74 5 The Kreps-Yan Theorem
Lemma 5.1.3.LetQbe a probability measure onFwhich is absolutely con-
tinuous w.r. toP. A locally bounded stochastic processSis a local martingale
under a probability measureQiff
EQ[(H·S)∞]=0, (5.1)for each admissible simple trading strategyH.
Proof.Let (τn)∞n=1be a sequence of finitely valued stopping times increasing
P-a.s. to infinity such that eachSτnis bounded.
Supposing that (5.1) holds true for each simple admissible integrand we
have to show that eachSτnis aQ-martingale. In other words, for eachn≥ 1
and each pair of stopping times 0≤σ 1 ≤σ 2 ≤τnwe have to show that
EQ[Sσ 2 |Fσ 1 ]=Sσ 1.This is tantamount to the requirement that for each Rd-valued Fσ 1 -
measurable, bounded functionhwe have
EQ[(h, Sσ 2 −Sσ 1 )] = 0,which holds true by assumption (5.1). HenceSis a localQ-martingale.
The proof of the converse implication, i.e., that the localQ-martingale
property ofSimplies (5.1) for each admissible integrand is straightforward
(compare Lemma 2.2.6).
For later use we note that the “=” in (5.1) may equivalently be replaced
by “≤”(or“≥”), asHis an admissible simple trading strategy iff−His so.
We define the subspaceKsimpleofL∞(Ω,F,P) of contingent claims, avail-
able at price zero via an admissible simple trading strategy, by
Ksimple={(H·S)∞|Hsimple, admissible}and byCsimplethe convex cone inL∞(Ω,F,P) of contingent claims dominated
by somef∈K
Csimple=Ksimple−L∞+={
f−k|f∈Ksimple,f∈L∞,k≥ 0}
.
Definition 5.1.4.Ssatisfies the no-arbitrage condition (NAsimple)withre-
spect to simple integrands, ifKsimple∩L∞+(Ω,F,P)={ 0 }(or, equivalently,
Csimple∩L∞+(Ω,F,P)={ 0 }).
As the following lemma shows there is no difference between an arbitrage
opportunity for simple admissible strategies and arbitrage opportunities for
admissible “buy and hold” strategies.
Lemma 5.1.5.Let the processH be a simple admissible strategy defined as
H=
∑n
i=1hiχ]]τi− 1 ,τi]]and yielding an arbitrage opportunity. In other words
we have(H·S)∞≥ 0 a.s. andP[(H·S)∞>0]> 0. Then there is a buy
and hold strategyK=h (^1) ]]σ 1 ,σ 2 ]]such thatK is admissible andK yields an
arbitrage opportunity.