8.3 Sigma-martingales and the Non-locally Bounded Case 143The subtle issue is the implication (ii) ⇒(i) of Theorem 8.3.1. The
first good news is that for the validity of Theorem 8.2.2, which asserts that
(NFLVR)implies the weak-star-closureness of the coneCdefined in (8.2), the
assumption of local boundedness is not needed: The proof of Theorem 8.2.2
does not use the local boundedness ofSand works in full generality. Hence
we may apply the Kreps-Yan Theorem 5.2.2 also under the assumptions of
Theorem 8.3.1 (ii) to find a probability measureQ∼Psuch thatEQ[f]≤0,
for eachf∈C. The set of these probability measures, i.e., the probability
measuresQ Psuch thatQ|C≤0isdenotedbyMesin Proposition 14.4.5.
Y.M. Kabanov [K 97] proposed the name “separating measures” for this set.
In the case ofSbeing (locally) bounded we have seen that, for a sepa-
rating measureQ, the semi-martingaleSis a (local)Q-martingale. We have
used this rather obvious fact (Lemma 5.1.3) to deduce Theorem 8.2.1 from
Theorem 8.2.2 above. But in the present case of a general semi-martingale
Sthis implication breaks down. The subsequent easy example illustrates the
situation.
Example 8.3.3.LetXbe a normally distributed real random variable with
meanμ∈Rand varianceσ^2 >0. Define the processS=(St)t≥ 0 by
St={
0 , for 0≤t< 1 ,
X, fort≥ 1.which we consider under its natural filtration (Ft)t≥ 0.
Observe that, for every admissible integrandH,wehaveH·S≡0. Ex-
pressing this property in prose: the onlyadmissibleway, i.e., with uniformly
bounded risk, to bet on the random variableX, is the zero bet. This follows
from the fact thatXis unbounded from below as well as from above.
Hence the coneK defined in (8.1) is reduced to zero, and the coneC
defined in (8.2) equals the negative orthantL∞−(Ω,F∞,P). It follows that
everyprobability measureQ Pis a separating measure. ButSis a sigma-
martingale w.r. toQ, iff it is a martingale w.r. toQ,iffEQ[X]=0.
The example shows that, if we allowSto have unbounded jumps, the sep-
arating measuresQare not necessarily sigma-martingale measures any more.
The important observation which will eventually prove Theorem 8.3.1 is the
following: the setMeσof measuresQ∼Psuch thatSis a sigma-martingale
underQis dense in the set of separating measuresMes(see Proposition 14.4.5)
which we restate here for convenience). Of course, this density assertion is a
more precise information than the assertion of Theorem 8.3.1 thatMeσis not
empty.
Proposition 8.3.4.Denote byMesthe set of probability measures
Mes={Q|Q∼Pand for eachf∈C:EQ[f]≤ 0 }.IfSsatisfies (NFLVR), then