EP[min((Lj·S)∞,1)]−

∑

m>m 0

2 m+1m^2 αm

≥EP[min((Lj·S)∞,1)]−

βm 0

2 m 0

(by the choice ofαm).

Because

lim inf

j→∞

EP[min((Lj·S)∞,1)]>

2

m 0

we can deduce that

lim inf

j→∞

EP[min((Lj·S)m 0 ,1)]>

2

m 0

−

βm 0

2 m 0

>

1

m 0

.

We may now suppose thatEP[min((Lj·S)m 0 ,1)]>m^10 for allj. Because of
the choice ofβmwe also see that

EP[min((Lj·S)−m 0 ,1)]

≥βmEP[min((Lj·S)+m 0 ,1)]≥βmEP[min((Lj·S)m 0 ,1)]>

βm

m 0

.

Let the setAjbe defined asAj={(Lj·S)m 0 < 0 }.
Because lim infj→∞min((Lj·S)∞,1) ≥ min(f 0 ,1) we also have that

lim infj→∞( (^1) Ajmin((Lj·S)∞,1))≥lim infj→∞( (^1) Ajmin(f 0 ,1)).
An application of Fatou’s lemma yields that