Sarah Kaakaï

Anis Matoussi

Achraf Tamtalini

Multivariate Optimized Certainty Equivalent Risk Measures and their Numerical Computation *

Keywords: Multivariate risk measures, Optimized certainty equivalent, Numerical methods, stochastic algorithms, risk allocations

We present a framework for constructing multivariate risk measures that is inspired from univariate Optimized Certainty Equivalent (OCE) risk measures. We show that this new class of risk measures verifies the desirable properties such as convexity, monotonocity and cash invariance. We also address numerical aspects of their computations using stochastic algorithms instead of using Monte Carlo or Fourier methods that do not provide any error of the estimation.

Introduction

One of the major concerns in finance is how to assess or quantify the risk associated with a random cashflow in the future. Starting with the pioneering work of [START_REF] Markowitz | Portfolio selection[END_REF], the risk associated with a random outcome was quantified by its variance. Then, [START_REF] Artzner | Coherent measures of risk[END_REF] published their famous seminal paper in which they introduce the theory of risk measures. In their paper, risk measures were defined as a map verifying certain properties, which are called "axioms", namely: Subadditivity, translation invariance, monotonocity and positive homogeneity. Such risk measures are called coherent risk measures. Many extensions have been proposed and studied in the literature after the introduction of the axiomatic approach. One important extension is the notion of convex risk measure developed by Föllmer and Schied (2002) and [START_REF] Frittelli | Putting order in risk measures[END_REF] where the subadditivity and positive homogeneity properties were replaced by the weaker property of convexity. The latter reflects the fact that diversification decreases the risk. In the banking industry, one of the most popular risk measures is the Value at Risk (VaR in short). This is due first, to its financial interpretation and second, to its easy and fast implementation. Indeed, VaR is defined as the minimal cash amount that needed to be added to a financial position in order to have a probability of losses below a certain threshold. Its computation amounts to the calculation of a quantile of the portfolio distribution. Nevertheless, VaR suffers from one drawback: it does not verify the convexity property. This has prompted the search for new examples of risk measures, the most prominent being the Conditional Value at Risk (CVaR), the entropic risk measure and the utility based risk measure (also known as shortfall risk measure). Some decision making problem based on utility functions are closely related to risk measures.

One can cite the optimized certainty equivalent (OCE) that was first introduced by Ben-Tal and Teboulle (1986). The idea behind the definition of OCE is as follows: Assume that a decision maker, with some utility function u, is expecting a random income X in the future and can consume a part of it at present. If he chooses to consume m dollars, the resulting present value of X is then P (X, m) := m + E[u(X -m)]. Hence, one can define the sure present value of X (i.e., its certainty equivalent) as the result of an optimal allocation of X between present and future consumption, that is the decision maker will try to find m that maximizes P (X, m). The main properties of the OCE were studied in [START_REF] Ben-Tal | An old-new concept of convex risk measures: the optimized certainty equivalent[END_REF] where it is showed that the opposite of the OCE provides a wide family of risk measures that verifies the axiomatic formalism of convex risk measures. They also proved that several risk measures, such as CVaR and the entropic risk measure, can be derived as special cases of the OCE by using particular utility functions (see also [START_REF] Cherny | Divergence utilities[END_REF]).

From a systemic point of view, the financial crisis of 2008 has demonstrated the need for novel approaches that capture the risk of a system of financial institutions. More precisely, given a network/system of d ∈ N different but dependent portfolios X := (X 1 , ..., X d ), we are interested in measuring/quantifying the risk carried by this system of portfolios. A classical approach consists in first aggregating the portfolios using some aggregation function Λ : R d → R and then apply some univariate risk measure applied to the aggregated portfolio. In practice, most of the times the aggregation function is just the sum of the components, i.e., Λ(x) = d i=1 x i . This will result in having a systemic risk measure of the form: R(X) = η(Λ(X)) = η( d i=1 X i ), where η is a univariate risk measure, such as the VaR, CVaR, entropic risk measure, etc. The mechanism behind this approach is also known as "Aggregate then Inject Cash" mechanism (see [START_REF] Biagini | A unified approach to systemic risk measures via acceptance sets[END_REF]). However, this approach suffers from one major drawback: While it quantifies the systemic risk carried by the whole system, it does not provide risk levels of each portfolio, and thus, one could not have a ranking of portfolios in terms of their systemic riskiness. One way to remediate to this, is to consider the reverse mechanism, that is to "Inject Cash then Aggregate". This consists in associating to each portfolio a risk measure and summing up the resulting risk levels. This results in considering systemic risk measures R(X) of the following form:

R(X) = d i=1 η i (X i )
, where η i 's are the univariate risk measures associated to each portfolio. Obviously, one could use the same univariate for all portfolios, that is η i = η, ∀i ∈ {1, ..., d}.

However, by doing so, we are assuming that the system is made of "isolated" portfolios with no interdependence structure, and hence, we might be overestimating or underestimating the systemic risk. This led several authors to look for approaches that address simultaneously the design of an overall risk measure and the allocation of this risk measure among the different components of the system. In this spirit, an extension of shortfall risk measures, introduced in Föllmer and Schied (2002), has been studied in [START_REF] Armenti | Multivariate shortfall risk allocation and systemic risk[END_REF] based on multivariate loss functions. However, one should note that, to ensure the existence of optimal allocation problem, these loss functions must verify a key property: permutation invariance. In other words, each component of the system is treated as if it has the same risk profile as all the other components and thus one cannot discriminate a particular component against one another. Moreover, classical risk measures such that the CVaR and the entropic risk measure cannot be recovered using multivariate shortfall risk measures, which limit their use in practice. We will see that, with our multivariate extension of OCE risk measure, the permutation invariance condition is no longer needed and by choosing the appropriate loss functions, we can retrieve most of the classical risk measures.

One of the major issues that arises when studying risk measures is their numerical approximation. The standard VaR can be computed by inverting the simulated empirical distribution of the financial position using Monte Carlo (see [START_REF] Glasserman | Monte Carlo Methods in Financial Engineering[END_REF] and [START_REF] Glasserman | Fast simulation of multifactor portfolio credit risk[END_REF]). An alternative method for computing VaR and CVaR is to use stochastic algorithms (SA). The rational idea behind this perspective comes from the fact that both VaR and CVaR are the solutions and the value of the same convex optimization problem as pointed out in [START_REF] Rockafellar | Conditional value-at-risk for general loss distributions[END_REF] and the fact that the objective function is expressed as an expectation. This was done in [START_REF] Bardou | Computing var and cvar using stochastic approximation and adaptive unconstrained importance sampling[END_REF], where they prove the consistency and the asymptotic normality of the estimators. In the same direction, in [START_REF] Kaakaï | Estimation of systemic shortfall risk measure using stochastic algorithms[END_REF], we extended the work of [START_REF] Dunkel | Stochastic root finding and efficient estimation of convex risk measures[END_REF] to approximate multivariate shortfall risk measures using stochastic algorithms. In [START_REF] Neufeld | Antonis papapantoleon[END_REF], they developed numerical schemes for the computations of univariate OCE using Fourier transform methods.

The outline of this paper is as follows: in section 1, we give the definition of multivariate OCE by introducing first the class of appropriate loss functions. Then, we show that this class of risk measures verifies the desirable properties. We also characterize the optimal solutions, give a dual representation and study the sensitivity with respect to external shocks.

Finally, section 2 treats the computational aspects of approximating multivariate OCE using 1. Multivariate OCE a deterministic scheme and a stochastic one.

(R d ) that X ≥ Y if P (X ≥ Y ) = 1.
To alleviate the notations, we will drop the reference to

R d in L 0 (R d ) whenever it is unnecessary. For Q = (Q 1 , ..., Q d ) a
vector of probabilities, we will write Q P if for all i = 1, ..., d, we have Q i P . In this section, we introduce the notion of multivariate Optimized Certainty Equivalent (OCE) and

give its main properties. The latter is an extension of univariate OCE that was introduced and studied in details in [START_REF] Ben-Tal | An old-new concept of convex risk measures: the optimized certainty equivalent[END_REF]. First, we start by giving the definition of a multivariate loss function that will be used in the rest of the paper. For the rest of the paper, the random vector X = (X 1 , ..., X d ) ∈ L 0 represents profits and losses of d portfolios.

Definition 1.1. A function l : R d → (-∞, ∞] is called a loss function, if it satisfies the following properties:

1. l is nondecreasing, that is if x ≤ y componentwise, then l(x) ≤ l(y).

2. l is lower-semicontinuous and convex.

3. l(0) = 0 and l(x) > d i=1 x i , ∀x = 0.

For integrability reasons, we will work in the multivariate Orlicz heart defined as:

M θ := {X ∈ L 0 : E[θ(λX)] < ∞, ∀λ > 0},
where θ(x) = l(|x|), x ∈ R d . On this space, we define the Luxembourg norm as:

||X|| θ := λ > 0, E θ |X| λ ≤ 1 .
Under the Luxembourg norm, M θ is a Banach lattice and its dual with respect to this norm is given by the Orlicz space L θ * :

L θ * := {X ∈ L 0 , E[θ * (λX)] < ∞, for some λ > 0}.
We also introduce the set of d-dimensional measure densities in L θ * , that is:

Q θ * := dQ dP := (Z 1 , ..., Z d ), Z ∈ L θ * , Z k ≥ 0 and E[Z k ] = 1 .
Note that for Q ∈ Q θ * and X ∈ M θ , dQ dP • X ∈ L 1 , thanks to Fenchel inequality and for the sake of simplicity, we will write E Q [X] := E[dQ/dP • X]. We refer to Appendix B in [START_REF] Armenti | Multivariate shortfall risk allocation and systemic risk[END_REF] for more details about multivariate Orlicz spaces. Definition 1.2. Assume l is a loss function. The multivariate OCE risk measure is defined for every X ∈ M θ as:

R(X) = inf w∈R d d i=1 w i + E[l(-X -w)] .
(1.1)

Example 1. When d = 1, we can recover some important convex risk measures such CVaR (also called Expected Shortfall or Average Value at Risk) and Entropic risk measure.

1. CVaR: Let α ∈ (0, 1) and take l(x) = 1 1-α x + , then the associated risk measure is the CVaR (see [START_REF] Rockafellar | Conditional value-at-risk for general loss distributions[END_REF]).

2. Polynomial loss function: For an integer γ > 1, the polynomial loss function is defined by: l(x) = ([1+x] + ) γ -1 γ . When γ = 2, the corresponding risk measure is the Monotone Mean-Variance (see Černỳ et al. (2012)).

3. Entropic risk measure: Fix λ > 0 and let l(x) := exp(λx)-1 λ . Then, the problem in (1.1) can be explicitly solved and the optimal w * and R(X) are given by:

w * = 1 λ log(E[e -λX ]), R(X) = w * = 1 λ log(E[e -λX ]).
Using univariate loss functions, we can construct multivariate loss functions in the following way: Given l 1 , ..., l d univariate loss functions and a nonnegative, convex and lowersemicontinuous function Λ : R → R + with Λ(0) = 0, one can define a multivariate loss function as follows:

l(x) := d i=1 l i (x i ) + Λ(x). (1.2)
It is easy to see that l verifies all the conditions in the definition 1.1. Note that by taking Λ the null function, the corresponding multivariate OCE boils down to a sum of univariate OCE. It is in this function Λ where the dependence between the different components in the system is taken into account. In this paper, we will focus on the following multivariate loss functions inspired from the univariate risk measures above:

l(x) = d i=1 e λ i x i -1 λ i + αe d i=1 λ i x i , λ i > 0, α ≥ 0, (1.3) l(x) = d i=1 ([1 + x i ] + ) θ i -1 θ i + α i<j ([1 + x i ] + ) θ i θ i ([1 + x i ] + ) θ j θ j , θ i > 1, α ≥ 0, (1.4) l(x) = d i=1 x + i 1 -β i + α i<j x + i 1 -β i x + j 1 -β j , 0 < β i < 1, α ≥ 0.
(1.5)

Multivariate OCE

In the next theorem, we show that the multivariate OCE is a convex risk measure as defined in Föllmer and Schied (2002).

Theorem 1.3. The function R in (1.1) is real valued, convex, monotone and cash invariant1 risk measure. In particular, it is continuous and subdifferentiable. If l is positive homogeneous, then R is too. Furthermore, it admits the following representation:

R(X) = max Q∈D l * {E Q [-X] -α(Q)}, (1.6)
where the penalty function α is defined for

Q = (Q 1 , ..., Q d ) P by: α(Q) = E l * dQ dP and D l * = {Q P, α(Q) < ∞} := dom(α).
Proof.

• R(X) ∈ R for all X ∈ M θ : Since M θ ⊆ L 1 , by the third property of loss functions, we have for every X ∈ M θ and w ∈ R d :

d i=1 w i + E[l(-X -w)] ≥ E[-d i=1 X i ] > -∞. R(X) < +∞ since for w = 0, we have E[l(-X)] < ∞. • Monotonicity: Let X, Y ∈ M θ such that X ≤ Y . Since l is non-decreasing, then E[l(-X -w)] ≥ E[l(-Y -w)] for every w ∈ R d , which in turn implies R(X) ≥ R(Y ).
• Convexity: Let X, Y ∈ M θ and λ ∈ [0, 1]. We have thanks to the convexity of l: R(λX

+ (1 -λ)Y ) = inf w∈R d { d i=1 w i + E[l(-X -w)]} = inf w 1 ,w 2 ∈R d d i=1 λw 1 i + (1 -λ)w 2 i + E[l(λ(-X -w 1 ) + (1 -λ)(-Y -w 2 ))] ≤ inf w 1 ∈R d inf w 2 ∈R d λ( d i=1 w 1 i + E[l((-X -w 1 )] + (1 -λ)( d i=1 w 2 i + E[l((-X -w 2 )]) = λR(X) + (1 -λ)R(Y ).
• Cash Invariance: Let m ∈ R d , we have: [START_REF] Cheridito | Risk measures on orlicz hearts[END_REF] or Theorem 1 in [START_REF] Biagini | On the extension of the namioka-klee theorem and on the fatou property for risk measures[END_REF].

R(X + m) = inf w∈R d d i=1 w i + E[l(-X -m -w)] = inf w∈R d d i=1 (w i + m i ) - d i=1 m i + E[l(-X -m -w)] = R(X) - d i=1 m i • Continuity and subdifferentiability: Since (M θ , || • || θ ) is a Banach space, this is a direct consequence of Theorem 4.1 in
• Positive homogeneity: If l is positive homogeneous, then by the definition of R(X), we have for λ > 0:

R(λX) = inf w∈R d d i=1 w i + E[l(-λX -w)] = inf w∈R d d i=1 w i + λE[l(-X - w λ )] = λ inf w∈R d d i=1 w i λ + E[l(-X - w λ )] = λ inf w∈R d d i=1 w i + E[l(-X -w)] = λR(X).
• Representation: First, because R is convex and continuous, Fenchel-Moreau theorem implies that:

R(X) = sup Y ∈L θ * {E[X • Y ] -R * (Y )} = max Y ∈L θ * {E[X • Y ] -R * (Y )} (1.7) where R * (Y ) = sup{E[X • Y ] -R(X), X ∈ M θ }, Y ∈ L θ * . Now, if Y 0, then by the bipolar theorem, there exists X 1 ∈ M θ such that X 1 ≥ 0 and E[X 1 • Y ] > 0.
Using the definition of R * (Y ), we get the following:

R * (Y ) = sup X∈M θ {E[X • Y ] -R(X)} ≥ sup λ>0 {λE[X 1 Y ] -R(λX 1 )} ≥ sup λ>0 {λE[X 1 Y ]} -R(0)} = +∞,
where the last inequality is due to the monotonicity of R. Therefore, the maximum can be taken over Y ≤ 0. For k ∈ {1, ..., d}, let X = (0, ..., x, ...) and x > 0. By the translation invariance property, we have R(

X k ) = R(0) -x. Consequently, R * (Y ) = sup X∈M θ {E[X • Y ] -R(X)} ≥ xE[Y k ] -R(0) + x = x(E[Y k ] + 1) -R(0). If E[Y k ] = -1, then by sending x to infinity, we get that R * (Y ) = ∞. Finally, this
shows that the maximum in (1.7) could be taken over

D θ * , i.e., R(X) = max Q∈D θ * {E[-dQ dP • 1. Multivariate OCE X] -R * (-Q)} = max Q∈D θ * {E Q [-X] -R * (-Q)}. Let us now explicit more the expression of R * (-Q) for Q ∈ D θ * : R * (-Q) = sup X∈M θ E - dQ dP • X -R(X) = sup X∈M θ E - dQ dP • X -inf m∈R d d i=1 m i + E[l(-X -m)] = sup X∈M θ E - dQ dP • X + sup m∈R d - d i=1 m i -E[l(-X -m)] = sup X∈M θ sup m∈R d E - dQ dP • X - d i=1 m i -E[l(-X -m)] = sup m∈R d sup X∈M θ E - dQ dP • X - d i=1 m i -E[l(-X -m)] = sup m∈R d - d i=1 m i + sup X∈M θ E - dQ dP • X -E[l(-X -m)] = sup m∈R d - d i=1 m i + sup W ∈M θ E - dQ dP • (W -m) -E[l(-W )] = sup m∈R d - d i=1 m i + E dQ dP • m + sup W ∈M θ E - dQ dP • W -E[l(-W )] = sup m∈R d d i=1 -m i + m i E dQ i dP + sup W ∈M θ E dQ dP • W -E[l(W )] = sup m∈R d 0 + sup W ∈M θ E dQ dP • W -l(W ) = sup W ∈M θ E dQ dP • W -l(W ) Note that, for W ∈ M θ , we have for Q ∈ D θ * , dQ dP • W ∈ L 1 , thanks to Fenchel inequality. Furthermore, since W i ≤ l(W ) ≤ θ(W ) and both θ(W ) and W i are in L 1 , we have l(W ) ∈ L 1 . This allows us to write in the lines above E[ dQ dP •W ]-E[l(W )] = E dQ dP • W -l(W )
. Now, we would like to interchange the expectation with the supremum. To this end, we use Corollary on page 534 of [START_REF] Rockafellar | Integrals which are convex functionals[END_REF] with L = M θ , L * = L θ * and F (x) = l(x). Note that l is a lower-semicontinuous proper convex function, and it is easy to verify that M θ and L θ * are decomposable in their sens, so that all the conditions needed to apply this Corollary are satisfied. We get finally that,

R * (-Q) = E l * dQ dP := α(Q).
Finally, since R(X) is finite, then the maximum can be taken over D l * instead of D θ * .

Definition 1.4. A risk allocation is any minimizer of (1.1). When it is uniquely determined, we denote it RA(X).

Theorem 1.5. Let l be a loss function. Then, for every X ∈ M θ , the set of risk allocations is non empty and bounded. Furthermore, risk allocations are characterized by the following first order condition:

1 ∈ E[∂l(-X -m * )]. (1.8)
Moreover, the supremum in (1.6) is attained for Z * such that Z * ∈ ∂l(-X -m * ) a.s. and

E[Z * ] = 1.
Proof. The arguments used in this proof are an extension of the univariate case. To prove that the set of risk allocations is non empty and bounded, it is sufficient to show that the objective function has no direction of recession thanks to Theorem 27.1(d) in [START_REF] Rockafellar | Convex Analysis[END_REF]. Let w = 0 and let f (w) :

= d i=1 w i + E[l(-X -w)]. We have, f 0 + (w) = lim r→∞ f (m + rw) -f (m) r = lim r→∞ m i + r w i + E[l(-X -m -rw)] -m i -E[l(-X -m)] r = w i + lim r→∞ E[l(-X -m -rw)] -E[l(-X -m)] r = w i + lim r→∞ E[l(-X -m -rw)] r .
Now, since l is convex and l(0) = 0, for λ > 1 we have 1 λ l(λx) ≥ l(x). This implies, together with Lebesgue's dominated convergence theorem and lower-semicontinuity of l

f 0 + (w) ≥ w i + lim r→∞ E l -X -m r -w = w i + E lim inf r→∞ l -X -m r -w ≥ w i + l(-w) > 0.
The last strict inequality is a consequence of the third property of l. So we have shown that for every w = 0, f 0 + (w) > 0, i.e., f has no direction of recession. We conclude that the set of minimizers is non empty bounded set. Moreover, we have m * ∈ argminf if and only if m * satisfies 0 ∈ ∂f (m * ). Using Theorem 4.47 in [START_REF] Shapiro | Lectures on Stochactic Programming: Modeling and Theory[END_REF], we can interchange the partial operator and the expectation sign leading to the following characterization of minimizers:

m * is a minimizer of f ⇔ 1 ∈ E[∂l(-X -m * )].
In the following, we prove that the maximum in (1.6) is attained for Z * ∈ ∂l(-X -m * ) a.s. and E[Z * ] = 1. We start by proving the existence of such Z * . Let m * be such that

1 ∈ E[∂l(-X -m * )]. Note that, for each z ∈ R d , if ν ∈ ∂l(z), then ν is nonnegative.
In fact, by definition, we have, l(x) ≥ l(z)

+ d i=1 ν i (x i -z i ), ∀x ∈ R d . So, if for some k ∈ 1, ..., d, ν k < 0, then choosing x = z -ne k < z
where e k is the k-th standard unit vector, we get that -nν k ≤ l(x) -l(z) ≤ 0. By sending n to +∞, we get a contradiction.

Therefore, since 1 ∈ E[l(-X -m * )], there exists a random variable Z * such that Z * ≥ 0 and

Z * ∈ ∂l(-X -m * ) a.s. and E[Z * ] = 1. Next, we will show that Z * ∈ D θ * , that is E[l * (Z * )] < ∞. Note that since Z * ∈ ∂l(-X -m * ),
we have that,

l * (Z * ) = Z * • (-X -m * ) -l(-X -m * ), a.s.
(1.9)

First, we will start by proving that Z * • X ∈ L 1 . Thanks to (1.9), we have

X • Z * + l * (Z * ) = -m * • Z -l(-X -m * ). Because X ∈ M θ , the right term of the previous equality is in L 1 . So, this shows that X • Z * + l * (Z * ) ∈ L 1 . Recall that l * (z) ≥ 0 for all z ∈ R d so that we have (X • Z * ) + ∈ L 1 . It remains to show that (X • Z * ) -∈ L 1 .
Using the convexity of l, we have the following inequality:

l(2(-X -m * )) ≥ l(-X -m * ) + Z * • (-X -m * ), a.s.
This in turn implies that

X • Z ≥ l(-X -m * ) -l(2(-X -m * )) -Z * • m * . The RHS of this inequality is in L 1 as X ∈ M θ . Hence, we get that (Z * • X) -∈ L 1 .
We are now able to say that all the terms in the RHS of (1.9) are in L 1 . We conclude that l * (Z * ) ∈ L 1 . Moreover, we have,

E[-X • Z * ] -E[l * (Z * )] = E[-X • Z * ] -E[Z * • (-X -m * ) -l(-X -m * )] = E[-X • Z * -Z * • (-X -m * ) + l(-X -m * )] = E[Z * • m * + l(-X -m * )] = R(X),
where we used the optimality of m * in the last equality. This completes the proof.

Example 2. The following example with a bidimensional loss function of exponential type as in Example 1, that is:

l(x 1 , x 2 ) = e λ 1 x 1 -1 λ 1 + e λ 2 x 2 -1 λ 2 + αe λ 1 x 1 +λ 2 x 2 , where λ 1 > 0, λ 2 > 0, α ≥ 0. If X ∼ N(0, Σ) with Σ = σ 2 1 ρσ 1 σ 2 ρσ 1 σ 2 σ 2 2
, then we can solve explicitly the optimal risk 1. Multivariate OCE allocations in (1.8) and to obtain

m * i =        λ i σ 2 i 2 , if α = 0, λ i σ 2 i 2 - 1 λ i ln(SC ij ), j = i, if α > 0, (1.10)
where the term SC ij , i = j, is the positive solution to the following second order equation:

αλ j exp(ρσ i σ j λ i λ j )X 2 + (1 + α(λ i -λ j ) exp(ρσ i σ j λ i λ j )) X -1 = 0
The risk measure could also be derived in explicit form:

R(X) = m 1 * + m 2 * + 2 -α λ 1 (SC 12 -1) = m 1 * + m 2 * + 2 -α λ 2 (SC 21 -1), (1.11)
Remark 1.6. The formula obtained in (1.10) is close to the one in Example 3.12 in [START_REF] Armenti | Multivariate shortfall risk allocation and systemic risk[END_REF]. It shows that the optimal allocations are disentangled into two components:

the first one is an individual contribution which takes the form of the entropic risk measure of X i and the second one is a systemic contribution which involves correlations between the two components of the system. This formula shows also an interesting feature: the partial differential of SRC with respect to ρ is always positive. This can be interpreted in the following way: the more correlated the system is, the riskier is. Note that this is not true in general and depends on the loss function l used.

Corollary 1.7. Let l a strictly convex loss function. Then,

RA(X + r) = RA(X) - d i=1 r i , for every X ∈ M θ and r ∈ R d .
If l is additionally positive homogeneous, then RA(λX) = λRA(X), for every X ∈ M θ and λ > 0.

Proof. Let X ∈ M θ and r ∈ R d . m : RA(X + r) is the unique solution of 1 ∈ E[∂l(-X -r - m)]. Setting w = r + m, we obtain that w satisfies 1 ∈ E[∂l(-X -w)],
which by uniqueness

Multivariate OCE

implies that w = RA(X), that is RA(X + r) = RA(X) -r. Let λ > 0, we have,

RA(λX) = argmin w w i + E[l(-λX -w)] = argmin w d i=1 w i + λE l(-X - w λ ) = argmin w d i=1 w i λ + λE l(-X - w λ ) = λ argmin w d i=1 w i + E[l(-X -w)] = λRA(X).
Now, we focus on the study of the sensitivity of our multivariate risk measure. We first give the definition of the marginal risk contribution of Y ∈ M θ to X ∈ M θ .

Definition 1.8. For X, Y ∈ M θ , we define the marginal risk contribution of Y to X as the sensitivity of the risk associated to X when an impact Y is applied as

R(X, Y ) := lim sup 0 R(X + Y ) -R(X) .
(1.12)

If R(X + Y ) admits a unique risk allocation RA(X + Y ) for small enough ≥ 0, then we define the risk allocation marginals of X with respect to the impact of Y as:

RA i (X; Y ) : lim sup 0 RA i (X + Y ) -RA i (X) , i = 1, ..., d.
(1.13) Theorem 1.9. Let X, Y ∈ M θ and assume that l is differentiable. Then,

R(X, Y ) = -E[Y • ∇l(-X -m * )] = - d i=1 E Q n * [Y n ], (1.14)
where m * is such that, E[∇l(-X -m * )] = 1, i.e. an infinimum for (1.1) and dQ * dP := ∇l(-X -m * ).

If furthermore, l is twice differentiable such that we can interchange the differentiation and

expectation of m → E[∇l(-X -m)] and M := E[∇ 2 l(-X -m * )] is invertible, then we have,
• There exists a unique m optimum of R(X + Y ) for small enough ≥ 0.

• As a function of , m is differentiable and we have

RA(X, Y ) = M -1 V, V := -E[∇ 2 l(-X -m * )Y ].
(1.15)

Proof. Take X, Y ∈ M θ and let m * be an infinimum for R(X). We have

R(X) = d i=1 m * i + E[l(-X -m * )] and E[∇l(-X -m * )] = 1. By the definition of R(X + Y ), we have R(X + Y ) -R(X) ≤ m * i + E[l(-X -Y -m * )] -m * i -E[l(-X -m * )] = E l(-X -Y -m * ) -l(-X -m * ) .
Using the convexity, monotonocity of l and the fact that -l(x) ≥ -

x i , x ∈ R d , for 0 < < 1 2 , we get that, l(-X -Y -m * ) -l(-X -m * ) ≤ l(-X -m * -(1 -)Y ) -l(-X -m * ) 1 - ≤ l(|X| + |m * | + (1 -)|Y |) -l(-X -m * ) 1 - ≤ 2 l(|X| + |m * | + |Y |) + d i=1 X i + m i .
Since X and Y are in M θ , the last term is bounded from above by a random variable which is in L 1 . Therefore, using Fatou's lemma, we obtain that,

lim sup 0 R(X + Y ) -R(X) ≤ E[-Y • ∇l(-X -m * )].
Now, using the representation given in Theorem 1.5

R(X + Y ) = max Q∈D l * E Q [-(X + Y )] - E[l * ( dQ dP )], and that R(X) = E Q * [-X] -E[l * ( dQ * dP )] with dQ * dP = ∇l(-X -m * ), we get, R(X + Y ) ≥ E -(X + Y ) • dQ * dP -l * dQ * dP = R(X) -E[Y • ∇l(-X -m * )],
Consequently, the other inequality follows:

lim sup 0 R(X + Y ) -R(X) ≥ -E[Y • ∇l(-X -m * )].
Second assertion is a direct application of Theorem 6 pp 34 in [START_REF] Fiacco | Nonlinear programming: sequential unconstrained minimization techniques[END_REF].

In the following Corollary, we explicit the impact of an independent exogenous shock in the case X and Y are independent.

Corollary 1.10. If X and Y are independent, then under assumptions of Theorem 1.9, we have,

R(X, Y ) = - d i=1 E[Y i ], RA(X, Y ) = -E[Y ].
(1.16)

Remark 1.11.

1. The equations in (1.14) and (1.15) are very interesting and show the relevance of the dual optimizer Q * . More precisely, (1.14) shows that the marginal risk contribution can be quantified thanks to the optimal probability Q * .

2. If only the value of portfolio i changes by a cash amount, that is Y i = c i and Y j = 0 for j = i, then the marginal risk contribution R(X, Y ) = -c i is exactly covered by the marginal risk allocation RA i (X, Y ) = -c i of portfolio i, whereas marginal risk allocations of other portfolios remain unchanged, i.e. RA j (X, Y ) = 0 for j = i. This property of full responsibility for one's own changes in financial position is known as causal responsibility (see [START_REF] Brunnermeier | Measuring and allocating systemic risk[END_REF]). In general, this is no longer true if Y i is a random variable, but in the particular case when Y i is independent of X, this property remains true as suggested by equation (1.16).

3. Equation (1.16) shows an interesting feature: Assume that two institutions i = j change their positions in opposite direction, that is Y i = -Y j , then the marginal risk contribution is zero, as if the portfolios compensate each other and a risk sharing mechanism take place.

Example 3. In this example, we illustrate the impact of an exogenous shock that may depend on X. More specifically, we consider a system with two portfolios X = (X 1 , X 2 ), an exogenous shock Y = (Y 1 , 0) impacting the first component only and a loss function of exponential type as in (1.3):

l(x 1 , x 2 ) = e λ 1 x 1 -1 λ 1 + e λ 2 x 2 -1 λ 2 + αe λ 1 x 1 +λ 2 x 2 .
As per Theorem 1.5, there exists a unique risk allocations m * . To alleviate the expressions, we denote the following:

             C X 1 := E[e λ 1 (-X 1 -m * 1 ) ], C X 2 := E[e λ 2 (-X 2 -m * 2 ) ], C X := E[e λ 1 (-X 1 -m * 1 )+λ 2 (-X 2 -m * 2 ) ], C X 1 Y := E[Y 1 e λ 1 (-X 1 -m * 1 ) ], C X 2 Y := E[Y 1 e λ 2 (-X 2 -m * 2 ) ], C XY := E[Y 1 e λ 1 (-X 1 -m * 1 )+λ 2 (-X 2 -m * 2 ) ].
The matrix M and vector V in Theorem 1.9 can be expressed thanks to the quantities above after some simple but lengthy computations (omitted here):

M = λ 1 C X 1 + αλ 2 1 C X αλ 1 λ 2 C X αλ 1 λ 2 C X λ 2 C X 2 + αλ 2 2 C X , V = -λ 1 C X 1 Y -αλ 2 1 C XY -αλ 1 λ 2 C XY .
The risk contribution marginal and risk allocations marginals follows:

R(X, Y ) = -C X 1 Y -αλ 1 C XY , (1.17) RA(X, Y ) = Common Factor × -C X 2 C X 1 Y -α(λ 1 C X 2 C XY + λ 2 C X C X 1 Y )) -α(λ 1 C X 1 C XY -λ 1 C X C X 1 Y ).
(1.18) (cf. [START_REF] Chen | Stochastic approximation procedures with randomly varying truncations[END_REF]). For the sake of simplicity, we will use the classical "constrained" version with a fixed compact set K as it has the same asymptotic behaviour as the one with projection "à la Chen". In [START_REF] Armenti | Multivariate shortfall risk allocation and systemic risk[END_REF], numerical schemes were developed to find optimal allocations for multivariate shortfall risk measures. They first estimated the different expectations using Monte Carlo/Fourier methods and then a root finding algorithm was used to find the optimum. Although this method shows good results of convergence and is quite fast, it has several drawbacks: It is sensitive to the starting point of the root finding algorithm and one has no control over the error of estimation. With SA, there is one major advantage over the former method: One could derive Central Limit Theorems (CLT) for the estimation and therefore obtain confidence intervals could be obtained for the estimators.

We will study the behaviour of SA algorithms for the different loss functions in example 1.

Recall that, for X ∈ M θ and under the assumption (A l ), there exists a unique risk allocation

m * solution of 1 = E[∇l(-X -m * )].
We fix K a hyperrectangle such that m * ∈ int(K) and we define for X ∈ M θ and m ∈ R d :

                     H 1 (X, m) := ∇l(-X -m) -1, h 1 (m) := E[H 1 (X, m)], σ 2 (m) := E[||H 1 (X, m) -h 1 (m)|| 2 ], m 2+p (m) := E[||H 1 (X, m) -h 1 (m)|| 2+p ||, p > 0, Σ(m) := E[(H 1 (X, m) -h 1 (m))(H 1 (X, m) -h 1 (m)) ].
(2.1)

We introduce the following set of assumptions:

(A a.s. ) i. n≥0 γ n = +∞ and n≥0 γ 2 n < ∞;

ii. h 1 is continuous on K;

iii. sup m∈K σ 2 (m) < ∞.

Theorem 2.1. Let (X n ) a sequence of random variables having the same law as X ∈ M θ and define the sequence (m n ) as follows:

m n+1 = Π K [m n + γ n H 1 (X n+1 , m n )] , m 0 ∈ L 0 , (2.2)
where Π K is the projection into K. Under (A l ) and (A a.s. ), we have, m n → m * a.s. as n → ∞.

Proof. Following the same arguments of Theorem 2.4 in [START_REF] Kaakaï | Estimation of systemic shortfall risk measure using stochastic algorithms[END_REF], the only limit point of the projected ODE associated to the algorithm in (2.2) is m * . Thus, we can use Theorem 2.1 in [START_REF] Kushner | Stochastic approximation and recursive algorithms and applications[END_REF] that argues that m n will converge to the limit point m * if we can verify their assumptions (A2.1)-(A2.5). Indeed, (A2.1) is guaranteed thanks to the assumption (A a.s. )-iii.. The other assumptions are verified thanks to (A a.s. )-ii..

Once we have an estimator of m * , it comes the question of estimating the multivariate OCE

R(X) = d i=1 m * i + E[l(-X -m * )].
A naive way consists in estimating R(X) in a two steps procedure:

• Step 1: Use the estimate m n from (2.2) to have a good approximation of m * .

• Step 2: Use another sample of X to approximate R(X) using Monte Carlo:

R(X) ≈ d i=1 m * + 1 n n k=1 l(-X k -m * ).
(2.3)

A natural way to avoid this two steps procedure is to use a companion procedure (CP) of the algorithm (2.2) and to replace the quantity m * in (2.3) by its estimate at step k -1, that is,

R n = 1 n n k=1 ( d i=1 m i k-1 ) + l(-X k -m k-1 ) .
Note that R n is a sequence of empirical means of non i.i.d. random variables that can be written also as:

R n+1 = R n - 1 n + 1 H 2 (X n+1 , R n , m n ), n ≥ 0, R 0 ∈ L 0 , (2.4)
where

H 2 (x, R, m) := R -F (x, m) := R - d i=1 m i + l(-x -m) .
We are now facing two procedures with different time steps: one for the estimation of m * and the other one for the estimation of R(X). In the following theorem, we prove the consistency of the second procedure using the same time step as the first one (γ n ), namely,

R n+1 = R n -γ n H 2 (X n+1 , R n , m n ), n ≥ 0, R 0 ∈ L 0 . (2.5)
To this purpose we need the following assumption:

(A CP ) ∀m ∈ K, l(-X -m) ∈ L 2 and m → E[|l(-X -m)| 2 ] is bounded around m * .
Theorem 2.2. Assume that assumptions (A l ), (A a.s. ) and (A CP ) hold and let (m n ) be given by (2.2) and (R n ) by (2.5). Then R n → R(X) a.s.

Proof. For n ∈ N, define the sequence (S n ) as:

S n = 1 n-1 k=0 (1 -γ k )
, S 0 = 0.

We have,

S n+1 = S n 1 -γ n = S n 1 + γ n 1 -γ n = S n + γ n S n+1 .
(2.6) Therefore using (2.5), we have,

S n+1 R n+1 = S n+1 (R n -γ n H 2 (X n+1 , R n , m n )) = S n R n + γ n S n+1 R n -γ n S n+1 H 2 (X n+1 , R n , m n ) = S n R n + γ n S n+1 R n -γ n S n+1 R n + γ n S n+1 F (X n+1 , m n ) = S n R n + γ n S n+1 F (X n+1 , m n ).
This implies for n ∈ N * ,

R n = 1 S n R 0 + 1 S n n-1 k=0 γ k S k+1 F (X k+1 , m k ). (2.7) First, we have log(S n ) = - n-1 k=0 log(1 -γ k ) ≥ n-1 k=0 γ k ,
and since by assumption, the RHS of the last inequality goes to ∞ as n → ∞, we deduce that S n → ∞ as n → ∞ and we get immediately that the first term of the RHS of (2.7) goes to 0 as n goes to ∞. Rewriting (2.7) by introducing f (m) := E[F (X, m)] and the martingale difference sequence δM k+1 = F (X k+1 , m k ) -f (m k ) with respect to the natural filtration

F k := σ(m 0 , X 1 , ..., X k ), we obtain, R n = 1 S n R 0 + 1 S n n-1 k=0 γ k S k+1 δM k+1 + 1 S n n-1 k=0 γ k S k+1 f (m k ).
(2.8)

Thanks to (2.6), we have n-1 k=0 γ k S k+1 = S n . Because f is convex (assumption (A l )-ii.) and therefore continuous at m * , Cesaro's Lemma implies that the third term in the previous equality converges to f (m * ) = R(X). The a.s. convergence of R n will follow from the a.s. convergence of the second term toward 0. Indeed, let us denote,

M γ n = n k=1 γ k-1 δM k Note that (M γ n ) is a F-martingale such that M γ ∞ = ∞ n=0 γ 2 n E[|δM n | 2 |F n-1 ].
But we also have,

E[|δM n | 2 |F n-1 ] ≤ E[|l(-X -m)| 2 ] |m=m n-1 ,
and assumption (A CP ) implies that

sup n≥1 E[|δM n | 2 |F n-1 ] < ∞, a.s.
Using the martingale convergence theorem, we get that (M γ n ) converges to some random variable. Finally, by Kronecker's Lemma we deduce that the second term of (2.8) converges to 0. This completes the proof.

The step sequence in (A a.s. )-i. is typically of the following form γ n = c n γ , where γ ∈ ( 1 2 , 1] and c is a positive constant. The choice of c plays a key role in the rate of convergence of SA algorithms. In order to circumvent problems related to the specification of the constant c, which are classical, we will use "averaging" techniques introduced by [START_REF] Ruppert | Stochastic approximation[END_REF] and [START_REF] Polyak | Acceleration of stochastic approximation by averaging[END_REF]. We introduce the following assumptions:

(A a.n. ) i. h 1 is continuously differentiable and let A := Dh 1 (m * ); ii. (H 1 (X n+1 , m n )1 |mn-m * |≤ρ ) is uniformly integrable for small ρ > 0;
iii. For some p > 0 and ρ > 0, sup

|m-m * |≤ρ m 2+p (m) < ∞; iv. Σ(•) is continuous at m * and Σ * := Σ(m * ) is positive definite.
The next theorem states the rate convergence of the average of the iterates of (RM) algorithm.

Theorem 2.3. Assume γ ∈ ( 1 2 , 1) and that assumptions (A l ), (A a.s. ) and (A a.n. ) hold. For any arbitrary t > 0, we define the (PR) sequence (m n ) as:

m n := γ n t n+t/γn-1 i=n m i , (2.9)
where any upper summation index is interpreted as its integer part. Then, we have

t γ n (m n -m * ) → N 0, A -1 Σ * (A -1 ) + O 1 t .
(2.10)

Proof. This is a consequence of Theorem 1.1 chapter 11 page 377 in [START_REF] Kushner | Stochastic approximation and recursive algorithms and applications[END_REF] if we could verify their assumption (A1.1). Thanks to Theorem 2.1 of chapter 10 in [START_REF] Kushner | Stochastic approximation and recursive algorithms and applications[END_REF], the condition (A1.1) is satisfied as soon as their conditions (A2.0)-(A2.7) hold. Assumption (A2.0) is automatically verified. (A2.1) is satisfied by (A a.n. )-ii.. (A2.2) is a consequence of Theorem 2.1. (A2.4) follows from Taylor's expansions and (A a.n. )-i.. (A2.5) follows from the fact that h 1 (m * ) = 0. (A2.6) is satisfied since m * is the optimum of a convex optimization problem. The first part and second parts of (A2.7) are guaranteed thanks to (A a.n. )-iii. and (A a.n. )-iv.. Finally, (A2.3) follows easily from Theorem 4.1 chapter 10 page 341 in [START_REF] Kushner | Stochastic approximation and recursive algorithms and applications[END_REF] since all their assumptions (A4.1)-(A4.5) are satisfied.

Remark 2.4. The previous CLT theorem states that under suitable conditions our average sequence is asymptotically normal with a corresponding covariance matrix that depends on Σ * and A. These quantities are unknown to us because, first, in general, they cannot be expressed in a closed form and second they depend on the optimum m * . So, in practice, these two quantities need to be approximated in order to derive confidence intervals. In the following proposition, we provide consistent estimators of these two quantities.

Proposition 2.5. Suppose (A l ), (A a.s. ) and

(A a.n. ) hold. If m → E[||H 1 (X, m)|| 4 ] is bounded around m * , then, Σ n := 1 n n-1 k=0 H 1 (X k+1 , m k ) H 1 (X k+1 , m k ) → Σ * a.s. as n → ∞.
(2.11)

Let A n be the matrix whose elements A n (i, j) for i, j ∈ {1, ..., d} are defined as follows:

A n (i, j) := 1 n n-1 k=0 H i 1 (X k+1 , m k + e j ) -H i 1 (X k+1 , m k ), then, lim →0 lim n→∞ A n = A a.s.
(2.12)

Proof. The proof of this proposition relies mainly on the martingale convergence theorem.

Let (δM k ) k∈N be the sequence defined as:

δM k := H 1 (X k+1 , m k ) H 1 (X k+1 , m k ) -Σ(m k ) -h 1 (m k ) h 1 (m k ), k ≥ 0.
(δM k ) k≥0 is a martingale difference sequence adapted to F and therefore the following sequence (M k ) k∈N * defined as:

M k := k i=1 δM i i , k ≥ 1, is a F-martingale. Furthermore, the boundedness of m → E[||H 1 (X, m)|| 4 ] around m * ,
assumptions (A a.s. )-ii. and (A a.n. )-iv. imply that:

sup k≥1 E[||δM n || 2 |F n ] < ∞ a.s.
Consequently, the martingale convergence theorem implies the existence of a finite ran-

dom variable M ∞ such that M n → M ∞ .
We then apply Kronecker's Lemma to get that

1 n n-1 k=0 δM k+1 → 0. Since, Σ n = 1 n n-1 k=0 δM k + 1 n n-1 k=0 Σ(m k ) + 1 n n-1 k=0 h 1 (m k ) h 1 (m k ),
we deduce that Σ n → Σ * . The proof of (2.12) follows using the same arguments above.

Remark 2.6.

1. Instead of averaging on all observations for the estimators above, we could average using only recent ones. This might improve the behaviour of these estimators.

2. If we denote

V n = (A n ) -1 Σ n ((A n ) -1
) , then we can obtain an approximate confidence interval for m * with a confidence level of 1 -α in the following form:

  m j n - V ,jj n tcn γ q α , m j n - V ,jj n tcn γ q α   , j ∈ {1, ..., d}, γ ∈ ( 1 2 , 1), (2.13)
where q α is the 1 -α 2 quantile of a standard random variable.

Numerical Analysis and Examples

In this section, we analyze and test the numerical methods developed in the previous section for the estimation of optimal allocations given by (2.9) and risk measures given by (2.5). The implementation was done on a standard computer using Python and we write CT for computational time. All the computations were run on a standard laptop with a processor Intel(R)

Core(TM) i7-9850H CPU @ 2.60GHz. The common parameters used in the computations are summarized in the following table :   Parameters Values n 500000 γ 0.8

t 10 c 1 10 -6
Table 1: Set of common parameters.

A first example

We start here by estimating optimal allocations and multivariate OCE for the first loss function in (1.3), that is:

l(x) = d i=1 e λ i x i -1 λ i + αe d i=1 λ i x i , λ i > 0, α ≥ 0.
We denote by λ the vector of λ i , i ∈ {1, ..., d}. First, we test our algorithms in the case d = 2 and the vector X having a Gaussian distribution, as optimal allocations are expressed in a closed form in this case (see (1.10)). This will allow us to test the efficiency of our algorithms. Three cases are considered: In the first case, we take α = 0 and λ = (1, 2), which as previously mentioned, corresponds to the entropic risk measure, a second one with α = 1 and λ = (1, 1), and finally a third one with α = 1, λ = (1, 2). As for the parameters for the normal distribution of X, we fix σ 1 = σ 2 = 1 and we take ρ ∈ {-0.9, -0.5, 0, 0.5, 0.9} for each of the three cases. The compact set K was set to [0, 3] 2 and the initial term m 0 = (0, 0). The table above summarizes the numerical results for the first case. The two columns CI1 and CI2 represent the confidence intervals of the (PR) estimators with a confidence level of 95%. Since α = 0, the exact optimal allocations do not depend on the correlation coefficient ρ. This explains why we obtain the same values for m * for different values of ρ.

The same remark goes for the estimators m n . Since λ 2 > λ 1 , we expect as per formula (1.10) that m 2 * > m 1 * . These numerical results suggest that the (PR) estimators m n as well as the (RM) estimator R n approximate very well the exact optimal allocations m * and the risk measure R(X). The width of the first confidence intervals (resp. second confidence intervals) is approximately 0.008 (resp. 0.02) which gives an accuracy of 1.5% (resp. 2%) for the first When taking the same values for λ 1 and λ 2 , the system becomes symmetric and we obtain the same optimal allocations for the first and second component. We also notice that optimal allocations and their estimators increase with the correlation coefficient ρ as it was expected from remark 1.6. Again, the estimators approximate well the optimal allocations and the risk measure. The accuracy of all confidence intervals is around ≈ 1%.

estimator m 1 n (resp. m 2 n ). ρ R n m 1 n m 2 n CI1 CI2 R(X) m 1 * m 2 * CT(s) -0.
ρ R n m 1 n m 2 n CI1 CI2 R(X) m 1 * m 2 * CT(s)
-0.9 1.6477 0.6194 1. 1275 [0.6152, 0.6237] [1.1184, 1.1366] In this final case, we take different values for λ 1 and λ 2 . Table 4 shows that the optimal allocations can be well approximated by the estimator in (2.9). This is also the case for the estimator R n . Again, the optimal allocations as well as the risk measure increase with the correlation coefficient (see Figure 1). All confidence intervals have an accuracy between 1% and 2% except the second confidence interval in the case ρ = 0.9 where the accuracy is a bit higher and is approximately around 4%.

X = µ + ZΓβ + √ ZΓ 1/2 Y, (3.1)
where Z ∼ IG(δ 2 , α 2 M N IG -β Γβ) and IG(χ, ψ) denotes the Inverse Gaussian distribution with parameters χ, ψ > 0 and Y ∼ N(0, I d ). Note that the random variable X|Z ∼ N(µ + ZΓβ, ZΓ), hence the name variance-mean mixture. The parameters of the MNIG distribution have natural interpretations. The parameter α M N IG is a shape parameter and controls the heaviness of the tails. The parameter β is a skewness vector parameter, δ is a scale parameter and µ is a vector translation parameter. Finally, the matrix Γ is assumed to be a positive semidefinite symmetric matrix and controls the degree of correlations between components and assumed to be such that det(Γ) = 1. In order for the MNIG to exist, the inequality α 2 M N IG > β Γβ must be satisfied. The cumulant generating function of the MNIG could be derived easily in a closed form of the parameters:

Φ X (t) = δ α 2 M N IG -β Γβ -α 2 M N IG -(β + it) Γ(β + it) + it µ.
This shows that the MNIG is infinitely divisible. Thus, we can easily evaluate the moments of this distribution. The mean vector and the covariance matrix Σ of X are given in the following:

E[X] = µ + δΓβ α 2 M N IG -β Γβ , (3.2) Σ = δ α 2 M N IG -β Γβ -1/2 Γ + α 2 M N IG -β Γβ -1 Γββ Γ . (3.3)
Note that due to the parameter β, even when µ = 0 (and Γ = I d resp.), the mean of the MNIG distribution is not null (the covariance matrix is not diagonal resp.). For more details about MNIG, we refer to Øigård et al. (2004).

In order to make the numerical analysis more realistic, we fitted, the parameters of the MNIG

Numerical Results

We will test our SA algorithms with a trivariate MNIG distribution for the polynomial loss functions. We recall that the polynomial loss function is given by:

l(x) = d i=1 ([1 + x i ] + ) θ i -1 θ i + α i<j ([1 + x i ] + ) θ i θ i ([1 + x i ] + ) θ j θ j , θ i > 1, α ≥ 0.
Since no closed formula is available to us in this case, we decided to use a Monte Carlo scheme as a benchmark to the SA method. This scheme consists in approximating the expectation in (1.11) by the corresponding Monte Carlo estimator and then to use Nelder-Mead algorithm as a minimization algorithm to find the optimal allocations. The compact set K for the SA method was set to [0, 2] 3 and α was taken to be equal to 1. First, we compare both methods in the case where the parameter θ was taken to be equal to θ = (2, 2, 2). Then, in a second case, we test both algorithms with the parameter θ = (1, 2, 3). The table 6 show that both methods give approximately the same values for the optimal allocations m * as well as the risk measure R(X). The values of the optimal allocations are approximately the same among the three components. This could be explained by the fact that the three components have almost the same distribution as shown in the figure 2 and the fact that we have taken θ = (2, 2, 2), so that the system becomes nearly symmetric. The Monte Carlo method is seven times faster that the SA method. However, with the Monte Carlo method, we do not have any confidence intervals and hence no control over the error of estimation. Moreover, since in the Monte Carlo method, we are using a deterministic minimization algorithm, it is sensitive to the initial values (Recall that we do not have this problem with the SA method). We do not have this problem with the SA method. 4 Appendix: Estimation of MNIG parameters

SA

Computational aspects

In this section, we give more details about the estimation of the MNIG parameters. The most conventional way to estimate the latters is the maximum likelihood estimation method.

However, in the case of MNIG, this method shows slow convergence due to the complexity of the likelihood. We therefore, propose here to use the Expectation Maximization (EM) algorithm which is known to be fast and accurate. The EM algorithm is a powerful tool that is used for maximum likelihood estimation for data containing "missing" values. This is suitable for distributions arising as mixtures which is the case of MNIG distributions where the mixing variable Z is unobserved. The EM algorithm is an iterative algorithm that consists of two steps at each iteration. Denoting θ = (δ, µ, β, α M N IG , Γ), X = (X 1 , ..., X N ) the observed data and Z = (Z 1 , ..., Z N ) the unobserved one, L(X, Z, θ) = log(P(X, Z)|θ) the complete data likelihood and θ n the estimate of θ at step n, we repeat the two following steps until some convergence criteria is verified:

• E-step : Compute Q(θ|θ n ) := E Z|X,θ n [L(X, Z, θ n )].
• M-step : choose

θ n+1 = argmax θ Q(θ|θ n ).
Next, we explicit the calculations of Q(θ|θ n ) in the E-step for the MNIG distribution. We have, by taking the constants away, L(X, Z, θ) = log (P(X, Z|θ))

= log (P(X|Z, θ)) + log (P(Z|θ))

= - d 2 N i=1 log(Z i ) - N 2 log(det(Γ)) - 1 2 N i=1 1 Z i (X i -µ -Z i Γβ) Γ -1 (X i -µ -Z i Γβ)+ N δ 2 (α 2 M N IG -β Γβ) + N log(δ) - 3 2 N i=1 log(Z i ) - 1 2 N i=1 δ 2 1 Z i + (α 2 M N IG -β Γβ)Z i .
In the following, we summarize the EM algorithm for the parameters estimation of MNIG distribution: The convergence properties of the EM algorithm are discussed in details in Algorithm 1: EM algorithm for parameters estimation of MNIG distribution Input: Observations (X 1 , ..., X N ), initial value θ 0 , tolerance tol and M number of iterations; 1 Set θ n = θ 0 and compute φ n , ζ n with (4.2) and (4.3); 2 Compute θ n+1 using in order (4.10), (4.12), (4.11), (4.13), (4.14) and (4. Output: Estimated parameters θ; [START_REF] Mclachlan | The EM algorithm and extensions[END_REF]. However, to avoid getting stuck in a local maximum, we will need to run the algorithm from several starting points to ensure that the obtained maximum is the global one. We can also combine the algorithm with other numerical methods, such as Newton-Raphson algorithm, to speed up the convergence.

Numerical aspects

We applied the EM algorithm described in the above subsection to a data set of daily log return of three European stock indices (CAC 40, BEL 20 and AEX) for a period from 12/05/2020 to 10/05/2022 obtained from the website of Euronext. The data set consisted of 514 observations. In order to test the behavior of the algorithm, several initial values were considered. Note that the conditional expectations in the E-step do not involve the parameters β and hence the convergence of the algorithm will not depend on the initial value of β. We fixed β = (0, 0, 0) and we stopped the iterations when ||θ n+1 -θ n || < tol for tol = 10 -5 and tol = 10 -10 . The values of the estimates of estimates for initial values were the same and are given in the following: α = 365.78, δ = 0.00373, β = (-64.28, 41.45, 7.35)

  F, P ) a probability space and we denote by L 0 (R d ) the space of Fmeasurable random vectors taking values in R d . For x, y in R d , we denote by || • || the Euclidean norm and x • y = x i y i . For a function f : R d → [-∞, ∞], we define f * the convex conjugate of f as f * (y) = sup x {x • y -f (x)}. The space L 0 (R d ) inherits the lattice structure of R d and hence, we can use the classical notations in R d in a P -almost-surely sens. We will say for example, for X, Y ∈ L 0

  5147 0.5049 0.9906 [0.5006, 0.5092] [0.9803, 1.0009] 1.5 0.5 1 70.0251 0.9 1.5264 0.5031 0.9970 [0.4988, 0.5074] [0.9867, 1.0073] 1.5 0.5 1 70.8549 Table 2: Numerical results: α = 0 and λ = (1, 2).

Figure 1 :

 1 Figure 1: R n , m 1 n and m 2 n as a function of ρ.

  In this example, we will be working with a Multivariate Normal Inverse Gaussian (MNIG) distribution for the vector X instead of a Gaussian distribution. The MNIG distribution yields a more flexible family of distributions that can be skewed and have fatter tails than the Gaussian distribution. For a fixed d, a MNIG distributed random variable is a variance-mean mixture of a d-Gaussian random variable Y with a univariate inverse Gaussian distributed mixing variable Z. The MNIG distribution has five parameters α M N IG > 0, β ∈ R d , δ > 0, µ ∈ R d and Γ ∈ R d×d and can be constructed as follows:

Figure 2 :

 2 Figure 2: of the log-returns of the three indices.

  15); 3 n = 0; 4 while ||θ n+1 -θ n || ≥ tol and n < M do 5 E-step: θ n ← θ n+1 and compute the new φ n and ζ n with (4.2) and (4.3); 6 M-step: Compute θ n+1 using in order (4.10), (4.12), (4.11), (4.13), (4.14) and (4.15); 7 n ← n + 1;

Table 4 :

 4 Numerical results: α = 1 and λ = (1, 2).

	1.6354 0.6202 1.1285 79.6036

Table 5 :

 5 Parameters sets for the MNIG.

	Parameters		MNIG
	α M N IG		365.78
	δ		0.00373
	β	(-64.28, 41.45, 7.35)
	µ	(0.00084, 0.00024, 0.00055)
			2.338 1.796 2.080 
	Γ	 	 1.796 2.327 2.088 
			2.080 2.088 2.555

distribution on the daily log-return of three European indices: CAC 40, BEL 20 and AEX.

The estimated parameters obtained using the Expectation Maximization (EM) algorithm, explained in details in Section 4, are summarized in the first column of the following table

5

.

Table 6 :

 6 Numerical results: Polynomial loss function with θ = (2, 2, 2) and MNIG distribution.

		CI-SA	Monte Carlo
	m * 1	0.31747 [0.31746, 0.31749]	0.31748
	m * 2	0.31748 [0.31746, 0.31750]	0.31745
	m * 3	0.31742 [0.31740, 0.31743]	0.31737
	R(X)	0.31336	0.31332
	CT(s)	141.20	28.07

Table 7 :

 7 Numerical results: Polynomial loss function with θ = (1, 2, 3) and MNIG distribution.

		SA	CI-SA	Monte Carlo
	m * 1	0.21996 [0.21995, 0.21998]	0.21994
	m * 2	0.25127 [0.25125, 0.25129]	0.25130
	m * 3	0.29929 [0.29927, 0.29931]	0.29926
	R(X)		0.37532	0.37529
	CT(s)		98.48	9.85

In the following sens: R(X+ m) = R(X) -d i=1 m i

The authors research is part of the ANR project DREAMeS (ANR-21-CE46-0002) and benefited from the support of the "Chair Risques Emergents en Assurance" under the aegis of Fondation du Risque, a joint initiative by Le Mans University and Covéa.

We notice the following:

• R(X, Y ) is disentangled into two components. The first one is the contribution of the first component in the risk contribution marginal and the second is a systemic contribution that is proportional to α. This same remark holds for RA 1 (X, Y ).

• The asymmetry of the shock on X 1 can be seen in the systemic contribution in RA 1 (X, Y ) and RA 2 (X, Y ). Indeed, we notice that both components are impacted by the shock and this is reflected by the term -αλ 1 C X 2 C X Y for the first component and -αλ 1 C X 1 C X Y for the second. However, there is a correction term proportional to λ 2 that is subtracted from the first component whereas another correction term proportional to λ 1 is added to the second component.

• In the case α = 0, i.e. without a systemic component, the risk marginal of the second portfolio is zero. This something we would expect as we applied a shock only on the first component. In other words, the first component takes full responsibility in this case.

In the rest of the paper, for every X ∈ M θ , we will assume the following:

Under assumption (A l ), there exists a unique risk allocation m * that is characterized through the following equation:

Computational aspects

In this section, we develop numerical schemes to compute the optimal risk allocations m * and R(X) using stochastic algorithms (SA). This is because the optimal allocations are solutions of a convex optimization problem whose objective function can be expressed as an expectation. Stochastic algorithms are generally used to find zeros of a certain function h that is unknown but could be approximated using some estimate. More specifically, SA algorithms take the following form:

and (γ n ) is a step sequence decreasing toward 0. This algorithm is known as Robbins-Monro algorithm (RM). For an overview of SA algorithms, we refer to [START_REF] Duflo | Algorithmes stochastiques[END_REF]. However, in order to be able to use classical convergence results of SA, we need a sub-linear growth over the function h (see for example condition (8) of Theorem 2.2 in [START_REF] Bardou | Computing var and cvar using stochastic approximation and adaptive unconstrained importance sampling[END_REF]), which in our case, considerably limits the choice of loss functions. To circumvent this condition, we will use a "constrained" variant where we force the iterations of the (RM) algorithm to remain in a certain compact K set that contains the optimal allocations. One could also use the well-known projection "à la Chen" algorithm based on reinitializations of the algorithm and taking larger compact sets each time the iteration goes out of the compact set Taking the conditional expectation on the both sides and denoting ζ n = (ζ n i ) i=1,...,N and φ n = (φ n i ) i=1,...,N , where

we get, again by removing the quantities that does not depend on θ,

The quantities φ n i and ζ n i can be derived from the fact that Z|X, θ follows a Generalized Inverse Gaussian distribution, i.e., Z|X, θ ∼ GIG -d+1 2 , q(X), α M N IG , where q is given as:

More precisely, they are given by,

K v is the modified Bessel function of the second kind with index v ∈ R. Having calculated Q(θ|θ n ), we now need to calculate the next term θ n+1 := argmax θ Q(θ|θ n ). This will be done by first calculating the gradient of Q.

To alleviate the expressions, we will denote φ

Setting the previous set of equations to 0, we obtain,

From the second equation we deduce that, Using this in the fourth equation, we obtain,

(4.12)

Now that µ and Γβ are explicitly known, denoting

n Γβ(Γβ) , we get from the fifth equation,

Going back to (4.11), we get,

Finally, using (4.9), α M N IG can be deduced as:

The number of iterations needed until convergence along with the computational (CT) time can be seen in 8.

Initial Values tol = 10 -5 tol = 10


Johann Stan
email: johann.stan@alcatel-lucent.com

An overview and research vision of the use of Semantic Technologies for User Modeling and Collaborative Tagging

This work aims at providing a review of recent work in semantic modeling of users. A second focus area is the tagging. Tagging links a user to a resource by a tag and we consider that this approach is suitable to partially answer the question: How semantic technologies can be leveraged to model social networks. Also this is a good reference work to see how users and resources can be interlinked and thus how a social network model can go beyond the traditional people-to-people model.

Introduction

The current developments in the field of the semantic web enables a new realm of applications. Semantic web made it possible to have the necessary tools to handle computer-understandable semantics. These tools, generally evolving from XML are used to enrich the description of web-pages. Languages like OWL (Web Ontology Language), RDF (Resource Description Framework), DAML+OIL are some of the most widely used languages. The advantage of these languages is the fact that they are machine readable and strongly related to Description Logics. A state-of-the art on this subject can be found in [START_REF] Baader | The Description Logic Handbook[END_REF]. The RDF [START_REF]Resource Description Framework (RDF) / W3C Semantic Web Activity[END_REF] language makes statements about resources in the form of triples: (subject, predicate, object). The subject denotes the resource and the predicate denotes the relationship between the subject and the object, which can be another resource or a literal. The OWL [START_REF]OWL Web Ontology Language Overview[END_REF] language is a subset of RDF which allows a wide range of inferences providing more semantics in the way of expressing meaning.

Towards Semantic User Representation

We examine in this section recent approaches and models that address the issue of user modeling. We intend to show how ontologies emerged from ad-hoc modeling techniques as a powerful way to describe the user. The reason whu we adress the representation of individuals is that this allows to better understand the collaborative aspects.

FOAF (Friend-of-a-Friend) [START_REF] Brickley | FOAF Vocabulary Specification 0.91[END_REF] is an ontology-based RDF vocabulary to describe people profiles, friends, affiliations, creations etcThe aim of FOAF is that of a completely decentralized machine-readable social network that is based on personal profiles. The profile contains mostly static data, like personal information, work history, links to contacts and services. The use of FOAF is currently way beyond popular social network communities, like LinkedIn. , which mostly addresses professional interactions, or more "fashionable playgrounds", like Facebook and its rival MySpace3 . FOAF is currently used only by research communities or semantic-web developers. A person begins by describing himself or herself using the foaf:Person class, listing key attributes such as name, gender and resources relating to them. They can also list their interests, and each person is uniquely identified by using the foaf:mbox propery containing their email address. An alternative identification propery is foaf:openid conforming to the OpenID perspective of using a unique single URI to establish teh identity. The person in question then moves on to describing their friends, each friend is an instance of the foaf:Person class. FOAF is both machine-readable, and humanreadable, and was adopted by LiveJournal, the blogging site, to offer the facility for each user to export their personal informatio. FOAF has seen a slow adoption by WEb 2.0 sites and services, this could be due to the lack of interest in the exportation of social information from one rival site to another and thus going beyond the walled gardens. It is, however, the most widely used specification for expressing personal and realtionship information within the Semantic Web community.

In their work, [START_REF] Carmagnola | Towards a Tag-Based User Model: How Can User Model Benefit from Tags?[END_REF] investigates how tagging allows to infer data about user preferences or interests, allowing to create a user model. Tagging is the process where users label or annotate different ressources (web-pages for example) with the objective to share, organize or diffuse them. This is a concept strongly related to the web 2.0 and social networking. The way users employ tags might give an insight on different issues like how interested he is in the given ressource and the type of tags used (many synonims for example) can infer subjective details like level of creativity. Tests and a deep analysis is needed to better unerstand relations between tagging and these high-level concepts (preference, interest). Research is currently in progress examining how ontologies like Wordnet allows to categorize and to infer automatically the type of tag and relationship with the user (matching). [START_REF] Hessling | Semantic User Profiles and their Applications in a Mobile Environment[END_REF] proposes an architecture where semantic user profiles are used in a peer-topeer mobile environment. In a ubiquitous mobile environment, service providers like a cinema are equipped with bluetooth-enabled devices to broadcast the service (movie being played for exemple). Passing people have mobile devices which store the owner's profile (interests, preferences, disinterests). Easy matching between user interests and service is necessary, and descriptive logic is considered a good approach. The user profile is relatively simple, consisting simply of the union of interests and disinterests: A common domain ontology for concepts in both services and profile description is used to make possible this semantic matching using a reasoner. The system is completely peer-to-peer, allowing independence, cost effectiveness, scalability and most importantly, privacy management, which is the result of the fact that profiles are stored on the mobile device.

[7] argues that techniques like RDF and OWL together with ontologies are the key elements in the developpement of the next generation of user profiles. The User Profile Ontology presented grew out from a quite simple model containing semantical contact information encoded in the RDF language.

In the proposed ontology 1, which is built around three imported ontologies (Person Ontology, containing classes relevant only to the user; Organisation Ontology, containing business oriented information and a Common Ontology, containing information relevant to both persons and organisations) personal information like e-mail, telephone, Instant Messaging identifier, physical addresses are uniquely identified by a GenericContactIdentifier class. Social interactions inherit properties from an Event class. These interactions are classified into voice, text, real-time, online communications. The address book of the user is stored in the class ContactGroup. The article presents well the fraemework, but the structure of the profile ontology is not clear in all aspects. The applicability of the Profile Ontology is tested with a Dynamic Address Book application, which maintains dynamically the contact list of a user based on the frequency of communications. Rules are defined to decide when to remove or add a contact according to a given calling frequency threshold. Results show that this is a promising approach, but further evaluation of the system is needed. The framework is realised using among others the Java based Jena engine.

[8] presents an application-independent user profile ontology. The objective is to create a "general, comprehensive and extensible" user model taking into account existing literature, user-and context models. The use of ontologies in user profiling is a known issue, however the problem with existing proposals is that they respond only to application specific needs, mainly in personalized information retrieval and web search. It is important to stress out that the proposed ontology deals only with the static profile of the user, not the dynamic or contextual one (like current position, occupation or terminal).

Before presenting the proposed ontology model, we collect all the necessary concepts for a user profile considered in the literature, based on the same article. This can give an insight of what is currently considered important in describing end-users. A first observation is that dyamic, context-aware concepts are not yet well coupled with the static ones. The proposed ontology (Figure 2), where the main class is Person uses these concepts and many others to create a profile applicable in any kind of domain or application. Therefore, no restrictions are present in the ontology, it is completely up to the developer to personalize it according to the specific needs of the project. However, some restrictions could have been added like the fact that a friend of a Person can only be a Friend, which is still domain independent and represents general knowledge.

[15] addresses the issue of modeling users in a context-aware "smart home" environment. Static and dynamic user profiles are distinguished. The Amigo project aims to develop services for a smart home environment, which offers proactivity to users according to current context or situation. A machine-understandable shared user-model is needed between different services to make a unified way of querying preferences and interests possible. As mentioned before, the user model is separated into two components:

-The context-aware static user profile: Tree-based representation of individual user preferences and personal data, grouped in agreement with user ontology representation in the system The context-aware dynamic user profile This profile learns user behaviour from history of activities, learning meaning the ability to recommend a given topic in a given situation (for example a movie when Bob is alone at home on Friday night). Interaction history is stored in the form "Context"-¿"User Action", where context is a set of environment descriptors and user action any kind of interaction with the "smart home". Two machine learning techniques, CBR (Case-based reasoning) and SVM (Support Vector Machines) were used to test how items can be classified into terms like "good" or "bad" for a given context. Test results indicate that CBR gives good performances in almost any situation, since the retrieval of items is based on the overall similarity of context descriptors. UPOS (User-Profile Ontology with Situation-Dependent Preferences Support) [START_REF] Sutterer | UPOS: User Profile Ontology with Situation-Dependent Preferences Support[END_REF] is part of the Spice project It is probably the newest user profile ontology, addressing both static and context-aware aspects. This ontology, defined in OWL, allows creating situation-dependent sub-profiles. A user has a profile and a context (location or activity) associated. The notion of condition is defined, which includes a user, an operator and a context-value. For example, a condition can be: "if the context of user Bob equals the MyOffice location...". According to this condition, a corresponding sub profile can be applied that contains all personalization indications for services (e.g. not to use sms). This approach was inspired by the ETSI human factors group, which defined guidelines for designing context-aware user profiles. The most important guideline is to structure the profile into sub-profiles, each containing user preferences that correspond to a specific situation, as seen in the previous example.

Linking peoples and objects with the use of annotations/tags

The representation of tags and tagging activity has been going through the same process as user profiles: semantic technologies proved to be an important component in linking concepts together in the web 2.0 era. Many studies have been performed in a lot of disciplines via innovative approaches. However, without consistent structures and semantics, contributions of these studies are unable to analyze the social phenomenon relating the folksonomies. In order to operate social ecosystems on the Web, we need various technical and social analyses for folksonomies as well as formal representation for adopting the results. The semantics of tagging data is primarily about an agreement on the meaning among people or a community in the social space. A common semantics provides a way to share tag representation among services. We now provide an overview of a number of existing efforts that had the common aim of representing the concepts and operations of tags and tagging. Gruber [START_REF] Thomas | Ontology of folksonomy: A mashup of apples and oranges[END_REF] describes tagging as a relation between:

-An object, i.e. the resource to be tagged. For example, a bookmark, a picture, a blog post etc.

-A tag, i.e. the tag associated with the resource -A tagger, i.e. the agent -more generally a person -that created the link between the tag and the object -A source, i.e. the space where the tagging action has been performed, e.g.: Flickr, del.icio.us.

Notably, Gruber defines the source as the scope of namespaces or universe of quantification for objects. This allows one to differentiate between tagging data from different systems and is the basis for collaborative tagging across multiple applications. P. Mika [START_REF] Mika | Ontologies Are Us: A Unified Model of Social Networks and Semantics[END_REF] already represented the tagging action from a theoretical point of view, but did not use this notion of source that Gruber introduces. Yet, while this model is widely commented, there is no currently available implementation to our best knowledge. It was also Gruber who defined the term ontology, as an explicit specification of a conceptualization of a domain [START_REF] Gruber | Toward principles for the design of ontologies used for knowledge sharing[END_REF].

Newman et al. [START_REF] Richard | Tag Ontology[END_REF] defined an ontology of tags and tagging, simply called the Tag Ontology 2, that describes the relationship between an agent, an arbitrary resource, and one or more tags. Thus, in his ontology, the three core concepts Taggers, Tagging, and Tags are used to represent the tagging activity. Contrary to Gruber [START_REF] Thomas | Ontology of folksonomy: A mashup of apples and oranges[END_REF], it does not represent the source of the tagging action. Yet, this ontology has been implemented (in OWL), is available on the Web, and is currently used in some projects such as Revyu.com, a review website combining Web 2.0 and Semantic Web technologies. Notably, in this ontology tags are represented as instances of the tags:Tag class which is assigned custom labels, i.e. the string representing the tag as seen by the user. Being instances of a class means that they are assigned a URI. URIs are a key feature of the Semantic Web, since, contrary to simple literals, they can be used as subject of triples, while literals can be only used as objects. This way, tags-identified by URIs can be linked together and people can semantically represent connection and similarities between tags. For this purpose the ontology introduces a tags:related property.

Yet, this relation does not have much semantics, since it does not define the nature of the relation, e.g. if this a linguistic variation or because it identifies a similar topic. Another limitation is that the ontology does not define any cardinality constraint on the number of labels a Tag can have. This can raise problems since it allows a Tag instance to have two completely disjoint labels (i.e. a Tag instance with labels "RDF" and "Paris"), which makes no sense from a tagging point of view. Still, this ontology reuses pre-defined Semantic Web vocabularies, making it compliant with existing standards. SKOS properties are used to model relations between tags and the Tag class itself inherits from skos:Concept. DublinCore is used to represent the date of a tagging action, with subproperties of dc:date. Finally, the ontology relies on FOAF to identify the tagger of a tagging action thanks to foaf:Person. SIOC (Semantically-Interlinked Online Communities [START_REF] Breslin | Semantically-Interlinked Online Communities[END_REF], http://sioc-project.org) represented on 3 is an ontology-based framework aimed at interconnecting online community sites and internet-based discussions. The idea is to enable crossplatform interoperability so that conversation spanning over multiple online media (e.g. blogs, forums, mailing lists...) can be unified into one open format. The interchange format expresses the information contained both explicitly and implicitly in internet discussion methods, in a machine-readable manner. A similar approach is proposed by the OPSN (Open Portable Social Network, http://www.opsn.net/) initiative which also covers notification and synchronization of contacts across platforms. However there is no existing implementation,and privacy control for personal published information seems not to have been addressed yet. Online communities allow web users to express their thoughts, gain feedback and critique and interact with individuals who share a similar interest. Modern web users all have some kind of participation in this realm: forums, chat rooms, newsgroups and Social Networking Sites (SNS). Each community can be considered a walled garden, without link to others. The SIOC project focuses on ways to integrate and merge these gardens, providing bridges between the knoweldge that exists there.

The SIOC ontology uses existing specifications such as FOAF and RSS to define classes and their properties. The ontology represents knowledge existing in online communities with six main classes: Site, forum, post, role, usergroup and user. Thus, a flexible range is offered for capturing knowledge in online communities. The concept sioc: User can be thought of as the central point for the ontology, sioc: User is a subclass of the foaf: OnlineAccount class. The SIOC ontology also defines properties that relate each of the classes to the user in a similar manner to how a user interacts within an online community. SIOC allows the capture of interactions between individuals, and is capable of expressing the role an individual plants in an online social space. The Social Semantic Cloud of Tags (SCOT) (Figure 4) ontology aims to describe the structure and the semantics of tagging data and to offer social interoperability of the data among heterogeneous sources [START_REF] Hak Lae Kim | Social Semantic Cloud of Tag: Semantic Model for Social Tagging[END_REF]. Both Tagcloud and Tag class in SCOT play a role to be able to represent social and semantic context of tagging, since both classes include users, tags, and resources and additional information to clarify tags' semantics. scot:TagCloud has properties that describe a certain user, tag spaces, number of tags, posts and co-occurrences and their frequencies, as well as updated information. The property scot:contains links scot:TagCloud to a set of scot:Tag instances. scot:Tag, as a subclass of tags:Tag from the Tag Ontology, describes a tag that is aggregated from individual tagging activities. SCOT allows the exchange of semantic tag metadata for reuse in social applications and enables interoperation amongst data sources, services, or agents in a tag space.

MOAT's -Meaning Of A Tag [START_REF] Passant | Combining Structure and Semantics for Ontology-Based Corporate Wikis[END_REF] (Figure 5)-goal is to provide a Semantic Web model to define the meaning of tags in a machine-readable way [START_REF] Tazari | Modeling User Context[END_REF]]. To achieve it, MOAT defines:

the global meanings of a tag, i.e. the list of all meanings the local meaning of a tag, i.e. the meaning of a tag in a particular tagging action. Moreover, MOAT introduces a social aspect that lets people share their tags -and their meanings -within a community by subscribing to a MOAT server, as they could do with the Annotea annotation server. They subscribe to a tag server in which they can share and update tag meanings, and use it when tagging content. When a user tag content, the client queries the server to retrieve tag meanings and let the user choose which one is the most relevant one, regarding the context.

Application Examples of the presented tag representation schemas

LODR (A Linking Open Data Tagging System) [START_REF] Passant | LODr -A Linking Open Data Tagging System[END_REF] is a prototype system based on the previously described MOAT vocabulary that allows to semantically enhance tags. Tagging is widely deployed, however the most important issue, that is the meaning of the tags, still needs to be resolved. The LODR system proposes to solve this issue by letting people give meaning to their tags using URIs of Semantic Web resourves from the Open Data initiative [START_REF] Bizer | Interlinking open data on the web[END_REF]. This means modeling facts as: "When I tag this picture 'apple', I mean http://dbpedia.org/resource/Apple Records, i.e. the record label, not the fruit". The system is based on a set of wrappers (currently available for 5 different services including Flickr and Slideshare), that parse the RSS feeds of user's data, extract items and related tags and translate it to RDF using SIOC and the Tagging Ontology. The data is then stored into a local triple-store and for each tagged item, the user can browse it and give meaning to its tags. To ease the process of chosing the right meaning, human-readable labels can be displayed instead of URIs. When no URI have been previously defined or when existing ones do not correspond to the meaning of the tag in the current context, a new URI can be added.

Comparison of the reviewed models

Our comparison will take into account several criteria. It is important to compare the modeling of the tagger in the different propositions. Several questions are to be considered: what user profile model is used to model the tagger? Does this model allow collaborative tagging, e.g. is the belonging of the user to a community considered? Another key question is the context of the tagger? What is the relationship of the tagger towards the resource? Also, we take into account the formal representation of the tags. What language is used to model association between tags, taggers and tags, tags and ressources? We also list existing implementations of the compared models and make a review of the applications. A key question that we will try to answer after this comparison is: which model could be efficient to model a linked community of individuals, objects and relationships, e.g. a social network that goes beyond the existing models, where relationships are declared explicitely. Indeed, objects can also link people: a photo, a video, a shared file etc.

Fig. 1 .

 1 Fig. 1. The User Profile Ontology (some concepts were deleted for a better visibility) viewed with the Jambalaya Protege plugin

Fig. 2 .

 2 Fig. 2. Tagging Ontology of Newman

Fig. 3 .

 3 Fig. 3. Semantically Interlinked Online Communities

Fig. 4 .

 4 Fig. 4. Semantically Interlinked Online Communities

Fig. 5 .

 5 Fig. 5. Part of the MOAT ontology

Table 1 .

 1 Concepts considered important in the literature for user profiling

	Concepts considered important in user profiling	References
	Static profile
	User identity, characteristics, capabilities, universal
	preferences, state of the user, application-specific
	preferences
	Dynamic profile (context)	[9]
	Current activity, location, terminal, motion state, orientation
	Parameters
	Personal information (name, address, birthday)
	General characteristics
	Physical factors, abilities, disabilities, education, occupation, hobbies
	Interests	[10] [11] [12] [13]
	Preferences	[14] [13]
	User expertise	[13]

www.linkedin.com, www.facebook.com, www.myspace.com

Model

Namespace Tagging is a ternary relation between a user (section 1), a resource and a tag. Several models were reviewed that intend to give a vocabulary and formmal representation of the tagging itseld with all the implications: the users, resources and tags.

In the following, we will review the AOA (Agent Oriented Abstraction) [?] model, which proposes a categorization of annotations/tags. AOA consideres four categories for annotations (semantic interpretation about a resource): 4 classes of knowledge annotations -Ontology (structured semantic knowledge bases/classes) -Communication (protocols of data exchange) -Safety/Integrity (reachability of an annotation, resssource) -Cognition These annotation classes refer in a more general way to capabilities of an agent to associate semantics to a knowledge according to these annotation classes.