Véronique Maume-Deschamps
email: veronique.maume@univ-lyon1.fr

AND KHALIL SAID Didier Rullière
email: didier.rulliere@univ-lyon1.fr

Khalil Said
email: khalil.said@universite-lyon.fr

MULTIVARIATE EXTENSIONS OF EXPECTILES RISK MEASURES

Keywords: Mathematics Subject Classification. 62H00, 62P05, 91B30 Multivariate risk measures, Solvency 2, Risk management, Risk theory, Dependence modeling, Capital allocation, Multivariate expectiles, Elicitability, Coherence properties, Stochastic approximation, Copulas

published or not. The documents may come

Introduction

The risk measure theory has a rich literature in continuous updating. The mathematical construction of different indicators and methods used in risk management has to be in accordance with the professional practices. In this spirit, the notion of coherence is particularly important in the literature of actuarial science. The coherence of univariate risk measures was introduced in the famous paper [START_REF] Artzner | Coherent measures of risk[END_REF], a generalization to set-valued risk measures was presented in [START_REF] Jouini | Vector-valued coherent risk measures[END_REF]. The different axioms chosen to characterize the measures coherence are justified by the economic importance of the properties they present in practice. From that, the coherence is not limited only to these axioms, but concerns all desirable properties from practitioners' point of view. [START_REF] Gneiting | Making and Evaluating Point Forecasts[END_REF] [START_REF] Gneiting | Making and Evaluating Point Forecasts[END_REF] has raised an important issue concerning the statistical coherence of the usual risk measures. Elicitability, according to him, is a natural property that must be satisfied by a measure to ensure the possibility of implementing backtesting procedures, which are necessary from a practical point of view. This work has implicitly challenged the notion of coherence, especially for measures that are not elicitable even if they are coherent. Several studies were presented recently on the subject. Some analyze this property in the context of risk measures, others are dedicated to finding a characterization of the measures that are both coherent and elicitable. The expectiles have therefore appeared in the literature as the only law invariant risk measures that meet this need, they are elicitable by construction and coherent for a threshold level range.

The concept of elicitability was studied in [START_REF] Ziegel | Coherence and elicitability[END_REF]. It is also examined in a purely mathematical framework in [START_REF] Steinwart | Elicitation and Identification of Properties[END_REF]. The elicitability of risk measures is analyzed by [START_REF] Bellini | On elicitable risk measures[END_REF] [START_REF] Bellini | On elicitable risk measures[END_REF], and in [START_REF] Wang | Elicitable distortion risk measures: A concise proof[END_REF]. Expectiles, as risk measures, were the subject of the works presented in [START_REF] Emmer | What is the best risk measure in practice? A comparison of standard measures[END_REF], [START_REF] Bellini | Generalized quantiles as risk measures[END_REF], and [START_REF] Bellini | Risk management with expectiles[END_REF].

The statistical importance of the elicitability property invites us to reconsider the mathematical construction of some risk measures. Indeed, risk management should be naturally based on a minimization of certain risk quantity or a score, defined according to user needs. From this point of view, it seems useful to integrate this vision since the choice of risk measures, and also to take it in account by insurers in decision making related to internal risk management, like capital allocation. In this paper, we present a construction of a new family of vector-valued risk measures.

It is firstly characterized by its elicitability, and then justified by its economic interpretations. The constructed measures are vectors of the same size as the risks. The literature presents a limited number of such measures, we cite as examples, the multidimensional VaR and CVaR introduced in [START_REF] Di Bernardino | Modélisation de la dépendance et mesures de risque multidimensionnelles[END_REF], and studied in [START_REF] Cousin | On multivariate extensions of value-at-risk[END_REF], [START_REF] Cousin | On multivariate extensions of conditional-tailexpectation[END_REF] and [START_REF] Di Bernardino | On multivariate extensions of the conditional value-at-risk measure[END_REF]. In practice, such measures can be used in risk allocation and for the measurement of systemic risk. There is also noteworthy contributions on risk measures for risk vectors including axiomatic characterization, studying coherence properties and also including concrete classes of examples in particular versions of multivariate VaR and CVaR, see [START_REF] Burgert | Consistent risk measures for portfolio vectors[END_REF], [START_REF] Rüschendorf | Law invariant convex risk measures for portfolio vectors[END_REF], [START_REF] Rüschendorf | Worst case portfolio vectors and diversification effects[END_REF] or [START_REF] Ekeland | Law invariant risk measures on L ∞ (R d )[END_REF]. In these works, the proposed multivariate risk measures remain real valued since they assign to each random vector of R d a real value. In our context, the multivariate extensions are R d valued, which means our risk measures give a risk level for each marginal, function of the overall risk represented by the random vector. The measures in this case, can be seen as a risk allocation. The aim of our construction is first to obtain measures that can be interpreted as a vector of solvency thresholds. We also want them to verify some coherence properties, especially elicitability, so we proposed a multivariate extension of expectile risk measures that are both elicitable and coherent in dimension 1.

The proposed measures, in this paper, are in general minimizers of strictly convex functions. Therefore, it is possible to use stochastic approximation algorithms to determine the minimum which is the desired measure. We shall use a version of Robbins-Monro's algorithm (1951) ) [START_REF] Robbins | A stochastic approximation method[END_REF] based on its multidimensional form proposed in [START_REF] Blum | Multidimensional stochastic approximation methods[END_REF]. This method allows approximating our measures in the general case, with quite satisfactory convergence except for the asymptotic levels where the number of observations is limited. We finish this work by a first introductory result on the asymptotic multivariate expectile. This paper is organized as follows. In a first section, we present the concept of multivariate elicitability that will constitute the basis of our multivariate extensions of expectiles. Section 2 is a presentation of some constructions of multivariate expectiles. We introduce a generalization of expectiles in higher dimension using norms. We examine in details two expectiles families, euclidean expectiles and matrix expectiles. We analyze the economic interpretation of the obtained measures, and we discuss what could be the best construction. We then focus on matrix expectiles. We study the different coherence properties that are satisfied by these measures in Section 3. In Section 4, using the Robbins-Monro's stochastic optimization algorithm, we present an approximation method of multivariate expectile in the general case. We give some numerical illustrations of this method. The last section is an introduction to the study of the asymptotic behavior of multivariate expectiles.

Elicitable risk measures

Since the publication of Gneiting's work, the elicitability notion, coming originally from the decision theory, occupies more importance in the new literature and research studies on the coherence properties of risk measures. In this section, we present a definition of elicitability. We analyze its importance as a property in a context of risk measures and we examine its possible extension in a multivariate case. The aim of this section is to present a general framework that will be the basis of a construction of a vector valued risk measures in the next section.

A statistic T is elicitable if it can be written, for any random variable X, as a minimizer of a scoring function denoted S T (X) = arg min

x E P [S(x, X)], X ∼ P.
The statistical utility of this property can be summarized in two important advantages:

• The ability to compare different statistical methods using the scoring function, and thus give meaning to the backtesting procedures, which are very important in finance and insurance. • The ability to produce forecasts and estimations by mean regression in the case of elicitable measures. For more details on the statistical importance of the elicitability property, we invite the reader to consult papers [START_REF] Gneiting | Making and Evaluating Point Forecasts[END_REF] and [START_REF] Emmer | What is the best risk measure in practice? A comparison of standard measures[END_REF].

1.1. Elicitability of risk measures. Elicitability is a desirable property of risk measures. The usual measures are not all elicitable. Variance and TVaR are not elicitable, proofs are presented in [START_REF] Lambert | Eliciting properties of probability distributions[END_REF] for the Variance and in [START_REF] Gneiting | Making and Evaluating Point Forecasts[END_REF] for TVaR. The mean is elicitable because it can be written as

E[X] = xdF (x) = arg min t {E[(X -t) 2 ]}.
Quantiles are also elicitable for any threshold level α ∈ [0, 1]. They can be written as a minimizer of the scoring functions

S α (x, y) = α(y -x) + + (1 -α)(x -y) + called Pinball functions q α (X) = min{x : F X (x) ≥ α} = arg min x∈R {E[α(X -x) + + (1 -α)(x -X) + ]}.
The natural question that arises is about the presence of any risk measures which are both coherent and elicitable. Expectiles are the only coherent as well as elicitable risk measure according to [START_REF] Bellini | On elicitable risk measures[END_REF] [START_REF] Bellini | On elicitable risk measures[END_REF]. The expectiles were introduced in the context of statistical regression models by [START_REF] Newey | Asymmetric Least Squares Estimation and Testing[END_REF] [START_REF] Newey | Asymmetric Least Squares Estimation and Testing[END_REF]. Definition 1.1 (Expectiles [START_REF] Newey | Asymmetric Least Squares Estimation and Testing[END_REF]). For a random variable X with finite order 2 moment, the expectile of level α is defined as

(1.1) e α (X) = arg min x∈R E[α(X -x) 2 + + (1 -α)(x -X) 2 + ],
where (x) + = max(x, 0).

These measures are elicitable by construction. For α = 1/2 expectile coincides with the mean. This risk measure is coherent for α 1/2. For α < 1/2, the expectile is super-additive and therefore not coherent.

The uniqueness of the minimum is guaranteed by the strict convexity of the scoring function. One can also define the expectiles using the optimality condition of the first order, as the unique solution of the equation

(1.2) αE[(X -x) + ] = (1 -α)E[(x -X) + ].
The above equation can also be written as

1 -α α = E[(X -x) + ] E[(x -X) + ] .
This makes the economic interpretation of expectile as measure clearer. The expectiles can be seen as threshold that provides a Profits/Loss ratio of value 1-α α .

The properties of expectile risk measures have been studied in several recent papers. We recall some of them. The proofs are presented in [START_REF] Emmer | What is the best risk measure in practice? A comparison of standard measures[END_REF] and in [START_REF] Bellini | Risk management with expectiles[END_REF].

• The expectile e α (X) is a strictly increasing function of α ∈ [0, 1];

• If the distribution of the random variable X is symmetrical with respect to a point x 0 then, e α (X) + e 1-α (X) 2 = x 0 ;

• The expectile e α (X) is a law invariant and positively homogeneous risk measure for all 0 < α < 1. It satisfies the invariance by translation; • The expectiles satisfy e α (-X) = -e 1-α (X), for all α ∈ [0, 1];

• The expectile e α (X) is stochastically strictly monotone function of X, which means if X ≤ Y, a.s and P(X < Y ) > 0, then e α (X) < e α (Y );

• The expectiles are sub-additive and then coherent for α ∈ [1/2, 1[, they are super additive for α < 1/2; • The expectiles are additive measures by linear dependence corr(X, Y ) = 1 ⇒ e α (X + Y ) = e α (X) + e α (Y ), but they are not comonotonically additive. The asymptotic behavior of Expectiles is studied in [START_REF] Bellini | Risk management with expectiles[END_REF]. The second order of this behavior is analyzed in [START_REF] Mao | Risk concentration based on expectiles for extreme risks under fgm copula[END_REF]. [START_REF] Bellini | Generalized quantiles as risk measures[END_REF] [START_REF] Bellini | Generalized quantiles as risk measures[END_REF] have introduced the generalized quantiles risk measures, which include Expectiles, defined as a minimizer of an asymmetric error

(1.3) x α (X) = arg min x∈R {αE[Φ + ((X -x) + )] + (1 -α)E[Φ -((X -x) -)]},
where Φ + and Φ -are a convex scoring functions. Expectiles corresponds to the case x → x 2 for both functions. Recently, [START_REF] Daouia | Estimation of tail risk based on extreme expectiles[END_REF] [START_REF] Daouia | Estimation of tail risk based on extreme expectiles[END_REF] proposed an estimation of the VaR and ES risk measures using Expectiles.

1.2. Multivariate elicitability. In order to overcome the lack of elicitability of some usual risk measures, several works have used multivariate versions of elicitability. [START_REF] Lambert | Eliciting properties of probability distributions[END_REF] [START_REF] Lambert | Eliciting properties of probability distributions[END_REF] introduced the concept of indirect elicitability which allows to consider variance as an elicitable measure via the elicitability of the couple (E[X], E[X 2 ]). The same kind of solution has been proposed in [START_REF] Fissler | Higher order elicitability and osband's principle[END_REF] for the elicitability of the couple (VaR, TVaR) to allow validation of TVaR's Backtesting procedures.

The multivariate context of risk management and the multidimensional nature of statistics, make the multivariate generalization of the elicitability property in higher dimension, a natural step of a significant utility in risk modelling. The idea is to build multidimensional measures

T : P → R d , that can be written in the form arg inf u∈U E[S(X, u)],
where X is a random vector in R d and u a vector of size d u which can be different from d and U ⊂ R du .

Several authors have published important contributions on this subject. The elicitability of vector statistics was studied, using multivariate scoring functions, in [START_REF] Osband | Providing Incentives for Better Cost Forecasting[END_REF]. [START_REF] Lambert | Eliciting properties of probability distributions[END_REF] [START_REF] Lambert | Eliciting properties of probability distributions[END_REF] introduced the notion of k-elicitability, replacing the elicitability property by a linear combination of elicitable functional.

Multivariate elicitability also has great importance in the study of learning machines, we cite as examples [START_REF] Frongillo | Vector-valued property elicitation[END_REF] [START_REF] Frongillo | Vector-valued property elicitation[END_REF] who study the elicitation of vector-valued measures using the concept of the separability of scoring functions. According to this work, a vector-valued statistics is elicitable, if it is the case for each component T i : P → R, and in this case, the vector is elicitable using the sum function of the univariate scoring functions S i . Separability of scoring functions remains a strong assumption, as example, [START_REF] Osband | Providing Incentives for Better Cost Forecasting[END_REF] shows that there is no separable scoring function that elicits a bivariate quantile.

In this paper, we use multivariate elicitability in a more general context. Our aim is the characterization of elicitability in the case of multidimensional risk measures of the same size as the risk vector. This vision is traduced by Definition 1.2.

Definition 1.2 (Elicitable vectorial measures). A vector-valued risk measure

T : P → R k is elicitable if there exists a scoring function s : R d × R k -→ R, such that T (X) ∈ arg inf x∈U ⊂R k E[s(X, x)],
for all random vector X.

In a recent paper, Fissler and Ziegel (2015) [START_REF] Fissler | Higher order elicitability and osband's principle[END_REF] studied this definition in the general case. When the scoring function is strictly convex Definition 1.2 becomes simpler due to the uniqueness of the minimum. In this paper, we will focus on this case.

We must distinguish between the multivariate elicitability as presented in Definition 1.2 and that of the k-elicitability introduced by Lambert et al. (2008). For example, the variance is considered as 2-elicitable since Var(X) = E[X 2 ] -E[X] 2 and the measures E[X 2 ],E[X] are elicitable, which does not necessarily imply the multivariate elicitability of the vector (E[X 2 ], E[X]).

In the context of capital allocation, the allocation may be considered as a vector-valued risk measure with k = d. In [START_REF] Maume-Deschamps | On a capital allocation by minimization of some risk indicators[END_REF], an axiomatic characterization of multivariate coherence of capital allocation methods is given. It deals with, in particular, the allocation by minimizing multivariate risk indicators, which can be seen as scoring functions. The allocation as measure is therefore elicitable in the sense of Definition 1.2.

Multivariate extensions of expectiles

Following the univariate approach, we present in this section some multivariate constructions of Expectile risk measures.

Let . be a norm on R d . We denote by (X) + the vector (X) + = ((X 1 ) + , . . . , (X d ) + ) T and by (X) -the vector (X) -= ((X 1 ) -, . . . , (X d ) -) T . We define the following scoring function

s α (X, x) = α (X -x) + 2 +(1 -α) (x -X) + 2 , for all x ∈ R d .
We call multivariate expectile any minimizer

x * ∈ arg min x∈R d E[s α (X, x)].
We consider the function φ :

R d × R d -→ R defined by φ(t, x) = α (t -x) + 2 +(1 -α) (x -t) + 2 .
It is easy to verify that the function φ(t, x) is strictly convex in x, therefore, E[φ(X, x)] = E[s α (X, x)] is also strictly convex in x. In this case, the uniqueness of the minimum is guaranteed, and the multivariate expectile of the vector X with a confidence level α is defined by

(2.1) e α (X) = arg min x∈R d E[α (X -x) + 2 +(1 -α) (x -X) + 2 ].
The obtained vector-valued risk measure is elicitable by construction. The choice of a common threshold α for all the components of X seems natural in insurance contexts, since the accepted risk level must be the same between all off them.

In order to illustrate the construction, we present two possible examples of multivariate expectiles families.

Euclidean Expectiles (L p -expectiles).

The L p norms on R d , . p , 1 ≤ p ≤ ∞ are potential candidates to construct multivariate expectiles. Definition 2.1 in this case becomes e α (X) = arg min

x∈R d E[α (X -x) + 2 p +(1 -α) (x -X) + 2 p ].
For a norm . p with p < +∞, e α (X) = arg min When p = 2, the expectile is not taking into account the dependence. Indeed, the L 2 -expectile is composed by the marginal univariate expectiles. e α (X) = arg min

x∈R d E[α d i=1 (X i -x i ) p + 2/p + (1 -α) d i=1 (x i -X i ) p + 2/p ].
x∈R d E α d i=1 (X i -x i ) 2 + + (1 -α) d i=1 (x i -X i ) 2 + = arg min x∈R d E d i=1 α(X i -x i ) 2 + + (1 -α)(x i -X i ) 2 + = (e α (X 1 ), . . . , e α (X d )) T .
Overall, for p ≥ 1, we can determine the optimal solution with the first order necessary conditions for optimality. We have for all k ∈ {1, . . . , d}

∇ k (E[s α (X, x)]) = -2αE (X -x) + p (X -x) + p-1 p (X k -x k ) p-1 + 1 1 {X k >x k } + 2(1 -α)E (x -X) + p (x -X) + p-1 p (x k -X k ) p-1 + 1 1 {X k <x k } .
The first order necessary conditions for optimality can be written

αE (X -x) + p (X -x) + p-1 p ∂(X -x) + ∂x (X -x) p-1 + = (1-α)E (x -X) + p (x -X) + p-1 p ∂(x -X) + ∂x (x -X) p-1 + .
The L p -expectile is the unique solution of the equation system (2.2)

αE (X -x) + p (X -x) + p-1 p (X k -x k ) p-1 + 1 1 {X k >x k } = (1 -α)E (x -X) + p (x -X) + p-1 p (x k -X k ) p-1 + 1 1 {X k <x k } ,
for all k ∈ {1, . . . , d}. For 2 < p < +∞, the equations system (2.2) can be written as follows

αE (X k -x k ) p-1 + (X -x) + p-2 p = (1 -α)E (X k -x k ) p-1 - (X -x) - p-2 p ∀k ∈ {1, . . . , d},
the dependence is taken into account by using the norm. In order to enlighten the economic interpretation, we write the previous system in the following form

E[(X k -x k ) + (X k -x k ) + (X-x) + p p-2 ] E[(X k -x k ) - (X k -x k ) - (X-x) -p p-2 ] = 1 -α α ∀k ∈ {1, . . . , d}.
The multivariate expectile can be considered as a vector of thresholds that makes a ratio relative gains / relative losses identical between all random variables X k . The multivariate expectile can be considered as a vector of thresholds that makes a ratio relative gains / relative losses identical between all random variables X k . The adjective "relative" reflects the presence of a weight power of each X k

(X k -x k ) + (X-x) + p p-2 ( (X k -x k ) - (X-x) -p p-2
) in the total aggregated by the norm. This interpretation is more intuitive in the case p = 3.

The case p = 1 is quite interesting, a new equivalent definition can be given to the L 1 -expectile using the first order condition of optimality (2.2). The L 1 -expectile is the unique solution in R d of the following system

αE[ (X -x) + 1 1 1 {X k >x k } ] = (1 -α)E[ (X -x) -1 1 1 {X k <x k } ], ∀k ∈ {1, . . . , d}.
or equivalently

(2.3) E[ (X -x) + 1 1 1 {X k >x k } ] E[ (X -x) -1 1 1 {X k <x k } ] = 1 -α α , ∀k ∈ {1, . . . , d},
which can be interpreted as a ratio of an activity participation in the positive scenarios and its participation in the negative ones. The multivariate expectile represents in this case, the vector of marginal levels that make this ratio constant for all the activities, and thereby, balance the risk level between the portfolio's components. In insurance, and considering the random variables as risks, the interest will be to minimize the ratio β = (1 -α)/α, which means to increase the level α.

From this point of view, the multivariate expectile may be a capital allocation tool. It can also be seen as vector of thresholds that balances the systemic risk of all the components of the risk vector.

On another side, the expectile vector can also be used as evaluation tool for some optimal reinsurance cases. In fact, in the Stop-Loss reinsurance, the reinsurer pays the part of a risk X i that exceeds a certain amount u i . For d Stop-Loss contracts, the multivariate expectile represent the vector of the amounts that makes constant, for all the contracts, the ratio of the participation of in the reinsurer expected loss and its participation of the insurer insureds expected loss. It can also give an idea about the impact of a new contract on the others, and help to propose an amount u i for the new contract.

Equation (2.3) can be written using bivariate expected values

(2.4) α d i=1 E[(X i -x i ) + 1 1 {X k >x k } ] = (1 -α) d i=1 E[(X i -x i ) -1 1 {X k <x k } ], ∀k ∈ {1, . . . , d}.
The system (2.4) can also be written using the bivariate Stop-Loss transform functions, introduced and studied for actuarial application in [START_REF] Hürlimann | On higher-degree bivariate stop-loss transforms, with applications[END_REF]. Remark that the L 1 -expectile takes into account only the bivariate dependence.

Matrix Expectiles (Σ-expectiles).

It is also possible to construct multivariate expectiles using matrices. The idea is to choose a real symmetric matrix Σ to define a multivariate scoring function.

Several usual matrices can be used to construct multivariate expectiles, as examples

• Mahalanobis expectiles: If Σ is the inverse matrix of the matrix of covariances, which is equivalent to the use of Mahalanobis distance, we can call the obtained vector Mahalanobis expectile. This constructions is more adapted to the random vectors of elliptic marginal distributions. • Correlated expectiles: This construction is based on the use of a correlation matrix. We can also use any other bivariate dependence measure as coefficients of the construction matrix.

Definition 2.1 (Multivariate matrix expectiles). Let X = (X 1 , . . . , X d ) T ∈ R d be a random vector such that E[|X i X j |] < +∞ for all (i, j) ∈ {1, . . . , d} 2 , and Σ = (π ij ) 1≤i,j≤d a real square matrix of order d, symmetric and positive semi-definite that verifies (1) For all i ∈ {1, . . . , d}, π ii = π i > 0;

(2) For all i, j ∈ {1, . . . , d}, π ii ≥ π ij . We call a Σ-expectile of X, all vector verifying e Σ α (X) ∈ arg min

x∈R d E[α(X -x) T + Σ(X -x) + + (1 -α)(X -x) T -Σ(X -x) -],
if the solution is unique, the multivariate expectile of the vector X is

(2.5) e Σ α (X) = arg min x∈R d E[α(X -x) T + Σ(X -x) + + (1 -α)(X -x) T -Σ(X -x) -].
This construction gives a more general framework that encompass several examples, especially the correlated expectiles, Mahalanobis expectile, and the L 1 euclidean expectile if Σ ij = 1, ∀i, j.

Under the coefficients positivity assumption of the construction matrix, the multivariate expectile is finally, the unique solution of the following system of equations

(2.6) α d i=1 π ki E[(X i -x i ) + 1 1 {X k >x k } ] = (1 -α) d i=1 π ki E[(x i -X i ) + 1 1 {x k >X k } ], ∀k ∈ {1, . . . , d}, equivalent to (2.7) α = d i=1 π ki E[(x i -X i ) + 1 1 {x k >X k } ] d i=1 π ki E[(X i -x i ) + 1 1 {X k >x k } ] + E[(x i -X i ) + 1 1 {x k >X k } ]
, ∀k ∈ {1, . . . , d},

or d i=1 π ki E[(X i -x i ) + 1 1 {X k >x k } ] d i=1 π ki E[(x i -X i ) + 1 1 {x k >X k } ] = 1 -α α , ∀k ∈ {1, . . . , d}.
In order to illustrate this construction, Figure 2 shows the impact of the multivariate approach compared with univariate expectile and the quantile and of the same threshold. It exemplifies also the impact of the correlation measure choice on the behavior of the obtained multivariate expectile. For an FGM bivariate model, we analyze the difference between a construction using a static matrix, illustrated by the L 1 -expectile, and using a dynamic dependence matrix. In this example, three matrix-expectiles are considered: Pearson's, Spearman's and Kendall's expectiles, with π ii = 1 and respectively

π ij = θ ij 4 π ij = θ ij 3 or π ij = 2θ ij 9 , ∀i = j ∈ {1, 2}
, where θ ij are the parameters of the FGM model. The impact is more important for the least risky branch. The curves order follows the order of the dependence measures. The more dependence, the further away from the univariate case the curve is.

The impact of matrix Σ choice is visible in the expectiles behavior in the presence of dependence. Figure 3 shows that the impact of dependence is not the same between the different constructions. The L 1 expectile seems to be an increasing function of the dependence level. At the opposite, the other expectiles are decreasing when the dependence weight increases. Other constructions are possible. We can define a geometric expectiles following the same ideas as for geometric quantiles. The Chaudhuri's approach (1996) [START_REF] Chaudhuri | On a geometric notion of quantiles for multivariate data[END_REF] and the Abdous and Theodorescu's one (1992) [START_REF] Abdous | Note on the spatial quantile of a random vector[END_REF] are easily adjustable for this purpose.

The choice of the construction may differ according to the practical goal. For example, the use of the correlation matrix reflects a choice of dependence modeling between risks. A matrix composed of tail dependence coefficients emphasizes the consideration of tail dependence. That is why we believe that the question of construction choice should be left open to the users, and it must take into account their applications needs. However, the simplest one remains the matrix expectiles. In fact, using positive semi-definite matrix with positive coefficients, the strict convexity of the scoring function is guaranteed, and the first order condition of optimality is easy to get as a system of equations linking transformed Stop-Loss functions. For these reasons, the rest of our study is focused on this expectiles family as defined in Definition 2.1.

Properties of multivariate expectiles

In this section we give some elementary properties of the multivariate expectiles. We then discuss the sub-additivity and monotonicity properties. In this subsection we present some of these properties. Proposition 3.1. For any order 2 random vector X = (X 1 , . . . , X d ) T in R d

(1) Positive homogeneity : for any non negative real constant a, e α (aX) = ae α (X).

(2) Invariance by translation : for any vector a = (a 1 , . . . , a d ) T in R d , e α (X + a) = e α (X) + a.

(3) Law invariance : for any order

2 random vector Y = (Y 1 , . . . , Y d ) T on R d such that (X i , X j ) L = (Y i , Y j ) for all (i, j) ∈ {1, . . . , d} 2 , then ∀α ∈ [0, 1], e α (X) = e α (Y ).
(4) Pseudo-invariance by linear transformations : We denote by e Σ α the multivariate expectile obtained using the matrix Σ.

For any vectors a = (a 1 , . . . , a d ) T ∈ R d and b = (b 1 , . . . , b d ) T ∈ (R + * ) d , e Σ α (V X + a) = V e V ΣV α (X) + a,
where V is the diagonal square matrix associated to b, V = Diag(b T ).

Proof. The proofs of the first 3 items are straightforward using (2.7). For (4), using (2), it is sufficient to prove that

e Σ α (V X) = V e V ΣV α (X).
We have e Σ α (V X) = arg min

x∈R d E[α(V X -x) T + Σ(V X -x) + + (1 -α)(V X -x) T -Σ(V X -x) -]
= arg min

x∈R d E[α(X -V -1 x) T + V T ΣV (X -V -1 x) + + (1 -α)(X -V -1 x) T -V T ΣV (X -V -1 x) -] = V e V T ΣV α (X) = V e V ΣV α (X),
because V is symmetric.

Proposition 3.2 (Symmetry with respect to α). For any order 2 random vector X = (X 1 , . . . , X d ) T in R d , and any α ∈ [0, 1], e α (-X) = -e 1-α (X).

Proof. We denote by x * = e 1-α (X) the multivariate expectile with threshold (1 -α) associated to the random vector X and x = e α (-X) the multivariate expectile threshold α associated to -X.

Using the optimality condition (2.6), x is the unique solution of the following equation system

α d i=1 π ki E[(-X i -x i ) + 1 1 {-X k >x k } ] = (1 -α) d i=1 π ki E[(x i + X i ) + 1 1 {x k >-X k } ], ∀k ∈ {1, . . . , d},
rewritten as

α d i=1 π ki E[(-x i -X i ) + 1 1 {X k <-x k } ] = (1 -α) d i=1 π ki E[(X i -(-x i )) + 1 1 {-x k <X k } ], ∀k ∈ {1, . . . , d}.
From that we deduce -x is the unique solution of the optimality condition corresponding e 1-α (X), so -x = x * .

Proposition 3.3 (Impact of independence). For a random vector X = (X 1 , . . . , X d ) T , such that ∃i ∈ {1, . . . , d} with π ij = 0, ∀j ∈ {1, . . . , d}\{i} e i α (X) = e α (X i ), where e i α (X) is the i th coordinate of e α (X), and e α (X i ) is the univariate expectile associated to the random variable X i .

Proof. Since π ij = 0 ∀j = i, the i th equation of the optimality system (2.7) is

αE[(X i -e i α (X)) + ] = (1 -α)E[(X i -e i α (X)) -]
, which is the optimality condition that define the univariate expectile of X i .

Proposition 3.4 (The support stability). For any order 2 random vector

X = (X 1 , . . . , X d ) T in R d e i α (X) ∈ [x i I , x i F ]
. where e i α (X) is the i th coordinate of e α (X), x i I = essInf (X i ) and x i F = essSup(X i ). Proof. In the case of infinite marginal support, the result is trivial. Let us assume the existence of k ∈ {1, . . . , d} such that x k F < +∞ and e k α (X) > x k F . The multivariate expectile is defined as e α (X) = arg min

x∈R d E[α(X -x) T + Σ(X -x) + + (1 -α)(X -x) T -Σ(X -x) -],
and the minimum in this case is

min x∈R d E[s α (X, x)] = min x∈R d E[α(X -x) T + Σ(X -x) + + (1 -α)(X -x) T -Σ(X -x) -] = (1 -α)   π k E[(e k α (X) -X k ) 2 + ] + 1≤i≤d,i =k π ik E[(e i α (X) -X i ) + (e k α (X) -X k ) + ]   + (1 -α) 1≤i≤d,i =k 1≤j≤d,j =k π ij E[(e i α (X) -X i ) + (e j α (X) -X j ) + ] + α 1≤i≤d,i =k 1≤j≤d,j =k π ij E[(X i -e i α (X)) + (X j -e j α (X)) + ], because E[(X k -e k α (X)) + (X i -e i α (X)) + ] = 0 for all i ∈ {1, . . . , d}. The function x -→ (1 -α) π k E[(x -X k ) 2 + ] + 1≤i≤d,i =k π ik E[(e i α (X) -X i ) + (x -X k ) + ] is non decreasing in x, considering e k α (X) > x * > x i F , for i ∈ {1, . . . , d}\{k} E[(X k -x * ) 2 + ] = E[(X k -x * ) + (X i -e i α (X)) + ] = 0, we have for x * = (e 1 α (X), . . . , e k-1 α (X), x * , e k+1 α (X), . . . , e d α (X)) E[s α (X, x * )] = (1 -α)   π k E[(x * -X k ) 2 + ] + 1≤i≤d,i =k π ik E[(e i α (X) -X i ) + (x * -X k ) + ]   + (1 -α) 1≤i≤d,i =k 1≤j≤d,j =k π ij E[(e i α (X) -X i ) + (e j α (X) -X j ) + ] + α 1≤i≤d,i =k 1≤j≤d,j =k π ij E[(X i -e i α (X)) + (X j -e j α (X)) + ] < E[s α (X, e α (X))] = min x∈R d E[s α (X, x)],
which is absurd. We deduce that e k α (X) ≤ x k F , ∀k ∈ {1, . . . , d}, and using the symmetry by α property, we get e k α (X) ≥ x k I , ∀k ∈ {1, . . . , d}.

Proposition 3.5 (Strong intern monotonicity). We consider the case of multivariate expectile constructed using a symmetric positive-definite matrix Σ = (π ij ) 1≤i≤j≤d of a positive coefficients. For any order 2 random vector X = (X 1 , . . . , X d ) T in R d , if there exists a couple (i, j) ∈ {1, . . . , d} 2 such that X i ≤ X j , a.s, π i ≤ π j , π ik ≤ π jk for all k ∈ {1, . . . , d}\{i, j}, and the couples (X i , X k ) and (X j , X k ) have the same copula for all k ∈ {1, . . . , d}\{i, j}, then

e i α (X) ≤ e j α (X), for all α ∈ [0, 1].
Proof. Let (X 1 , . . . , X d ) T be a random vector in R d . We suppose the existence of i and j such that X i ≤ X j , a.s. We denote by l α X i ,X j for all (i, j) ∈ {1, . . . , d} 2 the functions l

α X i ,X j (x i , x j ) = αE[(X i -x i ) + 1 1 {X j >x j } ] -(1 -α)E[(X i -x i ) -1 1 {X j <x j } ], for all (x i , x j ) ∈ R 2 , and by l α X i the function l α X i (x i ) = l α X i ,X i (x i , x i ).
The multivariate expectile is the unique solution of System (2.6) that can be written using the functions l α (3.1)

d i=1 π ki l α X i ,X k (x i , x k ) = 0 ∀k ∈ {1, . . . , d}.
The functions l α X i are non increasing for all i ∈ {1, . . . , d}, and l α X i ,X j (x i , x j ) are non increasing in x i for all (i, j) ∈ {1, . . . , d} 2 . On another side, for all (i, j) ∈ {1, . . . , d}

2 l α X i ,X j (x i , x j ) = E[(α(X i -x i ) + + (1 -α)(X i -x i ) -) 1 1 {X j ≥x j } ] -(1 -α)E[(X i -x i ) -],
functions l α X i ,X j are then non increasing in x j too. We deduce that if X i ≤ X j , a.s, then for all X k that has the same bivariate copula with both X i and X j , we have l

α X k ,X i (x k , x) ≤ l α X k ,X j (x k , x), ∀(x k , x). Now, we suppose that x j = e j α (X) < e i α (X) = x i . We have l α X k ,X i (x k , x i ) ≤ l α X k ,X i (x k , x j ) ≤ l α X k ,X j (x k , x j ), ∀k ∈ {1, . . . , d} \ {i, j}. The coefficients of Σ verify 0 ≤ π ik ≤ π jk , ∀k ∈ {1, .
. . , d}, so, from the system of optimality (3.1), we can deduce that

(3.2) d k=1,k =i,k =j π ik l α X k ,X i (x k , x i ) ≤ d k=1,k =i,k =j π jk l α X k ,X j (x k , x j ).
Since x i > x j , the almost sure dominance X i ≤ X j , a.s implies

1 1 {X i ≥x i ,X j ≤x j } = 0.
We deduce from that

E[(X i -x i ) + ] = E[(X i -x i ) + 1 1 {X j >x j } ] and E[(X j -x j ) -] = E[(X j -x j ) -1 1 {X i <x i } ],
hence, we obtain l α X i (x i ) < l α X i ,X j (x i , x j ) and l α X j ,X i (x j , x i ) < l α X j (x j ), which gives us

(3.3) π ij l α X i (x i ) + π ij l α X j ,X i (x j , x i ) < π jj l α X j (x j ) + π ij l α X i ,X j (x i , x j ). We have l α X i (x) ≤ l α X j (x), ∀x, because the function t -→ α(t -x) + -(1 -α)(x -t) + is non decreasing in t. Therefore (3.4) (π i -π ij )l α X i (x i ) ≤ (π j -π ij )l α X j (x j ), because 0 ≤ π i -π ij ≤ π j -π ij .
Finally, form (3.4) and (3.3)

(3.5) π i l α X i (x i ) + π ij l α X j ,X i (x j , x i ) < π j l α X j (x j ) + π ij l α X i ,X j (x i , x j )
, which is contradictory with (3.2) and the system of optimality (3.1). Lemma 3.6 (Impact of dependence). Let X = (X 1 , . . . , X d ) T be an order 2 For any order 2 random vector.

(1) If there exists (i, j) ∈ {1, . . . , d} 2 such that

(X i , X k ) L = (X j , X k ), π i = π j , π ik = π jk for all k ∈ {1, . . . , d}, then e i α (X) = e j α (X), for all α.
(2) If the vectors (X i , X j ) i = j have the same distribution, and there exists a positive constant π such that π ij = π for all i = j ∈ {1, . . . , d}, π i = π j for all i, j ∈ {1, . . . , d}, then (a) e α (X) = x * 1 1 R d , where x * is the solution of the following equation

α(E[(X-x) + ]+(d-1) π π 1 E[(X 2 -x 2 ) + 1 1 {X 1 >x 1 } ]) = (1-α)(E[(X-x) -]+(d-1) π π 1 E[(X 2 -x 2 ) -1 1 {X 1 <x 1 } ]).
(b) If the vector X is comonotonic then

e α (X) = e α (X)1 1 R d ,
where e α (X) is the univariate expectile of the random variable X. (c) If the bivariate common copula is denoted C then

C C ⇒ e C α (X) ≤ e C α (X), for all α ≥ 1/2
, where e C α (X) is the multivariate expectile associated to C. Recall that for all bivariate copulas C and

C C C ⇔ C(u, v) ≤ C (u, v)∀(u, v) ∈ [0, 1] 2 .
Proof. For (1) we just apply the law invariance property for the random vectors X = (X 1 , . . . , X d ) T and Y = (X 1 , . . . , X i-1 , X j , X i+1 , . . . , X j-1 , X i , X j+1 . . . , X d ) T . For (2 -a), using (1) e i α (X) = e j α (X) = x * , ∀i, j ∈ {1, . . . , d}. The multivariate expectile is the unique solution of Equation System (2.6), which can be written as

d i=1 π ki l α X i ,X k (x i , x k ) = 0 ∀k ∈ {1, . . . , d}, where, l α X i ,X j (x i , x j ) = αE[(X i -x i ) + 1 1 {X j >x j } ] -(1 -α)E[(X i -x i ) -1 1 {X j <x j }
], for all (x i , x j ) ∈ R 2 . These function are depending only of the bivariate distributions of (X i , X j ), so, in our case

l α X i ,X j (x * , x * ) = l α X 1 ,X 2 (x * , x * ), ∀i = j ∈ {1, . . . , d}. We denote by l α X i the function l α X i (x i ) = l α X i ,X i (x i , x i ).
It depends only on the marginal distribution of X i , and in our case

l α X i (x * ) = l α X (x * ) ∀i ∈ {1, .
. . , d}. With the previous notations, and using the optimality system, x * is the unique solution of the following equation

π 1 l α X (x * ) + (d -1)πl α X 1 ,X 2 (x * , x * ) = 0. (2 -b) is a direct application of (2 -a).
In fact, for the comonotonic case

l α X 1 ,X 2 (x * , x * ) = l α X (x *
), x * is then the unique solution of the equation l α X (x * ) = 0, which is exactly the univariate expectile definition.

For (2 -c), with l α C (x) = l α X 1 ,X 2 (x, x), we have l α C (x) = αE[(X 2 -x) + 1 1 {X 1 >x} ] -(1 -α)E[(X 2 -x) -1 1 {X 1 <x} ], then for α ≥ 1/2 l α C (x) = (α -1/2) E[(X 2 -x) + 1 1 {X 1 >x} ] + E[(X 2 -x) -1 1 {X 1 <x} ] + 1/2 E[(X 2 -x) + 1 1 {X 1 >x} ] -E[(X 2 -x) -1 1 {X 1 <x} ] = (α -1/2) E[(X 2 -x) + 1 1 {X 1 >x} ] + E[(X 2 -x) -1 1 {X 1 <x} ] + 1/2 E[(X 2 -x)1 1 {X 1 >x} ] -E[(X 2 -x) -] ,
we deduce that for all bivariate copulas C and

C if C C then (3.6) l α C (x) ≤ l α C (x), for all x.
We consider two bivariate copulas C and C such that C C , we denote by e C α (X) the multivariate expectile associated to C, and we suppose that e C α (X) < e C α (X). l α C is non increasing in x, so, using (3.6) we get

(3.7) l α C (e C α (X)) ≤ l α C (e C α (X)) ≤ l α C (e C α (X)
). On the other hand, l α X is strictly decreasing for continuous distributions, then

l α X (e C α (X)) < l α X (e C α (X)
), and consequently

(3.8) l α C (e C α (X)) < l α C (e C α (X)
). The inequalities (3.7) and (3.8) are contradictory, so e C α (X) ≤ e C α (X).

Proposition 3.7 (Derivatives with respect to α). Let X = (X 1 , . . . , X d ) T be a random vector of a continuous distribution in R d . We consider a multivariate expectile constructed using a positive semi-definite matrix, then the vector

X ∂α = ( ∂e 1 α (X) ∂α , . . . , ∂e d α (X)
∂α ) T composed of the derivatives by α satisfies the following system of equations

B k ∂x k ∂α + d i=1 γ ki ∂x i ∂α = A k , ∀k ∈ {1, . .

. , d},

where

A k = d i=1 π ki E[(X i -x i ) + 1 1 {X k >x k } ] + d i=1 π ki E[(x i -X i ) + 1 1 {X k <x k } ] ≥ 0, and
B k = f X k (x k ) d i=1,i =k π ki (αE[(X i -x i ) + | X k = x k ] + (1 -α)E[(x i -X i ) + | X k = x k ]) ,
for all k ∈ {1, . . . , d}, and

γ ki = απ ki P(X i > x i , X k > x k ) + (1 -α)π ki P(X i < x i , X k < x k ), ∀(i, k) ∈ {1, . . . , d} 2 .
Proof. We denote x i = e i α (X). From the system of optimality (2.7)

α d i=1 π ki E[(X i -x i ) + 1 1 {X k >x k } ] = (1 -α) d i=1 π ki E[(x i -X i ) + 1 1 {X k <x k } ], ∀k ∈ {1, . . . , d}. Thus ∀ k = 1, . . . , d A k = ∂x k ∂α d i=1 π ki (1 -α) ∂ ∂x k E[(x i -X i ) + 1 1 {X k <x k } ] -α ∂ ∂x k E[(X i -x i ) + 1 1 {X k >x k } ] + d i=1,i =k π ki ∂x i ∂α (1 -α) ∂ ∂x i E[(x i -X i ) + 1 1 {X k <x k } ] -α ∂ ∂x i E[(X i -x i ) + 1 1 {X k >x k } ] , (3.9)
where

A k = d i=1 π ki E[(X i -x i ) + 1 1 {X k >x k } ] + d i=1 π ki E[(x i -X i ) + 1 1 {X k <x k } ]. Moreover, ∀i = k = 1, . . . , d ∂ ∂x k E[(X i -x i ) + 1 1 {X k >x k } ] = +∞ x i -f X k (x k )P(X i > t | X k = x k )dt = -f X k (x k )E[(X i -x i ) + | X k = x k ],
and

∂ ∂x k E[(x i -X i ) + 1 1 {X k <x k } ] = f X k (x k )E[(x i -X i ) + | X k = x k ], ∂ ∂x i E[(X i -x i ) + 1 1 {X k >x k } ] = P(X i > x i , X k > x k ), ∀(i, k) ∈ {1, . . . , d} 2 , ∂ ∂x i E[(x i -X i ) + 1 1 {X k <x k } ] = P(X i < x i , X k < x k ), ∀(i, k) ∈ {1, . . . , d} 2 .
From (3.9), we deduce that X ∂α = ( ∂x 1 ∂α , . . . , ∂x d ∂α ) T satisfies the announced equation system.

In this subsection, we have shown that multivariate expectiles satisfy a set of desirable properties of multivariate risk measures that confirms their potential utility.

3.2. Discussion on coherence properties. If invariance by translation and positive homogeneity properties have natural extensions in higher dimension, the properties of sub-additivity and monotonicity can be defined in different ways (see e.g. [START_REF] Cascos | Multivariate risks and depth-trimmed regions[END_REF] for some constructions when the risk measure takes values in an abstract cone). In fact, the sub-additivity property is generalizable using different concepts. Using an internal regulation point of view, with an allocation concept, consider a vector valued risk measure ρ as sub-additive if for all random vector X = (X 1 , . . . , X d ) T in R d , and all subset S of D = {1, . . . , d} of cardinal |S|

i∈S ρ i (X) ≤ ρ 1   i∈S X i , X j∈D\S T   ,
where ρ i , i = 1 . . . , d denote the components of the vector ρ(X) and i∈S X i , X j∈D\S T is the

R d-|S|+1
random vector, composed of the random variables i∈S X i , X j∈D\S . This definition expresses that the measure takes into account the diversification gain inside the vector X. Another point of view is to consider ρ as sub-additive, if for all random vectors X and

Y on R d ρ(X + Y ) v ρ(X) + ρ(Y )
where v denotes an order on vectors, for example the componentwise order:

ρ i (X + Y ) ≤ ρ i (X) + ρ i (Y ), ∀i ∈ {1, . . . , d}.
The convexity property can be defined with the same approach, and as the multivariate expectiles are positively homogeneous, it is equivalent to the sub-additivity.

The monotonicity property can also be generalized using different definitions derived from several multivariate stochastic orders, like the supermodular one. For all random vectors X and Y on R d

X SOV Y ⇒ ρ(X) v ρ(Y )
where SOV is a multidimensional stochastic order and v denotes an order on vectors. One aim of multivariate risk measures is to quantify the systemic risk. In this perspective, the internal monotonicity property may be a sufficient requirement.

As already mentioned in this article, the elicitability property can be a basis of the definition of coherence. We could say that a multivariate risk measure taking values in R k is coherent if it satisfies:

(1) ρ is elicitable;

(2) For all R d risk vector X and a vector a ∈ R d , ρ(X + a) = ρ(X) + a;

(3) For all R d risk vector X and a real

α ∈ R + , ρ(α.X) = α.

ρ(X);

(4) There exists s : R d × R k -→ R a strictly F-consistent scoring function for ρ, where the class F of probability distributions is the domain of ρ, such that for all X, Y R d risk vectors

X cx Y ⇒ E F [s(X, ρ(X))] ≤ E F [s(Y, ρ(Y))], ∀F ∈ F;
(5) There exists s : R d × R k -→ R a strictly F-consistent scoring function for ρ, such that,

E F [s(X + Y, ρ(X + Y))] ≤ E F [s(X, ρ(X))] + E F [s(Y, ρ(Y))],
for all X, Y risk vectors in R d and for all F ∈ F. In the case of the proposed multivariate expectiles, only (4) is not proved in this paper, and (5) can replace the sub-additivity property. In the univariate case, it is well known that the expectile is consistent with convex stochastic order, a proof is to be found in [START_REF] Jakobsons | Dependence uncertainty bounds for the expectile of a portfolio[END_REF].

Stochastic estimation

In general, the multivariate expectiles cannot be calculated directly, but their estimation is possible using noisy observations. In this section, we present a stochastic approximation method which is adapted to the multivariate expectiles. We mainly use some usual stochastic optimization and root finding algorithms.

The multivariate expectile obtained using a positive semi-definite matrix Σ = (π ij ) 1≤i≤j≤d , is the unique solution of system of equations (2.6)

α d i=1 π ki E[(X i -x i ) + 1 1 {X k >x k } ] = (1 -α) d i=1 π ki E[(x i -X i ) + 1 1 {x k >X k } ], ∀k ∈ {1, . . . , d},
which has also the form

φ(x) = E[Φ(x, X)] = 0, where Φ(., X) is a function of R d on R d , with Φ k (x, X) = d i=1 π ki α(X i -x i ) + 1 1 {X k >x k } -(1 -α)(x i -X i ) + 1 1 {x k >X k , k ∈ {1, . . . , d}.
The stochastic approximation methods are iterative algorithms of the following form

x n+1 = x n + γ n Φ(x n , X n+1 ),
where (γ n ) is a deterministic sequence of steps which satisfies some further specified conditions, and (X n ) is a sequence of independent and identically distributed random vectors, obtained from the same distribution of a generic random variable X.

Robbins-Monro's algorithm.

To obtain the multivariate expectile, we use the Robbins-Monro's algorithm. We denote by x the desired expectile. The idea of the algorithm is to define a sequence (x n )

x n+1 = x n + γ n Z n+1 , where x 0 ∈ R d is the starting point, and Z n+1 is an observed random variable which satisfies φ(x n ) = E[Z n+1 | F n ] with F n the σ-algebra of information present in the time n

F n = σ(x 0 , Z 1 , x 1 , . . . , x n-1 , Z n ).
Here, we use the version of the Robbins-Monro's theorem as presented in [START_REF] Fraysse | Estimation récursive dans certains modeles de déformation[END_REF] (Theorem 1.3.1).

Theorem 4.1 (Robbins-Monro). Under the following assumptions:

(1) φ is a continuous function;

(2) For all x = x * , (xx * ) T φ(x) < 0;

(3) For al n ≥ 0,

E[Z n+1 | F n ] = φ(x n ), a.s.; (4) There exists K > 0 such that E[ Z n+1 2 | F n ] ≤ K 1+ x n -x * 2 a.s.;
(5) The sequence (γ n ) is decreasing to 0 and satisfies

+∞ n=0 γ n = +∞ et +∞ n=0 γ 2 n < +∞,
the sequence (x n ) defined by

x n+1 = x n + γ n Z n+1 ,
converges almost surely to x * , solution of φ(x * ) = 0.

The first two assumptions deal with the regularity of φ, they guarantee the uniqueness of the solution. Assumptions 3 and 4 are related to the observations sequence, they respectively guarantee its close distance to the exact value and its variance control. The last assumption on the sequence of steps is required to achieve convergence. Other modified versions of the algorithm are proposed in the literature when these assumptions are not satisfied. 4.2. Application for multivariate expectiles. For a random vector X of continuous marginal distributions, the function φ defined by φ(x) = E[Φ(x, X)] is clearly continuous. On the other hand, the multivariate expectile x * is the minimum of a strict convex function on R d of gradient -2φ(x), it satisfies then for all

x ∈ R d \ {x * } < -2φ(x) + 2φ(x * ), x -x * > = < -2φ(x), x -x * > > 0,
we deduce from that, for all x = x * , (xx * ) T φ(x) < 0.

The assumptions of Robbins-Monro's Theorem 4.1 on φ thus satisfied.

Consider the sequence of independent and identically distributed random vectors (X n ) L = X. We define the sequence of observations (Z n ) for each iteration n ≥ 1 by

Z (k) n = d i=1 π ki α(X (i) n -x (i) n-1 ) + 1 1 {X (k) n >x (k) n-1 } -(1 -α)(x (i) n-1 -X (i) n ) + 1 1 {X (k) n <x (k) n-1 }
for all k ∈ {1, . . . , d}, where x (i) denote the i th coordinate of x. This previous sequence satisfies for all k ∈ {1, . . . , d} and all n ≥ 0

E[Z (k) n+1 | F n ] = φ k (x n ), a.s., hence ∀n ≥ 0, E[Z n+1 | F n ] = φ(x n ), a.s..
Using the triangle inequality of the absolute value as norm, and then the Hölder's inequality (p = q = 2), by setting

K = 2 max (α, 1 -α) 2 d k=1 d i=1 |π ik | 2 max ( X -x * 2 , 1) we obtain E[ Z n+1 2 | F n ] ≤ K 1+ x n -x * 2 a.s..
Robbins-Monro's Theorem 4.1 assumptions on the observations sequence are satisfied. The theorem is then relevant for any sequence of steps (γ n ), decreasing to 0, chosen such that

+∞ n=0 γ n = +∞ et +∞ n=0 γ 2 n < +∞.
A natural choice of the sequence of steps is γ n = 1/n or 1/n κ with 1/2 < κ < 1. More generally, the choice of this sequence can be made by adjusting the constants κ, a, b such that γ n = a/(b + n) κ . Using some numerical illustrations, we will discuss the impact of this choice on the speed of algorithm's convergence.

4.3. Numerical illustrations. We consider a simple bivariate exponential model. X 1 ∼ exp(β 1 ) and X 2 ∼ exp(β 2 ). Firstly, we examine under the independence assumption the L 1 -expectile case (π ij = 1, ∀(i, j) ∈ {1, 2} 2 ). The optimality system is explicit and composed of the two following equations

(2α -1) 1 β i e -β i x i -(1 -α) x i - 1 β i = (1 -α) x j - 1 β j (1 -e -β j x j ) (1 -e -β i x i ) -α 1 β j e -β j x j e -β i x i , for (i, j) ∈ {(1, 2
), (2, 1)}. The exact solution can be determined using numerical optimization methods. For our illustration, we use the Newton-Raphson's multidimensional algorithm. The graphical analysis of the error's normality is shown in Figure 5. The impact of the sequence of steps on the speed of convergence is illustrated in Figure 6. The convergence is not very sensitive to the starting point choice.

The Robbins-Monro's algorithm convergence is studied in [START_REF] Lelong | Etude asymptotique des algorithmes stochastiques et calcul du prix des options Parisiennes[END_REF]. Two CLT are presented, one for the choice of 1/n as sequence of steps and the other in case of sequences of the form γ n = γ/n κ , where κ is a constant and 1/2 < κ < 1. We consider now a bivariate Pareto independent model. The two random variables are of Pareto distribution, X i ∼ P a(a, b i ), i ∈ {1, 2}, such that a > 1 and b i > 0 for all i ∈ {1, 2}. The system of optimality of L 1 -expectile is explicit. The expectile (x 1 , x 2 ) T is the unique solution of the following system l α X i (x i ) = -l α X j ,X i (x j , x i ), (i, j) ∈ {(1, 2), (2, 1)}, where,

l α X i (x i ) = (2α -1) b i a -1 b i b i + x i a-1 -(1 -α) x i - b i a -1 , and l α X j ,X i (x j , x i ) = b j a -1 b i b i + x i a -(1 -α) b j b j + x j a-1 -(1-α) 1 - b i b i + x i a x j - b j a -1 .
The Newton-Raphson's method is used to get the exact solution. Figure 7 is an illustration of the difference in convergence between two different levels α = 0.7 and α = 0.99 of the expectile. Convergence is not very satisfactory for values of α close to 1. The algorithm is not efficient to estimate the asymptotic expectile. A study of the asymptotic behavior of the expectile seems necessary, particularly in cases where there is no analytical solution. The next section is devoted to the asymptotic expectiles.

For the dependence case, we consider a random vector X = (X 1 , X 2 ) T of exponential marginals X i ∼ E(β i ), i = 1, 2 and bivariate FGM copula, as dependence structure, of parameter θ ∈

[-1, 1]. The expression of FGM bivariate copula is C FGM θ (u, v) = uv[1 + θ(1 -u)(1 -v)], ∀(u, v) ∈ [0, 1] 2 .
The FGM copula is presented in details in [START_REF] Nelsen | An Introduction to Copulas[END_REF] (Example 3.12., section 3.2.5). We recall that it is only a weak dependence structure that cannot take into account extreme dependencies. The bivariate distribution function is given by

F X 1 ,X 2 (x 1 , x 2 ) = C FGM θ (F X 1 (x 1 ), F X 2 (x 2 )) = F X 1 (x 1 )F X 2 (x 2 )[1 + θ FX 1 (x 1 ) FX 2 (x 2 )] = (1 -e -β 1 x 1 )(1 -e -β 2 x 2 ) + θ(1 -e -β 1 x 1 )(1 -e -β 2 x 2 )e -β 1 x 1 e -β 2 x 2 .
The L 1 -expectile is the unique solution of the following optimality system

(2α -1) 1 β j e -β j x j -(1 -α) x j - 1 β j = (1 -α)(1 -e -β j x j )(1 -θe -β j x j ) x i - 1 β i (1 -e -β i x i ) + (1 -α) θ 2 (1 -e -β j x j )e -β j x j 2x i - 1 β i (1 -e -2β i x i ) -α 1 β i e -β i x i e -β j x j 1 + θ(1 - e -β i x i 2 )(1 -e -β j x j ) ,
for (i, j) ∈ {(1, 2), (2, 1)}. Figure 8 presents the result obtained for β 1 = 0.05 (red) and β 2 = 0.25 (blue), in cases of positive and negative dependence.

Asymptotic behavior

Let (X 1 , . . . , X d ) T be a random vector in R d . The support of each random variable

X i is denoted [x i I , x i F ], where x i I ∈ R ∪ {-∞} and x i F ∈ R ∪ {+∞}.
We denote by X F the vector (x 1 F , . . . , x d F ) T , and by X I the vector (x 1 I , . . . , x d I ) T . We recall the definition of functions l α X i ,X j for all (i, j) ∈ {1, . . . , d} 2 as follows (5.1) l α X i ,X j (x i , x j ) = αE[(X i -x i ) + 1 1 {X j >x j } ] -(1 -α)E[(X i -x i ) -1 1 {X j <x j } ], for all (x i , x j ) ∈ R 2 . We also denote by l α X i the function l α X i (x i ) = l α X i ,X i (x i , x i ). The System of optimality 2.6 can be written using these function as (5.2) d i=1 π ki l α X i ,X k (x i , x k ) = 0 ∀k ∈ {1, . . . , d}.

We focus now on the asymptotic behavior of multivariate expectiles. The construction matrix Σ is supposed constituted of positive coefficients (π ij ≥ 0, ∀(i, j) ∈ {1, . . . , d} 2 ).

X i ) + 1 1 {x k >X k } ] d i=1 π ki E[(X i -x i ) + 1 1 {X k >x k } ] + E[(x i -X i ) + 1 1 {x k >X k } ]
.

Proof. It is sufficient to prove that lim α-→1 e α (X) = X F , and we deduce the limit for α -→ 0 using the property of symmetry by α e α (-X) = -e 1-α (X).

For simplicity, we make the proof for π ij = 1∀(i, j) ∈ {1, . . . , d} 2 . The generalization is straightforward.

Taking if necessary a convergent subsequence, we consider that the limit lim e i α (X) < x i F }.

We suppose firstly J ∞ = ∅. Then, ∀ i, j ∈ J∞ = {1, . . . , d}, lim α-→1 l α X i (x i ) > 0 and lim α-→1 l α X i ,X j (x i , x j ) > 0. That is absurd, because it is contradictory with the system of optimality (5.2). From that, we deduce J ∞ = ∅. There exists at least one k ∈ {1, . . . , d} such that lim α-→1 e k α (X) = x k F . We have lim α-→1 l α X i ,X k (e i α (X), e k α (X)) = 0, ∀i ∈ J∞ , and lim α-→1 i∈J∞ l α X i ,X k (e i α (X), e k α (X)) = -lim α-→1 i∈J∞

(1 -α)E[(X ie i α (X)) -1 1 {X k <e k α (X)} ] = 0, by (5.2). We deduce for i = k lim α-→1 l α X i (e i α (X)) = -lim α-→1

(1 -α)E[(X ke i α (X)) -] = 0, ∀i ∈ J ∞ . (5.2)then leads to (5.4) lim α-→1 l α X k (e k α (X)) = -lim α-→1

(1 -α)E[(X ke k α (X)) -] = 0, ∀k ∈ J ∞ .

If we assume that J∞ = ∅, there exists ∈ {1, . . . , d} such that lim α-→1 e α (X) < x F . In this case, lim α-→1 l α X i ,X (e i α (X), e α (X)) = E[(X i -lim

α-→1 e i α (X)) + 1 1 {X > lim α-→1
e α (X)} ] ∈ R + \{+∞}, ∀i ∈ J∞ , and using (5.4) lim α-→1 l α X i ,X (e i α (X), e α (X)) = -lim α-→1

(1 -α)E[(X ie i α (X)) -1 1 {X < lim α-→1 e α (X)} ] = 0, ∀i ∈ J ∞ , because E[(X ie i α (X)) -1 1 {X < lim α-→1 e α (X)} ] ≤ E[(X ie i α (X)) -, ] and lim α-→1

(1 -α)E[(X ie i α (X)) -] = 0, for all i ∈ J ∞ . Finally, using the th equation of optimality system (5.2), we conclude that

E[(X i -lim α-→1 e i α (X)) + 1 1 {X > lim α-→1
e α (X)} ] = 0, ∀i ∈ J∞ , and in particularly that E[(X -lim α-→1 e α (X)) + ] = 0, that is contradictory with the assumption lim α-→1 e α (X) < x F . We deduce therefore that J∞ = ∅. The second part of Proposition 5.1 is straightforward in the case of infinite supports.

The multivariate expectile tends to the vector of the marginal endpoints when α → 1. The asymptotic behavior models the situation of extreme risk, hence the practical importance of its study, especially in insurance.

Conclusion

In this paper, we have presented different approaches to construct some multivariate risk measures. The starting point of these methods was the elicitability property. In a second time, we have chosen a specific construction using matrices to study its coherence properties. Multivariate expectiles are obtained using positive semi-definite matrices with positive coefficients, they take into account the dependence structure and the marginal distributions. We also proposed a stochastic approximation method for this family of measures, based on the Robbins-Monro's algorithm. For asymptotic expectile levels α, the approximation does not provide relevant information on the behavior of expectile vector. A natural perspective of this work is to develop a theoretical analysis and an estimation procedure of asymptotic expectiles.

Figure 1

 1 Figure 1 is an illustration of the contour lines of the scoring function in the bivariate case ψ(z) = α (z) + 2 p +(1 -α) (z) - 2 p , for different level of α (α = 0.05, 0.5, 0.95) and different euclidean norms (L 1 , L 2 and L 10 ).

Figure 1 .

 1 Figure 1. L p -expectile: Contour lines of the scoring function

Figure 2 .

 2 Figure 2. Impact of construction's choice in FGM bivariate model with dependence parameter θ = 1 (left X 1 ∼ E(0.05), right X 2 ∼ E(0.25)).

Figure 3 .

 3 Figure 3. Impact of dependence in FGM bivariate model for α = 0.95 (left X 1 ∼ E(0.05), right X 2 ∼ E(0.25)).

3. 1 .

 1 Elementary properties. Several desirable properties are satisfied by multivariate expectiles.

Figure 4

 4 presents the result obtained for β 1 = 0.05 (red) and β 2 = 0.25 (blue). The exact value of the bivariate expectile (20.02, 3.22) is represented by the straight dotted lines of the same colors, respectively. In order to ensure the robustness of the result, we use an average of 100 outputs of the algorithm.

Figure 4 .

 4 Figure 4. Convergence of the algorithm: L 1 -expectile, Exponential independent model (red β 1 = 0.05, blue β 2 = 0.25).

Figure 5 .

 5 Figure 5. The algorithm's error, L 1 -expectile, Exponential independent model.

Figure 6 .

 6 Figure 6. Impact of the choice of the sequence of steps on the convergence.

Figure 7 .

 7 Figure 7. Convergence of the algorithm, L 1 -expectile, Pareto independent model (a = 2, red b 1 = 10, blue b 2 = 20).

Figure 8 .

 8 Figure 8. Convergence of the algorithm: L 1 -expectile α = 0.85, FGM model (left θ = -1, right θ = 1, red β 1 = 0.05, blue β 2 = 0.25).

Proposition 5 . 1 (

 51 Asymptotic Expectiles). For any order 2 random vectorX = (X 1 , . . . , X d ) T in R d , where E[|X i |] < +∞ for all i ∈ {1, . . . , d}, lim α-→1 e α (X) = X F ,and lim α-→0 e α (X) = X I . If in addition, all supports are infinite, then for all k ∈ {1, . . . , d} lim x k -→+∞ α(x 1 , . . . , x d ) = 1, and lim x k -→-∞ α(x 1 , . . . , x d ) = 0, where (5.3) α(x 1 , . . . , x d ) = d i=1 π ki E[(x i -

  α-→1e α (X) exists. We define considerJ ∞ := {i ∈ {1, . . . , d}| lim α-→1 e i α (X) = x i F }. Its complementary set is denoted J∞ J∞ := {i ∈ {1, . . . , d}|i / ∈ J ∞ } = {i ∈ {1, . . . , d}| lim α-→1