The Mathematics of Arbitrage

192 9 Fundamental Theorem of Asset Pricing

By Lemma 9.8.1 there aregn∈conv{fn,fn+1...}such thatgn→ga.e.
whereg:Ω→[0,∞]. AlsoP[g>0]>0. Ifgn=λn 0 fn+···+λnkfn+kis the
convex combination, let us putKn=λn 0 Hn+···+λnkHn+k. Clearly

(a)‖(Kn·S)−∞‖→0and
(b) (Kn·S)∞ → g:Ω→[0,∞].

SinceP[g>0]>0, this is a contradiction to(NFLVR)with simple inte-
grands.

Lemma 9.7.4.The set

G=

{ n

∑

k=0

(STk+1−STk)^2

∣

∣

∣

∣

∣

0 ≤T 0 ≤T 1 ≤...≤Tn+1<∞stopping times

}

is bounded inL^0.

Proof.For 0≤T 0 ≤T 1 ≤...≤Tn+1<∞stopping times put:

H=− 2

∑n

k=0

STk (^1) ]]Tk,Tk+1]].
Because|S|≤1wehavethatHis bounded by 2. Also

(H·S)∞=

∑n

k=0

(STk+1−STk)^2 −ST^2 n+1+ST^20

and hence (H·S)∞≥−1. The same calculation applied to the sequence of
stopping times min(T 1 ,t),...,min(Tn,t) yields (H·S)t≥−1 and therefore
sup 0 ≤t(H·S)−t ≤1. The preceding lemma now implies thatGis bounded in
L^0.

Proof of Theorem 9.7.2.We have to show that ifHnis a sequence of simple
predictable processes such thatHn→0 uniformly overR+×Ω, then (Hn·
S)∞→0 in probability. By the Bichteler-Dellacherie theorem this implies the
classical definition of a semi-martingale. (In [P 90] this property is used as the
definition of a semi-martingale). It is of course, sufficient to show that the
sequence (Hn·S)∞is bounded inL^0. If this were not true then there would
exist a subsequence of simple integrands, still denoted by (Hn)n≥ 1 , such that

(a)Hn→0 uniformly overR+×Ω;
(b)P[(Hn·S)∞≥n]≥ε>0.
(c) EachHncan be written as

Hn=

∑Nn

k=0

fkn (^1) ]]Tkn,Tkn+1]]