Some classes of multivariate risk measures 65We now introduce the concept ofFr ́ echet space:Rdenotes theFr ́ echet space
given the margins, that isR(F 1 ,F 2 )is the class of all bivariate distributions with
given marginsF 1 ,F 2. The lower Frechet bound ́ XofXis defined byFX(t):=
max{F 1 (t 1 )+F 2 (t 2 )− 1 , 0 }and the upper Fr ́echet bound ofX,X,isdefinedby
FX(t):=mini{Fi(ti)},wheret=(t 1 ,t 2 )∈R^2 andi= 1 ,2. The following theorems
summon up some known results about stochastic orders. For a more interested reader,
we cite [5].
Theorem 1.LetX,Ybe bivariate random variables, whereX,Y∈R(F 1 ,F 2 ). Then:
XUOY ⇔ YLOX ⇔ XSMY ⇔ XCY.
This result is no longer true when multivariate random variables are considered with
n≥3.
Theorem 2.LetX,Ybe bivariate random variables inR(F 1 ,F 2 ). The following
conditions are equivalent:
i) XSMY;
ii) E[f(X)]≤E[f(Y)]for every increasing supermodular function f ;
iii) E[f 1 (X 1 )f 2 (X 2 )]≤E[f 1 (Y 1 )f 2 (Y 2 )]for all increasing functions f 1 ,f 2 ;
iv)πX(t)≤πY(t) ∀t∈R^2.
3 Multivariate distorted risk measures
Distorted probabilities have been developed in the theory of risk to consider the
hypothesis that the original probability is not adequate to describe the distribution
(for example to protect us against some events). These probabilities generate new
risk measures, called distorted risk measures, see for instance [4, 12, 13].
In this section we try to deepen our knowledge about distorted risk measures in
the multidimensional case. Something about this topic is discussed in [9], but here
there is not a representation through complete mathematical results.
We can define the distortion risk measure in the multivariate case as:
Definition 3.Given a distortion g, which is a non-decreasing function such that g:
[0,1]→[0,1], with g( 0 )= 0 and g( 1 )= 1 , a vector distorted risk measure is the
functional: Rg[X]=
∫+∞
0 ...
∫+∞
0 g(
SX(x))
dx 1 ...dxn.We note that the functiong
(
SX(x))
:Rn+→[0,1] is non-increasing ineach com-
ponent.
Proposition 1.The properties of the multivariate distorted risk measures are the
following: P1-P6, and P7, P8 if g is concave.
Proof.P1 and P2 follow immediately from Definition 3, P3 follows recalling that
SX+a(t)=SX(t−a), P4 is a consequence of the fact thatScX(t)=SX
(t
c)
,P5
follows from the relationship between multivariate stochastic orders and P6 is given
by
∫b 1
0 ...∫bn
0 g(^1 )dtn...dt^1 =bn...b^1 =
b ̄. P7 follows from this definition of