170 9 Fundamental Theorem of Asset Pricing
From now onQwill be fixed. SinceS is bounded it is a special semi-
martingale and its canonical decomposition (with respect toQ) will be de-
noted asS=M+A,whereMis the local martingale part andAis of finite
variation and predictable. The symbolsMandAare from now on reserved
for this decomposition.
The next lemma is crucial in the proof of the main theorem. It is used to
obtain bounds onHn·M. Because we shall need such an estimate also for other
integrands we state it in a more abstract way. Forλ>0, letHλbe the convex
set of 1-admissible integrandsHwith the extra property‖(H·S)∗‖L (^2) (Q)≤λ.
Lemma 9.4.8.Forλ> 0 the set of maximal functions{(H·M)∗|H∈Hλ}
is bounded inL^0 (Q).
Proof.Fixλ>0 and abbreviate the setHλbyH. The semi-martingalesH·S
whereHis inH, are special (with respect toQ) because their maximal func-
tions are inL^2 (Q). Therefore, by Theorem 9.2.2, the canonical decomposition
ofH·Scomes from the decompositionS=M+Ai.e.,H·Mis the local
martingale part ofH·SandH·Ais the predictable part of finite variation.
Because the proof of the lemma is rather lengthy let us roughly sketch the
idea, which is quite simple. IfKnis a sequence inHsuch that (Kn·M)∗is un-
bounded in probability, thenKn·Ais also unbounded and — keeping in mind
thatKn·Ais predictable — using good strategies we might take advantage
of positive gains. This turns out to be possible as the calculations will show
that the gains coming from the predictable partAin the long run overwhelm
the possible losses coming from the martingale partM. This will contradict
the property(NFLVR). Very roughly speaking, the gains coming from the
predictable partAadd up proportionally in time, whereas the expected losses
from the martingale part only add up proportionally to
√
time. These phe-
nomena are due to the orthogonality of martingale differences, whereas the
variation of the predictable part over the union of two intervals is the sum of
the variations over each interval.
Let us now turn to the technicalities. If{(H·M)∗|H∈H}in not bounded
inL^0 , there is a sequence (Kn)n≥ 1 inH,aswellasα>0, such that for all
n≥1wehaveQ[(Kn·M)∗>n^3 ]> 8 α.FromtheL^2 bound on (H·S)∗and
Tchebycheff’s inequality we deduce thatQ[supt|(Kn·S)t|>n]≤λ
2
n^2 and for
nlarge enough (sayn≥N) this expression is smaller thanα 3 .Foreachnwe
now defineTnas
Tn=inf{
t∣
∣|(Kn·M)t|≥n^3 or|(Kn·S)t|≥n}.If we now define the integrandLn=n^12 Kn (^1) [[ 0,Tn]]we obtain that
(i) Ln·Mare local martingales
(ii) Q[(Ln·M)∗≥n]≥Q[(Kn·M)∗≥n^3 ]−Q[(Kn·S)∗≥n]≥ 8 α−λ
2
n^2 ≥^7 α
for alln≥N.
(iii)Ln·Mis constant afterTn.