5.1 A General Framework 73Throughout this chapter we callHadmissibleif, in addition, the stopped
processSτnand the functionsh 1 ,...,hnare uniformly bounded.
This definition is a well-known building block for developing a stochastic
integration theory (see, e.g., [P 90]). It has a clear economic interpretation in
the present context: at timeτi− 1 an investor decides to adjust her portfolio in
the assetsS^1 ,...,Sj,...,Sdby fixing her investment in assetSjto behji(ω)
units; we allowhjito have arbitrary sign (holding a negative quantity means
borrowing or “going short”), and to depend on the random elementωin an
Fτi− 1 -measurable way, i.e., using the information available at timeτi− 1 .The
funds for adjusting the portfolio in this way are simply financed by taking the
appropriate amount from (or putting into) the “cash box”, modelled by the
num ́eraireS^0 ≡1 (compare Sect. 2.1). The investor holds this portfolio fixed
up to timeτi. During this period the value of the risky stocksSj,j=1,...,d,
changed fromSjτi− 1 (ω)toSτji(ω) resulting in a total gain (or loss) given by the
random variable (hi,Sτi−Sτi− 1 ). At timeτi,fori<n, the investor readjusts
the portfolio and at timeτnshe liquidates the portfolio, i.e., converts all her
positions into the num ́eraire. Hence the random variable (H·S)τn=(H·S)∞
models the total gain (in units of the num ́eraireS 0 ) which she finally, i.e., at
timeτn, obtained by adhering to the strategyH; the process (H·S)tmodels
the gains accumulated up to timet.
The concept of a simple trading strategy is designed in a purely algebraic
way, avoiding limiting procedures in order to be on safe grounds.
The next crucial ingredient in developing the theory is the proper gener-
alisation of the notion of an equivalent martingale measure.
Definition 5.1.2.AprobabilitymeasureQonFwhich is equivalent (resp.
absolutely continuous with respect) toPis called an equivalent(resp.abso-
lutely continuous)local martingale measure,ifSis a local martingale under
Q.
We denote byMe(S)(resp.Ma(S)) the family of all such measures, and
say thatSsatisfies the condition of the existence of an equivalent local mar-
tingale measure (EMM) ifMe(S)=∅.
Note that, by our assumption of local boundedness ofS,wehavethatS
is a localQ-martingale iffSτis aQ-martingale for each stopping timeτsuch
thatSτis uniformly bounded (compare Chap. 7).
Why did we use the notion of a local martingale instead of the more famil-
iar notion of a martingale? This is simplythe natural degree of generality. The
subsequent straightforward lemma (whose proof is an obvious consequence of
the chosen concepts and left to the reader) shows that this notion does the
job just as well as the notion of a martingale for the present purpose of no-
arbitrage theory. Last but not least, the restriction to the notion of martingale
measures would lead to a different version of the general version of the funda-
mental theorem of asset pricing (Theorem 2.2.7 above), as may be seen from
easy examples (see [DS 94a], compare also [Y 05]).