12.3 The No-Arbitrage Property and Immediate Arbitrage 239and only if for each predictableRd-processf, such that‖f(t, ω)‖is either 0
or 1 , the relationdV f=0impliesf′dA=0.
Remark 12.2.4.IfSis a semi-martingale with values inRd, then the bracket
[S, S] and (if it exists) also the bracket〈S, S〉define increasing processes with
values in Pos(Rd). The fact that values are taken in Pos(Rd) is a reformulation
of the Kunita-Watanabe inequalities:
|d[Si,Sj]|≤√
d[Si,Si]d[Sj,Sj],
|d〈Si,Sj〉| ≤√
d〈Si,Si〉d〈Sj,Sj〉.12.3 The No-Arbitrage Property
and Immediate Arbitrage
We now turn to the main theme of the paper, a detailed analysis of the notion
ofno-arbitrage. We start with an easy lemma, which turns out to be very
useful. It shows that the general case of an arbitrage may be reduced to two
special kinds of arbitrage.
Lemma 12.3.1.If the cadl ag semi-martingaleS does not satisfy the no-
arbitrage property with respect to general admissible integrands then at least
one of the two following statements holds
(i) There is anS-integrable strategyHand a stopping timeT,P[T<∞]> 0
such thatHis supported by[[T,T+1[[,H·S≥ 0 and(H·S)t> 0 ,for
t>T.
(ii)There is anS-integrable 1 -admissible strategyK,ε> 0 and two stopping
timesT 1 ≤T 2 such thatT 2 <∞on{T 1 <∞},P[T 2 <∞]> 0 ,K=
K (^1) ]]T 1 ,T 2 ]]and(K·S)T 2 ≥εon the set{T 2 <∞}.
Proof.LetSallow arbitrage and letHbe a 1-admissible strategy that produces
arbitrage, i.e., (H·S)∞≥0 with strict inequality on a set of strictly positive
probability. We now distinguish two cases. Either the processH·Sis never
negative or the processH·Sbecomes negative with positive probability. In
the first case letT=inf{t|(H·S)t> 0 }.
Let (θn)∞n=1be a dense in ]0,1[ and letH ̃=
∑∞
n=1^2−nH (^1) [[T,T+θ
ninft(H·S)t<− 2 ε. We thank an anonymous referee for correcting a slip in a pre-
vious version of this paper at this point.
In the second case we first look forε>0 such thatP[inft(H·S)t<− 2 ε]>
- We then defineT 1 asthefirsttimetheprocessH·Sgoes under− 2 ε, i.e.
T 1 =inf{t|(H·S)t<− 2 ε}.On the set{T 1 <∞}we certainly have that the processH·Shas to gain at
least 2ε. Indeed at the end the processH·Sis positive and therefore the time