The Representation of Natural Risk Statistics on the Space of Bounded Sequences by Coherent Risk Statistics

In this paper, we present a representation theorem of natural risk statistics on the space of infinite and bounded real sequences via coherent risk statistics.


Introduction
Heyde et al. [4].(2006) proposed the notion of natural risk statistics on the vector space of finite samples, instead of random variables as in the classical approach, first introduced in [2] by Artzner et al.(1999).By weakening the subadditivity axiom, these measures of risk are consistent with industrial practice and have some interesting features which make them consistent with scenario analysis used in practice.Assa et al. [3].(2010) presented a generalization of natural risk statistics to the space of infinite and bounded sequences l ∞ , motivated by their work, we introduce an alternative proof of the main result in [3] which states that a weak-star lower semicontinuous risk statistics on l ∞ is natural if and only if, it can be represented as a supremum over a family of convex combinations of order statistics.

Preliminary Notes
In this section we recall some definitions and results regarding coherent and natural risk statistics.
, where E is a locally convex topological vector space, the Fenchel-Legendre transform of f is the function 1ρ is invariant by translation and positively homogeneous: The next theorem is the well-known representation theorem in [2] of coherent risk measures, stated here in l ∞ , where ) Let ρ be a function on l ∞ .The function ρ is a coherent risk statistics if and only if, ρ(X) = sup w∈W w, X ∀X ∈ l ∞ where we have Now, we present the definition of a natural risk statistics on l ∞ , as it was extended in [3], from IR n to l ∞ .Definition 2.6 ([3]) A function ρ : l ∞ → IR is a natural risk statistics, if: 1ρ is invariant by translation and positively homogeneous:

Main Results
We define the set .We consider for every element X ∈ l ∞ the set s X = {x 0 = lim sup X} ∪ {x i : x i ≥ x 0 } with multiplicity (s X contains all the copies of the components of X which are larger than x 0 ) and the element (Λ(X)) i = max i s X , (Λ(X)) i−1 > x 0 x 0 , o.w. of l ∞ where max i s X denotes the i-th biggest number of s X .
Lemma 3.1 Let ρ : l ∞ → IR be a natural risk statistics and ρ : l ∞ → IR be a coherent risk statistics, we have ρ Remark 3.2 A real-valued coherent or natural risk statistics on l ∞ is Lipschitz continuous and convex.Hence, it is continuous in the strong topology which is the topology induced by the supremum norm, then it is weak lower semicontinuous, but not necessary weak-star lower semicontinuous on l ∞ .Now, we are in a position to state a relationship between coherent risk statistics and natural risk statistics.Theorem 3.3 A function ρ : l ∞ → IR is a natural risk statistics if and only if, there exists a coherent risk statistics ρ : -Considering the theorem 2.3 and the remark 3.2, ρ could be represented as follows: ρ(X) = sup v∈V v, X where V is a convex and closed subset of the -We construct ρ as for every element X of l ∞ ρ(X) = sup v∈V v, X where V = {X ∈ l ∞ : X ≥ 0} ∩ V .The set V is non empty because ρ(X) < ∞ ∀X ∈ D.Moreover, by compactness of V the supremum of v, X over V is achieved ∀X ∈ l ∞ , then ρ is well defined.ρ is convex, as a maximum of a collection of convex functions sup v∈V v, X .
-Now we will show that ρ(X) = ρ(X) ∀X ∈ D .If X ∈ F r(D), we consider the sequence X k ∈ D • ) such that lim k→+∞ X k = X in the usual supremum norm of l ∞ , by the fact that ρ is continuous and that V is compact, we obtain ρ(X) = lim • Suppose that ρ is a coherent risk statistics.It is clear that ∀(λ, c) ∈ IR + × IR Λ(λX + c) = λΛ(X) + c, which means that Λ is invariant by translation and positively homogeneous, also Λ is naturally permutation invariant.if X ≤ Y then lim sup X = x 0 ≤ lim sup Y = y 0 and max i s X ≤ max i s Y ∀i = 1, 2, 3, ..., so, (Λ(X)) i ≤ (Λ(Y )) i which means that Λ is increasing.
The next corollary presents the main theorem stated in [3].
Corollary 3.4 Let ρ be a real-valued function on l ∞ .ρ is a weak-star lower semicontinuous natural risk statistics if and only if, ρ(X) = sup w∈W w, Λ(X) ∀X ∈ l ∞ where W ⊆ {w ∈ l ∞ : w ≥ 0, ∞ i=1 w i = 1} is a closed, convex set of l 1 .
Proof.Combining the theorem 2.5, the theorem 3.3 and the remark 3.2 we obtain the result.
the interior of D and Fr(D) its frontier or boundary, χ D is the indicator function of D, i.e., χ D (X) = 0 , X ∈ D +∞ , o.w.