Dual representations for systemic risk measures based on acceptance sets
aa r X i v : . [ q -f i n . M F ] O c t Dual representations for systemic risk measures based on acceptance sets
Maria Arduca
Department of Statistics and Quantitative Methods, University of Milano-Bicocca
Pablo Koch-Medina, Cosimo Munari
Center for Finance and Insurance and Swiss Finance Institute, University of Zurich, Switzerland
October 25, 2019
Abstract
We establish dual representations for systemic risk measures based on acceptance sets in a generalsetting. We deal with systemic risk measures of both “first allocate, then aggregate” and “first aggre-gate, then allocate” type. In both cases, we provide a detailed analysis of the corresponding systemicacceptance sets and their support functions. The same approach delivers a simple and self-containedproof of the dual representation of utility-based risk measures for univariate positions.
Keywords : systemic risk, macroprudential regulation, risk measures, dual representations
The study of risk measures for multivariate positions was first developed by Burgert and R¨uschendorf [8],R¨uschendorf [33], Ekeland and Schachermayer [15], and Ekeland et al. [14] and extended to the set-valuedcase by Jouini et al. [26], Hamel and Heyde [24], Hamel et al. [25], and Molchanov and Cascos [30]. Inthis literature, multivariate positions are typically interpreted as (random) portfolios of financial assets.In recent years, there has been significant interest in extending the theory of risk measures to a systemicrisk setting, in which multivariate positions represent the (random) vector of future capital positions,i.e. assets net of liabilities, of financial institutions. In this setting, one considers a system of d financialinstitutions whose respective capital positions at a fixed future date is represented by a random vector X = ( X , . . . , X d ) . The bulk of the literature assumes that a macroprudential regulator specifies an “aggregation function”Λ : R d → R by means of which the system is summarized into a single (univariate) aggregated position Λ( X ). Thesimplest aggregation function is given by Λ( x ) = P di =1 x i and corresponds to aggregating the entire systeminto a single consolidated balance sheet. The regulator also specifies a set A of “acceptable” aggregatedpositions: The level of systemic risk of the financial system is deemed acceptable whenever Λ( X ) belongsto the prescribed acceptance set A . Two main classes of systemic risk measures based on aggregationfunctions and acceptance sets have been studied in the literature.A first branch of the literature adopts a so-called “first allocate, then aggregate” approach, which isthe macroprudential equivalent of the fundamental idea introduced in the context of microprudential1egulation by Artzner et al. [4]: To ensure that the financial system has an acceptable level of systemicrisk, the macroprudential regulator can require each of the member institutions to raise a suitable amountof capital. Such a requirement is represented by a vector m = ( m , . . . , m d ) ∈ R d , where m i correspondsto the amount of capital raised by institution i . This leads to a systemic risk measure of the form ρ ( X ) = inf ( d X i =1 m i ; m ∈ R d , Λ( X + m ) ∈ A ) . The quantity ρ ( X ) corresponds to the minimum amount of aggregate capital that needs to be injectedinto the financial system to ensure acceptability. This type of systemic risk measures has been studiedin Feinstein et al. [18], Armenti et al. [3], Ararat and Rudloff [2], and Biagini et al. [5, 6] (where randomallocations of the aggregate capital requirement are also considered).A second branch of the literature advocates a “first aggregate, then allocate” approach and studiessystemic risk measures of the form e ρ ( X ) = inf { m ∈ R ; Λ( X ) + m ∈ A} . In this case, the quantity e ρ ( X ) represents the minimal amount of capital that has to be added to theaggregated position to reach acceptability. In contrast to ρ , the operational interpretation of e ρ is notstraightforward since it is unclear how much each of the member institutions should contribute to theaggregate amount of required capital or, if the outcome of the above risk measure is interpreted as abail-out cost, which institution should obtain which amount. Such systemic risk measures have beenstudied in Chen et al. [9], Kromer et al. [27], and Ararat and Rudloff [2].The main objective of this note is to establish dual representations for the above systemic risk measuresin a general setting with special emphasis on systemic risk measures of the “first allocate, then aggregate”type. By doing so, we provide a unifying perspective on the existing duality results in the literature.More precisely, we consider an arbitrary probability space (Ω , F , P ) and assume that the multivariatepositions belong to a space of d -dimensional random vectors X that is in duality with another space of d -dimensional random vectors X ′ through the pairing( X, Z ) d X i =1 E P [ X i Z i ]for X ∈ X and Z ∈ X ′ . This setup is general enough to cover all the interesting examples encountered inthe literature. Dual representations for ρ have been studied by Armenti et al. [3] and Biagini et al. [6] inthe setting of Orlicz hearts and acceptance sets based on (multivariate) utility functions and by Ararat andRudloff [2] in the setting of bounded random variables with only mild restrictions on the acceptance set.The strategy in [3] is to apply Lagrangian techniques while that of [2] is to rely on the dual representationof e ρ , which is tackled by using Fenchel-Moreau techniques for composed maps. Similarly to [6], the pointof departure of this note is to observe that ρ can be written as ρ ( X ) = inf { π ( m ) ; m ∈ R d , X + m ∈ Λ − ( A ) } where the “acceptance set” Λ − ( A ) and the “cost functional” π : R d → R are given byΛ − ( A ) = { X ∈ X ; Λ( X ) ∈ A} , π ( m ) = d X i =1 m i . This shows that ρ belongs to the class of “multi-asset risk measures” introduced in Frittelli and Scan-dolo [20] and thoroughly studied in Farkas et al. [16]. Under suitable conditions on Λ and A , the generalduality results obtained in those papers yield the representation ρ ( X ) = sup ( σ ( Q , . . . , Q d ) − d X i =1 E Q i [ X i ] ; Q , . . . , Q d ≪ P , (cid:18) d Q d P , . . . , d Q d d P (cid:19) ∈ X ′ ) σ ( Q , . . . , Q d ) = inf X ∈ Λ − ( A ) d X i =1 E Q i [ X i ] . The map σ corresponds to the (lower) support function of the systemic acceptance set Λ − ( A ) andplays a fundamental role in the dual representation. The main technical contribution of this note is toprovide a detailed analysis of these objects. In particular, we devote some effort to obtain a more explicitdescription of σ in terms of the primitives Λ and A . It is worth noting that, in the systemic setting,the closedeness of the acceptance set Λ − ( A ) does not necessarily imply the lower semicontinuity of ρ ,which is a necessary condition for ρ to admit a dual representation. Hence, it is important to provideconditions on the primitives Λ and A to ensure that ρ is lower semicontinuous in the first place. To thiseffect, we rely on the abstract results in Farkas et al. [16], where an effort was made to derive propertiesof risk measures, such as lower semicontinuity, from the properties of the underlying acceptance sets. Inparticular, the duality results for acceptance sets obtained there build the starting point for our analysisof the systemic acceptance set Λ − ( A ).On the other side, dual representations for e ρ have been studied by Chen et al. [9] in a finite-dimensionalsetting, by Ararat and Rudloff [2] in the setting of bounded random vectors, and by Kromer et al. [27] atour level of generality. In line with those papers, our starting point is to observe that e ρ can be expressedas a standard cash-additive risk measure (namely the cash-additive risk measure associated with A , whichwe denote by ρ A ) applied to the aggregated univariate position as e ρ ( X ) = ρ A (Λ( X )) . However, instead of working out the Fenchel-Moreau representation of a composition of maps, we exploitthe standard dual representation for ρ A to derive the desired representation in a direct way. In this case,conditions to ensure the lower semicontinuity of e ρ are also easier to formulate.Finally, to illustrate the convenience of our approach to duality based on acceptance sets, we provide asimple and self-contained proof of the dual representation for utility-based risk measures for univariatepositions, which can be viewed as special systemic risk measures where d = 1 and Λ is a von Neumann-Morgenstern utility function. In this section, we recall the necessary terminology and notation from probability theory and convexanalysis and describe our setting. For a general presentation of convex duality we refer to Aliprantis andBorder [1] and Z˘alinescu [34].
Throughout, we fix a probability space (Ω , F , P ). We denote by L the vector space of (equivalence classeswith respect to almost-sure equality of) random variables, i.e. Borel-measurable functions X : Ω → R . Asusual, we do not distinguish explicitly between an equivalence class and any of its representatives. Forevery p ∈ [1 , ∞ ] we denote by L p the space of p -integrable random variables if p < ∞ and the space ofbounded random variables if p = ∞ .A (nonzero) linear subspace L ⊂ L is said to be admissible if it is a Banach lattice (with respect to thealmost-sure partial order) such that L ∞ ⊂ L ⊂ L . In this case, we set L ′ := { Z ∈ L ; E [ | XZ | ] < ∞ , ∀ X ∈ L} , where E denotes the expectation with respect to P . Note that we always have L ∞ ⊂ L ′ ⊂ L .3 xample 2.1. The class of admissible spaces contains Orlicz spaces, which include the standard examplesencountered in the literature. A nonconstant function
Φ : [0 , ∞ ) → [0 , ∞ ] is said to be an Orlicz functionif it is convex, nondecreasing, right-continuous at , and satisfies Φ(0) = 0 . The convex conjugate of Φ isthe function Φ ∗ : [0 , ∞ ) → [0 , ∞ ] given by Φ ∗ ( t ) := sup s ∈ [0 , ∞ ) { st − Φ( s ) } . It is easy to see that Φ ∗ is also an Orlicz function. The Orlicz space associated with Φ is L Φ := (cid:26) X ∈ L ; E (cid:20) Φ (cid:18) | X | λ (cid:19)(cid:21) < ∞ for some λ ∈ (0 , ∞ ) (cid:27) . The corresponding Orlicz heart is defined by H Φ := (cid:26) X ∈ L ; E (cid:20) Φ (cid:18) | X | λ (cid:19)(cid:21) < ∞ for every λ ∈ (0 , ∞ ) (cid:27) . These spaces are Banach lattices with respect to the Luxemburg norm k X k Φ := inf (cid:26) λ ∈ (0 , ∞ ) ; E (cid:20) Φ (cid:18) | X | λ (cid:19)(cid:21) ≤ (cid:27) . The norm dual of L Φ cannot be identified with a subspace of L in general. However, the norm dual of H Φ can always be identified with L Φ ∗ provided that Φ is finite valued (otherwise H Φ = { } ). We saythat Φ satisfies the ∆ condition if there exist r ∈ (0 , ∞ ) and k ∈ (0 , ∞ ) such that Φ(2 s ) < k Φ( s ) forevery s ∈ [ r, ∞ ) . It is well-known that, if Φ is ∆ , then L Φ = H Φ . In a nonatomic setting the converseimplication also holds.Every L p space, with p ∈ [1 , ∞ ] , can be viewed as an Orlicz space. Indeed for every p ∈ [1 , ∞ ) we have L Φ = L p if we set Φ( s ) = s p for every s ∈ [0 , ∞ ) . In this case, the Luxemburg norm coincides with thestandard L p norm and we have L Φ = H Φ . Moreover, we have L Φ = L ∞ if we define Φ( s ) = 0 for s ∈ [0 , and Φ( s ) = ∞ otherwise. In this case, the Luxemburg norm coincides with the standard L ∞ norm andwe have H Φ = { } .The following statements hold; see e.g. Edgar and Sucheston [13] or Meyer-Nieberg [29]:(1) L = L Φ is admissible and L ′ = L Φ ∗ ;(2) L = H Φ is admissible if Φ is finite valued, in which case L ′ = L Φ ∗ ;(3) L = L p is admissible if p ∈ [1 , ∞ ] , in which case L ′ = L pp − (with the convention = ∞ and ∞∞ = 1 ). Fix m ∈ N . We always consider the standard inner product h· , ·i : R m × R m → R defined by h a, b i := m X i =1 a i b i . We denote by L ( R m ) the vector space of all m -dimensional random vectors. We say that a linear subspace L ⊂ L ( R m ) is admissible whenever L = L × · · · × L m with admissible L , . . . , L m ⊂ L . Note that, being the product of Banach lattices, the space L is also aBanach lattice. In particular, the lattice operations on L are understood component by component. Asabove, we define L ′ := L ′ × · · · × L ′ m . L , L ′ ) is equipped with the bilinear form ( ·|· ) : L × L ′ → R given by( X | Z ) := E [ h X, Z i ] = m X i =1 E [ X i Z i ] . The coarsest topology on L making the linear functional X ( X | Z ) continuous for every Z ∈ L ′ isdenoted by σ ( L , L ′ ). Similarly, the coarsest topology on L ′ making the linear functional Z ( X | Z )continuous for every X ∈ L is denoted by σ ( L ′ , L ). Equipped with these topologies, L and L ′ arelocally-convex topological vector spaces (which are also Hausdorff because the above form is separating).The following standard notions from convex analysis will be freely used throughout the note. For a subset S ⊂ L ( R m ) we denote by S + , respectively S ++ , the set of all random vectors in S with componentsthat are almost-surely nonnegative, respectively strictly-positive. The domain of finiteness of a map f : L → [ −∞ , ∞ ] is defined by dom( f ) := { X ∈ L ; f ( X ) ∈ R } . The (lower) support function of a (nonempty) set
A ⊂ L is the map σ A : L ′ → [ −∞ , ∞ ) defined by σ A ( Z ) := inf X ∈A E [ h X, Z i ] . Its domain of finiteness is denoted by bar( A ) and called the barrier cone , i.e.bar( A ) := dom( σ A ) = { Z ∈ L ′ ; σ A ( Z ) > −∞} . If A is σ ( L , L ′ )-closed and convex, the Hahn-Banach Theorem yields the representation A = \ Z ∈L ′ { X ∈ L ; E [ h X, Z i ] ≥ σ A ( Z ) } = \ Z ∈ bar( A ) { X ∈ L ; E [ h X, Z i ] ≥ σ A ( Z ) } . (2.1)We say that A is monotone if A + L + ⊂ A . In this case, we have bar( A ) ⊂ L ′ + .Consider a nonconstant map f : L → ( −∞ , ∞ ]. The convex conjugate of f is the map f ∗ : L ′ → ( −∞ , ∞ ]given by f ∗ ( Z ) := sup X ∈L { E [ h X, Z i ] − f ( X ) } . The Fenchel-Moreau Theorem states that, if f is convex and σ ( L , L ′ )-lower semicontinuous, then f ( X ) = sup Z ∈L ′ { E [ h X, Z i ] − f ∗ ( Z ) } for every X ∈ L . We will find it convenient to use the concave counterpart to convex duality. Here,consider a nonconstant map g : L → [ −∞ , ∞ ). The concave conjugate of g is the map g • : L ′ → [ −∞ , ∞ )defined by g • ( Z ) := inf X ∈L { E [ h X, Z i ] − g ( X ) } = − ( − g ) ∗ ( − Z ) . Hence, the Fenchel-Moreau Theorem implies that, if g is concave and σ ( L , L ′ )-upper semicontinuous, then g ( X ) = inf Z ∈L ′ { E [ h X, Z i ] − g • ( Z ) } for every X ∈ L . The indicator function of a set A ⊂ L is the map δ A : L → [0 , ∞ ] given by δ A ( X ) := ( X ∈ A , ∞ otherwise . Note that, for every Z ∈ L ′ , we have σ A ( Z ) = ( − δ A ) • ( Z ) = − δ ∗A ( − Z ).5 .2 Financial systems and systemic risk We consider a one-period economy in which uncertainty at the terminal date is modeled by the probabilityspace (Ω , F , P ). In this economy, we assume the existence of a financial system consisting of d memberinstitutions (for completeness we also allow for the case d = 1). The possible terminal capital positions,i.e. assets net of liabilities, of these d institutions belong to an admissible space X = X × · · · × X d ⊂ L ( R d ) . For every X = ( X , . . . , X d ) ∈ X the random variables X , . . . , X d correspond to the capital positionsof the various member institutions. Since X contains all bounded random vectors, the space R d canbe naturally viewed as a linear subspace of X . We denote by e the constant random vector with allcomponents equal to 1, i.e. e := (1 , . . . , ∈ R d . The impact of the financial system on systemic risk is measured through an impact map S : X → E where E is a suitable admissible subspace of L . Hence, for every X ∈ X , the random variable S ( X ) isinterpreted as an indicator of the systemic risk posed by X ; see Remark 2.5. Definition 2.2.
We say that S is admissible if it satisfies the following five properties:(S1) Discrimination : S is not constant;(S2) Normalization : S (0) = 0;(S3) Monotonicity : S ( X ) ≥ S ( Y ) for all X, Y ∈ X such that X ≥ Y ;(S4) Concavity : S ( λX + (1 − λ ) Y ) ≥ λS ( X ) + (1 − λ ) S ( Y ) for all X, Y ∈ X and λ ∈ [0 , Semicontinuity : The map X E [ S ( X ) W ] is σ ( X , X ′ )-upper semicontinuous for every W ∈ E ′ + .The next proposition provides a number of sufficient conditions for the technical assumption (S5) tohold. Recall that the lattice operations on X are performed component by component. Here, we use thestandard notation for the limit superior of a sequence of random variables. Definition 2.3.
We say that S has the Fatou property if for every sequence ( X n ) ⊂ X and every X ∈ X X n → X a.s. , sup n ∈ N | X n | ∈ X = ⇒ S ( X ) ≥ lim sup n →∞ S ( X n ) . We say that S is surplus invariant if S ( X ) = S (min( X, X ∈ X . Proposition 2.4.
Assume that (S3) and (S4) hold. Then, (S5) holds in any of the following cases:(i) X ′ i is the norm dual of X i for every i ∈ { , . . . , d } .(ii) X i = L Φ i with Φ ∗ i being ∆ (e.g. X i = L ∞ ) for every i ∈ { , . . . , d } and S has the Fatou property.(iii) S is surplus invariant and has the Fatou property. roof. Throughout the proof fix W ∈ E ′ + and define a functional ϕ W : X → R by setting ϕ W ( X ) := E [ S ( X ) W ] . Note that ϕ W is concave and nondecreasing by (S3) and (S4). Assume that (i) holds. In this case, wecan apply the Extended Namioka-Klee Theorem from Biagini and Frittelli [7] to infer that ϕ W is uppersemicontinuous (in fact, continuous) with respect to the norm topology on X . As the space X ′ coincideswith the norm dual of X by assumption, it follows from Corollary 5.99 in Aliprantis and Border [1] that ϕ W is also σ ( X , X ′ )-upper semicontinuous.We make some preliminary observations before proceeding with the proof of (ii) and (iii) . First, wenote that the Fatou property of S implies that ϕ W is sequentially upper semicontinuous with respectto order convergence, i.e. dominated almost-sure convergence. Indeed, consider a sequence ( X n ) ⊂ X that converges almost surely to some X ∈ X and such that sup n ∈ N | X n | ≤ M for some M ∈ X . Since | S ( X n ) | ≤ max( | S ( M ) | , | S ( − M ) | ) for every n ∈ N by (S3), it follows from the Fatou property of S andfrom the Fatou Lemma that ϕ W ( X ) ≥ E h lim sup n →∞ S ( X n ) W i ≥ lim sup n →∞ E [ S ( X n ) W ] = lim sup n →∞ ϕ W ( X n ) , as claimed. Second, Theorem 2.6.4 in Meyer-Nieberg [29] tells us that, for every i ∈ { , . . . , d } , theorder-continuous dual of X i , i.e. the space of linear functionals that are continuous with respect to orderconvergence, coincides with X ′ i . This implies that the order-continuous dual of X also coincides with X ′ .Denote by X ∼ n the order-continuous dual of X . We establish (S5) by showing that the upper semicontinuityof ϕ W with respect to order convergence implies its σ ( X , X ∼ n )-upper semicontinuity.Assume that (ii) holds. If d = 1, the desired assertion follows from Theorem 4.4 in Delbaen and Owari [11](see also Theorem 3.2 in Delbaen [10] for the bounded case and Theorem 3.7 in Gao et al. [22] for the Orliczcase in a nonatomic setting). This result can be extended to a multivariate setting by using the resultsin Leung and Tantrawan [28]. We use their notation and terminology. Observe first that the constantvector e is a strictly-positive element in X ∼ n . Second, note that all the spaces X i ’s are monotonicallycomplete by Theorem 2.4.22 in [29], admit a special modular by Example 3.1 in [28], and their normduals are order continuous by Remark 3.5 in [11]. This implies that X is also monotonically complete,admits a special modular, and its norm dual is order continuous. As a result, we can apply Theorem3.4 in [28] to conclude that X satisfies property (P1) of that paper. This property implies that everyconcave functional on X that is upper semicontinuous with respect to order convergence, as our ϕ W , isautomatically σ ( X , X ∼ n )-upper semicontinuous. This delivers the desired result.Finally, assume that (iii) holds. In this case, the functional ϕ W is surplus invariant in the sense of Gaoand Munari [23]. Since ϕ W is concave and upper semicontinuous with respect to order convergence, wecan apply Theorem 21 in [23] to infer that ϕ W is σ ( X , X ∼ n )-upper semicontinuous. As above, this deliversthe desired result. Remark 2.5. (i) In the literature, the impact map is typically derived from an aggregation functionΛ : R d → R by setting S ( X ) = Λ( X ) for every X ∈ X . We refer to the literature cited in the introduction fora discussion of concrete examples. Clearly, the choice of Λ limits the choice of the space E since, forinstance, one needs to ensure that the random variables Λ( X ) ’s are integrable. This is typically doneeither by working in a space of bounded positions or by working in an Orlicz space where the Orliczfunctions are defined in terms of Λ . To avoid having to worry about this aspect, we have defined theimpact map as a map between abstract spaces.(ii) If Λ is assumed to be nonconstant, nondecreasing, concave, and to satisfy Λ(0) = 0 , then the corre-sponding S clearly fulfills properties (S1)-(S4). Moreover, as Λ is automatically continuous by concavity, S has automatically the Fatou property. Hence, we can use Proposition 2.4 to ensure property (S5).
7e now assume that the regulator has defined acceptable levels of systemic risk by specifying a set
A ⊂ E called the acceptance set : The financial system with capital positions X ∈ X is deemed to have anacceptable level of systemic risk if the systemic risk indicator S ( X ) belongs to A . Definition 2.6.
We say that A is admissible if it satisfies the following properties:(A1) Discrimination : S − ( A ) is a nonempty proper subset of X ;(A2) Normalization : 0 ∈ A ;(A3)
Monotonicity : A + E + ⊂ A ;(A4) Convexity : λ A + (1 − λ ) A ⊂ A for every λ ∈ [0 , Closedness : A is σ ( E , E ′ )-closed.The next proposition highlights a variety of situations where assumption (A5) is always satisfied. Definition 2.7.
We say that A is Fatou closed if for every sequence ( U n ) ⊂ A and every U ∈ E U n → U a.s. , sup n ∈ N | U n | ∈ E = ⇒ U ∈ A . We say that A is law invariant if for every U ∈ A and every V ∈ E with the same probability distributionas U we have V ∈ A . Moreover, we say that A is surplus invariant if for every U ∈ A and every V ∈ E such that min( V,
0) = min( U,
0) we have V ∈ A . Proposition 2.8.
Assume that (A3) and (A4) hold. Then, (A5) holds in any of the following cases:(i) E ′ is the norm dual of E and A is norm closed.(ii) E = L Φ with Φ ∗ being ∆ (e.g. E = L ∞ ) and A is Fatou closed.(iii) E = L Φ with (Ω , F , P ) nonatomic and A is law invariant and Fatou closed.(iv) A is surplus invariant and Fatou closed.Proof. The desired assertion holds under (i) by Theorem 5.98 in Aliprantis and Border [1]; under (ii) byTheorem 4.1 in Delbaen and Owari [11] (see also Theorem 3.2 in Delbaen [10] in the bounded case andTheorem 3.7 in Gao et al. [22] in the Orlicz case in a nonatomic setting); under (iii) by Corollary 4.6 inGao et al. [21]; under (iv) by Theorem 8 in Gao and Munari [23].
In this section we focus on systemic risk measures of “first allocate, then aggregate” type. After discussingsome conditions for their representability, we establish a general dual representation and provide a de-tailed analysis of the properties of the corresponding systemic acceptance sets and “penalty functions”.Throughout the section we fix an admissible impact map S and an admissible acceptance set A .8 .1 The systemic risk measure ρ “First allocate, then aggregate”-type systemic risk measures are defined as follows (we adopt the usualconvention inf ∅ = ∞ ): Definition 3.1.
We define the map ρ : X → [ −∞ , ∞ ] by setting ρ ( X ) := inf ( d X i =1 m i ; m ∈ R d , S ( X + m ) ∈ A ) . (3.1)The above map determines the minimum amount of aggregate capital that can be allocated to the memberinstitutions to ensure that the level of systemic risk of the financial system is acceptable. We start byobserving that (3.1) can be rewritten as ρ ( X ) = inf { π ( m ) ; m ∈ R d , X + m ∈ S − ( A ) } , π ( m ) = d X i =1 m i . (3.2)As a result, ρ belongs to the broad class of risk measures introduced in Frittelli and Scandolo [20] andthoroughly studied in Farkas et al. [16]. We exploit this link in a systematic way. The first propositioncollects some basic properties of the “systemic acceptance set” S − ( A ) and of the risk measure ρ . Proposition 3.2. (i) The set S − ( A ) is monotone, convex, σ ( X , X ′ ) -closed, and contains .(ii) The systemic risk measure ρ is nonincreasing, convex, and satisfies ρ (0) ≤ . Moreover, ρ satisfiesthe multivariate version of cash-additivity, i.e. ρ ( X + m ) = ρ ( X ) − d X i =1 m i for every X ∈ X and every m ∈ R d .Proof. (i) It is straightforward to prove that S − ( A ) contains 0 and that it is monotone and convex. Toshow σ ( X , X ′ )-closedeness, it is enough to recall that bar( A ) ⊂ E ′ + and use (2.1) to get S − ( A ) = { X ∈ X ; S ( X ) ∈ A} = { X ∈ X ; E [ S ( X ) W ] ≥ σ A ( W ) , ∀ W ∈ bar( A ) } . The claim follows immediately from (S5). (ii)
The stated properties of ρ are straightforward; see also Lemma 2 in Farkas et al. [16]. ρ In order to admit a dual representation, the risk measure ρ needs to be proper and lower semicontinuous.We highlight a number of sufficient conditions for this to be the case. We start with a simple characteri-zation of properness provided we already know that ρ is lower semicontinuous. Recall that, by definition, ρ is proper if it never attains the value −∞ and is finite at some point. Proposition 3.3. If ρ is σ ( X , X ′ ) -lower semicontinuous, then ρ is proper if and only if ρ (0) > −∞ .Proof. We know that ρ (0) < ∞ by Proposition 3.2. As a result, the above equivalence follows from thefact that a σ ( X , X ′ )-lower semicontinuous convex map that assumes the value −∞ cannot assume anyfinite value; see e.g. Proposition 2.2.5 in Z˘alinescu [34].9n contrast to the standard univariate (cash-additive) case, the closedeness of S − ( A ) does not implythe lower semicontinuity of ρ ; see Example 1 in Farkas et al. [16]. The purpose of the next result is toprovide a number of sufficient conditions for ρ to be lower semicontinuous. The last two conditions areparticularly easy to verify and often satisfied in the literature. Proposition 3.4.
The following statements hold:(i) Assume that Ω is finite. If ρ (0) > −∞ , then ρ is finite valued and continuous.(ii) Assume that X ′ i is the norm dual of X i for every i ∈ { , . . . , d } . If ρ is finite valued, then ρ is σ ( X , X ′ ) -lower semicontinuous.(iii) Assume that X ′ i is the norm dual of X i for every i ∈ { , . . . , d } . If S − ( A ) has nonempty interiorin the norm topology and ρ (0) > −∞ , then ρ is finite valued and σ ( X , X ′ ) -lower semicontinuous.(iv) Set M := (cid:8) m ∈ R d ; P di =1 m i = 0 (cid:9) . If S − ( A ) ∩ M = { } , then ρ is proper and σ ( X , X ′ ) -lowersemicontinuous.(v) If A ∩ R − = { } and S ( m ) ∈ ( −∞ , for every nonzero m ∈ M , then ρ is proper and σ ( X , X ′ ) -lower semicontinuous.Proof. (i) Since Ω is finite, e is an interior point of X + . Then, the desired result follows from Proposition1 in Farkas et al. [16]. (ii) It follows from the Extended Namioka-Klee Theorem in Biagini and Frittelli [7] that ρ is lower semi-continuous (in fact, continuous) with respect to the norm topology. Then, ρ is also lower semicontinuouswith respect to σ ( X , X ′ ) by virtue of Corollary 5.99 in Aliprantis and Border [1]. (iii) Note that e is a strictly-positive element of X , i.e. for every Z ∈ X ′ + \ { } we have E [ h e, Z i ] = d X i =1 E [ Z i ] > . Proposition 2 in [16] implies that ρ is finite valued so that (ii) can be applied. (iv) For every X ∈ X it is not difficult to show that ρ ( X ) = inf n r ∈ R ; X + rd e ∈ S − ( A ) + M o ;see Lemma 3 in [16]. Then, it follows from Proposition 3.2 that S − ( A ) + M − rd e ⊂ { X ∈ X ; ρ ( X ) ≤ r } ⊂ cl (cid:18) S − ( A ) + M − rd e (cid:19) for every r ∈ R , where cl denotes the closure operator with respect to σ ( X , X ′ ). To establish thedesired lower semicontinuity, we show that S − ( A ) + M is σ ( X , X ′ )-closed. To this effect, recall fromProposition 3.2 that S − ( A ) is convex and σ ( X , X ′ )-closed. Moreover, M is a finite-dimensional vectorspace and S − ( A ) ∩ M = { } . The closedness criterion in Dieudonn´e [12] now implies that S − ( A ) + M is σ ( X , X ′ )-closed. Properness follows from Proposition 3.3. (v) Let m ∈ M . By assumption, we have S ( m ) ∈ A if and only if m = 0. This yields S − ( A ) ∩ M = { } and the desired statement immediately follows from point (iv) .10 .3 The dual representation of ρ We have already mentioned that, in view of (3.2), the risk measure ρ belongs to the class of risk measuresstudied in Farkas et al. [16]. The general results established in that paper can be exploited to derive a dualrepresentation for ρ . This also follows from the general dual representation in Frittelli and Scandolo [20]. Definition 3.5.
We denote by C the convex subset of X ′ + defined by C := { Z ∈ X ′ + ; E [ Z ] = · · · = E [ Z d ] = 1 } . Theorem 3.6. If ρ is proper and σ ( X , X ′ ) -lower semicontinuous, then bar( S − ( A )) ∩ C 6 = ∅ and ρ ( X ) = sup Z ∈C { σ S − ( A ) ( Z ) − E [ h X, Z i ] } for every X ∈ X . The supremum can be restricted to C ∩ X ′ ++ provided that bar( S − ( A )) ∩ C ∩ X ′ ++ = ∅ .Proof. Note that the cost functional π in equation (3.2) is defined on R d ⊂ X . It is easy to see that, forevery Z ∈ X ′ , the functional X E [ h X, Z i ] is a positive extension of π to X if and only if Z belongsto C . Since ρ is proper and σ ( X , X ′ )-lower semicontinuous, it follows from Proposition 6 in [16] thatthe barrier cone of S − ( A ) contains positive linear extensions of the cost functional π to X , i.e. we havebar( S − ( A )) ∩ C 6 = ∅ . The desired representation is now a consequence of Theorem 3 in [16].Now, assume we find Z ∗ ∈ bar( S − ( A )) ∩ C ∩ X ′ ++ and take any element Z ∈ bar( S − ( A )) ∩ C . For every X ∈ X and every λ ∈ (0 ,
1) we have λZ ∗ + (1 − λ ) Z ∈ C and λ ( σ S − ( A ) ( Z ∗ ) − E [ h X, Z ∗ i ]) + (1 − λ )( σ S − ( A ) ( Z ) − E [ h X, Z i ]) ≤ sup Z ′ ∈C∩X ′ ++ { σ S − ( A ) ( Z ′ ) − E [ h X, Z ′ i ] } by concavity of σ S − ( A ) . Letting λ tend to 0 and taking a supremum over Z yields ρ ( X ) ≤ sup Z ′ ∈C∩X ′ ++ { σ S − ( A ) ( Z ′ ) − E [ h X, Z ′ i ] } . The converse inequality is clear. This establishes the last assertion and concludes the proof.
Remark 3.7. (i) We highlight the link between the dual representation in Theorem 3.6 and the stan-dard Fenchel-Moreau representation; see also Remark 17 in Farkas et al. [16]. To see it, note that themap − σ S − ( A ) ( −· ) + δ C ( −· ) is convex and lower semicontinuous and, if ρ is proper and σ ( X , X ′ ) -lowersemicontinuous, it satisfies ρ ( X ) = sup Z ∈X ′ { E [ h X, Z i ] + σ S − ( A ) ( − Z ) − δ C ( − Z ) } for every X ∈ X by Theorem 3.6. From the Fenchel-Moreau Theorem it follows that for every Z ∈ X ′ ρ ∗ ( Z ) = − σ S − ( A ) ( − Z ) + δ C ( − Z ) = ( sup X ∈ S − ( A ) E [ h X, Z i ] if Z ∈ −C , ∞ otherwise . (ii) The dual elements in C can be naturally identified with d -dimensional vectors of probability measures on (Ω , F ) that are absolutely continuous with respect to P or, in case they have strictly-positive components,equivalent to P . This allows to reformulate the above dual representation in terms of probability measures.More concretely, denote by Q ( P ) , respectively Q e ( P ) , the set of all d -dimensional vectors of probability easures over (Ω , F ) that are absolutely continuous with respect to P , respectively equivalent to P . Forevery Q = ( Q , . . . , Q d ) ∈ Q ( P ) and for every X ∈ X we set d Q d P := (cid:18) d Q i d P , . . . , d Q d d P (cid:19) , E Q [ X ] := E (cid:20)(cid:28) X, d Q d P (cid:29)(cid:21) = d X i =1 E Q i [ X i ] . Moreover, for every Q ∈ Q ( P ) we define σ ( Q ) := σ S − ( A ) (cid:18) d Q d P (cid:19) = inf X ∈ S − ( A ) E Q [ X ] . If ρ is proper and σ ( X , X ′ ) -lower semicontinuous, then for every X ∈ X we can write ρ ( X ) = sup Q ∈Q ( P ) , d Q d P ∈X ′ { σ ( Q ) − E Q [ X ] } . We can replace Q ( P ) by Q e ( P ) in the above supremum provided that bar( S − ( A )) ∩ C ∩ X ′ ++ = ∅ . The condition bar( S − ( A )) ∩ C ∩ X ′ ++ = ∅ is necessary to be able to restrict the domain in the above dualrepresentation to strictly-positive dual elements. In the terminology of convex analysis, this conditionrequires that the convex set S − ( A ) admits a strictly-positive supporting functional that belongs to thespecial set C . In the next proposition we show that this always holds if the acceptance set A is supportedby a strictly-positive functional and the impact map S is bounded above by a strictly-increasing affinefunction of the consolidated capital position. Proposition 3.8.
Assume that X i = E for every i ∈ { , . . . , d } . Moreover, suppose that bar( A ) ∩ E ′ ++ = ∅ and there exist a ∈ (0 , ∞ ) and b ∈ R such that S ( X ) ≤ a d X i =1 X i + b for every X ∈ X . Then, bar( S − ( A )) ∩ C ∩ X ′ ++ = ∅ .Proof. Take W ∈ bar( A ) ∩ E ′ ++ and set Z = ( aW, . . . , aW ) ∈ C ∩ X ′ ++ . Then, we easily see that σ S − ( A ) ( Z ) = inf X ∈ S − ( A ) E [ h X, Z i ] ≥ inf X ∈ S − ( A ) E [( S ( X ) − b ) W ] ≥ σ A ( W ) − b E [ W ] > −∞ . This delivers the desired assertion. S − ( A ) Through the support function of the “systemic acceptance set” S − ( A ), the dual representation of thesystemic risk measure ρ in Theorem 3.6 highlights the dependence on the two fundamental underlyingingredients: The impact map S and the acceptance set A . The aim of this subsection is to provide a dualdescription of the systemic acceptance set by using “penalty functions” that are related to (but differentfrom) the support function σ S − ( A ) and to investigate the main properties of these maps. Our analysis isbased on the following definition. Definition 3.9.
We define two maps α, α + : X ′ → [ −∞ , + ∞ ] by setting α ( Z ) := sup W ∈ bar( A ) n σ A ( W ) + inf X ∈X { E [ h X, Z i ] − E [ S ( X ) W ] } o ,α + ( Z ) := sup W ∈ bar( A ) ∩ ( E ′ ++ ∪{ } ) n σ A ( W ) + inf X ∈X { E [ h X, Z i ] − E [ S ( X ) W ] } o . emark 3.10. (i) It is easy to see that α and α + are different in general. For example, if d > and X i = E for every i ∈ { , . . . , d } and we set S ( X ) = P di =1 X i for every X ∈ X and A = E + , then we have α = − δ D = − δ D∩ ( X ′ ++ ∪{ } ) = α + where D = { Z ∈ X ′ + ; Z = · · · = Z d } .(ii) The above maps belong to the class of maps α K : X ′ → [ −∞ , + ∞ ] defined by α K ( Z ) := sup W ∈K n σ A ( W ) + inf X ∈X { E [ h X, Z i ] − E [ S ( X ) W ] } o , where K is a convex cone in bar( A ) such that λ K + (1 − λ ) bar( A ) ⊂ K for every λ ∈ [0 , . This will allowus to prove properties for α and α + simultaneously. In fact, all properties of α and α + we will considerare shared by the entire class. The next theorem records the announced dual representation of the systemic acceptance set and showswhy the above maps are natural “penalty functions”.
Theorem 3.11.
The systemic acceptance set S − ( A ) can be represented as S − ( A ) = \ Z ∈X ′ { X ∈ X ; E [ h X, Z i ] ≥ α ( Z ) } . If bar( A ) ∩ E ′ ++ = ∅ , then S − ( A ) can also be represented as S − ( A ) = \ Z ∈X ′ { X ∈ X ; E [ h X, Z i ] ≥ α + ( Z ) } . Proof.
Let
K ⊂ bar( A ) be a convex cone as in Remark 3.10. Note that, by concavity of σ A , we canequivalently rewrite the representation (2.1) applied to A as A = \ W ∈K { U ∈ E ; E [ U W ] ≥ σ A ( W ) } . Now, for each W ∈ K ⊂ E ′ + we consider the functional ϕ W : X → R defined by ϕ W ( X ) := E [ S ( X ) W ] . As remarked in the proof of Proposition 2.4, the functional ϕ W is concave by (S3) and (S4) and σ ( X , X ′ )-upper semicontinuous by (S5). Hence, it follows from the Fenchel-Moreau Theorem that ϕ W ( X ) = inf Z ∈X ′ { E [ h X, Z i ] − ( ϕ W ) • ( Z ) } for every X ∈ X . As a result, we obtain S − ( A ) = { X ∈ X ; S ( X ) ∈ A} = { X ∈ X ; E [ S ( X ) W ] ≥ σ A ( W ) , ∀ W ∈ K} = { X ∈ X ; E [ h X, Z i ] − ( ϕ W ) • ( Z ) ≥ σ A ( W ) , ∀ W ∈ K , ∀ Z ∈ X ′ } = \ Z ∈X ′ n X ∈ X ; E [ h X, Z i ] ≥ sup W ∈K { σ A ( W ) + ( ϕ W ) • ( Z ) } o = \ Z ∈X ′ { X ∈ X ; E [ h X, Z i ] ≥ α K ( Z ) } . This delivers the desired representation when applied to K = bar( A ) and K = bar( A ) ∩ ( E ′ ++ ∪ { } ).13he next proposition collects some properties of the maps α and α + and shows the relation between them.Here, we denote by dom( α ) the domain of finiteness of α (similarly for α + ). In addition, we denote by clthe closure operator with respect to the topology σ ( X ′ , X ). Proposition 3.12.
The maps α, α + : X ′ → [ −∞ , ∞ ] satisfy the following properties (the statementsabout α + require that bar( A ) ∩ E ′ ++ = ∅ ):(i) α and α + take values in the interval [ −∞ , .(ii) α and α + are concave and positively homogeneous.(iii) α + ≤ α with equality on dom( α + ) .(iv) dom( α + ) ⊂ dom( α ) ⊂ cl(dom( α + )) ⊂ X ′ + .Proof. Throughout the proof we fix a convex cone
K ⊂ bar( A ) as in Remark 3.10. The desired assertionswill follow by taking K = bar( A ) and K = bar( A ) ∩ ( E ′ ++ ∪ { } ). (i) The representation of the systemic acceptance set S − ( A ) established in the proof of Theorem 3.11yields α K ( Z ) ≤ σ S − ( A ) ( Z ) ≤ Z ∈ X ′ . (ii) To show that α K is concave, set for all Z ∈ X ′ and W ∈ E ′ Φ( Z, W ) := inf X ∈X { σ A ( W ) + E [ h X, Z i ] − E [ S ( X ) W ] } . Being the infimum over the parameter X of a function that is clearly jointly concave in Z and W , we seethat Φ is itself jointly concave. Since α K ( Z ) = sup W ∈K Φ( Z, W )for every Z ∈ X ′ , we infer that α K is concave. To show that α K is positively homogeneous, note first that0 always belongs to K , so that α K (0) ≥
0. Together with point (i) , this implies that α K (0) = 0. Finally,for Z ∈ X ′ and λ ∈ (0 , ∞ ) we have α K ( λZ ) = sup W ∈K n σ A ( W ) + inf X ∈X { λ E [ h X, Z i ] − E [ S ( X ) W ] } o = λ sup W ∈K (cid:26) σ A (cid:18) λ W (cid:19) + inf X ∈X (cid:26) E [ h X, Z i ] − E (cid:20) S ( X ) 1 λ W (cid:21)(cid:27)(cid:27) = λ sup W ∈K n σ A ( W ) + inf X ∈X { E [ h X, Z i ] − E [ S ( X ) W ] } o = λα K ( Z ) , where we used that K is a cone. This shows that α K is positively homogeneous. (iii) It is clear that α + ≤ α . To show that α + = α on dom( α + ), take Z ∈ dom( α + ) and note that α ( Z ) = sup W ∈ bar( A ) Φ( Z, W ) , α + ( Z ) = sup W ∈ bar( A ) ∩E ′ ++ Φ( Z, W ) . Take W ∗ ∈ bar( A ) ∩ E ′ ++ such that Φ( Z, W ∗ ) is finite. For each W ∈ bar( A ) set W λ = λW + (1 − λ ) W ∗ for λ ∈ [0 , W λ ) ⊂ bar( A ) ∩ E ′ ++ , so that α + ( Z ) ≥ Φ( Z, W λ ) ≥ λ Φ( Z, W ) + (1 − λ )Φ( Z, W ∗ ) λ ↑ −−→ Φ( Z, W ) . Taking a supremum over W delivers α + ( Z ) ≥ α ( Z ).14 iv) Note that dom( α + ) ⊂ dom( α ) by point (iii) . Since α ≤ σ S − ( A ) as proved in point (i) , we also havedom( α ) ⊂ bar( S − ( A )) ⊂ X ′ + . As X ′ + is σ ( X ′ , X )-closed, it remains to show that dom( α ) ⊂ cl(dom( α + )).To this effect, let Z ∈ dom( α ) and note that Φ( Z, W ) must be finite for some W ∈ bar( A ). Take Z ∗ ∈ dom( α + ) and W ∗ ∈ bar( A ) ∩ E ′ ++ such that Φ( Z ∗ , W ∗ ) is finite. Then, for every λ ∈ [0 ,
1] we have α + ( λZ + (1 − λ ) Z ∗ ) ≥ Φ( λZ + (1 − λ ) Z ∗ , λW + (1 − λ ) W ∗ ) ≥ λ Φ( Z, W ) + (1 − λ )Φ( Z ∗ , W ∗ ) > −∞ by the joint convexity of Φ. The claim follows by letting λ ↑ The case where S is induced by ΛAs mentioned in Remark 2.5, the bulk of the literature has focused on the case where the impact functionis based on an aggregation function Λ : R d → R . The last part of this subsection is devoted to providean equivalent formulation of α and α + in this situation. We focus on the positive cone X ′ + because bothmaps take nonfinite values elsewhere. For ease of notation, for every Z ∈ X ′ + we set E + ( Z ) := d [ i =1 { Z i > } ∈ F . Proposition 3.13.
Assume that X is closed with respect to multiplications by characteristic functions,i.e. for every X ∈ X and E ∈ F we have ( E X , . . . , E X d ) ∈ X . Moreover, consider a nonconstant,nondecreasing, concave function Λ : R d → R satisfying Λ(0) = 0 and assume that S ( X ) = Λ( X ) for every X ∈ X . Then, the following statements hold for every nonzero Z ∈ X ′ + :(i) We have bar( A ) ∩ { W ∈ E ′ + ; W > on E + ( Z ) } 6 = ∅ and α ( Z ) = sup W ∈ bar( A ) , W > on E + ( Z ) (cid:26) σ A ( W ) + E (cid:20) { W > } Λ • (cid:18) ZW (cid:19) W (cid:21)(cid:27) . (ii) If bar( A ) ∩ E ′ ++ = ∅ , then α + ( Z ) = sup W ∈ bar( A ) ∩E ′ ++ (cid:26) σ A ( W ) + E (cid:20) Λ • (cid:18) ZW (cid:19) W (cid:21)(cid:27) In both cases, the ratio ZW is understood component by component.Proof. Let
K ⊂ bar( A ) be a convex cone as in Remark 3.10 and fix a nonzero element Z ∈ X ′ + . Inspiredby Ararat and Rudloff [2], we invoke Theorem 14.60 in Rockafellar and Wets [32] to getinf X ∈X { E [ h X, Z i ] − E [Λ( X ) W ] } = E h inf x ∈ R d { xZ − Λ( x ) W } i (3.3)for every W ∈ K (this result requires that X be closed with respect to multiplications by characteristicfunctions). Recall that K ⊂ E ′ + and note that for every W ∈ K we haveinf x ∈ R d { xZ − Λ( x ) W } = Λ • (cid:0) ZW (cid:1) W on { W > } , { W = 0 } ∩ ( E + ( Z )) c , −∞ on { W = 0 } ∩ E + ( Z ) . (3.4)It follows from the definition of α K and (3.3) that α K ( Z ) = sup W ∈K n σ A ( W ) + E h inf x ∈ R d { xZ − Λ( x ) W } io . W ∈ bar( A ) with P ( { W = 0 } ∩ E + ( Z )) > α K ( Z ) = sup W ∈K , W > on E + ( Z ) n σ A ( W ) + E h inf x ∈ R d { xZ − Λ( x ) W } io = sup W ∈K , W > on E + ( Z ) n σ A ( W ) + E h { W > } inf x ∈ R d { xZ − Λ( x ) W } io = sup W ∈K , W > on E + ( Z ) n σ A ( W ) + E h { W > } Λ • (cid:0) ZW (cid:1) W io , where we used (3.4) in the last equality. The desired assertions follow by taking K = bar( A ) and K = bar( A ) ∩ ( E ′ ++ ∪ { } ). σ S − ( A ) As we have already noticed, the dual representation in Theorem 3.6 depends on the impact map S and the acceptance set A through the support function of the systemic acceptance set S − ( A ). Thegoal of this subsection is to provide an equivalent description of the support function that relies on the“penalty functions” α and α + . This is a direct consequence of the results in the preceding subsection.Here, we denote by usc( α ) the σ ( X ′ , X )-upper semicontinuous hull of α , i.e. the smallest σ ( X ′ , X )-uppersemicontinuous map dominating α (similarly for α + ). Theorem 3.14.
The support function σ S − ( A ) can be represented as σ S − ( A ) = usc( α ) . If bar( A ) ∩ E ′ ++ = ∅ , then σ S − ( A ) can also be represented as σ S − ( A ) = usc( α + ) . Proof.
Let
K ⊂ bar( A ) be a convex cone as in Remark 3.10. In the proof of Theorem 3.11 we establishedthat S − ( A ) = \ Z ∈X ′ { X ∈ X ; E [ h X, Z i ] ≥ α K ( Z ) } . (3.5)This implies that α K ≤ σ S − ( A ) . It follows from the σ ( X ′ , X )-upper semicontinuity of σ S − ( A ) that wealso have usc( α K ) ≤ σ S − ( A ) . In particular, usc( α K ) never takes the value ∞ . Moreover, note thatusc( α K )(0) ≥ α K (0) = 0. As a result, Proposition 2.2.7 in Z˘alinescu [34] tells us that usc( α K ) inheritsconcavity and positive homogeneity from α K . Note that α K can be replaced by usc( α K ) in (3.5). Since theonly σ ( X ′ , X )-upper semicontinuous map σ : X ′ → [ −∞ , ∞ ) that is concave and positively homogeneousand satisfies S − ( A ) = \ Z ∈X ′ { X ∈ X ; E [ h X, Z i ] ≥ σ ( Z ) } is precisely the support function of S − ( A ), see e.g. Theorem 7.51 in Aliprantis and Border [1], weconclude that usc( α K ) = σ S − ( A ) must hold. The desired assertions follow by taking K = bar( A ) and K = bar( A ) ∩ ( E ′ ++ ∪ { } ).It is natural to ask whether taking the upper semicontinuous hull in Theorem 3.14 is redundant in thesense that α and/or α + are upper semicontinuous in the first place and, hence, coincide with the supportfunction σ S − ( A ) . As illustrated by the following example, the answer is negative in general.16 xample 3.15. Let (Ω , F , P ) be nonatomic and consider the pairs given by ( X , X ′ ) = ( L ∞ ( R d ) , L ( R d )) and ( E , E ′ ) = ( L ∞ ( R ) , L ( R )) . Fix λ ∈ (0 , and for every U ∈ L ( R ) define the Value at Risk and
Expected Shortfall of U at level λ by VaR λ ( U ) := inf { m ∈ R ; P ( U + m < ≤ λ } , ES λ ( U ) := 1 λ Z λ VaR µ ( U ) dµ. Define S : X → E and
A ⊂ E by setting S ( X ) = d X i =1 min( X i , , A = { U ∈ E ; ES λ ( U ) ≤ } . It is immediate to see that S − ( A ) = X + , so that σ S − ( A ) = − δ X ′ + = − δ L ( R d ) . To determine α , take any Z ∈ X ′ + and recall from Theorem 4.52 in F¨ollmer and Schied [19] that σ A = − δ bar( A ) , bar( A ) = (cid:26) W ∈ E ′ + ; W ≤ λ E [ W ] (cid:27) . As a result, we infer that α ( Z ) = sup W ∈E ′ + , W ≤ E [ W ] λ inf X ∈X E " d X i =1 ( X i Z i − min( X i , W ) = sup W ∈E ′ + , W ≤ E [ W ] λ inf X ∈X + E " d X i =1 X i ( W − Z i ) . Now, if Z j is not bounded for some j ∈ { , . . . , d } , then P ( W − Z j < > for every W ∈ bar( A ) and inf X ∈X + E " d X i =1 X i ( W − Z i ) ≤ inf n ∈ N E [ n { W − Z j < } ( W − Z j )] = −∞ . In this case, we have α ( Z ) = −∞ . Otherwise, if Z is bounded, set W = max i ∈{ ,...,d } k Z i k ∞ ∈ bar( A ) and observe that ≥ α ( Z ) ≥ inf X ∈X + E " d X i =1 X i ( W − Z i ) = 0 . In conclusion, we have α = − δ X + = − δ L ∞ + ( R d ) . Since L ∞ ( R ) = L ( R ) when the underlying probability space is nonatomic, we conclude that σ S − ( A ) = α .The same conclusion holds for α + as well (note that bar( A ) ∩ E ′ ++ = ∅ ). This follows because, byProposition 3.12, we always have α + ≤ α . Alternatively, we can repeat the above argument and find that α + = α in our situation. Remark 3.16.
By combining the dual representation in Theorem 3.6 and the representation of σ S − ( A ) obtained in Theorem 3.14, we see that ρ ( X ) = sup Z ∈C { usc( α )( Z ) − E [ h X, Z i ] } = sup Z ∈X ′ { usc( α )( Z ) − δ C ( Z ) − E [ h X, Z i ] } (3.6) for every X ∈ X . If the equality σ S − ( A ) = α holds, then we can drop the upper-semicontinuous hull inthe representation (3.6) and obtain ρ ( X ) = sup Z ∈C { α ( Z ) − E [ h X, Z i ] } = sup Z ∈X ′ { α ( Z ) − δ C ( Z ) − E [ h X, Z i ] } (3.7)17 or every X ∈ X . One may wonder whether the “simplified” representation (3.7) holds even if the equality σ S − ( A ) = α does not hold. Note that usc( α ) − δ C is concave and σ ( X ′ , X ) -upper semicontinuous and that α − δ C is concave. As a result, the “simplified” representation holds if and only if usc( α − δ C ) = usc( α ) − δ C . The same holds with α + instead of α (provided that bar( A ) ∩ E ′ ++ = ∅ ). It is unclear whether this equalityholds without additional assumptions on S and A because, in general, it is not possible to take an indicatorfunction out of an upper-semicontinuous hull. For example, consider the simple situation where Ω = { ω } and d = 2 . In this case, we have the identification ( X , X ′ ) = ( R , R ) . Consider the concave and positivelyhomogeneous function f and the convex closed set C defined by f = − δ D , D = { z ∈ R ; 0 ≤ z < z } ∪ { (0 , } , C = { z ∈ R ; z = z = 1 } = { (1 , } . Then, it is easy to see that usc( f − δ C ) = − δ ∅ = − δ { (1 , } = usc( f ) − δ C . σ S − ( A ) = α to hold We know from Theorem 3.14 that the support function of the systemic acceptance set S − ( A ) alwayscoincides with the upper semicontinuous hull of the penalty function α . However, as illustrated byExample 3.15, there are simple situations where the map α fails to be upper semicontinuous and, hence,the equality σ S − ( A ) = α does not hold. In this subsection we establish a variety of sufficient conditionsfor this equality to hold. Clearly, one could also ask when σ S − ( A ) = α + , which would automaticallyimply the statement for α . While it is easy to find examples where this holds, none of the conditions inthis section apply to α + .As a first step, we highlight that the desired equality can be equivalently expressed in terms of a suitableminimax problem. Lemma 3.17.
Let Z ∈ X ′ and define a map K : X × E ′ → [ −∞ , ∞ ] by setting K Z ( X, W ) := σ A ( W ) + E [ h X, Z i ] − E [ S ( X ) W ] . The following statements are equivalent:(a) σ S − ( A ) = α .(b) α is σ ( X ′ , X ) -upper semicontinuous.(c) For every Z ∈ X ′ we have inf X ∈X sup W ∈E ′ K Z ( X, W ) = sup W ∈E ′ inf X ∈X K Z ( X, W ) . Proof.
The equivalence between (a) and (b) is clear by Theorem 3.14. To establish equivalence with (c) ,fix Z ∈ X ′ and note that α ( Z ) = sup W ∈E ′ inf X ∈X K Z ( X, W )by definition of α . It remains to show that σ S − ( A ) ( Z ) = inf X ∈X sup W ∈E ′ K Z ( X, W ) . f Z : X → ( −∞ , ∞ ] defined by f Z ( X ) := E [ h X, Z i ] + δ S − ( A ) ( X )and F Z : X × E → ( −∞ , ∞ ] defined by F Z ( X, U ) := E [ h X, Z i ] + δ A− S ( X ) ( U ) . Note that for every X ∈ X the map F Z ( X, · ) is convex and lower semicontinuous and satisfies( F Z ( X, · )) ∗ ( W ) = sup U ∈E { E [ U W ] − F Z ( X, U ) } = sup U ∈E , U + S ( X ) ∈A { E [ U W ] − E [ h X, Z i ] } = sup V ∈E { E [( V − S ( X )) W ] − E [ h X, Z i ] } = − σ A ( − W ) − E [ h X, Z i ] + E [ S ( X )( − W )]= − K Z ( X, − W )for every W ∈ E ′ . As F Z ( X,
0) = f Z ( X ) for every X ∈ X , we can apply Fenchel-Moreau to get σ S − ( A ) ( Z ) = inf X ∈X f Z ( X ) = inf X ∈X sup W ∈E ′ { E [0 W ] − ( F Z ( X, · )) ∗ ( W ) } = inf X ∈X sup W ∈E ′ K Z ( X, W ) . This concludes the proof.The preceding lemma shows that, for every Z ∈ X ′ , the identity σ S − ( A ) ( Z ) = α ( Z ) is equivalent tothe existence of a saddle value for the function K Z . Unfortunately, the standard minimax theorems, seee.g. Fan [17], rely on compactness assumptions that do not hold in our setting. The remainder of thissubsection is devoted to showing a number of situations where the identity holds or, equivalently, theabove minimax problem has a solution. The linear case
We start by proving the desired equality in the simple case where the impact map is given by theaggregated, or consolidated, capital position of all the d financial institutions. In this case, there is norestriction on the acceptance set. Proposition 3.18.
Assume that X i = E for every i ∈ { , . . . , d } . If S ( X ) = P di =1 X i for every X ∈ X ,then α = σ S − ( A ) .Proof. First of all, we show that for every Z ∈ X ′ + we have σ S − ( A ) ( Z ) = ( σ A ( Z ) if Z = · · · = Z d , −∞ otherwise . To see this, assume first that P ( Z i > Z j ) > i, j ∈ { , . . . , d } and for every n ∈ N definea random vector X n ∈ X by X nk = − n { Z i >Z j } if k = i,n { Z i >Z j } if k = j, . Since S ( X n ) = 0 ∈ A for every n ∈ N , we clearly have σ S − ( A ) ( Z ) ≤ inf n ∈ N E [ h X n , Z i ] = inf n ∈ N n E [ { Z i >Z j } ( Z j − Z i )] = −∞ . Z = · · · = Z d and note that, in this case, we have σ S − ( A ) ( Z ) = inf X ∈ S − ( A ) E [ S ( X ) Z ] = σ A ( Z ) . This proves the above claim. Now, for every Z ∈ X ′ + note that α ( Z ) = sup W ∈ bar( A ) ( σ A ( W ) + inf X ∈X E " d X i =1 X i ( Z i − W ) = ( σ A ( Z ) if Z = · · · = Z d ∈ bar( A ) , −∞ otherwise . This yields the desired assertion.
The conic case
Next, we deal with the case where S is positively homogeneous and A is a cone. In this case, we firstshow that α is given by a suitable indicator function and provide a general sufficient condition for theequality between σ S − ( A ) and α . At a later stage, we apply this general condition to a variety of concretesituations. Lemma 3.19.
Assume that S is positively homogeneous and A is a cone. Then, we have α = − δ D for D := { Z ∈ X ′ + ; ∃ W ∈ bar( A ) : E [ h X, Z i ] ≥ E [ S ( X ) W ] , ∀ X ∈ X } . Proof.
Clearly, for every Z ∈ D there exists W Z ∈ bar( A ) such thatinf X ∈X { E [ h X, Z i ] − E [ S ( X ) W Z ] } = E [ h , Z i ] − E [ S (0) W Z ] = 0 . As a result, for every Z ∈ D we have 0 ≥ α ( Z ) ≥ σ A ( W Z ) + 0 = 0, showing that α ( Z ) = 0. Now, fix Z ∈ X ′ \D and observe that, for every W ∈ bar( A ), we find X W ∈ X such that E [ h X W , Z i ] < E [ S ( X W ) W ].Then, inf X ∈X { E [ h X, Z i ] − E [ S ( X ) W ] } ≤ inf n ∈ N { E [ h nX W , Z i ] − E [ S ( nX W ) W ] } = inf n ∈ N { n ( E [ h X W , Z i ] − E [ S ( X W ) W ]) } = −∞ . This implies that α ( Z ) = −∞ and concludes the proof. Lemma 3.20.
Assume that S is positively homogeneous and A is a cone. Moreover, assume that S ( e ) ∈ R + \ { } and that bar( A ) ∩ { W ∈ L ( R ) ; k W k ≤ } is σ ( E ′ , E ) -compact. Then, σ S − ( A ) = α .Proof. Recall that σ S − ( A ) = α holds if and only if α is σ ( X ′ , X )-upper semicontinuous. Hence, byLemma 3.19, it suffices to show that D is σ ( X ′ , X )-closed. To this effect, take a net ( Z γ ) ⊂ D convergingto some Z ∈ X ′ in the topology σ ( X ′ , X ). Note that Z ∈ X ′ + . By definition of D , for each γ we find W γ ∈ bar( A ) such that E [ h X, Z γ i ] ≥ E [ S ( X ) W γ ]for every X ∈ X . To establish the desired closedness, it is enough to show that ( W γ ) admits a subnetthat converges to some element of bar( A ) in the topology σ ( E ′ , E ). Note that bar( A ) = { σ A ≥ } byconicity of A , showing that bar( A ) is σ ( E ′ , E )-closed. Since bar( A ) ⊂ E ′ + , we see that E [ h e, Z γ i ] ≥ E [ S ( e ) W γ ] ≥ ,
20r equivalently E [ h e, Z γ i ] S ( e ) ≥ E [ W γ ] ≥ , for every γ . Since E [ h e, Z γ i ] → E [ h e, Z i ], the net ( W γ ) is bounded in L ( R ) and, hence, by using thecompactness assumption, it admits a convergent subnet in the topology σ ( E ′ , E ). In view of the σ ( E ′ , E )-closedness of bar( A ), we infer that the limit belongs to bar( A ). This concludes the proof.The next proposition describes a number of situations where we can ensure the above compactness con-dition and, thus, we can establish that σ S − ( A ) = α . Proposition 3.21.
Assume that S is positively homogeneous and A is a cone. Moreover, assume that S ( e ) ∈ R + \ { } . Then, σ S − ( A ) = α in each of the following cases:(i) Ω is finite.(ii) A is polyhedral, i.e. there exist W , . . . , W n ∈ E ′ + and a , . . . , a n ∈ R such that A = n \ i =1 { U ∈ E ; E [ U W i ] ≥ a i } . (iii) A is induced by Expected Shortfall, i.e. there exists λ ∈ (0 , such that A = { U ∈ E ; ES λ ( U ) ≤ } . Proof. (i)
In the case that Ω is finite, the space E ′ is finite dimensional and the compactness condition inLemma 3.20 is clearly satisfied because bar( A ) = { σ A ≥ } is always σ ( E ′ , E )-closed. (ii) If A is polyhedral, then it is easy to see that bar( A ) is a finitely-generated convex cone, i.e. thereexist W , . . . , W n ∈ E ′ + such thatbar( A ) = ( n X i =1 λ i W i ; λ , . . . , λ n ∈ [0 , ∞ ) ) . Note that for all λ , . . . , λ n ∈ [0 , ∞ ) we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i W i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = n X i =1 λ i k W i k . As a result, bar( A ) ∩ { W ∈ L ( R ) ; k W k ≤ } is easily seen to be σ ( E ′ , E )-compact and we can applyLemma 3.20 to get the desired result. (iii) If A is induced by Expected Shortfall as in Example 3.15, thenbar( A ) = (cid:26) W ∈ E ′ + ; W ≤ λ E [ W ] (cid:27) . As a result, we easily see thatbar( A ) ∩ { W ∈ L ( R ) ; k W k ≤ } ⊂ { W ∈ L ∞ + ( R ) ; W ≤ λ − } . Since the set bar( A ) ∩ { W ∈ L ( R ) ; k W k ≤ } is σ ( L ∞ ( R ) , L ( R ))-closed, it follows from the Banach-Alaoglu Theorem that it is even σ ( L ∞ ( R ) , L ( R ))-compact. As E ⊂ L ( R ), we automatically have σ ( E ′ , E )-compactness and we may conclude by applying Lemma 3.20.21 he case where the image of S intersects the interior of A As a final step, we follow Rockafellar [31] to establish the identity σ S − ( A ) = α under a suitable interioritycondition, which also appears in Armenti et al. [3] and Biagini et al. [6]. Proposition 3.22. (i) If there exists X ∗ ∈ X such that S ( X ∗ ) belongs to the σ ( E , E ′ ) -interior of A ,then α = σ S − ( A ) .(ii) Assume that E ′ is the norm dual of E . If there exists X ∗ ∈ X such that S ( X ∗ ) belongs to the norminterior of A , then α = σ S − ( A ) .Proof. (i) By assumption, we find a σ ( E , E ′ )-neighborhood of zero U ⊂ E such that S ( X ∗ ) + U ⊂ A . Now,fix an element Z ∈ X ′ and define a map ψ Z : E → [ −∞ , ∞ ] by setting ψ Z ( U ) := inf X ∈X F Z ( X, U ) . Here, we have adopted the notation introduced in the proof of Lemma 3.17. It is easy to verify that F Z is jointly convex and, hence, ψ Z is convex. Note that ψ Z ( U ) ≤ F Z ( X ∗ , U ) = E [ h X ∗ , Z i ]for every U ∈ U , so that ψ Z is bounded from above on U . In view of Lemma 3.17, the desired assertionfollows from Theorem 17 in Rockafellar [31] (by taking ϕ = ψ Z and F = F Z in the notation of that result). (ii) Since the norm topology on E is compatible with our bilinear form on E × E ′ under the assumptionthat E ′ is the norm dual of E , we can repreat the same argument as in (i) by exploiting the fact thatTheorem 17 in [31] holds under any compatible topology. In this short section we turn to systemic risk measures of “first aggregate, then allocate” type and theirdual representations. Throughout the section we fix an admissible impact map S and an admissibleacceptance set A . e ρ “First aggregate, then allocate”-type systemic risk measures are defined as follows. Definition 4.1.
We define a map e ρ : X → [ −∞ , ∞ ] by setting e ρ ( X ) = inf { m ∈ R ; S ( X ) + m ∈ A} . The difference with respect to ρ is that, instead of injecting capital into the system in order to reach anacceptable level of systemic risk, one looks at the minimum level of the chosen systemic risk indicator thatensures acceptability. In particular, if the impact function is expressed in monetary terms, then e ρ ( X ) canbe interpreted as a bail-out cost for the “aggregated position” S ( X ). For a thorough presentation of thistype of systemic risk measures we refer to the literature cited in the introduction.In what follows, we exploit the fact that e ρ can be expressed as the composition between the impact mapand the standard cash-additive risk measure ρ A : E → [ −∞ , ∞ ] given by ρ A ( X ) := inf { m ∈ R ; X + m ∈ A} . The next result records the key properties of e ρ . In particular, differently from the systemic risk measure ρ , we show that e ρ is always lower semicontinuous under our standing assumptions on the impact mapand the acceptance set. 22 roposition 4.2. The systemic risk measure e ρ is convex, σ ( X , X ′ ) -lower semicontinuous, and satisfies e ρ (0) ≤ . Moreover, e ρ is proper if and only if e ρ (0) > −∞ if and only if A ∩ ( − R + ) = − R + .Proof. Convexity is clear by composition. To show lower semicontinuity, note that ρ A is σ ( E , E ′ )-lowersemicontinuous by the σ ( E , E ′ )-closedness of A . Now, take r ∈ R and note that { X ∈ X ; e ρ ( X ) ≤ r } = S − ( { U ∈ E ; ρ A ( U ) ≤ r } ) . Following the argument in the proof of Proposition 3.2 we can show that the above set is σ ( X , X ′ )-closed,which delivers the desired lower semicontinuity. To show properness, observe first that e ρ (0) ≤ S (0) = 0 ∈ A . The above equivalence can now be established as in the proof of Proposition 3.3. e ρ The purpose of this subsection is to derive a dual representation of e ρ and to compare it with the dualrepresentation of ρ . In this case, the acceptability test is performed on S ( X ) and the acceptance set is A . This suggests to rely on the dual representation of ρ A in order to achieve in a straightforward waythe desired dual representation of e ρ . The following maps are the fundamental ingredients of the desiredrepresentation. Definition 4.3.
We define two maps e α, e α + : X ′ → [ −∞ , + ∞ ] by setting e α ( Z ) := sup W ∈ bar( A ) , E [ W ]=1 n σ A ( W ) + inf X ∈X { E [ h X, Z i ] − E [ S ( X ) W ] } o , e α + ( Z ) := sup W ∈ bar( A ) ∩ ( E ′ ++ ∪{ } ) , E [ W ]=1 n σ A ( W ) + inf X ∈X { E [ h X, Z i ] − E [ S ( X ) W ] } o . Remark 4.4.
The above maps belong to the class of maps e α K : X ′ → [ −∞ , + ∞ ] defined by e α K ( Z ) := sup W ∈K , E [ W ]=1 n σ A ( W ) + inf X ∈X { E [ h X, Z i ] − E [ S ( X ) W ] } o , where K is a convex cone in bar( A ) such that λ K + (1 − λ ) bar( A ) ⊂ K for every λ ∈ [0 , ; see alsoRemark 3.10. This will allow us to prove properties for e α and e α + simultaneously. In fact, all propertiesof e α and e α + we will consider are shared by the entire class. Before we establish the desired dual representation we highlight some relevant properties of the abovemaps and point out their relationship with the penalty functions α and α + . Here, we denote by dom( e α )the domain of finiteness of e α (similarly for e α + ). Moreover, we denote by cl the closure operator withrespect to the topology σ ( X ′ , X ). Proposition 4.5.
The maps e α, e α + : X ′ → [ −∞ , ∞ ] satisfy the following properties (the statements about e α + require that bar( A ) ∩ E ′ ++ = ∅ ):(i) e α and e α + take values in the interval [ −∞ , .(ii) e α and e α + are concave.(iii) dom( e α + ) ⊂ dom( e α ) ⊂ cl(dom( e α + )) ⊂ X ′ + .(iv) α is the smallest positively homogeneous map dominating e α , i.e. for every Z ∈ X ′ α ( Z ) = sup λ> e α ( λZ ) λ . v) α + is the smallest positively homogeneous map dominating e α + , i.e. for every Z ∈ X ′ α + ( Z ) = sup λ> e α + ( λZ ) λ . Proof. (i)-(ii), (iv)-(v)
Let
K ⊂ bar( A ) be a convex cone as in Remark 4.4. It is clear that α K ( Z ) = sup λ> e α K ( λZ ) λ for every Z ∈ X ′ . In particular, e α K ≤ α K . It follows from the proof of Proposition 3.12 that e α K takesvalue into [ −∞ , α K in that result can be repeated to establishthe concavity of e α K . The desired assertions follow by taking K = bar( A ) and K = bar( A ) ∩ ( E ′ ++ ∪ { } ). (iii) The assertion can be proved by repeating the proof of the corresponding statement in Proposition 3.12.We record the announced dual representation of e ρ in the next result. Theorem 4.6. (i) If e ρ is proper, then we have e ρ ( X ) = sup Z ∈X ′ + { e α ( Z ) − E [ h X, Z i ] } for every X ∈ X . The supremum can be restricted to X ′ ++ provided that dom( e α ) ∩ X ′ ++ = ∅ .(ii) Assume that bar( A ) ∩ E ′ ++ = ∅ . If e ρ is proper, then we have e ρ ( X ) = sup Z ∈X ′ + { e α + ( Z ) − E [ h X, Z i ] } for every X ∈ X . The supremum can be restricted to X ′ ++ provided that dom( e α + ) ∩ X ′ ++ = ∅ .Proof. Let
K ⊂ bar( A ) be a convex cone as in Remark 4.4. Note that the dual representation (2.1)applied to A yields A = \ W ∈K { U ∈ E ; E [ U W ] ≥ σ A ( W ) } = \ W ∈K , E [ W ]=1 { U ∈ E ; E [ U W ] ≥ σ A ( W ) } , (4.1)where we used the positive homogeneity of σ A (together with the fact that K ⊂ E ′ + ). As a result, forevery U ∈ E we get ρ A ( U ) = sup W ∈K , E [ W ]=1 { σ A ( W ) − E [ U W ] } . Using the notation introduced in the proof of Theorem 3.11, we immediately get e ρ ( X ) = sup W ∈K , E [ W ]=1 { σ A ( W ) − E [ S ( X ) W ] } = sup W ∈K , E [ W ]=1 sup Z ∈X ′ + { σ A ( W ) − E [ h X, Z i ] + ( ϕ W ) • ( Z ) } = sup Z ∈X ′ + sup W ∈K , E [ W ]=1 { σ A ( W ) − E [ h X, Z i ] + ( ϕ W ) • ( Z ) } = sup Z ∈X ′ + { e α K ( Z ) − E [ h X, Z i ] } for every X ∈ X . If, in addition, dom( e α K ) ∩ X ′ ++ = ∅ , then we get e ρ ( X ) = sup Z ∈X ′ ++ { e α K ( Z ) − E [ h X, Z i ] } for every X ∈ X by the same argument used to reduce the domain of the supremum in the proof ofTheorem 3.6. The desired assertions now follow by taking K = bar( A ) and K = bar( A ) ∩ ( E ′ ++ ∪ { } ).24 emark 4.7. (i) As in Remark 3.7, we highlight the link between the dual representation in Theorem 4.6and the standard Fenchel-Moreau representation. We claim that, if e ρ is proper, then e ρ ∗ ( Z ) = − usc( e α )( − Z ) = − usc( e α + )( − Z ) for every Z ∈ X ′ (where the last equality holds provided that bar( A ) ∩ E ′ ++ = ∅ ). Here, we have denotedby usc( e α ) the σ ( X ′ , X ) -upper semicontinuous hull of e α (similarly for e α + ). To see this, note first that e ρ ( X ) = sup Z ∈X ′ { e α ( Z ) − E [ h X, Z i ] } = sup Z ∈X ′ { usc( e α )( Z ) − E [ h X, Z i ] } for every X ∈ X . The left-hand side equality holds because e α = −∞ outside X ′ + by Proposition 4.5.The right-hand side equality follows from Theorem 2.3.1 in Z˘alinescu [34]. Since usc( e α ) is concave and σ ( X ′ , X ) -upper semicontinuous, the desired claim is a consequence of the Fenchel-Moreau Theorem. Theargument for e α + is identical.(ii) The dual elements in the above representation can be identified with d -dimensional vectors of probabil-ity measures on (Ω , F ) that are absolutely continuous (or equivalent) with respect to P up to a normalizingvector that collects their expectations. This allows to express the above representation in terms of proba-bility measures. Indeed, for every w ∈ R d + define Q w ( P ) := { Q ∈ Q ( P ) ; Q i = P if w i = 0 , ∀ i ∈ { , . . . , d }} , Q we ( P ) = Q e ( P ) ∩ Q w ( P ) , where we have used the notation from Remark 3.7. Then, if e ρ is proper, we easily see that e ρ ( X ) = sup w ∈ R d + , Q ∈Q w ( P ) , d Q d P ∈X ′ (cid:26)e α (cid:18) w d Q d P , . . . , w d d Q d d P (cid:19) − d X i =1 w i E Q i [ X i ] (cid:27) for every X ∈ X . We can replace Q w ( P ) by Q we ( P ) in the above supremum provided that dom( e α ) ∩X ′ ++ = ∅ .The same holds with e α + instead of e α (provided that bar( A ) ∩ E ′ ++ = ∅ ). The condition dom( e α ) ∩ X ′ ++ = ∅ is needed to restrict the domain in the above dual representationto strictly-positive dual elements (similarly for e α + ). We conclude this section by providing a sufficientcondition for this to hold; see also Proposition 3.8. Proposition 4.8.
Assume that X i = E for every i ∈ { , . . . , d } . Moreover, suppose that bar( A ) ∩ E ′ ++ = ∅ and there exist a ∈ (0 , ∞ ) and b ∈ R such that S ( X ) ≤ a d X i =1 X i + b for every X ∈ X . Then, dom( e α + ) ∩ X ′ ++ = ∅ (and, a fortiori, dom( e α ) ∩ X ′ ++ = ∅ ).Proof. Take W ∈ bar( A ) ∩ E ′ ++ and note that we can always assume that E [ W ] = 1 by conicity of bar( A ).Setting Z = ( aW, . . . , aW ) ∈ X ′ ++ , we easily see that e α ( Z ) ≥ e α + ( Z ) ≥ σ A ( W ) + inf X ∈X { E [ h X, Z i ] − E [ S ( X ) W ] } ≥ σ A ( W ) − b E [ W ] > −∞ . This delivers the desired assertion. 25
Risk measures based on univariate utility functions
In this final section we provide a simple proof of the dual representation of shortfall risk measures, seeTheorem 4.115 in F¨ollmer and Schied [19], that uses our general strategy to obtain dual representations.For ease of comparison, we focus on bounded positions.Throughout the entire section we fix a nonconstant, concave, increasing function u : R → R , which isinterpreted as a standard von Neumann-Morgenstern utility function. We fix u ∈ R such that u ( x ) > u for some x ∈ R and define a map ρ u : L ∞ → [ −∞ , ∞ ] by ρ u ( X ) := inf { m ∈ R ; E [ u ( X + m )] ≥ u } . Theorem 5.1.
The risk measure ρ u is convex and σ ( L ∞ , L ) -lower semicontinuous. Moreover, ρ u ( X ) = sup Q ≪ P (cid:26) E Q [ − X ] + sup λ> (cid:26) λ (cid:18) u + E (cid:20) u • (cid:18) λ d Q d P (cid:19)(cid:21)(cid:19)(cid:27)(cid:27) for every X ∈ L ∞ .Proof. It is well-known that ρ u is convex and σ ( L ∞ , L )-lower semicontinuous. To establish the aboverepresentation, note that ρ u can be viewed as a “first allocate, then aggregate”-type systemic risk measurecorresponding to the specifications d = 1 , ( X , X ′ ) = ( L ∞ , L ) , ( E , E ′ ) = ( L ∞ , L ) , S ( X ) = u ( X ) , A = { U ∈ L ∞ ; E [ U ] ≥ u } . First of all, note that bar( A ) = R + and σ A ( λ ) = λu for every λ ∈ R + . Since bar( A ) ∩ E ′ ++ is nonempty,we can work with α + ; see Definition 3.9. It follows from Proposition 3.13 that α + ( Z ) = sup W ∈ bar( A ) ∩E ′ ++ (cid:26) σ A ( W ) + E (cid:20) u • (cid:18) ZW (cid:19) W (cid:21)(cid:27) = sup λ> (cid:26) λu + E (cid:20) u • (cid:18) Zλ (cid:19) λ (cid:21)(cid:27) = sup λ> (cid:26) λ ( u + E [ u • ( λZ )]) (cid:27) for every nonzero Z ∈ X ′ + . The representation of S − ( A ) in Theorem 3.11 yields S − ( A ) = \ Z ∈X ′ + \{ } { X ∈ X ; E [ XZ ] ≥ α + ( Z ) } = \ Q ≪ P (cid:26) X ∈ X ; E Q [ X ] ≥ α + (cid:18) d Q d P (cid:19)(cid:27) , where we used that dom( α + ) ⊂ X ′ + and that α + is positively homogeneous; see Proposition 3.12. Itremains to observe that ρ u ( X ) = inf { m ∈ R ; X + m ∈ S − ( A ) } for every X ∈ X . References [1] Aliprantis, Ch.D., Border, K.C.:
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