172 9 Fundamental Theorem of Asset Pricing
Tchebycheff’s inequality now yieldsQ[(Ln·M)∗ (^1) Bc ≥n]≤α^2 which
impliesQ[Bc∩{(Ln·M)∗≥n}]≤α^2 ≤αand hence
Q[B]≥Q[(Ln·M)∗≥n]−Q[Bc∩{(Ln·M)∗≥n}]> 7 α−α=6α.
Forn≥Nandi=1,...,kn, the random variablesfn,iare bounded in
L^2 (Q)-norm by 2 but inL^0 (Q) they satisfy the lower boundQ[|fn,i|≥1]>
6 α. This will allow us to obtain a lowerL^0 (Q) estimate forfn,i−.Letβ=α^2
andBn,i={fn,i− ≥α}. We will show thatQ[Bn,i]>β.
The martingale property implies that
EQ[fn,i−]=EQ[fn,i+]=
EQ[|fn,i|]
2
3 α.
Therefore asfn,i− is bounded byαoutsideBn,i:
EQ[fn,i− (^1) Bn,i]≥EQ[fn,i−]−α> 2 α.
On the other hand the Cauchy-Schwarz inequality gives
EQ[fn,i− (^1) Bn,i]≤‖fn,i‖L (^2) (Q)Q[Bn,i]
(^12)
≤ 2 Q[Bn,i]
(^12)
.
Both inequalities show thatQ[Bn,i]>α^2 =β.
We now turn toLn·A. BecauseLn·S=Ln·M+Ln·Aand we know
thatLn·Sis small and the negative parts ofLn·Mare big, we can deduce
that positive parts inLn·Aare also big. Let us formalise this idea: from the
definition ofλwe infer that for alli
‖(Ln·S)Tn,i−(Ln·S)Tn,i− 1 ‖L (^2) (Q)≤
2 λ
n^2
.
Tchebycheff’s inequality implies
Q
[
∣
∣(Ln·S)Tn,i−(Ln·S)Tn,i− 1∣
∣≥^2 λ
n]
≤
(
2 λ
n^2) 2
n^2
4 λ^2=n−^2.BecauseQ[((Ln·M)Tn,i−(Ln·S)Tn,i− 1 )−≥α]>βwe necessarily have
Q
[
(Ln·A)Tn,i−(Ln·A)Tn,i− 1 ≥α−^2 nλ]
β−n−^2 and this holds for all
i≤knandn≥N.
We will now construct a strategy that allows us to take profit of thesekn
positive differences. The processLn·Ais of bounded variation. The Hahn de-
composition of this measure, see the discussion preceding Definition 9.2.7, pro-
duces a partition ofR+×Ω in two predictable setsBn+andB−non which this
measure is respectively positive and negative. The processes (Ln (^1) Bn+·A)and
(−Ln (^1) Bn−·A) are therefore increasing. LetRnbe the processLn (^1) B+n∩[[ 0,Tn,kn]].
The process (Rn·A)=(Ln (^1) Bn+∩[[ 0,Tn,kn]]·A)satisfies
(Rn·A)Tn,i−(Rn·A)Tn,i− 1 ≥(Ln·A)Tn,i−(Ln·A)Tn,i− 1