On fairness of systemic risk measures

In our previous paper “A unified approach to systemic risk measures via acceptance sets” (Mathematical Finance, 2018), 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 prove a dual representation of a particular subclass of such systemic risk measures and the existence and uniqueness of the optimal allocation related to them. We also introduce an associated utility maximisation problem which has the same solution as the minimisation problem associated to the systemic risk measure. In addition, the optimiser in the dual formulation provides a risk allocation which is fair from the point of view of the individual financial institutions. The case with exponential utilities which allows explicit computation is treated in detail.


Introduction
Consider a vector X = (X 1 , . . . , X N ) ∈ L 0 ( , F, P; R N ) of N random variables denoting a configuration of risky (financial) factors at a future time T associated to a system of N financial institutions/banks. One of the first proposals in the framework of risk measures to measure the systemic risk of X, see Chen et al. [16], was to consider the map ρ(X) := inf{m ∈ R : (X) + m ∈ A}, (1.1) where : R N → R is an aggregation rule that aggregates the N -dimensional risk factors into a univariate risk factor, and A ⊆ L 0 ( , F, P; R) is an acceptance set of real-valued random variables. As within the framework of univariate monetary risk measures, systemic risk might again be interpreted as the minimal cash amount that secures the system when it is added to the total aggregated system loss (X), given that (X) allows a monetary loss interpretation. Note, however, that in (1.1), systemic risk is the minimal capital added to secure the system after aggregating individual risks. It might be more relevant to measure systemic risk as the minimal cash amount that secures Here, the amount m i is added to the financial position X i of institution i ∈ {1, . . . , N} before the corresponding total loss (X + m) is computed (we refer to Armenti et al. [3], Biagini et al. [7] and Feinstein et al. [27]). One of the main novelties of our paper [7] was the possibility of adding to X not merely a vector m = (m 1 , . . . , m N ) ∈ R N of deterministic cash amounts, but more generally a random vector Y ∈ C for some given class C. In particular, the main example considered in [7], and studied further in this paper, is given by choosing the aggregation function Y n ∈ R , (1.4) where the subspace L ⊆ L 0 ( , F, P; R N ) will be specified later. Here, the notation N n=1 Y n ∈ R means that N n=1 Y n is P-a.s. equal to some deterministic constant in R, even though each single Y n , n = 1, . . . , N, is a random variable. Under these assumptions, the systemic risk measure considered in [7] takes the form u n (X n + Y n ) ≥ B (1.5) 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, while the total capital requirement N n=1 Y n is determined today, contrary to (1.2), the individual allocation Y i (ω) to institution i does not need to be decided today, but in general depends on the scenario ω realised at time T . This total cash amount ρ(X) is computed today through the formula N n=1 ρ n (X) = ρ(X), where each ρ n (X) ∈ R is the risk allocation of each bank, as explained in Definition 1.2 below. Thus, one prominent example that can be modelled by considering random allocations is the default fund of a CCP 1 that is liable for any participating institution. We come back to this mechanism in Sect. 5.
By considering scenario-dependent allocations, we are also taking into account possible dependencies among the banks, as the budget constraints in (1.5) do not depend only on the marginal distribution of X, as it would happen for deterministic Y n . Definition 1.1 A scenario-dependent allocation Y X = (Y n X ) n=1,...,N ∈ C is called a systemic optimal allocation for ρ(X) defined in (1.5) if it satisfies ρ(X) = N n=1 Y n X and E[ N n=1 u n (X n + Y n X ))] ≥ B.
As two of the main results of the paper, -we study in Sect. 3 the dual formulation of the systemic risk measure (1.5) as where Q := (Q 1 , . . . , Q N ), the penalty function α B and the domain D are specified in Sect. 3. In particular, we establish existence and uniqueness of the optimiser Q X ∈ D of (1.6).
-we show in Sect. 4 existence and uniqueness of the systemic optimal allocation Y X for the systemic risk measure (1.5).
We now associate to the risk minimisation problem (1.5) a related utility maximisation problem that plays a central role in this paper, namely (1.7) If we interpret N n=1 u n (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) to express the dependence on the minimal level of expected utility B ∈ R and maximal budget level A ∈ R, respectively. We shall see in Sect. 4.1 that B = π A (X) if and only if A = ρ B (X), and in these cases, the two problems π A (X) and ρ B (X) have the same unique solution Y X . From this, we infer that once a level ρ(X) of total systemic risk has been determined, then -the systemic optimal allocation Y X for ρ maximises the expected system utility among all random allocations of total cost less than 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. Definition 1. 2 We say that a vector (ρ n (X)) n=1,...,N ∈ R N is a systemic risk allocation of ρ(X) if it fulfils N n=1 ρ n (X) = ρ(X).
The requirement N n=1 ρ n (X) = ρ(X) is known as the "full allocation" property; see for example Brunnermeier and Cheridito [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 no longer have such a canonical way to determine ρ n (X); however, we shall provide evidence that a good choice is for n = 1, . . . , N, (1.8) where Q X is the optimiser of the dual problem (1.6). To this end, suppose a probability vector Q = (Q 1 , . . . , Q N ) is given for the system and consider an alternative formulation of the systemic utility maximisation problem in terms of the valuation provided by Q, namely π Q (X) = π Q A (X) := sup E Note that in (1.9) (as well as in (1.10) below), the allocation Y belongs to a vector space L of random variables (introduced later) without requiring that Y ∈ C R (which would mean that the componentwise sum is equal to a deterministic quantity). Thus for π Q (X), we maximise the expected systemic utility among all Y ∈ L satisfying the budget constraint N n=1 E Q n [Y n ] ≤ A. Similarly, we can introduce a systemic risk measure in terms of the vector Q of probability measures by For ρ Q (X), we thus look for the minimal systemic cost N n=1 E Q n [Y n ] among all Y ∈ L under the acceptability constraint E[ N n=1 u n (X n + Y n )] ≥ B. A priori, ρ and ρ Q defined in (1.5) and (1.10) are quite different objects: even if they both subsume the same systemic budget constraint, ρ is defined only through the computation of the cash amount N n=1 Y n ∈ R, while in ρ Q the risk is defined by calculating the value (or the cost) of the random allocations, N n=1 E Q n [Y n ]. A similar comparison applies to π and π Q . Remark 1. 3 To better understand the above comparison, we make an analogy with the classical (univariate) utility maximisation from terminal wealth in securities markets. Let K := {(H.S) T : H admissible}, where (H.S) T is the stochastic integral, and let U(x) = sup{E [u(x + K)] : K ∈ K} be the utility from the initial wealth x ∈ R when optimally investing in the securities S adopting admissible strategies H . In this case, there is no need to introduce a cost operator, as we are investing in replicable contingent claims having by definition initial value x. On the other hand, ≤ 0} is the optimal utility function when a probability vector Q is given. A priori, the two problems are of different nature, unless one shows (see [6]) that for a particular probability measure Q x , the two problems have the same value and U(x) = U Q x (x) = min Q∈M U Q (x), where M is the set of martingale measures. From the mathematical point of view, once the minimax martingale measure Q x is determined, U Q x (x) is easier to solve than U(x), and the solution to U Q x (x) can then be used to find the solution to U(x). Also for the financial application, one may use Q x to compute the fair price (see [21] and [23, Remark 3.2.2]) of a contingent claim C by computing E Q x [C].
In view of the analogy in the above remark, we also prove in this paper that (ii) all four problems have the same (unique) solution Y X when A := ρ B (X); (1.11) (iv) and where the domain D is defined in (3.3) below and replaces, in analogy with utility maximisation, the set of martingale measures.
Hence ρ Q X B is a valid alternative to ρ B (same value and solution), and this justifies its use to compute the systemic risk. In addition, (1.11) shows that the operator assigned by E Q X [·] evaluates the risk component Y n X of the optimal allocation according to ρ B (not only to ρ Q X B ) and proves that the definition in (1.8) provides indeed a systemic risk allocation for ρ(X). In Sect. 5, we further elaborate on this interpretation, we study in detail the properties of the systemic risk probability vector Q X , and we provide in particular for the marginal risk contribution the formula We also discuss certain properties inferred from the above results that argue for the fairness of the systemic risk allocation. Based on the above exposition, we structure the remaining part of the paper as follows. In Sect. 2, we introduce the technical setting within Orlicz spaces and the main assumptions, and we show that our optimisation problems are well posed. In Sect. 3, we study the dual representation (1.6) of the systemic risk measure. Notably, existence and uniqueness of the dual optimiser Q X are proved in Proposition 3.1; see also Corollary 4.13 in Sect. 4. In Sect. 4, we deal with existence and uniqueness of solutions of the primal problems (1.5), (1.7) and (1.9), (1.10). To guarantee existence, we need to enlarge the environment and consider appropriate spaces of integrable random variables. In Sect. 5, we derive cash-additivity and risk marginal contribution properties of the systemic risk measure ρ(X), and fairness properties of the optimal allocations ρ n (X). The case with exponential utilities and grouping of institutions is treated in detail in Sect. 6, where additional sensitivity and monotonicity properties are established as well.
We conclude this section with a literature overview on systemic risk. In Craig and von Peter [20], Boss et al. [12] and Cont et al. [19], one can find empirical studies on banking networks, while interbank lending has been studied via interacting diffusions and a mean-field approach in several papers like Fouque and Sun [30], Fouque and Ichiba [28], Carmona et al. [15], Kley et al. [37], Battiston et al. [5]. Among the many contributions on systemic risk modelling, we mention the classical contagion model proposed by Eisenberg and Noe [26], the default model of Gai and Kapadia [33], the illiquidity cascade models of Gai and Kapadia [32], Hurd et al. [36] and Lee [39], the asset fire sale cascade model by Cifuentes et al. [18] and Caccioli et al. [14], as well as the model in Weber and Weske [45] that additionally includes cross-holdings. Further works on network modelling are Amini et al. [1], Rogers and Veraart [43], Amini et al. [2], Gleeson et al. [34], Battiston and Caldarelli [4], Detering et al. [24] and Detering et al. [25]. See also the references therein. For an exhaustive overview on the literature on systemic risk, we refer the reader to the recent volumes of Hurd [35] and of Fouque and Langsam [29].

The setting
We now introduce the setting and discuss some fundamental properties of our systemic risk measures. Given a probability space ( , F, P), we consider the space of random vectors The measurable space ( , F) is fixed throughout the paper and does not appear in the notations. Unless we need to specify a different probability, we also suppress P from the notations and simply write L 0 (R N ). In addition, we 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. We assume that L 0 (R N ) is equipped with the componentwise order relation, i.e., 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.

Orlicz setting
We consider systemic risk measures defined on Orlicz spaces; see Rao and Ren [40, Chap. III, Sect. 3.4 and Chap. IV, Sects. 4.2 and 4.4] 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 by Biagini and Frittelli [9] that the Orlicz setting is natural to embed utility maximisation problems, as the natural integrability condition E[u(X)] > −∞ is implied by E[φ(X)] < +∞; see below. Univariate convex risk measures on Orlicz spaces have been introduced and studied by Cheridito and Li [17] and Biagini and Frittelli [10].
Let u : R → R be a concave and increasing function with lim x→−∞ u(x) is a strict Young function, meaning that 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 dual of M φ is the Orlicz space L φ * , where the convex conjugate φ * of φ defined by φ * (y) := sup x∈R (xy − φ(x)), y ∈ R, is also a strict Young function. Note that . In addition, from the Fenchel inequality xy ≤ φ(x) + φ * (y), we obtain for any probability measure Q P that and we immediately deduce that dQ dP ∈ L φ * implies L φ ⊆ L 1 (Q; R).

Assumptions and some properties of ρ
We consider systemic risk measures ρ : as in (1.5), where the notation E[ N n=1 u n (X n + Y n )] ≥ B also implicitly means that N n=1 u n (X n + Y n ) ∈ L 1 (P) and the linear space C R was introduced in (1.4). Note that there is no loss of generality in assuming u n (0) = 0 (simply replace B with B − N n=1 u n (0)). The following are standing assumptions for the rest of the paper.
Also, from the Fenchel inequality u n (X) ≤ X dQ dP + v n ( dQ dP ) P-a.s., we immediately deduce that if X ∈ L 1 (Q) and E[v n ( dQ dP )] < ∞ for some probability measure Q P, then E[u n (X)] < +∞. Some further useful properties of v n are collected in Lemma A.5. Item 4) in Assumption 2.2 is related to the reasonable asymptotic elasticity condition on utility functions, which was introduced by Schachermayer [44]. The assumption in 4), even though quite weak (see [8,Sect. 2.2]), is fundamental to guarantee the existence of solutions to classical utility maximisation problems (see [44] and [8]). In this paper, it is necessary in Sect. A.3 and for the results of Sect. 4.  (b) Furthermore, we have for X ∈ dom(ρ) that If there exists an optimal allocation We complete this subsection by introducing one relevant example for the set of admissible random elements, which we denote by C (n) .
(2.4) Definition 2.5 models a cluster C = (C 1 , . . . , C h ) of financial institutions which is a partition of {X 1 , . . . , X N }. The constraint on Y is that the components of Y must sum up to a real number in each element C i of the cluster, i.e., j : 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 I m is fixed by n. It is then easily seen that C (n) is a linear space containing R N and closed with respect to convergence in probability. 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 can have only one group, h = 1, and C (n) = C R ∩ M is the largest possible class corresponding to an arbitrary random injection Y ∈ M with the only constraint N n=1 Y n ∈ R.

Dual representation of ρ
We 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 (3.1) is a consequence of Proposition 2.4, Theorem A.2 and Propositions 3.9 and 3.11 in [31], taking into consideration that C is a convex cone, the dual space of the Orlicz heart M is the Orlicz space L * and M = dom(ρ). Note that by Theorem A.2, the dual elements ξ ∈ L * + are positive, but a priori not normalised. However, we get E[ξ n ] = 1 by taking Y = ±e j ∈ R N and using N n=1 (ξ n (Y n ) − Y n ) ≤ 0 for all Y ∈ C, so that ξ j (1) − 1 ≤ 0 and ξ j (−1) + 1 ≤ 0 imply ξ j (1) = 1. This shows the form of the domain D in (3.3).
Proposition 3.1 guarantees the existence of a maximiser Q X to the dual problem (3.1) and that α B (Q X ) < +∞. Uniqueness is proved in Corollary 4.13 below.

Definition 3.2 Fix any
A vector Q of probability measures 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, because 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 N n=1 E Q n [Y n ] should at most be equal to N n=1 Y n , for any "fair" valuation operator E Q [·], which is the case if dQ dP ∈ D. There exists a simple relation among ρ B , ρ Q B and α B (Q) defined in (2.3), (1.10) and (3.2), respectively.

Proposition 3.3 We have
and

6)
where Q X is a solution of the dual problem (3.1).

Proposition 3.4 If α B (Q) < +∞, the penalty function in (3.2) can be written as
and E[v n (λ dQ n dP )] < ∞ for all n and all λ > 0. In addition, the infimum is attained in (3.7), i.e., dQ n dP v n λ * dQ n dP , (3.8) where λ * > 0 is the unique solution of the equation 2 Example 3.5 Consider the grouping of Definition 2.5. As C (n) is a linear space containing R N , the dual representation (3.1) applies. In addition, we have in each group that ±(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 ∈ I m , of the dual elements are all the same, i.e., Q i = Q j for all i, j ∈ I m , and the representation (3.1) becomes and If we have only one single group, all components of a dual element Q ∈ D are the same.
is a systemic risk allocation as in Definition 1.2, i.e., Example 3.6 Consider u n : R → R, u n (x) = −e −α n x /α n , α n > 0, for each n and let B < 0. Then v n (y) = 1 α n ln y. From the first order condition (3.9), we obtain that the minimiser is where H (Q n |P) := E[ dQ n dP ln dQ n dP ] is the relative entropy.

Existence of solutions
In this section, we deal with existence and uniqueness of optimal allocations for ρ B (X) and the other related primal optimisation problems introduced in Sect. 1.
Throughout this section, we assume X ∈ M and that Q = (Q 1 , . . . , Q N ) satisfies Q n P, dQ dP ∈ L * and α B (Q) < +∞, or equivalently ρ Q B (X) > −∞. Recall from Proposition 3.4 that this implies E[v n (λ dQ n dP )] < +∞ for all n and all λ > 0. Set where the inclusions follow from Remark 2.1 and dQ dP ∈ L * .
Without loss of generality, we may assume that u i (0) = 0, 1 ≤ i ≤ N , and observe that then When the utility functions u n are of exponential type, the Orlicz heart M is sufficiently large and contains the optimal allocation Y X to ρ B (X); see Sect. 6. This of course also happens for general utility functions on a finite probability space. As shown in Sect. 4.3, in general, we cannot expect to find the solution Y Q for the problem ρ Q B (X) in the space M , but only in the larger space L 1 (Q), and this motivates the introduction of several extended problems. Let B ∈ R and define Analogously, we define π Q A (X), π Q A (X) and π A (X) for A ∈ R by using the optimisation (1.9). We show in (4.8) and (4.9) below that these extensions from M to integrable random variables do not change the optimal values.
In order to prove the existence of an optimal allocation for ρ B (X), we proceed in several steps. In Theorem 4.10, we first prove the existence of a solution Y Q ∈ L 1 (Q) for ρ Q B (X). Then in Proposition 4.11, we show that when it exists, the optimiser to ρ B (X) or to ρ B (X) coincides with Y Q X ∈ L 1 (Q X ). The next key step is to show the existence of Y ∈ L 1 (P) which is, as specified in Theorem 4.14, a candidate solution to the extended problem and then to prove that Y ∈ L 1 (Q X ). In a final step (see Theorem 4.19, Proposition 4.22 and Corollary 4.23), we prove that ρ B (X) = ρ B (X) and that the above Y ∈ L 1 (P, Q X ), hereafter denoted withỸ X , is an optimiser of the extended problem ρ B (X) and hence coincides with Y Q X .

On ρ B (X) and π A (X)
Recall that under Assumption 2.2, C is a convex cone so that if Y ∈ C, then Y + δ ∈ C for every deterministic δ ∈ R N . Note that ρ Q B (X) < +∞ and π Q A (X) > −∞.

(X) and there exists a solution to one of the two problems π A (X) or ρ B (X), then it is the unique solution to both problems.
Proof (a) "⇐" Let A = ρ B (X) and suppose first that π A (X) > B. Then there exists Suppose now that π A (X) < B. Then there must exist δ > 0 such that we have Due to E[u n (Z n )] > −∞ for all Z ∈ M and the continuity of u n , we may select we obtain a contradiction.
(b) This follows in the same way as "⇒" in (a), replacing M with L 1 (P, Q X ).
(c) Suppose there exists Y ∈ C 0 ∩ M which is a solution to problem (1.5). As A := ρ B (X), then N n=1 Y n = A and the constraint in problem (1.7) is fulfilled for Y. By (a), B = π A (X) ≥ E[ N n=1 u n (X n + Y n )] ≥ B and we deduce that Y is a solution to problem (1.7). Suppose there exists Y ∈ C 0 ∩ M which is a solution to problem (1.7) and set B := π A (X). Then E[ N n=1 u n (X n + Y n )] = B and the constraint in problem (1.5) is fulfilled for Y. By (a), and we deduce that Y is a solution to problem (1.5). As ρ B (X) admits at most one solution by Proposition 2.4, the same must be true for π A (X).
Proof Use step by step the same arguments as in the proof of Proposition 4.1, re- The uniqueness in (c) is a consequence of Remark 4.9.
When using Q = Q X , we have already proved that ρ B (X) = ρ Q X B (X). Similarly:

On the optimal values
The main contribution of this section is to show that the optimal values coincide, see (4.8) and (4.9) below, and that, see (4.11) below, where U n (a n ) and a ∈ R N . In the sequel, we write U Q n n (a n ) when we need to emphasise the dependence on Q n . Note that W ∈ M φ n implies W ∈ L 1 (Q n ) and the problem (4.3) is well posed. Due to the monotonicity and concavity of u n , the function U n is monotone increasing, concave and continuous on R and we may replace in its definition the inequality with an equality sign. However, in general, the solution to (4.3) only exists on a larger domain, as suggested by the well-known result reported in Proposition A.6. This leads us to introduce the auxiliary problems U n (a n ) := sup{E[u n (X n + W )] : W ∈ L 1 (Q n ), E Q n [W ] ≤ a n }, where L 1 (P, Q n ) is defined as in (4.1). The following proposition is a multi-dimensional version of well-known utility maximisation problems. Its proof is based on the extended Namioka-Klee theorem and deferred to Appendix A.4.

Proposition 4.4 We have that
U n (a n ) = U n (a n ) = U n (a n ) < +∞, if U n (a n ) < u n (+∞), then U n : R → R is differentiable, and We now show that the optimal values are the same.

Lemma 4.5 Let
and which is a contradiction. Hence (4.7) holds true. Note that Indeed, take Y ∈ M and let a n := E Q n [Y n ] ∈ R and Z n := Y n − a n ∈ M φ n . Then U n (a n ), (4.10) which shows the first equality in (4.8). Then π A (X) are consequences of (4.4) and the decompositions analogous to the one just obtained for π

Proposition 4.6 Let
U n (a n ) = N n=1 U n (a n * ) and N n=1 a n * = A. (4.11) Proof Fix δ > 0 and let a m = (a 1 m , . . . , a N m ) m∈N be an approximating sequence for the supremum in (4.11). Then N n=1 U n (a n m ) ≥ π Q A (X) − δ =: C and N n=1 a n m = A for large enough m. Then (4.11) is a consequence of the continuity of U n and of Lemma 4.7 below, which guarantees that a m belongs to a compact set in R N .
We now turn to the uniqueness of the solution to problem (3.2). The proof is in Appendix A.4 and uses the same arguments as in the proof of Proposition 2.4. (4.12) and there exists at most one Z ∈ L 1 (P, Q) satisfying

Lemma 4.8 The penalty function can be written as
Remark 4.9 From (4.9) and (3.5), we have Hence with a proof similar to the one of Lemma 4.8, we may replace the inequality with an equality sign in the budget constraint in the definition of ρ

On the solution of ρ Q and comparison of solutions
where λ * is the unique solution to (3.9).
The integrability conditions hold thanks to the results stated in Appendix A.3. From (3.5) and the expression (3.8) for the penalty, we compute We show that Y n Q satisfies the budget constraint by Lemma A.5 and (3.9). Finally, from (4.9), it follows that ρ Q B (X) = ρ Q B (X), and Remark 4.9 implies uniqueness.
When solutions to both problems ρ B (X) and ρ Q X B (X) exist, they coincide.

Proposition 4.11
Let Y X ∈ C 0 ∩ M be the optimal allocation for ρ B (X) and Q X a solution to the dual problem (3.1).
as Y X ∈ C and Q X ∈ D. From (4.14), (3.5), (3.4) and (4.17), we deduce that As Y X satisfies (4.16), the definition of ρ which shows together with (4.18 From (4.19) and (4.20), we then deduce that As both X + Y X and X + Y Q X satisfy the budget constraints associated to α B (Q X ) in (4.13), this implies that α B (Q X ) is attained by both X + Y X and X + Y Q X . The uniqueness shown in Lemma 4.8 allows us to conclude that Y X = Y Q X .
Remark 4.12 Theorem 4.19 below proves the existence ofỸ X ∈ C 0 ∩ L 1 (P, Q X ) satisfying (4.16)-(4.18) withỸ X instead of Y X . Then the above proof shows that Y X = Y Q X . Similarly, Corollary 4.13 below holds for suchỸ X ∈ C 0 ∩ L 1 (P, Q X ).
We now show that the maximiser of the dual representation is unique.
As v n is invertible, we conclude that λ *

A first step
We first show that ρ B reaches its infimum at some Y ∈ L 1 (P; R N ).

Remark 4.15
We note that the random vector Y in Theorem 4.14 satisfies all the conditions for being the optimal allocation for ρ B (X), except for the integrability condition Y ∈ M , which is replaced by Y ∈ L 1 (P; R N ). Furthermore, Y = lim k→∞ Y k P-a.s. for Y k ∈ C 0 ∩ M . If we assume 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 (2.4), then Y also belongs to C 0 , but in general not to C (as M is in general not closed for P-a.s. convergence). A special case is when the cardinality of is finite and the set C is closed for P-a.s. convergence; under these assumptions, Y belongs to C and Y = Y X = Y Q X . In Sect. 4.4.2, we show when Y also belongs to C 0 ∩ L 1 (Q X ; R N ).
Proof of Theorem 4.14 Take a sequence (V k ) k∈N k∈N 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 then, because of the constraint N n=1 V n k = c k , both N + ∞ and N − ∞ must be nonempty. From Lemma A.1 (a), Jensen's inequality and (4.2) give which is a contradiction as the second sum in the last term is not bounded from above. Hence our minimising sequence (V k ) k∈N has bounded L 1 (P; R N )-norm and we may apply a Komlós compactness argument as in [22,Theorem 1.4]. Applying this to the sequence (V k ) k∈N ⊆ C, we can find for all k some Y k ∈ conv(V i , i ≥ k) ⊆ C, as C is convex, such that (Y k ) converges P-a.s. to some Y ∈ L 1 (P; R N ). 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 u n (X n +V n k )] ≥ B, also the Y k satisfy this constraint and therefore and from ρ B (X) ≤ N n=1 Y n k ≤ c k , we then deduce that N n=1 Y n = ρ B (X). We now show that Y also satisfies the budget constraint. If all utility functions are bounded from above, this is an immediate consequence of Fatou's lemma In the general case, recall first that the sequence (V k ) is bounded in L 1 (P; R N ), and the argument used in (4.22) shows that We now need to exploit the Inada condition at +∞. Applying Lemma A.1 (b) to the utility functions u n , assumed null in 0, we get Plugging X + Y into the expression above and applying Fatou's lemma, we have As the term b(ε) cancels in the above inequality, we conclude that for all ε > 0, and since sup k∈N X + Y k 1 < ∞, we obtain E[ N n=1 −u n (X n +Y n )] ≤ −B so that Y satisfies the constraint.

Second step: the optimal allocation is in
We now prove further integrability properties of the random vector Y in Theorem 4.14.

Lemma 4.17
The random vector Y in Theorem 4.14 satisfies Y + ∈ L 1 (Q X ).
Proof In Theorem 4.14, we have proved the existence of Y ∈ L 1 (P; By passing to a subsequence, we may assume that N n=1 Y n k ↓ ρ B (X). Let j ∈ {1, . . . , N}. Fatou's lemma gives From the proof of Theorem 4.14, we know that (X n + Y n k ) k∈N is L 1 (P)-bounded for all n = 1, . . . , N, and thus

By Remark 2.1, it then follows that
where we recall that by assumption, X j ∈ M φ j ⊆ L 1 (Q j X ). From (4.25) and (4.26) together with (4.24), the claim follows.

The final step
For our final result on existence, we need one more assumption.

Definition 4.18 We say that C 0 is closed under truncation
and for all m ≥ m Y , we have Note that in Definition 2.5, the set C (n) 0 is closed under truncation.

Theorem 4.19
Let C = C 0 ∩ M and suppose that C 0 ⊆ C R is closed for convergence in probability and closed under truncation. For any X ∈ M , there exists so thatỸ X is the solution to the extended problem ρ B (X).
Proof Take asỸ X the vector Y in Theorem 4.14, which belongs to L 1 (P, Q X ) by Theorem 4.14 and Lemmas 4.16 and 4.17, and to C 0 as C 0 is closed for convergence in probability and Y = lim m→∞ Y m P-a.s. and (Y m ) ⊆ C 0 . Comparing Theorem 4.19 with Theorem 4.14, we see that it remains to prove ρ B = ρ B and holds asỸ X fulfils the budget constraints of ρ Q X B (X).

Proposition 4.20 Suppose that C 0 is closed under truncation. Then
Proof Fix Y ∈ C 0 ∩L 1 (Q X ; R N ) and consider Y m for m ∈ N as in (4.27), where without loss of generality, we assume m Y = 1. Note that N n=1 Y n m = c Y (= N n=1 Y n ).

By boundedness of Y m and (4.27), we have
The map ρ B is defined on M , but the admissible claims Y belong to the set C 0 ∩ L 1 (P, Q X ) not included in M . As L 1 (P, Q X ) ⊆ L 1 (P; R N ) by the same argument as in the proof of Proposition 2.4, we can show that ρ B (X) > −∞ for all X ∈ M . By the same argument as in the proof of Proposition 2.4 and by (2.1), we also deduce that ρ B (X) < +∞ for all X ∈ M , so that the function ρ B : M → R is convex and monotone decreasing on its domain dom( ρ) = M . From Theorem A.2, we then know that the penalty functions of ρ B and ρ B are defined as . We then have that because N n=1 (E Q n X [Z n ] − Z n ) ≤ 0 for all Z ∈ C 0 ∩ L 1 (P, Q X ) as shown in Proposition 4.20. The last equality follows from (4.12). The opposite inequality is trivial as

Proposition 4.22
If C 0 is closed under truncation, then

A (X) exist and coincide with
and Q X is the unique solution to the dual problem (3.1).
Proof From (4.28), (4.9), (4.8), (3.6) and Corollary 4.3, we already know that (4.29) and (4.30) hold true when A := ρ B (X). By Theorem 4.19, there exists a solutioñ Y X ∈ C 0 ∩L 1 (P, Q X ) to ρ B (X) and by Proposition 4.11 and Remark 4.12, it coincides with the unique solution Y Q X for ρ Q X B (X). By (4.15), and then Y Q X =Ỹ X ∈ C 0 ∩ L 1 (P, Q X ) proves thatỸ X is also the solution for ρ Q X B (X). From (4.29) and (4.30), we know that B = π . Therefore Proposition 4.2 (d) shows thatỸ X is the unique solution to π Q X A (X) and π Q X A (X).

Additional properties of Q X and fair risk allocation
In this section, we provide additional properties for the systemic risk measure ρ(X) from (1.5) and for the systemic risk allocations ρ n (X) = E Q n X [Y n X ], n = 1, . . . , N, from (1.8). We argue that the choice of Q X as systemic vector of probability measures is fair from the point of view of both the system and the individual banks.

Cash-additivity and marginal risk contribution
In this section, we provide a sensitivity analysis of ρ(X) with respect to changes in the positions X, which also shows the relevance of the dual optimiser Q X . We first show that ρ(X) is cash-additive. Recall from (1.3). for all V such that εV ∈ W C for all ε ∈ (0, 1].

Example 5.2
For the set C (n) in Definition 2.5, ρ is cash-additive on W C (n) = C (n) . The latter equality holds because we are not imposing any restrictions on the vector d = (d, . . . , d m ) ∈ R m which determines the grouping.

Remark 5.3 Under Assumption 2.2, we have R N ⊆ W C and then (5.1) holds for all
The marginal risk contribution d dε ρ(X+εV)| ε=0 was also considered in [13] and [3] and is an important quantity which describes the sensitivity of the risk of X with respect to the impact V ∈ L 0 (R N ). The property (5.1) cannot be immediately generalised to the case of random vectors V as N n=1 V n / ∈ R in general. In the following, we obtain the general local version of cash-additivity, which extends (5.1) to a random setting.
Proposition 5.4 Let X and V ∈ M . Let Q X be the solution to the dual problem (3.1) associated to ρ(X) and assume that ρ(X+εV) is differentiable with respect to ε at ε = 0, and that dQ X+εV dP Proof As the penalty function α B does not depend on X, (3.4) yields where the equality between (5.3) and (5.4) 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 5. 5 We emphasise that the generalisation (5.2) of (5.1) holds because we are computing the expectation with respect to the vector Q X . The assumptions of Proposition 5.4 are satisfied for exponential utility functions, which are considered in Sect. 6.

Interpretation and implementation of ρ(X)
Going back to the definition (1.5), we see that ρ(X) represents the minimal total cash amount needed to make the system acceptable at time T . For notational simplicity, we write in the sequel Y X for the solution of ρ B (X), i.e., do not distinguish Y X and Y X . As already mentioned in Sect. 1 and as a result of Proposition 4.1, one economic justification for ρ is that the optimal allocation Y X of ρ(X) maximises the expected system utility among all random allocations of cost less than or equal to ρ(X).
We notice also that the class C may determine the level of risk sharing (as explained below in (b)) between the banks, ranging from no risk sharing in the case C = R N of deterministic allocations to the case C = C R of full risk sharing, and other constraints in between as in the Definition 2.5 of grouping. We now discuss two features of our systemic risk measure.

Implementation of the scenario-dependent allocation
(a) In practice, the scenario-dependent allocation can be described as a default fund as in the case of a CCP (see [3]). The amount ρ(X) is collected at time 0 according to some systemic risk allocation ρ n (X), n = 1, . . . , N, which satisfies N n=1 ρ n (X) = ρ(X). Then at time T , this exact same amount is redistributed among the banks according to the optimal scenario-dependent allocations Y n X satisfying N n=1 Y n X = ρ(X), so that the fund acts as a clearing house, assuming that each bank fulfils its commitment.
(b) An alternative interpretation and implementation of the scenario-dependent allocation more in the spirit of monetary risk measures is in terms of capital requirements together with a risk sharing mechanism. Consider again a given systemic risk allocation ρ n (X), n = 1, . . . , N. At time 0, a capital requirement ρ n (X) is imposed on each bank n = 1, . . . , N. Then at time T , a risk sharing mechanism takes place: each bank provides (if negative) or collects (if positive) the amount Y n X − ρ n (X), assuming as before that each bank fulfils its commitment. Note that in sum, the financial position of bank n at time T is X n + ρ n (X) + (Y n X − ρ n (X)) = X n + Y n X as required. This risk sharing mechanism is made possible by the clearing property N n=1 (Y n X − ρ n (X)) = 0, which follows from N n=1 Y n X = ρ(X) and the full risk allocation requirement N n=1 ρ n (X) = ρ(X). The incentive for a single bank to enter in such a mechanism is made clear below after we introduce the choice of a fair risk allocation in Sect. 5.3.
Total risk reduction and dependence structure of X From a system-wide point of view, considering the optimal random allocation Y X implies a reduction of the total amount needed to secure the system (compared with the optimal deterministic allocation). This reduction is also a consequence of our framework of scenario-dependent allocations that allows taking into account the dependence structure of X. An example showing these features can be found in [7,Example 7.1]. If the aggregation function is a sum of utility functions as in (1.3), one can see directly that the dependence structure of X is taken into account from the constraint E[ N n=1 u n (X n + Y n )] ≥ B in (1.5), which depends only on the marginal distributions of X in the case of deterministic Y n .

Fair systemic risk allocation ρ n (X)
We now address the problem of choosing a systemic risk allocation (ρ n (X)) n=1,...,N in R N (or individual contributions at time zero) as introduced in Definition 1.2. Note that in our setting, besides providing a ranking of the institutions in terms of their systemic riskiness, a risk allocation ρ n (X) can be interpreted as a capital contribution/requirement for institution n in order to secure the system. From (5.2), we see that already appeared as a multivariate valuation operator, and on the other hand, we have obtained in (4.20) that the minimiser Y X and the maximiser Q X of the dual problem satisfy n = 1, . . . , N, gives a systemic risk allocation.
Any vector Q = (Q n ) n=1,...,N of probability measures gives rise to a valuation operator E Q [·] and to the systemic risk measure ρ Q given by (1.10). Note, however, that in (1.10), the clearing condition N n=1 Y n = ρ(X) is not guaranteed since the optimisation is there performed over all Y ∈ M . Now, using the valuation E Q X [·] given by the dual optimiser, we know by Proposition 4.11 that the optimal allocation in (1.10) fulfils the clearing condition Y X ∈ C R , and is in fact the same as the optimal allocation for the original systemic risk measure in (1.5). From (4.19) and (4.20), we obtain which shows that the valuation by E Q X [·] agrees with the systemic risk measure ρ(X). This supports the introduction of E Q X [·] as a suitable systemic valuation operator. The essential question for a financial institution is now whether its allocated share of the total systemic risk given by the risk allocation (E Q 1 , is fair. With the choice Q = Q X , Corollary 4.3, Lemma 4.5 and (4.11) lead to (5.5) Choose A = ρ B (X). Then Proposition 4.2 and the fact that Y X is then the solution of π = A, and (5.5) can be rewritten as This means that by using Q X for valuation, the system utility maximisation in (1.9) reduces to individual utility maximisation for the banks without the "systemic" constraint Y ∈ C, i.e., to The optimal allocation Y n X and its value E Q n X [Y n X ] can thus be considered fair by the nth bank as Y n X maximises its individual expected utility among all random allocations (not constrained to be in C R ) with value E Q n X [Y n X ]. In particular, it is clear then that for individual banks, it is more advantageous to use random rather than cash-valued allocations as the supremum will be larger, as previously stated in Sect. 5.2 (a) and (b). This finally argues for the fairness of the risk allocation ) as fair valuation of the optimal scenario-dependent allocation (Y 1 X , . . . , Y N X ).

The exponential case
In this section, we focus on a relevant case under Assumption 2.2, that is, we set C = C (n) , see Definition 2.5 and Example 3.5, and choose u n (x) = −e −α n x /α n , α n > 0, n = 1, . . . , N, as in Example 3.6. Then v n (y) = 1 α n (y ln y − y) and v n (y) = 1 α n ln y. We select B < N n=1 u n (+∞) = 0. Under these assumptions, For a given partition n and allocations C (n) , we can explicitly compute the value ρ(X), the unique optimal allocation of (6.1) and the unique optimiser Q X of the corresponding dual problem (3.10). Note that in the present exponential case, the vectorỸ X = Y X ∈ (M exp ) N is the solution for ρ(X) and ρ(X).
The vector Q X of probability measures with densities is the solution of the dual problem (3.10), i.e.,

5)
and is a systemic risk allocation as in Definition 1.2. Proof By (3.11), we note that Q X defined in (6.4) belongs to D. Using Q X and selecting λ * = − B β from Example 3.6, it is easy to verify that the random variable Y n X := −X n − v n (λ * dQ n X dP ) from Corollary 4.23 coincides with the expression in (6.3) and n∈I m Y n . A priori, these equations are not sufficient to prove that (Y X , Q X ) are indeed the solutions to the primal and dual problems, as one needs to know that one of the two is indeed an optimiser of the corresponding problem. The proof that Y X defined in (6.3) is the optimiser of ρ(X) uses the Lagrange method and several estimates of lengthy computations; it is omitted. 3 Assuming that Y X is the optimiser of the problem associated to ρ, so that we have ρ(X) = Y I = d m , we now prove (6.5). First notice that By (3.13), α B (Q X ) can be rewritten as Then (3.12) concludes the proof.

Remark 6.3
Note that if we arbitrarily change the components of the vector X, but keep fixed the components in one given subgroup, say I m 0 , then the risk measure ρ(X) will of course change, but d m 0 and Y k m 0 for k ∈ I m 0 remain the same.

Sensitivity analysis
Let X ∈ (M exp ) N , V ∈ (M exp ) N and set V m := k∈I m V k for m = 1, . . . , h. We consider a perturbation εV, ε ∈ R, and perform a sensitivity analysis. Consider the optimal allocations Y i X+εV and the solution Q X+εV of the dual problem associated to ρ(X + εV); see (6.4). By (6.3) and (6.2), we have Proposition 6.4 Let ρ be the systemic risk measure defined in (6.1). Then we have: 2) The local causal responsibility is

4) The marginal risk allocation of institution n ∈ I m is
5) The sensitivity of the penalty function is

6) The systemic marginal risk contribution is
Proof The proof is the result of lengthy computations and is omitted. 4 The interpretation of the above formulas is not simple because we are dealing with the systemic probability measure Q m X and not with the "physical" measure P. Think of the difference between the physical measure P and a martingale measure. If we replace Q m X with P, none of the results of Proposition 6.4 will hold in general. The first term E Q m X [−V n ] in (6.6) or (6.7) is easy to interpret: is the contribution to the marginal risk allocation of bank n regardless of any systemic influence. The sign of the increment V n in the first term of (6.6) 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 terms are present.
To understand the other terms in (6.6) or (6.7), take V =V j e j with j = n. Then the first term in (6.6) disappears (V n = 0) and we obtain To fix ideas, suppose that Cov Q m X (V j , X n ) < 0 and examine for the moment only the contribution of 1 β m Cov Q m X (V j , X n ). This component does not depend on the specific α n , but it depends on the dependence structure between (V j , X n ). If the systemic risk probability Q m X attributes negative correlation to (V j , X n ), then from the systemic perspective, this is good (independently of the sign of V j ); indeed, a decrement in bank j is balanced by bank n, and vice versa. If bank n is negatively correlated (as seen by Q m X ) with the increment of bank j , then the risk allocation of bank n should decrease. Therefore, bank n takes advantage of this as its risk allocation is reduced ( 1 β m Cov Q m X (V j , X n ) < 0). Since the overall marginal risk allocation of the group m is fixed (and equal to E Q m , someone else has to pay for this advantage to bank n. This is the last term in (6.7), which is discussed next.
For the third component in (6.7), we distinguish between the systemic component − 1 , which only depends on the aggregate group X m , and the systemic relevance 1 α n of bank n. The systemic quantity is therefore distributed among the various banks according to 1 α n . In addition, this term must compensate for the possible risk reduction (the second term in (6.7)) as the overall risk allocation to group m is determined by . Finally, 1) and 6) express the same property (which holds in general, as shown in Proposition 5.4) for one group or for the entire system, respectively.

Monotonicity
Another desirable fairness property is monotonicity. If C 1 ⊆ C 2 ⊆ C R , then we have 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 unconstrained 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 should decrease, i.e., ρ n 1 (X) ≥ ρ n 2 (X) for each n = 1, . . . , N.
The opposite would clearly be perceived as unfair. In the next result (see in particular (6.11)), we prove that (6.8) holds true in the context of the Definition 2.5 of grouping when the risk allocation ρ n (X) = E Q n X [Y n X ] is computed using Q X . If we were to select a vector of probability measures R different from Q X to compute the risk allocation with the formula E R n [Y n X ], the property (6.8) would be lost in general. For a given partition n and C = C (n) , let Y k r , k ∈ I r , r = 1, . . . , h, be the corresponding optimal allocations of the primal problem (6.1) and Q r X , r = 1, . . . , h, the solutions of the corresponding dual problem (3.10) (in this section, we suppress the label X from the optimal allocation Y X to ρ(X)).
Consider for some m ∈ {1, . . . , h} a nonempty subgroup I m of the group I m and set I m := I m \I m . Then the h + 1 groups I 1 , I 2 , . . . , I m−1 , I m , I m , I m+1 , . . . , I h correspond to a new partition n . The optimal allocations of the primal problem (6.1) with C = C (n ) coincide with Y k r , k ∈ I r , for r = m. For r = m, i ∈ I m , we have the following.

Proposition 6.5 Denote by (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 (i.e., with C = C R ∩ (M exp ) N ), we conclude that

11)
where Q X is the unique solution of the dual problem with C = C R ∩ (M exp ) N .
Proof Given the subgroup I m , define β m := k∈I m 1 α k . Then the 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's inequality, we obtain Then (6.10) and (6.11) follow directly by (6.9).
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 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, N n=1 x n m ≤ 0 so that for each fixed (large) m,

Appendix
Using the inequalities of Lemma A.  Since R N ⊆ C, this gives m1 ∈ C and {Y ∈ C : (X + Y) ∈ A} = ∅ so that ρ(X) < +∞. Hence ρ : M → R and then convexity and monotonicity are straightforward. The remaining properties in (a) are a consequence of Theorem A.2 below and the fact that M is a Banach space.
To prove (b), we claim that if E[ (X + Y)] > B, then Y ∈ C cannot be optimal, i.e., Indeed, the continuity of u n and E[u n (Z n )] > −∞ for all Z ∈ M imply the ex- We now show uniqueness by way of contradiction. Suppose ρ(X) is attained by two distinct Y 1 ∈ C and Y 2 ∈ C so that P[Y For λ ∈ [0, 1], set Y λ : and for λ ∈ (0, 1), where we used that u j is strictly concave and P[Y We have that f is convex and decreasing with respect to the componentwise order. Let f * (ξ ) be its convex conjugate for ξ ∈ L * . We assume that ξ ≡ 0. By the Fenchel inequality E[Zξ ] ≤ f (Z)+f * (ξ ), we obtain for all Z ∈ A and λ > 0 that Hence By the definition of the convex Fenchel conjugate and the fact that M is a product space, we have where we have used (2.2), and therefore To prove (3.7), we only need to show that there is no duality gap in (A.4), i.e., if α B (ξ ) < +∞, then Observe that by the definition of f * , we have for each λ > 0 that As ξ ≡ 0 and M is a linear space, we have sup Z∈M E[−ξ Z] = +∞ and therefore We claim that Assuming (A.6), we may immediately conclude that We now prove (A. 6 .7)] is bounded from above in a neighbourhood of 0. This is easily verified by showing the existence of an element Z 0 ∈ M such that u → F (Z 0 , u) is bounded from above in a neighbourhood of 0. This concludes the proof of (3.7). To prove (3.8), we set ξ n := dQ n dP ≥ 0 a.s. From Lemma A.5 below, v n is strictly convex with v n (+∞) = +∞, v n (0+) = u n (+∞), lim z→+∞ v n (z) z = +∞ because of Assumption 2.2, 2) and v n is continuously differentiable. As u n (+∞) = 0 and u n (−∞) = +∞, we get v n (0) = −∞ and v n (+∞) = +∞. Set η = 1 λ ∈ (0, +∞) and consider the differentiable function F : (0, +∞) → R defined by Then α B (ξ ) = inf η>0 F (η) and (3.9) can be rewritten as Note that if η * > 0 is the (unique, see below) solution to (A.9), then inserting η * into F (η) immediately gives (3.8).
Next, using the integrability conditions provided by Lemma A.4 below, we show the existence of a solution η * > 0 of (A.9). First we consider η → +∞. Since Hence lim inf η→+∞ F (η) > 0. We now look at η → 0 and find The convexity of v n implies that for any fixed z 0 > 0 and z > z 0 ,

A.3 Auxiliary results for existence
The following auxiliary result is standard and can be found in many articles on utility maximisation; see for example [8,Lemma 18]. Recall that we are working under Assumption 2.2, 4).

A.4 Proofs for Sect. 4.2
Proof of Proposition 4.4 From M φ n ⊆ L 1 (P, Q n ) ⊆ L 1 (Q n ), we clearly have U n (a n ) ≤ U n (a n ) ≤ U n (a n ) ≤ u n (+∞) so that if U n (a n ) = u(+∞), then U n (a n ) = U n (a n ) = U n (a n ) = u n (+∞). (A.10) By the Fenchel inequality, we get E[u n (X n + W )] ≤ λ(E Q n [X n ] + E Q n [W ]) + E v n λ dQ n dP and hence U n (a n ) ≤ U n (a n ) ≤ U n (a n ) ≤ inf λ>0 λ(E Q n [X n ] + a n ) + E v n λ dQ n dP < +∞ (A. 11) as E[v n (λ dQ n dP )] < +∞. Therefore (4.4) is a consequence of (A.10) and (4.6).
To show (4.6), we consider the integral functional I : M φ n → R defined by I (X n ) = E[u n (X n )]. It is finite-valued, monotone increasing and concave on M φ n (as E[u n (X n )] ≤ u n (E[X n ]) < +∞), and therefore by Theorem A.2, it is normcontinuous on M φ n . We can then follow the well-known duality approach (see for example [11]), as follows.
When U n (a n ) < u n (+∞), we have U n (a n ) = inf λ>0 (V n (λ) + λa n ) from (4.6), which shows that U n and V n are conjugate to each other, i.e., we have V n (λ) = sup a n >0 U n (a n ) − λa n .
Let a ∈ K and take i with a i = min{a 1 , . . . , a N }. Note that for all j = 1, . . . , N, we have a j ≤ A − (N − 1)a i because N n=1 a n ≤ A. Assume that a i < R. Then B ≤ N n=1 U n (a n ) ≤ U i (a i ) 1 + n =i U n (A − (N − 1)a i ) U i (a i ) which is a contradiction. Therefore 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. This proves the claim.
Let X ∈ M and consider the function F (δ) := E[ N n=1 u n (X n + Y n − δ)] with δ ∈ R. If Y ∈ M , then F is finite-valued and concave on R, hence continuous on R (see the discussion at the beginning of Sect. 4.2). However, when Y ∈ L 1 (Q) satisfies E[ N n=1 u n (X n + Y n )] > B (with the understanding that u n (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 u n (X n + Y n − δ)] > −∞ for δ > 0.
Lemma A.7 If X ∈ M and Z ∈ L 1 (Q) satisfy E[ N n=1 u n (X n + Z n )] > B, then there exists Z ∈ L 1 (Q) which satisfies N n=1 E Q n [ Z n ] < N n=1 E Q n [Z n ] and E[ N n=1 u n (X n + Z n )] = B.
Proof Set A n := {X n + Z n > k n } and let k n ∈ R satisfy P[A n ] > 0 and Q n [A n ] > 0. For any δ > 0, consider the random variable Z ∈ L 1 (Q) with Z n := Z n − δ1 A n and define G(δ) := E[ N n=1 u n (X n + Z n − δ1 A n )]. Then u n (X n + Z n )1 A c n + E Finally, uniqueness follows from an argument similar to the one applied at the end of the proof of Proposition 2.4, replacing N n=1 Y n with N n=1 E Q n [Y n ].