174 9 Fundamental Theorem of Asset Pricing
From Q[
(Rn·A)Tn,i−(Rn·A)Tn,i− 1 ≥α−^2 nλ]
β−n−^2 and from
Corollary 9.8.7 we deduce that
Q
[
(Rn·A)Tn,kn≥
kn
2(
α−
2 λ
n)
(
β−n−^2)
]
>
β−n−^2
2.
It follows thatQ
[
(Vn·A)Tn,kn≥1
2
(
α−2 λ
n)
(
β−n−^2)
]
>
β−n−^2
2
−Q[Un<∞]or
Q[
(Vn·A)∞≥(
α
2−
λ
n)
(
β−n−^2)
]
>
β−n−^2
2− 64 α1
√
n.
Since(α
2 −λ
n)
(β−n−^2 ) tends toγ=αβ 2 we obtain that fornlarge enough,
sayn≥N′
Q[
(Vn·A)∞≥
γ
2]
>
β
4.
Let us now look at (Vn·S)∞=(Vn·M)∞+(Vn·A)∞. The first term
(Vn·M)∞tends to zero inL^2 (Q). Indeed
‖(Vn·M)∞‖L (^2) (Q)≤
1
kn∥
∥(Rn·M)Tn,k
n∥
∥
L^2 (Q)≤^2
1
√
kn→ 0.
The second term satisfiesQ[
(Vn·A)∞>γ 2]
β 4.
Tchebycheff’s inequality therefore implies that fornlarge enough, say
n≥N′′we have
Q
[
(Vn·S)∞>γ
4]
≥
β
4−Q
[
(Rn·M)Tn,kn>γ
4]
≥
β
8.
The functionsgn=(Vn·S)∞have their negative parts going to zero in
the norm ofL∞. This is a contradiction to Corollary 9.3.8.
The next step in the proof is to obtain convex combinations Ln ∈
conv{Hn;n≥ 1 }so that the local martingalesLn·Mconverge in the semi-
martingale topology. lf we knew that the elementsHn·Mwere bounded in
L^2 (Q) then we could proceed as follows: by taking convex combinations the
elementsHncan be replaced by elementsLnsuch thatLn·Mconverge in the
L^2 (Q)-topology, whence in the semi-martingale topology. Afterwards we then
should concentrate on the processesLn·A. Unfortunately we do not dispose
of such anL^2 (Q)-bound but only aL^0 -bound and a slightly more precise
information given by the preceding lemma. It suggests that we should stop
the local martingalesHn·Mwhen they cross the levelc>0, apply Corol-
lary 9.2.4 to control the final jumps inL^2 (Q) and apply someL^2 -argument
on the so obtainedL^2 -bounded martingales. Afterwards we should take care
of the remaining parts and letctend to∞. Again the idea is simpler than