The Mathematics of Arbitrage

9.7 Simple Integrands 191

particular this theorem shows that in the Main Theorem 9.1.1, the hypothesis
that the price process is a semi-martingale is not a big restriction. The theorem
is a version of [AS 93, Theorem 8]. The proof follows the same lines but control
inL^2 -norm is replaced by other means. The theorem only uses conditions that
are invariant under the equivalent changes of measure. The context of the
following theorem is therefore more natural than the same theorem stated in
anL^2 -environment. We, however, pay a price by requiring the processSto
be locally bounded. A counter-example will show that the local boundedness
cannot be dropped.

Theorem 9.7.2.LetS:R+×Ω:→Rbe an adapted cadl ag process. IfSis
locally bounded and satisfies the no free lunch with vanishing risk property for
simple integrands, thenSis a semi-martingale.

The proof requires some intermediate results that have their own merit.
SinceSis locally bounded there is an increasing sequence of stopping times
(τn)n≥ 1 such that each stopped processSτn is bounded andτn→∞a.s..
To prove thatSis a semi-martingale it is sufficient to prove that eachSτn
is a semi-martingale. We therefore may and do suppose thatSis bounded.
To simplify notation we suppose that|S|≤1. In the following lemmas it is
always assumed thatSsatisfies(NFLVR)for simple integrands.

Lemma 9.7.3.Under the assumptions of Theorem 9.7.2, letHbe a family
of simple predictable integrands each bounded by 1 , i.e.|Ht(ω)|≤ 1 for allt
andω∈Ω.If
{
sup 0 ≤t(H·S)−t

∣

∣H∈H

}

is bounded inL^0 ,then

{

sup 0 ≤t(H·S)+t

∣

∣H∈H} is also bounded inL^0.

Proof.Suppose that the set{sup 0 ≤t(H·S)+t |H∈H}is not bounded in L^0.
This implies the existence of a sequencecn→∞,Hn∈Handε>0 such that
P[sup 0 ≤t(H·S)+t >cn]>ε.TakeKsuch thatP[sup 0 ≤t(H·S)−t <−K]<ε 2
for allnand allH∈Hand define the stopping times

T

′

n=inf{t|(H

n·S)t≥cn},

Un=inf{t|(Hn·S)t<−K}.

Clearly (Hn·S)t≥−K−2on[[0,Un]] s i n c e e a c hHnis bounded by 1 and
|S|≤1. TakeTn=min(T

′

n,Un) and observe that

P[(Hn·S)Tn≥cn, sup

0 ≤t≤Tn

(Hn·S)t≤K+2]≥ε 2.

Take nowδn→0sothatδncn→∞and remark that

(a) (δnHn (^1) [[ 0,Tn]]·S)−∞≤δn(K+2)
(b)fn=(δnHn (^1) [[ 0,Tn]]·S)∞satisfiesP[fn≥δncn]≥ε 2.