13.1 Introduction 253use finitely additive measures in order to describe the closure of the space of
bounded workable contingent claims.
Part of the results were obtained when the first named author was visiting
the University of Tsukuba in January 1994 and when the second author was
visiting the University of Tokyo in January 1995. Discussions with Professor
Kusuoka and Professor Shirakawa are gratefully acknowledged.
The setup in this paper is the usual setup in mathematical finance. A prob-
ability space (Ω,F,P) with a filtration (Ft) 0 ≤tis given. The time set is sup-
posed to beR+, the other cases, e.g. finite time interval or discrete time
set, can easily be imbedded in our more general approach. The filtration is
assumed to satisfy the “usual conditions”, i.e. it is right continuous andF 0
contains all null sets ofF.
A price processS, describing the evolution of the discounted price ofd
assets, is defined onR+×Ω and takes values inRd. We assume that the
processSis locally bounded, e.g. continuous. As shown under a wide range
of hypothesis, the assumption thatSis a semi-martingale follows from arbi-
trage considerations, see Chap. 9 and references given there. We will therefore
assume that the processSis a locally bounded semi-martingale. In order to
avoid cumbersome notation and definitions, we will always suppose that mea-
sures are absolutely continuous with respect toP. Stochastic integration is
used to describe outcomes of investment strategies. When dealing with more
dimensional processes it is understood that vector stochastic integration is
used. We refer to Protter [P 90] and Jacod [J 79] for details on these matters.
Definition 13.1.1.AnRd-valued predictable processHis calleda-admissible
if it isS-integrable, ifH 0 =0, if the stochastic integral satisfiesH·S≥−a
and if(H·S)∞= limt→∞(H·S)texists a.s.. A predictable processHis called
admissible if it isa-admissible for somea.
Remark 13.1.2.We explicitly required thatH 0 = 0 in order to avoid the
contribution of the integral at zero.
The following notations will be used:K={(H·S)∞|His admissible}
Ka={(H·S)∞|His a-admissible}
C 0 =K−L^0 +
C=C 0 ∩L∞.The basic Theorem 9.1.1 above uses the concept of no free lunch with vanishing
risk,(NFLVR)for short. This is a rather weak hypothesis of no-arbitrage
type and it is stated in terms ofL∞-convergence. The(NFLVR)property is
therefore independent of the choice of the underlying probability measure, i.e.
it does not change if we replacePby an equivalent probability measureQ.
Only the class of negligible sets comes into play. We also recall the definition
of the property of no-arbitrage,(NA)for short.