9.2 Definitions and Preliminary Results 159Throughout the paper, with the exception of Sect. 9.7,Swill be a fixed
semi-martingale. As mentioned in the introductionSrepresents the discounted
price of a financial asset.
Definition 9.2.7.Let a be a positive real number. AnS-integrable predictable
processHis calleda-admissibleifH 0 =0and(H·S)≥−a(i.e. for all
t≥0:(H·S)t≥−aalmost everywhere).H is called admissible if it is
admissible for somea∈R+.
Given the semi-martingaleSwe denote, in a similar way as in [Str 90], by
K 0 the convex cone inL^0 , formed by the functions
K 0 ={
(H·S)∞
∣
∣
∣Hadmissible and (H·S)∞= limt→∞(H·S)texists a.s.}
.
ByC 0 we denote the cone of functions dominated by elements ofK 0 i.e.
C 0 =K 0 −L^0 +.WithCandKwe denote the corresponding intersections with
the spaceL∞of bounded functionsK=K 0 ∩L∞andC=C 0 ∩L∞.ByC
we denote the closure ofCwith respect to the norm topology ofL∞and by
C
∗
we denote the weak-star-closure ofC.Definition 9.2.8.We say that the semi-martingaleSsatisfies the condition
(i) no-arbitrage(NA) ifC∩L∞+={ 0 }
(ii)no free lunch with vanishing risk(NFLVR) ifC∩L∞+={ 0 }.
It is clear that(ii)implies(i). The no-arbitrage property(NA)is equiv-
alent toK 0 ∩L^0 +={ 0 }and has an obvious interpretation: there should be
no possibility of obtaining a positive profit by trading alone (according to an
admissible strategy): it is impossible to make something out of nothing with-
out risk. It is well-known that in general the notion(NA)is too restrictive to
imply the existence of an equivalent martingale measure forS, see Sect. 9.7.
Compare also to the results in [DMW 90] and [S 94, Remark 4.11].
The notion(NFLVR)is a slight generalisation of(NA).If(NFLVR)is
not satisfied then there is af 0 inL∞+ not identically 0, as well as a sequence
(fn)n≥ 1 of elements inC, tending almost surely tof 0 such that for allnwe
have thatfn≥f 0 −^1 n. In particular we havefn≥−n^1. In economic terms
this amounts to almost the same thing as(NA), as the risk of the trading
strategies becomes arbitrarily small. See also Proposition 9.3.7 below.
We emphasize that the setCand hence the properties(NA)and(NFLVR)
are defined usinggeneral admissible predictable processesH.Thisisamore
general definition than the one usually taken in the literature and used by
the authors in previous papers (see [S 94, D 92]). These classical concepts
were defined using simple integrands or/and integrands with bounded sup-
port. In these cases we will say thatSsatisfies(NA) for simple integrands,
(NFLVR) for integrands with bounded support,etc.Thesenotionswillreap-
pear in Sect. 9.7, where we will emphasize on the differences between these
notions.