et
be a probability space. It is natural, after results
(
Incomplete market ask
) and
(
Incomplete market bid
), to consider a
mapping
where the
is some set of absolutely continuous with respect to
measures.
Proposition
The coherent measure has the following properties:
Proposition
Proposition
1. (Subadditivity)
.
2. (Monotonicity)
.
3. (Positive homogeneity)
for
.
4. (Translation invariance).
.
5. (Fatou property).
in
.
Note that if
is finite then the representation (
Coherent
measure
) is exactly the formula (
Support
function
). Hence, for the finite
the representation (
Coherent measure
)
completely describes all functions that have the property 1 and 3.
Furthermore, if we introduce a notion of maximal set
(in inclusion sense) for a measure
then, given
and
as in (
Coherent measure
), taking double
convex dual of
reveals that the
is a closure of the convex hull of the original
.
Another description for
is
where the
is the set of all absolutely continuous measures with respect to
.
In the case of general
there is a following representation theorem.
Proposition
A measure
satisfies the conditions 15 above iff there exists a non empty
such that the (
Coherent measure
) holds.
Note that the theorem holds for
.
There reason may be easily understood by looking at derivations around the
formula (
Support function
). To extend those
derivations to the general case we need the duality to the space
in order to make sense of all the scalar products
.
Such duality needs to have reflexivity. The
has such dual space
while
,
endowed with the topology of convergence in probability, does not have such
structure.
