306 14 The FTAP for Unbounded Stochastic Processes
In the general case, i.e. when the processS is not necessarily locally
bounded, the set of admissible integrands might be restricted to the zero inte-
grand, compare Proposition 14.3.7 above. Below we will show that also in this
case the above equality remains valid, at least for positive random variablesg.
This result does not immediately follow from the results in Chap. 11.
Another approach to the problem is to enlarge the concept of admissible
integrands in a similar way as was done in [S 94] and Chap. 15. Here the idea is
to allow for integrandsHthat are such that the processH·Sis controlled from
below by an appropriate functionw, the so-calledw-admissible integrands.
We will generalise the above duality equality to the setting of such integrands
and we will see that even in the locally bounded case this generalisation yields
some new results.
If we want to control a processH·Sfrom below by a functionwthen, of
course, the problem is thatwcannot be too big, as this would allow doubling
strategies and therefore arbitrage. Alsowcannot be too small because this
could imply that the only such integrandHis the zero integrand. This idea
is made precise in the following definitions ofw-admissible integrands and of
feasible weight functions.
Definition 14.5.1.Ifw≥ 1 is a random variable, if there isQ 0 ∈Meσ
such thatEQ 0 [w]<∞,ifais a non-negative number, then we say that the
integrandHis(a, w)-admissible if for each elementQ∈Meσand eacht≥ 0 ,
we have(H·S)t≥−aEQ[w|Ft]. We simply say thatHisw-admissible if
His(a, w)-admissible for some non-negativea.
Remark 14.5.2.If we putw= 1 we again find the usual concept of admissi-
ble integrands. Of course, we could have defined the concept ofw-admissible
integrands for general non-negative functionsw. We, however, required that
w≥1, so that the admissible integrands become automaticallyw-admissible.
The idea in fact is to allow unbounded functionswand therefore there seems
to be no gain in introducing functionswthat are too small. Requiring that
w≥1 is by no means a restriction compared to the seemingly more general
requirement ess inf(w)>0.
Remark 14.5.3.The present notion of admissible integrand is more suitable
for our purposes than the one introduced in Chap. 15.
The next lemma, based on a stability property of the setMeσ,showsthat
in the inequality (H·S)t ≥−EQ[w|Ft], it does not harm to restrict to
elementsQ∈Meσsuch thatEQ[w]<∞.
Lemma 14.5.4.Letw≥ 0 be such thatEQ 0 [w]<∞for someQ 0 ∈Meσ.
Suppose that ,for someQ∈Meσ,t≥ 0 and some real constantk,theset
A={EQ[w|Ft]≤k}has positive probability, then there isQ 1 ∈Meσsuch
that we haveEQ 1 [w|Ft]=EQ[w|Ft]a.s. on the setAandEQ 1 [w]<∞.