The Mathematics of Arbitrage

12.3 The No-Arbitrage Property and Immediate Arbitrage 243

The proof of the theorem is based on the following lemma:

Lemma 12.3.8.If (12.1) holds almost surely, then for anya, ε, η > 0 we can
find 0 <δ<ε 2 and ana-admissible integrandHwith

H=H (^1) ]]δ,ε]],
∫ε
δ
|Hs′dA|s+
∫ε
δ
Hs′d〈M, M〉sHs<2+a,
P[(H·S)ε≥1]≥ 1 −η.
Proof of the lemma.Fixa, ε, η >0andletR≥max{η^8
(1+a
a

) 2

,(1 +a)^2 }.
Since (i) is satisfied almost surely, we have that

Klim↗∞

δ↘ 0

P

[∫ε

δ

(^1) {|h|≤K}h′td〈M, M〉tht≥R

]

=1.

Hence we can find aK>0anda0<δ<ε 2 such that

∞>

∫ε

δ

(^1) {|h|≤K}h′td〈M, M〉tht≥R
on aFε-measurable set Λ withP[Λ]≥ 1 −η 2 .Let
T=inf

{

t> 0

∣

∣

∣

∣

∫t

δ

(^1) {|h|≤K}h′td〈M, M〉tht≥R

}

∧ε

and letH=1+Rah (^1) ]]δ,T]] (^1) {|h|≤K}.Then
∫ε
0
Hs′d〈M, M〉sHs≤
(1 +a)^2
R
and ∫ε
0
|HsdA|s≤(1 +a) a.s..
ThereforeHisS-integrable. Moreover, (H·A)ε=1+aon Λ.
Since‖H·M‖^22 =E[
∫ε
0 H
′
sd〈M, M〉sHs]≤
(1+a)^2
R we obtain from Doob’s
inequality together with Tchebycheff’s inequality (both in theirL^2 -version)
P[(H·M)∗≥a]≤ 4
(1+a
a
) (^21)
R≤
η
2. (12.2)
We now localiseHto bea-admissible. Let
T 2 =inf{t> 0 |(H·M)t<−a}∧T.
ThenT 2 =Ton{(H·M)∗≤a}and from (12.2) we obtain