The Mathematics of Arbitrage

9.3 No Free Lunch with Vanishing Risk 161

the existence ofα>0 such thatP[(Hn·S)∞≥n]>α>0. The sequencefn=
min

( 1

n(H

n·S)∞, 1 )is inC,P[fn=1]>α>0and‖f−
n‖∞≤
1
n. By taking
convex combinations we may takegn∈conv{(fn,fn+1,...}that converge a.s.
tog:Ω→[0,1]. (We can use Lemma 9.8.1, but a simpler argument inL∞
can do the job, compare [S 94, Remark 3.4]). ClearlyE[g]≥αand therefore
P[g>0] =β≥α>0. By Egorov’s theoremgn→guniformly on a set Ω′of

measure at least 1−β 2. The functionshn=min(gn, (^1) Ω′) are still in the setC
andhn→g (^1) Ω′in the norm topology ofL∞.SinceP[g (^1) Ω′>0]≥β 2 >0we
obtain a contradiction to(NFLVR).

Proposition 9.3.2.IfSis a semi-martingale satisfying (NFLVR), then for
each admissibleH the function(H·S)∗=sup 0 ≤t|(H·S)t|is finite almost
everywhere and the set{(H·S)∗|H 1 -admissible}is bounded inL^0.
Proof.If the set is not bounded, we can find a sequence of 1-admissible in-
tegrandsHn, stopping timesTnandα>0 such thatP[Tn<∞]>α> 0
and (Hn·S)Tn>non{Tn<∞}. For each natural numberntaketnlarge
enough so thatα<P[Tn≤tn] and observe that forKn=Hn (^1) [[ 0,min(Tn,tn)]]
we have thatKnis of bounded support andP[(Kn·S)∞>n]>α>0,
a contradiction to Proposition 9.3.1.

We now prove the main result of this section. It extends from [S 94, Propo-
sition 4.2] to the present case of a general semi-martingaleS.
Theorem 9.3.3.IfSis a semi-martingale satisfying (NFLVR), then forH
admissible the limit(H·S)∞= limt→∞(H·S)texists and is finite almost
everywhere.
Proof.We will mimic the proof of the martingale convergence theorem of
Doob. The classical idea of considering upcrossings through an interval [β, γ]
may in mathematical finance be interpreted as the well-known procedure:
“Buy low, sell high”. We may suppose thatH is 1-admissible and hence
(H·S)∗ =sup 0 ≤t|(H·S)t|<∞almost surely by Proposition 9.3.2. We
therefore only have to show that lim inft→∞(H·S)t= lim supt→∞(H·S)t
a.s.. Suppose this were not the case and that P[lim inft→∞(H·S)t <
lim supt→∞(H·S)t]>0. Takeβ<γandα>0sothatP[lim inft→∞(H·S)t<
β<γ<lim supt→∞(H·S)t]>α. We will construct finite stopping times
(Un,Vn)n≥ 1 , such that
(1)U 1 ≤V 1 ≤U 2 ≤V 2 ≤...≤Un≤Vn≤Un+1≤...
(2)Ln=
∑n
k=1H^1 ]]Uk,Vk]]is (1 +β)-admissible
(3)P[(Ln·S)∞>n(γ−β)]>α 2.
The existence of such a sequence clearly violates the conclusion of Propo-
sition 9.3.2 and this will prove the Theorem.
The stopping times are constructed by induction. Take (εn)n≥ 1 strictly
positive and such that the sum

∑

n≥ 1 εn<

α

100 .LetAbe the set defined as