The Mathematics of Arbitrage

72 5 The Kreps-Yan Theorem

We first will make a technical assumption, namely that the processSis
locally bounded, i.e., that there exists a sequence (τn)∞n=1of stopping times,
increasing a.s. to +∞, such that the stopped processesStτn=St∧τnare uni-
formly bounded, for eachn∈N(We refer to Sect. 7.2 below for unexplained
notation). Note that continuous processes — or, more generally, cadlag pro-
cesses with uniformly bounded jumps — are locally bounded. This assumption
will be very convenient for technical reasons. At the end of Chap. 8 we shall
indicate, how to extend this to the general case of processes, which are not
necessarily locally bounded.
We have chosenT=[0,∞[ for the time index set in order to assume full
generality; of course this also covers the case of a compact intervalT=[0,T],
which is relevant in most applications, when assuming thatStis constant, for
t≥T. The use ofT=[0,∞[ as time index set also covers the case of discrete
time (either in its finite versionT={ 0 , 1 ,...,T}, or in its infinite version
T=N). Indeed, it suffices to restrict to processesSwhich are constant on
[n− 1 ,n[, for each natural numbernand only jump at timesn∈T.
We shall always assume that the filtration (Ft)∞t=0satisfies the usual condi-
tions i.e. it is right continuous andF 0 contains all null sets ofF∞.Furthermore
the processShas a.s. cadlag trajectories.
How to define the trading strategiesH, which played a crucial role in the
preceding sections? A very elementary approach, corresponding to the role of
step functions in integration theory, is formalised by the subsequent concept.
The reader will notice that the definitions in this chapter are variants of
a more general situation to be handled in Chap. 7 and later.

Definition 5.1.1.(compare, e.g., [P 90]) For a locally bounded stochastic pro-
cessSwe call anRd-valued processH=(Ht)∞t=0asimple trading strategy
(or, speaking more mathematically, asimple integrand), ifHis of the form

H=

∑n

i=1

hiχ]]τi− 1 ,τi]],

where0=τ 0 ≤τ 1 ≤...≤τn<∞are finite stopping times andhiare
Fτi− 1 -measurable,Rd-valued functions. We then may define, similarly as in
Definition 2.1.4, the stochastic integralH·Sas the stochastic process

(H·S)t=

∑n

i=1

(

hi,Sτi∧t−Sτi− 1 ∧t

)

=

∑n

i=1

∑d

j=1

hji

(

Sτji∧t−Sτji− 1 ∧t

)

, 0 ≤t<∞,

and its terminal value as the random variable

(H·S)∞=

∑n

i=1

(

hi,Sτi−Sτi− 1

)

.