On Fairness of Systemic Risk Measures

In our previous paper"A Unified Approach to Systemic Risk Measures via Acceptance Set", we have introduced a general class of systemic risk measures that allow random allocations to individual banks before aggregation of their risks. In the present paper, we address the question of fairness of these allocations and propose a fair allocation of the total risk to individual banks. We show that the dual formulation of the minimization problem identifying the systemic risk measure provides a valuation of the random allocations, which is fair both from the point of view of the society/regulator and from the individual financial institutions. The case with exponential utilities which allows for explicit computation is treated in details.


Introduction
Consider a vector X = (X 1 , . . . , X N ) ∈ L 0 (R N ) of N random variables denoting a configuration of risky factors at a future time T associated to a system of N entities/banks.
In the framework of Risk Measures, one of the first proposals, see [16], to measure the systemic risk of X was to consider the map where Λ : R N → R, is an aggregation rule that aggregates the N -dimensional risk factors into a univariate risk factor, and is a one-dimensional acceptance set. Systemic risk can again be interpreted as the minimal cash amount that secures the system when it is added to the total aggregated system loss Λ(X). The interpretation of (1) is that the systemic risk is the minimal capital needed to secure the system after aggregating individual risks.
It might be more relevant to measure systemic risk as the minimal capital that secures the aggregated system by injecting the capital into the single institutions before aggregating their individual risks. This way of measuring systemic risk can be expressed by Here, the amount m i is added to the financial position X i of institution i ∈ {1, · · · , N } before the corresponding total loss Λ(X +m) is computed (we refer to [3], [7] and [24]).
The main novelty of our paper [7] was the possibility of adding to X not merely a vector m = (m 1 , · · · , m N ) ∈ R N of cash, but, more generally, a random vector Y ∈ C in a class C such that where the subspace L ⊆ L 0 (R N ) will be specified later. Here, the notation N n=1 Y n ∈ R means that N n=1 Y n is equal to some deterministic constant in R, even though each single Y n , n = 1, · · · , N , is a random variable.
Then, the general systemic risk measure considered in [7] can be written as and can still be interpreted as the minimal total cash amount N n=1 Y n ∈ R needed today to secure the system by distributing the cash at the future time T among the components of the risk vector X. However, contrary to (2), in general the allocation Y i (ω) to institution i does not need to be decided today but depends on the scenario ω that has been realized at time T . This corresponds to the situation of a lender of last resort who is equipped with a certain amount of cash today and who will allocate it according to where it serves the most depending on the scenario that has been realized at T . Of course, in general, the use of scenario dependent allocation Y as in (3) reduces, in comparison to the deterministic case in (2), the minimal amount of capital ρ(X) needed to secure the system. Restrictions on the possible distributions of cash are given by the class C, as shown in the Example 1.
Definition 1 (i) We say that the scenario dependent allocation Y X = (Y n X )n ∈ C is a systemic optimal allocation for ρ(X), defined in (3), if it satisfies Λ(X + Y X ) ∈ A and ρ(X) = N n=1 Y n X .
(ii) We say that a vector (ρ n (X))n ∈ R N is a systemic risk allocation of ρ(X) if N n=1 ρ n (X) = ρ(X).
Even though, as mathematicians, we like well defined and sharp definitions, the analysis of a system of financial institutions suggests that the concept of fairness is a multi-faceted notion.
The aim of this paper is to analyze in detail the systemic risk measure in (3). In addition to several technical aspects regarding such systemic risk measures, we will answer the following main questions about fairness of risk allocations: 1. When is the systemic valuation ρ(X) and its optimal allocation Y X fair from the point of view of the whole system?
2. When is a systemic risk allocation (ρ n (X))n ∈ R N of ρ(X) fair from the point of view of the whole system? 3. When are the systemic optimal allocation Y X ∈ C and the systemic risk allocation (ρ n (X))n ∈ R N associated to ρ(X), fair from the point of view of each individual bank ?
We provide answers to these questions in the following introductory section without entering in the mathematical details of our analysis which will be provided in the subsequent sections. The optimal solution to the dual problem of the primal problem (3) will play a crucial role. It is a vector of probability measures Q X = (Q 1 X , · · · Q N X ) which will provide the fair valuation of the optimal random allocations through the formula ρ(X) = N n=1 E Q n X [Y n X ]. Existence and uniqueness of Q X is proved in Proposition 4 and Corollary 4. In Section 3 we introduce the setting of the paper and the main assumptions, and we show that our optimization problems are well posed. The main results are collected in Section 4, where we first present our results in the Orlicz space setting (introduced in Section 3.2) and in Section 5, where the existence of the optimal solution to ρ(X) (Theorem 7), as well as other technical existence results, are provided.
To guarantee existence, we need to enlarge the environment and consider appropriate spaces of integrable random variables. For this reason, we point out, in the course of the paper, those results that admit an extension to the larger setting. The case with exponential utilities and grouping of banks will be treated in details in Section 6, where meaningful sensitivity properties will be established as well.
In the rest of the paper, we shall assume that the aggregation function Λ is of the form Λ(x) = N n=1 un(xn) for utility functions un, n = 1, · · · , N .

Fairness of systemic risk measures and allocations
The main objective of this paper is to discuss various aspects of fairness of the systemic risk measures ρ(X), random allocations Y ∈ C, and risk allocations of the total systemic risk among individual banks. In this introductory section, we explain and motivate the various fairness properties, both from the point of view of the society/regulator and from the individual financial institutions. Precise definitions and statements, as well as detailed proofs, will be given in the course of the paper. For the remaining of this section, we assume that the infimum of the systemic risk measure is attained for an optimal (random) allocation Y X = (Y 1 X , · · · , Y N X ) ∈ C, which will turn out to be unique. Existence of such minimizer is proved in Section 5. Note that (4) is a particular case of (3), where the function Λ is the sum of the utility functions un and A is the particular acceptance set A = {Z ∈ L 0 (R), E[Z] ≥ B} for a given constant B.
We first introduce the related optimization problem so that, if we interpret N n=1 un(X n + Y n ) as the aggregated utility of the system after allocating Y, then π(X) can be interpreted as the maximal expected utility of the system over all random allocations Y ∈ C such that the aggregated budget constraint N n=1 Y n ≤ A holds for a given constant A. In the following, we may write ρ(X) = ρ B (X) and π(X) = π A (X) in order to express the dependence on the minimal level of expected utility B ∈ R and on the maximal budget level A ∈ R, respectively.
We will see in Section 4.2 that B = π A (X) if and only if A = ρ B (X), (6) and, in these cases, the two problems π A (X) and ρ B (X) have the same unique optimal solution Y X . From this, we infer that once a level ρ(X) of total systemic risk has been determined, the optimal allocation Y X of ρ maximizes the expected system utility among all random allocations of cost less or equal to ρ(X). Once the total systemic risk has been identified as ρ(X), the second essential question is how to allocate the total risk to the individual institutions. Recall that a vector (ρ 1 (X), · · · , ρ N (X)) ∈ R N is called a systemic risk allocation (SRA) of ρ(X) if N n=1 ρ n (X) = ρ(X). For deterministic allocation, this property is known as the "Full Allocation" property, see for example [13].
In the case of deterministic allocations Y ∈ R N , i.e. C = R N , the optimal deterministic Y X represents a canonical risk allocation ρ n (X) := Y n X . For general (random) allocations Y ∈ C ⊂ C R , we then follow the natural approach to consider risk allocations of the form ρ n (X) := E Q n [Y n X ] for n = 1, · · · , N, where Q = (Q 1 , · · · , Q N ) is a vector of probability measures with N n=1 E Q n [Y n X ] = ρ(X). In that way, ρ n (X) = E Q n [Y n X ] can be understood as a systemic risk valuation of Y n X . Note that in our setting, besides providing a ranking in terms of systemic riskiness, a risk allocation ρ n (X) can be interpreted as a capital requirement for institution n in order to fund the total amount ρ(X) of cash needed. In this sense, the vector Q allows for the monetary interpretation of a systemic pricing operator to determine the price (or cost) of (future) random allocations of the individual institutions . Obviously,  it is of high interest to identify fairness criteria, acceptable both by the society and  by the individual financial institutions, for such systemic valuation measures and their  corresponding risk allocations. Now, consider the situation where a valuation (or cost) operator Q = (Q 1 , · · · , Q N ) is given for the system. Then, a natural alternative formulation of the systemic risk measure and the related utility maximization problem in terms of the valuation provided by Q is Note that in (8) and (9) the allocation Y is not required to belong to C R (that is adding up to a deterministic quantity) but to a vector space L = M Φ of random variables introduced later. Thus, for the systemic risk measure ρ Q (X), we look for the mini- Similarly as in (6), we will see in Section 4.2 that and the two problems π Q A (X) and ρ Q B (X) have the same unique optimal solution.
The specific choice of a systemic valuation is the central question of this paper.
It will turn out that the optimizer Q X = (Q 1 X , · · · Q N X ) of the dual problem of (4), presented in detail in Section 4.1 and in Corollary 4, provides a systemic risk allocation Furthermore, by Proposition 13, Y X is the unique optimal allocation to ρ B (X) and (11). Similarly, π A (X) and π Q X A (X) in (12) have the same optimal solution cash ρ(X) but also the individual cash amounts Y ∈ R N allocated to the institutions are already known today (i.e., they are deterministic). Such a risk measure only depends on the marginal distributions of X as can be seen from the constraint (4) with Y n deterministic. However, ignoring potential dependencies among the banks might be over-conservative and too costly. By considering scenario-dependent allocations Y ∈ C R N (and by that considering the dependencies among the banks as was shown through examples in [7]), the consequential reduction of the overall cost of securing the system is beneficial to the society. Additionally, the requirement Y ∈ C and C ⊆ C R is important from the society's perspective as it guarantees that the cash amount ρ(X) determined today is sufficient to cover the allocations Y at time T in any possible scenario. There might be cross-subsidization (in the sense of a risk exchange) among the banks at time T , but N n=1 Y n = ρ(X) means that the system clears and no additional external injections (or withdrawals) are necessary at time T . In that sense, the requirement Y ∈ C ⊆ C R is fair from the society/regulator's perspective. Furthermore and most importantly, by (5) and (10), the optimal allocation Y X maximizes the expected systemic utility among all allocations with total cost less or equal to A = ρ B (X).
Next, consider the systemic risk valuation using Q X . To explain one of the features of Q X , observe first that ρ in (3) keeps the classical cash additivity property m n for all m ∈ R N and all X, which is a global property, see Section 4.3 for details. The local version associated to (13) is The expression d dε ρ(X+εm)| ε=0 represents the sensitivity of the risk of X with respect to the impact m ∈R N and was named the marginal risk contribution by [3]. However, such property can not be immediately generalized to the case where m ∈ R N is replaced by random vectors V, in particular when N n=1 V n is not a constant.
If the positions change from X to X + εV j e j , where e j is the jth unit vector and V j is a random variable, then, we show in Section 4.3 that the riskiness of the entire system changes linearly by which shows that Q X can be naturally introduced as a systemic risk valuation operator. Now, given a systemic risk valuation Q, one is naturally led to the specification (8) for a systemic risk measure. Note, however, that in (8) the clearing condition N n=1 Y n = ρ(X) is not guaranteed since the optimization is performed over all Y ∈ M Φ . Using the valuation with Q X is then fair from the society/regulator's point of view since, by Proposition 13, the optimal allocation in (8) fulfills the clearing condition Y ∈ C R , and is in fact the same as the optimal allocation of the original systemic risk measure in (4). From (73) and (74) we obtain which also shows that the selection of E Q X [·] as the valuation functional is as fair as computing ρ(X) as the infimum of N n=1 Y n , for admissible Y, and supports the definition of Q X as the systemic probability measure.
Fairness from the perspective of the individual institutions. The essential question for a financial institution is whether its allocated share of the total systemic risk determined by the risk allocation ( For the banks, the clearing condition Y ∈ C R is not relevant. Instead, given a vector (8) is more relevant. Thus, by choosing Q = Q X , the requirements from both the society and the banks are reconciled as seen from (11). Furthermore, with the choice Q = Q X , we have by (12) see Lemma 3 for details. Choosing A = ρ B (X), we obtain by (10) and the fact that, (16) can be rewritten as This means that by using Q X for valuation, the system utility maximization in (9) reduces to individual utility maximization problems for the banks without the "systemic" constraint Y ∈ C: The optimal allocation Y n X and its value E Q n X [Y n X ] can thus be considered fair by the n th bank, as Y n X maximizes the individual expected utility of bank n among all random allocations (not constrained to be in C R ) with value E Q n X [Y n X ]. This finally argues for the fairness of the risk allocation ( ) as fair valuation of the optimal allocation (Y 1 X , · · · , Y N X ).
Another desirable fairness property is monotonicity. It is clear that if C 1 ⊆ C 2 ⊆ C R , then ρ 1 (X) ≥ ρ 2 (X) for the corresponding systemic risk measures The two extreme cases occur for C 1 := R N (the deterministic case) and C 2 := C R (the unconstraint scenario dependent case). Hence we know that when going from deterministic to scenario-dependent allocations the total systemic risk decreases. It is then desirable that each institution profits from this decrease in total systemic risk in the sense that also its individual risk allocation decreases: The opposite would clearly be perceived as unfair. This is discussed in the exponential setting of Section 6.2, where we show that (17) holds when ρ n 1 (X) := Y n 1 and ρ n 2 (X) := (where Y n j is the optimal allocation to the systemic risk measure ρ j (X) associated to C j , so that Y n 1 ∈ R is deterministic, and Q 2 is the systemic probability measure associated to ρ 2 (X)). By using a probability measure R different from Q 2 to compute the risk allocation ρ n (17) is lost in general.
Additional fairness properties related to the systemic probability measure Q X are addressed in Section 6.1, Proposition 18.
We conclude this Section with a literature overview on systemic risk. In [19], [12] and [18] one can find empirical studies on banking networks, while interbank lending has been studied via interacting diffusions and mean field approach in several papers like [28], [26], [15], [35], [5]. Among the many contributions on systemic risk modeling, we mention the classical contagion model proposed by [23], the default model of [31], the illiquidity cascade models of [30], [34] and [37], the asset fire sale cascade model by [17] and [14], as well as the model in [42] that additionally includes cross-holdings. Further works on network modeling are [1], [40], [2], [32], [4], [21] and [22]. See also the references therein. For an exhaustive overview on the literature on systemic risk we refer the reader to the recent volumes of [33] and of [27].
The measurable space (Ω, F) will be fixed throughout the paper and will not appear in the notations. Unless we need to specify a different probability, we will also suppress P from the notations and simply write L 0 (R N ). In addition, we will sometimes suppress R d , d = 1, ..., N, in the notation of the vector spaces, when the dimension of the random vector is clear from the context. When Q = (Q 1 , ..., Q N ) is a vector of probability measures on (Ω, F), we set L 1 (Q) := {X = (X 1 , . . . , X N ) | X n ∈ L 1 (Q n ), n = 1, · · · , N }. Unless differently stated, all inequalities between random vectors are meant to be P-a.s. inequalities. A vector X = (X 1 , . . . , X N ) ∈ L 0 denotes a configuration of risky factors at a future time T associated to a system of N entities. We assume that L 0 (R N ) is a vector lattice equipped with the order relation Let C R be the linear space Here we use the notation N n=1 Y n ∈ R to denote that N n=1 Y n is equal to some deterministic constant in R, even though each single Y n , n = 1, · · · , N , is a random variable. By following [7], we consider systemic risk measures where the map is an aggregation rule that aggregates the N -dimensional risk factor into a univariate risk factor, A ⊆L 0 (R) is a one dimensional acceptance set and the set C of admissible random elements satisfies C ⊆ C R ∩ L, where is a vector subspace containing R N , that will be specified in the sequel.
Example 1 We now introduce one relevant example for the set of admissible random elements, which we denote C (n) .
Definition 2 Set n 0 = 0. For h ∈ {1, · · · , N } , let n := (n 1 , · · · , n h ) ∈ N h , with n m−1 < nm for all m = 1, · · · , h and n h := N , represent some partition of {1, · · · , N }. We set Im := {n m−1 + 1, · · · , nm} for each m = 1, · · · , h. The cardinality of each group is denoted with Nm := nm − n m−1 . We introduce the following family of allo- For a given n := (n 1 , · · · , n h ), the values (d 1 , · · · , d h ) may change, but the number of elements in each of the h groups Im is fixed by the partition n. It is then easily seen that C (n) is a linear space containing R N and closed with respect to convergence in probability. Beside the obvious interpretation of the restrictions imposed to the elements Y ∈C (n) , we point out that the family C (n) admits two extreme cases: (i) the strongest restriction occurs when h = N, i.e. we consider exactly N groups, and in this case C (n) = R N corresponds to the deterministic case; (ii) on the opposite side, we have only one group h = 1 and C (n) = C R ∩ L is the largest possible class, corresponding to arbitrary random injection Y ∈L with the only constraint N n=1 Y n ∈ R.

Assumptions and properties of ρ
We now specify further properties of systemic risk measures of the form (20) under some additional, but still general hypotheses. In the sequel we will always work under the following Assumption 1 As C is a convex cone containing R N , Y + δ ∈ C for every Y ∈ C and any deterministic δ ∈ R N . Under Assumption 1, a systemic risk measure of the form (20) can be written as Note that there is no loss of generality in assuming that un(0) = 0 (simply replace B with B − N n=1 un(0)), and that a natural selection for B is B := N n=1 un(0). In this case ρ(0) ≤ 0. The proof of the following proposition, which exploits the behavior of un at −∞, is given in Appendix A.1.
The domain of ρ is defined by If, in addition, for each n, un : R → R is strictly concave and there exists an optimal allocation Y X = {Y n X } n ∈ C 0 ∩ L of ρ(X), then it is unique.
We now show uniqueness by contradiction. Suppose that ρ(X) is attained by two distinct Y 1 ∈ C and Y 2 ∈ C, so that P(Y j 1 = Y j 2 ) > 0 for some j. Then we have and for λ ∈ (0, 1) where we used that u j is strictly concave and P(Y j 1 = Y j 2 ) > 0. This is a contradiction with ρ(X) = ρ = (X) and (23).
Remark 1 (Extension to L 1 (Q)) The extension of Proposition 2 to the case where Y ∈ C 0 ∩ L 1 (Q) (instead of Y ∈ C 0 ∩ L) would a priori require Assumption (85), as in this case we can not guarantee E[un(Z n )] > −∞ for all Z ∈ L 1 (Q), see Remark 14. However, we will obtain uniqueness also for Y ∈ C 0 ∩ L 1 (Q), based on the uniqueness of the solution to ρ Q X (X), see Remark 5, and on Remark 11.

Orlicz setting
We now study some important properties of systemic risk measures of the form (22) in a Orlicz space setting, see [38] for further details on Orlicz spaces. This presents several advantages. From a mathematical point of view, it is a more general setting than L ∞ , but at the same time it simplifies the analysis, since the topology is order continuous and there are no singular elements in the dual space. Furthermore, it has been shown in [10] that the Orlicz setting is the natural one to embed utility maximization problems, Let u : R → R be a concave and increasing function satisfying lim x→−∞ , it is finite valued, even and convex on R with φ(0) = 0 and lim x→+∞ φ(x) x = +∞. The Orlicz space L φ and Orlicz Heart M φ are respectively defined by and they are Banach spaces when endowed with the Luxemburg norm. The topological is also a strict Young function. Note that and we immediately deduce that dQ Given the utility functions u 1 , · · · , u N : R → R satisfying Assumption 1, with associated Young functions φ 1 , · · · , φ N , we define and consider Proof The equality M Φ = dom(ρ), so that ρ : M Φ → R, follows from Proposition 2, the definition of M Φ in (25), and (26). The remaining properties are a consequence of Proposition 2, Theorem 9 in Appendix and the fact that M Φ is a Banach space.
In the following section we will start presenting our results in the Orlicz space setting. When the utility functions un are of the exponential type, the Orlicz Heart M Φ is sufficiently large and it contains the optimal allocation Y X to ρ(X), see Section 6. This of course also happens in the case of general utility functions on a finite probability space. However, for arbitrary utility functions and a general probability space, the existence technical results established in Section 5 require a larger space of integrable random variables.

Dual representation of ρ
We now investigate the dual representation of systemic risk measures of the form (22).
We will frequently identify the density dQ dP with the associated probability measure Q P.
where the penalty function is given by (ii) Suppose that ±(e i 1 A − e j 1 A ) ∈ C for all i, j and all A ∈ F. Then Proof The dual representation (28) is a consequence of Proposition 3, Theorem 9 and of Propositions 3.9 and 3.11 in [29], taking into consideration that C is a convex cone, the dual space of the Orlicz Heart M Φ is the Orlicz space L Φ * and M Φ = dom(ρ).
Note that from Theorem 9 we know that the dual elements ξ ∈ L Φ * + are positive but a priori not normalized. However, we obtain E[ξ n ] = 1 by taking as Y = ±e j ∈ R N , and This shows the form of the domain D in (30). Furthermore: Proposition 4 guarantees the existence of a maximizer Q X to the dual problem (28) and that α Λ,B (Q X ) < +∞. Uniqueness will be proved in Corollary 4.
Definition 3 Let X ∈M Φ . An optimal solution of the dual problem (28) is a vector of probability measures Q X = (Q 1 The probability measures Q having density in D could be viewed, in the systemic N -dimensional one period setting, as the counterpart of the notion of (P-absolutely continuous) martingale measures. Indeed, as Y ∈ C 0 ⊆ C R , N n=1 Y n ∈ R is the total amount to be allocated to the N institutions and then the total cost or value should at most be equal to N n=1 Y n , for any "fair" valuation operator Q, that is dQ dP ∈ D. There exists a simple relation among ρ B , ρ Q B and α Λ,B (Q) defined in (22), (8), and (29), respectively. and where Q X is an optimal solution of the dual problem (28).
which proves (32). Then from (32) and (31) we deduce Then (33) shows that which means that ρ B is the most conservative among those risk measures ρ Q B defined through fair valuation operators dQ dP ∈ D. In this respect, the probability measure Q X plays, in the theory of systemic risk measure, an analogous role played by the minimax martingale measure in the theory of contingent claim valuation in incomplete markets, see [6] for details.
We now turn our attention to the uniqueness of the optimal solution to the problem (29). The proof employs the same arguments used in the proof of Proposition 2.
Lemma 1 If each un is strictly concave and α Λ,B (Q) < +∞, then there exists at Indeed, −∞ < c(Q) ≤c = (Q) and assume by contradiction that c(Q) < c = (Q). By definition of c(Q), there exist ε > 0 and Z ∈M Φ such that E[Λ(Z)] > B and N n=1 E Q n [Z n ] ≤ c(Q) + ε<c = (Q), which contradicts (35). For the uniqueness, let us suppose that c(Q) is attained by two distinct Z 1 ∈ M Φ and Z 2 ∈ M Φ , so that P(Z j 1 = Z j 2 ) > 0 for some j. Then we have where we used the strict concavity of u j and P(Z j 1 = Z j 2 ) > 0. This is a contradiction with c(Q) = c = (Q) and (35).
Remark 4 (Extension to L 1 (Q)) It is possible to extend Lemma 1 to the case where Z ∈ L 1 (Q) by applying the simple argument stated in Remark 14. So, if each un is strictly concave, there exists at most one Z ∈L 1 (Q) satisfying (34).
Remark 5 (Uniqueness) Suppose that each un is strictly concave. The existence of an optimizer Y Q for the problem ρ Q B (X) will be proved in Section 5.2. The uniqueness of Example 2 Consider the grouping Example 1. As C (n) is a linear space containing R N , the dual representation (28) applies. In addition in each group we have ±(e i 1 A − e j 1 A ) ∈ C (n) for all i, j in the same group and for all A ∈ F . Therefore, in each group the components Q i , i ∈ Im, of the dual elements are all the same, i.e., Q i = Q j , for all i, j ∈ Im, and the representation (28) becomes If we have only one single group, all components of a dual element Q ∈ D are the same.

Remark 6
Consider the grouping Example 1. Let Q = (Q 1 , · · · , Q n ) n=1,··· ,N be a vector of probability measures with the property that in each group the components is a systemic risk allocation as in Definition (1), i.e., Indeed, for such a vector (Q 1 , · · · , Q m ) of probability measures we have Returning to our general setting, from now on, we work under the following two assumptions, with the understanding that Assumption 3 will hold with respect to the probability measures (Q or Q X ) involved in the statements of the results.
Assumption 2 In addition to Assumptions 1, we assume that for any n = 1, · · · , N , un : R → R is increasing, strictly concave, differentiable and satisfies the Inada condi- . This assumption, even though quite weak (see [8] Section 2.2), is fundamental to guarantee the existence of the optimal solution to classical utility maximization problems (see [41] and [8]).
Assumption 3 For any n = 1, · · · , N , vn and Q n P satisfy From the Fenchel inequality Proposition 6 When α Λ,B (Q) < +∞, then the penalty function in (29) can be written as and E vn λ dQ n dP < ∞ for all n and all λ > 0.
Proof In Appendix A.
Next, thanks to the integrability conditions provided by Lemma 9, we show the existence of the solution η * > 0 of (40). First we consider η → +∞.
Hence lim inf η→+∞ F (η) > 0. We now look at η → 0: The convexity of vn implies that for any fixed z 0 > 0 and z > z 0 By the continuity of F we obtain the existence of the solution η * > 0 for (40). Uniqueness follows from the strict convexity of F .

Fairness in the details
We now turn to the details of the introductory Section 2 and establish important relations between primal problems (4) and (5), and problems (8) and (9). Note that in this section, we do not assume the existence of an optimizer for problems (4) or (5). We work under Assumptions 2 and 3.
Let A ∈ R, B ∈ R. As un is increasing, in both problems (4) and (5) we may replace the inequality in the constraints with an equality, and due to strict concavity the solution, if it exists, is unique (see Proposition 2 and Remark 5). Recall that under Assumptions 1, C is a convex cone and therefore, if Y ∈ C, then Y + δ ∈ C for every deterministic δ ∈ R N .
, and in this case the unique optimal solution, if it exists, is the same for the two problems π A (X) and ρ B (X).
Proof ⇐) Let A = ρ B (X) and suppose first that π A (X) > B. Then there must exist Y ∈ C such that N n=1Ỹ n ≤ A and E N n=1 un(X n +Ỹ n ) > B. By continuity of un, then, there exists ε > 0 and Y : Suppose that there exists Y ∈ C that is the optimal solution of problem (4). As and we deduce that Y is an optimal solution of problem (5).
Define ηε := inf a > 0 : E N n=1 un(X n + Y n ε + a N ) ≥ B and note that ηε ↓ 0 if ε ↓ 0. Select ε > 0 such that ηε < δ. Then, for any 0 < β < δ − ηε we have Suppose that there exists Y ∈ C that is the optimal solution of problem (5) and set B := π A (X). Then E N n=1 un(X n + Y n ) = B and the constraint in problem (4) is fulfilled for Y. Hence, A = ρ B (X) ≤ N n=1 Y n ≤ A and we deduce that Y is an optimal solution of problem (4). As ρ B (X) admits at most one solution by Proposition 2, the same must be true for π A (X). Now consider the situation where a valuation operator Q = (Q 1 , · · · , Q N ) such that dQ dP ∈ L Φ * is given for the system. Note that ρ Q B (X) < +∞ and π Q A (X) > −∞.
Then, similarly as in Proposition 8, we obtain Proposition 9 B = π Q A (X) < +∞ if and only if A = ρ Q B (X) > −∞, and the two problems have the same optimal solution Y Q .
Proof ⇐) Let Y be an optimal solution of problem (8) and set A : Then N n=1 E Q n [Y n ] = A and the constraint in problem (9) is fulfilled for Y. Hence, un(X n + Y n ) , and therefore, Y is an optimal solution of problem (9).
⇒) Let Y be an optimal solution of problem (9) and set B := π Q A (X) < +∞. Then we have E N n=1 un(X n + Y n ) = B and the constraint in problem (8) is and Y is an optimal solution of problem (8).
Remark 8 (Extension to L 1 (Q)) By applying the simple argument stated in Remark 14, Proposition 9 also holds when Y ∈ M Φ is replaced by Y ∈ L 1 (Q).
Recall that we denote by Q X = (Q 1 X , · · · Q N X ) an optimizer of the dual problem of (4), presented in detail in Section 4.1. The key relation (11) was shown in Proposition 5. We now prove the other key relation (12).

On local cash additivity and marginal risk contribution
We now show when the systemic risk measures of the form (20) are cash additive and local cash additive.

Lemma 2 Define
Then the risk measure ρ defined in (20) is cash additive on W C , i.e., Z n for all Z ∈ W C and all X ∈ L.
Proof Let Z ∈ W C . Then W := Z + Y ∈ C ⊆ C R for any Y ∈ C. For any X ∈ L it holds ρ( Example 4 In case of the set C (n) in Example 1, ρ is cash additive on Note that equality (42) holds since we are assuming no restrictions on the vector d = (d, · · · , dm) ∈ R m , which determines the grouping. If for example, we restrict d to have non negative components, then it is no longer true that W C (n) = C (n) .

Corollary 2
For the systemic risk measures of the form (20) we have: for all V such that εV ∈W C for all ε ∈ (0, 1]. Proof It follows from Lemma 2 which gives ρ(X+εV) = ρ(X) − ε N n=1 V n . The expression d dε ρ(X+εV)| ε=0 represents the sensitivity of the risk X with respect to the impact V ∈L 0 (R N ). In the case of a deterministic V := m ∈ R N , it was called marginal risk contribution in [3]. Such property cannot be immediately generalized to the case of random vectors V, also because in general N n=1 V n / ∈ R. In the following, we obtain the general local version of cash additivity, which extends the concept of marginal risk contribution to a random setting. In particular, (44) shows how the change in one component affects the change of the systemic risk measure.
Proposition 10 Let V ∈M Φ and X ∈M Φ . Let Q X be the optimal solution to the dual problem (28) associated to ρ(X) and assume that ρ(X+εV) is differentiable with respect to ε at ε = 0, and dQ X+εV Proof As the penalty function α Λ,B does not depend on X, by (31) we deduce d dε where the equality between (45) and (46) is justified by the optimality of Q X and the differentiability of ρ(X+εV), while the last equality is guaranteed by the convergence of dQ X+εV dP .

Remark 10
We emphasize that the generalization (44) of (43) holds because we are computing the expectation with respect to the systemic probability measure Q X . A relevant example where the assumptions of Proposition 10 hold is provided in Section 6.

Existence of the optimal solutions
Throughout the entire Section 5, we assume X ∈ M Φ and that Q = (Q 1 , ..., Q N ) satisfies Q n P, dQ dP ∈ L Φ * and α Λ,B (Q) < +∞, or equivalently ρ Q B (X) > −∞. Recall from Proposition 6 that this implies E vn λ dQ n dP < +∞ for all n and all λ > 0. Set where the inclusions follows from Remark 2 and dQ dP ∈ L Φ * . In order to prove the existence of the optimal solutions for ρ Q X B (X) and ρ B (X), we will proceed in several steps. As shown in Section 5.2, in general, we can not expect to find the optimal solution Y Q to the problem ρ Q B (X) in the space M Φ , but only in the larger space L 1 (Q). We first prove the existence of Y ∈ L 1 (P), which is the candidate solution, as specified in Theorem 5, to an extended problem. We already know that the optimal allocation to ρ B (X), when it exists, coincides with Y Q X ∈ L 1 (Q X ). So in a second step (see Theorem 7 and Corollary 5) we show that the optimal solution Y X to the extended problem ρ Q X B (X) = ρ B (X) exists and Y Q X = Y X ∈ L 1 (P; Q X ). W.l.o.g. we may assume that u i (0) = 0, 1 ≤ i ≤ N and observe that then

On the utility maximization problem
For a n ∈ R consider the problem: If we need to emphasize the dependence on Q n we write U Qn n (a n ). Note that E[un(X n + W )] ≤ un(E[X n + W ]) < +∞ for all X n , W ∈ M φn ⊆ L 1 (P; R). The conditions X n , W ∈ M φn imply that E[u(X n + W )] > −∞, which in turn implies that U (a n ) > −∞. As dQ dP ∈ L Φ * , then W ∈ M φn implies W ∈ L 1 (Q n ) and the problem (49) is well posed. Due to the monotonicity and concavity of un, Un is monotone increasing, concave and continuous on R and we may replace, in the definition of Un, the inequality with the equality sign. However, in general the optimal solution to (49) will only exist on a larger domain, as suggested by the well known result reported in Proposition 20. This leads to introduce the auxiliary problems: where L 1 (P, Q n ) is defined as in (47).

(51)
If Un(a n ) < un(+∞) then Un(a n ) = inf λ>0 λ E Q n [X n ] + a n + E vn λ dQ n dP (53) where the optimal solution Y n Q ∈ L 1 (Q n ) is given by un(X n + Y n Q ) ∈ L 1 (P) and λn > 0 is the unique solution of E Q n [X n ] + a n + E Q n v n λn dQ n dP = 0.
By the Fenchel inequality we get

Moreover
and similarly for ρ Q B (X) and ρ Q B (X). As shown in (60), the extension to L 1 (Q) does not increment the optimal value of π Q A (X). In addition (60) justifies equation (16) in Section 2.

Lemma 3 Let
Un(a n ) = π Q A (X) = π Q A (X), Proof Clearly, +∞ > π Q A (X) ≥ π Q,= A (X). By contradiction suppose that π Q A (X) > π Q,= A (X) and take ε > 0 such that π Q A (X) − ε > π Q,= A (X). By definition of π Q A (X) A (X), a contradiction. Hence (59) holds true. Note that Indeed, just take Y ∈M Φ and let a n := E Q n [Y n ] ∈ R and Z n := Y n − a n ∈ M φn . Then Un(a n ), which shows the first equality in (60). Then π Q A (X) = π Q A (X) = π Q A (X) are consequences of (51) and the decompositions analogous to the one just obtained for π Q A (X) in (62). From Proposition 9, its extension to L 1 (Q) in Remark 8 and ρ Q B (X) > −∞, we easily deduce (61).

Lemma 4 For arbitrary constant
Un(a n ) ≥ B .
Then K is a bounded closed set in R N .
Proof For N = 1 it is true. Let N > 1. First we prove that, for all j = 1, ..., N , Now suppose that for some k ∈ {1, ..., N }, u k (+∞) = +∞. Then Proposition 11 shows that U k (a k ) < +∞ = u k (+∞), U k > 0, U k (−∞) = +∞, U k (+∞) = 0. By l'Hopital's rule, for all j = 1, ..., N we obtain again From (64) and (65) we deduce that (63) holds true. We conclude that for any constant B there exists a constant R such that for all j = 1, ..., N and a < R Let a ∈ K and let i be such that a i = min{a 1 , ..., a N }. Note that a j ≤ A − (N − 1)a i for all j = 1, ..., N because N n=1 a n ≤ A holds. Assume that a i < R. Then which is a contradiction. Thus a j ≥ R for all j = 1, ..., N , and then also a j ≤ A − (N − 1)R for all j = 1, ..., N because N n=1 a n ≤ A holds. This proves the claim.
Proposition 12 Let A := ρ Q B (X) and π Q A (X) < +∞. There exists an optimal solution a * ∈ R N to the problem (60), namely Un(a n ) = N n=1 Un(a n * ) and N n=1 a n * = A. Corollary 3 Let A := ρ Q B (X) and suppose that for each n, Un(a n * ) < un(+∞), with the notation of Proposition 12. Then Un(a n * ) = where Y n Q ∈ L 1 (Q n ) is given by (55). Therefore, under the assumption Un(a n * ) < un(+∞), Y Q is the optimal solution to both extended problems π Q A (X) and ρ Q B (X).
Due to u(−v (y)) = v(y) − yv (y) (see Lemma 10) we also deduce from (55) that so that the vector {λn} n=1,...,N of positive numbers solves the equation above, that should be compared with (39). We will show in Theorem 4 that all components λn are equal, when in (56) the value a n is replaced by the optimal a n * .

On the optimal solution of ρ Q and comparison of optimal solutions
Theorem 4 Suppose that α Λ,B (Q) < +∞. Then the random vector Y Q given by where λ * is the unique solution to (39), satisfies Y n so that Y Q = Y Q is the optimal solution to the extended problem ρ Q B (X).
Proof Note that ρ Q B (X) > −∞, as α Λ,B (Q) < +∞. The integrability conditions hold thanks to the results stated in Appendix A.3. From (32) and the expression (38) for the penalty, we compute: We show that Y n Q satisfies the budget constraint: due to u(−v (y)) = v(y) − yv (y) (see Lemma 10) and (39). Finally, ρ Q B (X) = ρ Q B (X) follows from (61) and the uniqueness shown in Remark 5 proves that Y Q = Y Q .
When both solutions to the problems ρ B (X) and ρ Q X B (X) exist, then they coincide.

Proposition 13
Let Y X ∈ C 0 ∩ M Φ be the optimal allocation to ρ B (X), Q X be an optimal solution to the dual problem (28). Then: Proof Note that Y X satisfies: as Y X ∈ C and Q X ∈ D. From the definition of Y X , from (68), (32) and (31) we deduce that which shows, together with (72), that From (73) and (74) we then deduce As both (X + Y X ) and (X + Y Q X ) satisfy the budget constraints associated to α Λ,B (Q X ) in equation (34), this implies that α Λ,B (Q X ) is attained by both (X+Y X ) and (X + Y Q X ). The uniqueness shown in Lemma 1 allows us to conclude that Y X = Y Q X .
Remark 11 (Extension to L 1 (Q X )) We will show in Section 5 the existence of an optimal solution Y X to the problem ρ Q B (X), namely Y X ∈ C 0 ∩ L 1 (P, Q X ) satisfies (70), (71) and (72). Then the above proof and Remark 4 show that Y X = Y Q X , even for Y X ∈ C 0 ∩ L 1 (P, Q X ). Similarly, the following Corollary holds also for such Y X ∈ C 0 ∩ L 1 (P, Q X ).
We now show that the maximizer of the dual representation is unique.
Corollary 4 Suppose that there exists an optimal allocation Y X to ρ B (X). Then the optimal solution Q X = (Q 1 X , · · · , Q N X ) of the dual problem (28) is unique.
As v n is invertible, we conclude that λ * 1 dQ n 1 dP = λ * 2 dQ n 2 dP , P a.s., which then implies

5.3
On the existence of the optimal allocation to ρ(X)

A first step
and a sequence {Y k } k∈N ⊂ C such that E N n=1 un(X n +Y n k ) ≥ B and Y k → Y P-a.s.

Remark 12
Recall that C := C 0 ∩ M Φ and that C 0 ⊆ C R represents the effective constraint on the admissible injections, except for the integrability restriction expressed by M Φ . Assume further that C 0 is closed in L 0 (P), which is a reasonable assumption and holds true if C = C (n) , in which case C (n) 0 is defined in (21). Then the random vector Y in Theorem 5 would also belong to C 0 , but in general not to C (as M Φ is in general not closed for P-a.s. convergence). The conclusion is that Y satisfies all the conditions for being the optimal allocation to ρ B (X), with the only exception for the integrability condition Y ∈M Φ , which is replaced by Y ∈ L 1 (P; R N ). In the next subsection we will show when such Y also belongs to C 0 ∩ L 1 (Q X ; R N ).
It is now evident that when the cardinality of Ω is finite and the set C is closed for P-a.s. convergence, then the random vector Y in Theorem 5 belongs to C and The sequence (V k ) k∈N is bounded for the L 1 (P; R N ) norm if and only if so is the sequence (X + V k ) k∈N . Given the following decomposition in positive and negative part we define the index sets: and, similarly, N − ∞ and N − b for the negative part. We can split the expression (75) as If the sequence (X + V k ) k∈N is not L 1 (P; R N ) -bounded, then one of the sets N + ∞ or N − ∞ must be nonempty and therefore, because of the constraint which is a contradiction, as the second term that multiplies A in not bounded from above. Hence we exclude that our minimizing sequence (V k ) k∈N has unbounded L 1 (P; R N ) norm and we may apply a Komlós compactness argument, as stated below in Theorem 6, with E = R N . Applying this result to the sequence (V k ) k∈N ∈ C, we can find a Observe that by construction N n=1 Y n k is P-a.s. a real number and, as a consequence, so is N n=1 Y n . As E N n=1 un(X n +V n k ) ≥ B, also the Y k satisfy such constraint and therefore ρ B (X) ≤ N n=1 Y n k .
For any fixed k we compute We now show that Y also satisfies the budget constraint. In case that all utility functions are bounded from above, this is an immediate consequence of Fatou Lemma, since In the general case, recall first that the sequence V k is bounded in L 1 (P; R N ), and the argument used in (76) shows that Now we need to exploit the Inada condition at +∞. Applying the Lemma 12 to the utility functions un, assumed null in 0, we get Replacing X + Y in the expression above, applying Fatou Lemma we have As the term b(ε) simplifies in the above inequality, we conclude that for all ε > 0 so that Y satisfies the constraint.
Theorem 6 (Theorem 1.4 [20]) Let E be a Banach reflexive space and (f k ) k∈N ⊆ L 1 ((Ω, F, P); E) := L 1 be a sequence with bounded L 1 norms. Then there exists a sequence (g k ) k∈N and g 0 in L 1 such that g k ∈ conv(f i , i ≥ k) and g k − g E → 0 P−a.s., as k → ∞.

Second
Step: The optimal allocation to ρ(X) in L 1 (Q X ) Proof Applying (48) and φ j (x) := −u j (−|x|), note that for each fixed where we used Jensen inequality and X + Y ∈ L 1 (P; R N ). This yields ( Since, by assumption, X j ∈ M φj ⊆ L 1 (Q j X ), then also ((X j +Y j ) − +X j ) + ∈ L 1 (Q j X ) and so

Lemma 6
The random vector Y in Theorem 5 satisfies Y + ∈L 1 (Q X ).
Proof We proved in Theorem 5 the existence of Y satisfying ρ B (X) = N n=1 Y n ∈ R with Y ∈ L 1 (P; R N ), E N n=1 un(X n +Y n ) ≥ B and Y is the P-a.s. limit of By passing to a subsequence, w.l.o.g we may assume (78) where we used Jensen inequality and the fact that Y k satisfies N n=1 E [un(X n + Y n k )] > B. From the proof of Theorem 5 we know that (X n + Y n k ) k∈N is L 1 (P)-bounded for all n = 1, ..., N , and thus

By Remark 2 it then follows that
and thus where we recall that by assumption X j ∈ M φj ⊆ L 1 (Q j X ). From (78) and (79) together with (77) the claim follows.
For our final result on the existence we need one more assumption.

Definition 4
We say that C 0 is closed under truncation if for each Y ∈ C 0 there exists In Definition 2, the set C (n) 0 is closed under truncation.
Theorem 7 Let C = C 0 ∩ M Φ and suppose that C 0 ⊆ C R is closed for the convergence in probability and closed under truncation. For any X ∈ M Φ there exists Y X ∈ C 0 ∩ so that Y X is the optimal solution to the extended problem ρ B (X).
Proof The optimal solution Y X coincides with the vector Y in Theorem 5, which belongs to L 1 (P; Q X ), by Theorem 5, Lemma 5, Lemma 6, and to C 0 , as C 0 Ym → Y P-a.s. and C 0 is closed for the convergence in probability. Comparing Theorem 7 with Theorem 5 we see that it remains to prove ρ B = ρ B (Proposition 15) and holds as Y X fulfills the budget constraints of ρ Q X B (X).

Proposition 14
Suppose that C 0 is closed under truncation. Then Proof Let Y ∈ C 0 ∩ L 1 (Q X ; R N ) and consider Ym for m ∈ N as defined in (80), where w.l.o.g. we assume m Y = 1. Note that N n=1 Y n m = c Y (= N n=1 Y n ) for all m ∈ N. By boundedness of Ym and (80), we have Ym ∈ C 0 ∩ M Φ for all m ∈ N. Further, Ym → Y Q X -a.s. for m → ∞ , and thus, since |Ym| ≤ max{|Y|, |c Y |} ∈ L 1 (Q X ; R N ) for all m ∈ N, also Ym → Y in L 1 (Q X ; R N ) for m → ∞ by dominated convergence.
We then obtain The map ρ B is defined on M Φ but the admissible claims Y belongs to the set C 0 ∩ L 1 (P; Q X ), not included in M Φ . As L 1 (P; Q X ) ⊆ L 1 (P; R N ) with the same argument used in the proof of Proposition 1, we can show that ρ B (X) > −∞ for all X ∈M Φ . By the same argument in the proof of Proposition 2 and by (26) we also deduce that ρ B (X) < +∞ for all X ∈ M Φ , so that is convex and monotone decreasing on its domain dom( ρ) = M Φ . From Theorem 9, we then know that the penalty functions of ρ B and ρ B are defined as: In order to prove ρ B = ρ B we first show that α Λ,B (Q) = α Λ,B (Q). Set Proof In the proof, we will suppress the labels Λ and B from the penalty functions. From (29) and (61), note that the penalty function can also be written as Let Q ∈ D(L 1 (P; Q X )) and recall that X ∈ M Φ ⊆ L 1 (P; Q X ), so that W := X + Z ∈L 1 (P; Q X ) for X ∈ M Φ and Z ∈L 1 (P; Q X ). Set E [Λ(X + Z)] = E N n=1 un(X n +Z n ) . We then have that The opposite inequality is trivial, as ρ B ≤ ρ B implies Proposition 15 If C 0 is closed under truncation, then Proof We know that ρ B : M Φ → R is convex and monotone decreasing. By definition, Under the truncation assumption, in Proposition 14 we proved that Q X ∈ D(L 1 (Q X )) ⊆ D(L 1 (P; Q X )) and Lemma 7 shows that then α Λ,B (Q X ) = α Λ,B (Q X ).
Then, by Theorem 9, From Lemma 3, Proposition 5, and Corollary 1 we already know that, for A = ρ B (X), the optimal values satisfy From Theorem 7, Lemma 3 and by the same arguments applied in Proposition 13, Corollary 4 and Remark 11 we conclude: Under the same assumptions of Theorem 7, we have ρ B (X) = ρ B (X). The unique optimal solutions to the extended problems ρ Q X B (X), and Q X is the unique optimal solution to the dual problem (28).
Remark 13 Under the Assumption (85) and if C 0 is closed under truncation then Indeed, (84) is a consequence of Proposition 15 and of the equivalence B = π A (X) iff A = ρ B (X), that can be shown similarly as in Proposition 8, by using the Assumption (85) and Remark 14.
, δ ∈ R, is finite valued and concave on R, hence continuous on R. However, when Y ∈ L 1 (Q) satisfies E N n=1 un(X n + Y n ) > B (with the understanding that un(X n +Y n ) ∈ L 1 (P) for each n), it is not any more evident if F is continuous on R, as one has to guarantee that E N n=1 un(X n + Y n − δ) > −∞, for δ > 0. Set An := {X n + Y n > kn} and let kn ∈ R satisfy P(An) > 0 and Q n (An) > 0. For any δ > 0, (Y n − δ1 An )n ∈ L 1 (Q) and one has . This argument works when (Y n − δ1 An )n is not required to belong to C R .
(ii) Consider the following assumption on the utility function u at −∞: Such assumption is clearly satisfied if lim x→−∞ then, under (85), F (δ) is finite valued and continuous on R and (Y n − δ)n ∈ C R if so is Y.
The systemic risk measure (22) becomes For a given partition n and allocations C (n) , we can explicitly compute the unique optimal allocation Y of (86) and the corresponding systemic risk Theorem 8 For m = 1, · · · , h, and for k ∈ Im we have that where Xm = k∈Im X k and Proof In Appendix A.
Remark 15 Note that if we arbitrarily change the components of the vector X, but keep fixed the components in one given subgroup, say Im 0 , then the risk measure ρ(X) will of course change, but dm 0 and Y k m0 for k ∈ Im 0 remain the same. (89)

Proposition 17
The vector Q X of probability measures with densities given by (89) is the optimal solution of the dual problem (36), i.e., , m = 1, · · · , h, n ∈ Im, is a systemic risk allocation, as in Definition 1.
Proof First note that By (41), α Λ,B (Q X ) can be rewritten as Remark 6 concludes the proof.

Sensitivity analysis
Let X ∈M Φ , V ∈M Φ and set V m := k∈Im V k , for m = 1, · · · , h. We consider a perturbation εV, ε ∈ R, and perform a sensitivity analysis in the exponential case.
Consider the optimal allocations Y i X+εV and the optimal solution Q X+εV of the dual problem associated to ρ(X + εV), see (89). By (88) and (87) we have Marginal risk allocation of institution n ∈ Im: d dε 5. Sensitivity of the penalty function: 6. Systemic marginal risk contribution: The proof is postponed to the Appendix. The interpretation of these formulas is not simple because we are dealing with the systemic probability measure Q m X and not with the "physical" measure P. Indeed, Q m X is the "artificial" measure that emerges from the dual optimization (think of the difference between the physical measure P and a martingale measure). To fix the idea, let us take V with only one component different from 0, so that we write V =V j e j . From Item 1 (or Item 6), we see that In the following discussion, we have to keep in mind that Q m X already represents the systemic view of the system. If we replace Q X with P, none of the results of Proposition 18 will hold in general.

Remark 16
We now comment on the results of Proposition 18.
The first term E Q m X [−V n ] in (93) or (94) is easy to interpret: it is not a systemic contribution, as it only involves the increment V n in the (same) bank n. If we sum over all n in the same group, we obtain from (93) or (94) Responsibility for the whole group, but not for the single bank inside each group. Note that the sign of the increment V n in the first term of (93) is here relevant: an increment (positive) corresponds to a risk reduction, regardless of the dependence structure. If V is deterministic, the marginal risk allocation to bank n is exactly E Q m X [−V n ] = −V n and no other correction terms are present.
To understand the other terms in (93) or (94), take V =V j e j with j = n. In this way, the first term in (93) disappears (V n = 0) and we obtain d dε To fix the ideas, suppose that COV Q m The interpretation of the monotonicity condition (98) was already formulated at the end of Section 2. Its generalization in the context of h groups is formulated below in (96).
For r = m, i ∈ I m , we have the following.

Proposition 19
Define with Y i m , i ∈ I m , the optimal allocation to the primal problem with C = C (n ) . Then In particular, if the group I m consists of only one single element {i}, then Y i m is deterministic and If we compare the deterministic optimal allocation Y * (corresponding to C = R N ) with the (random) optimal allocations Y associated to one single group ( where Q X is the unique optimal solution of the dual problem associated to C = C R ∩M Φ .
Proof Given the subgroup I m , define with βm = k∈Im 1 α k . Then the optimal value with respect to C (n ) is given by Summing the components of the solutions relative to C (n) over k ∈ I m , we get Using Jensen inequality we obtain We have that (97) We now prove that . Since N n=1 x n m ↓ −∞, there must exist n 0 ∈ {1, · · · , N } and a subsequence x hm such that x n 0 hm ↓ −∞ as m → +∞. With an abuse of notation, denote again such subsequence x hm with xm. Then we have x n 0 m ↓ −∞. If there exists another coordinate n 1 ∈ {1, · · · , N } \n 0 such that lim infm→∞ x n 1 m = −∞, take the subsequence x km such that x n 1 km ↓ −∞. By diagonal procedure, we obtain one single sequence denoted again by xm such that x n 0 m ↓ −∞ and x n 1 m ↓ −∞, as m → +∞. We may adopt this procedure (at most N times) also in the case lim sup m→∞ x n 2 m = +∞ for some coordinate n 2 . At the end, we will obtain one single sequence xm and three disjoint sets of coordinate indices N − , N + , N * such that where K is a constant independent of m. We know that N − = ∅, since n 0 ∈ N − (but the other two sets N + and N * may be empty). Since N n=1 x n m ↓ −∞, we deduce that, for large m, where we use (100) in inequality (101) and C := n∈N + (c + bd n ) + n∈N − (An + and n ) + n∈N * un(d n + K) is independent of m. Then (a − b) n∈N − x n m ↓ −∞, as m → +∞, since a > b by Lemma 8 and x n m ↓ −∞ for each n ∈ N − . This concludes the proof.

Remark 17
Condition ρ(X) > −∞ is essentially a condition on the behavior of Λ : R N → R at −∞. Note that if the condition lim x→−∞ un(x) x = +∞ is not satisfied, there might be a problem. Take N = 2 and the increasing concave functions we cannot control Λ(x) as in (101).

A.2 Orlicz setting
We now recall an important result for the characterization of systemic risk measures of the form (22) on the Orlicz Heart.
Theorem 9 (Theorem 1, [9]) Suppose that L is a Fréchet lattice and ρ : L → R ∪ {+∞} is convex and monotone decreasing. Then 1. ρ is continuous in the interior of dom(ρ), with respect to the topology of L, 2. ρ is subdifferentiable in the interior of dom(ρ), 3. for all X ∈ int(dom(ρ)) where L * is the dual of L (for the topology for which L is a Frechet lattice), L * + = {Q ∈ L * | Q is positive} and α : L * → R ∪ {+∞} , defined by Θn(Z n ) + B, and observe that We have that f is convex and decreasing with respect to the order relation (18). Let f * (ξ) be its convex conjugate, for ξ ∈ L Φ * . We assume that ξ = 0. By the Fenchel inequality we obtain for all Z ∈ A and λ > 0, By definition of the convex Fenchel conjugate and the fact that M Φ is a product space, we have where we have used (27), and therefore We need only to prove that there is no duality gap in (102), i.e., if α Λ,B (ξ) < +∞ then Observe that, by the definition of f * , we have for each λ > 0 As ξ is not identically equal to 0 and M Φ is a linear space, we have sup Z∈M Φ {E[−ξZ]} = +∞ and therefore We claim that Assuming (104), we may immediately conclude that see (2.8) in [39], and the associated Lagrangian, see (4.4) in [39], Then (105) can be rewritten as As f : M Φ → R is convex decreasing and finite valued, Theorem 9 guarantees that it is continuous on M Φ (for the M Φ -norm). Therefore, see Example 1 on pages 7 and 22 in [39], the function F is closed convex in (Z, u). Then the absence of duality gap, expressed by (106) follows from Theorems 17 and 18 of [39], provided that the (convex) optimal value function, defined in (4.7) [39], is bounded from above in a neighborhood of 0. Clearly, it is sufficient to show the existence of an element Z 0 ∈ M Φ such that u → F (Z 0 , u) is bounded from above in a neighborhood of 0. The assumption Λ(+∞) > B guarantees the existence of Z 0 ∈ M Φ such that N n=1 E[un(Z n 0 )] > B (take Z n 0 equal to some large enough constant), i.e., f (Z 0 ) := N n=1 E[−un(Z n 0 )] + B < 0. Set 0 < δ < |f (Z 0 )|. Hence for all u ∈ R such that |u| < δ we have f (Z 0 ) < u and F (Z 0 , u) = E[ξZ 0 ] < +∞, as Z 0 ∈ M Φ and ξ∈ L Φ * .

Remark 18
In [25], (103) is deduced, by different means, in a L ∞ (R) setting and in the onedimensional case. In [3], (103) is obtained, by different means, in the multi-dimensional deterministic case, i.e. in R N .

A.3 Auxiliary results for existence
The following auxiliary results are standard and can be found in many articles on utility maximization. Recall that we are working under Assumptions 2 and 3.

Proof
The assumption guarantees the existence of a constant K > 0, which depends on ε, such that un(x) ≤ εx for x ≥ K and all n. Hence with associated Lagrangian L(d, Y, Z,µ) : The problem boils down to solve the system ∇L = 0, taking derivatives with respect to each (d, Y, Z, µ).
Consider now the general case h > 1. We have We compute the Gateaux derivative in the direction V ∈ M Φ : where in (108) we can apply the Dominated Convergence Theorem by using estimations similar to the ones in Remark 19. We now show that φ Y (V) is also the Fréchet derivative of M Φ , i.e., that We have and we obtain where we use twice the Hölder inequality. Since To conclude the proof, it is then sufficient to substitute Y of the form (88) in φ Y (V) to verify that φ Y (V) = 0 for all V ∈ M Φ . By (90) we then have By De L'Hôpital it follows Hence by (112), (114), and (115) we get 6. It follows by (109).

Remark 19
In (111) Note that in this case M φ 0 ⊆ L 2 (R), hence and because Xm, V m ∈ M φ 0 . We can conclude that f (Xm, V m)Zm is in L 1 (R).