Law-Invariant Return and Star-Shaped Risk Measures

This paper presents novel characterization results for classes of law-invariant star-shaped functionals. We begin by establishing characterizations for positively homogeneous and star-shaped functionals that exhibit second- or convex-order stochastic dominance consistency. Building on these characterizations, we proceed to derive Kusuoka-type representations for these functionals, shedding light on their mathematical structure and intimate connections to Value-at-Risk and Expected Shortfall. Furthermore, we offer representations of general law-invariant star-shaped functionals as robustifications of Value-at-Risk. Notably, our results are versatile, accommodating settings that may, or may not, involve monotonicity and/or cash-additivity. All of these characterizations are developed within a general locally convex topological space of random variables, ensuring the broad applicability of our results in various financial, insurance and probabilistic contexts.


Introduction
Over the past decades, a large literature has developed the theory of monetary risk measures -monotone and cash-additive functionals -and analyzed their applications in a variety of fields including economics, finance, insurance, operations research and statistics.More recently, [4] introduced return risk measures -monotone and positively homogeneous functionals -and first results in the development of their static and dynamic theory and applications were obtained in [2,4,5,26,27].Whereas monetary risk measures provide absolute assessments of risk, return risk measures provide relative assessments of risk, evocative of the distinct roles played by absolute and relative risk aversion measurements.The positive homogeneity property of return risk measures was relaxed by [26,27] asserting the more general star-shapedness property.
Law-invariant risk measures -also referred to as law-determined or distribution-invariant risk measures -play an important role in the theory and applications of risk measures. 1his is due to their simplicity, tractability and statistical appeal, being statistical functionals.Many well-known risk measures are specific examples of law-invariant risk measures (e.g., Value-at-Risk, Expected Shortfall, the entropic risk measure and the p-norm).Law-invariant representations of coherent, convex, quasi-convex and quasi-logconvex risk measures have been derived in [9,21,25,26].
In this paper, we establish new characterization results for classes of law-invariant starshaped functionals.The existing literature offers general representations for law-invariant convex and quasi-convex functionals that do not require monotonicity and/or cash-additivity; see the recent [3].For a comprehensive discussion of non-monotone preferences and their applications, see [1].There is, however, a notable gap in the literature for star-shaped functionals, which we aim to fill.In [26], representations for law-invariant quasi-logconvex star-shaped risk measures are derived.Furthermore, in [8], representations for law-invariant monetary star-shaped risk measures are obtained, and in [24] the latter results are extended to allow for cash-subadditivity.By contrast, our representation results for law-invariant star-shaped functionals do not rely on the properties of monotonicity, quasi-logconvexity and/or cash-(sub)additivity. Furthermore, whereas [8,24,26] primarily focus on L ∞ as the space of random variables, we derive our results in a general locally convex topological space of random variables.
The generality and versatility of our setting -both in terms of properties that our functionals may satisfy and in terms of the spaces they are defined on -make some of the mathematical proofs intricate.Furthermore, while in the case of convex functionals the properties of law-invariance and convex-order stochastic dominance consistency are equivalent, provided that the functionals exhibit lower semicontinuity, such equivalence does not hold in general for star-shaped functionals, even not when assuming lower semicontinuity.Therefore, it becomes necessary to separately examine and analyze these two properties in the case of star-shaped functionals.
That is, the primary contribution of this paper is to bridge the gap between existing general representations for law-invariant convex and quasi-convex functionals on the one hand and those pertaining to star-shaped functionals on the other hand.We first establish representations for second-order and convex-order stochastic dominance consistent (SSDand CSD-consistent, for short) star-shaped functionals.We also showcase the robustness of our proof strategies: when we impose the additional axioms of monotonicity and cashadditivity, our results align with those in [8], but within a (much) broader space than L ∞ .Second, we unveil novel representations à la Kusuoka [25] pertaining to SSD-and CSDconsistent star-shaped functionals, including their monetary variants that are previously unexplored in the literature.These characterization results may be viewed as a contribution of independent interest.Value-at-Risk and Expected Shortfall are pivotal building blocks in these representations.Third, we extend our analysis to general law-invariant star-shaped functionals as robustifications of Value-at-Risk.We demonstrate the validity of these representations even without the need for monotonicity or cash-additivity.Furthermore, in the case of law-invariant star-shaped risk measures that are also cash-additive, we present a more precise representation compared to that in Theorem 5 of [8], resulting in a reduced set over which minimization takes place.We demonstrate that these three sets of results extend to positively homogeneous functionals, thus generalizing [2,26], and yielding in particular new representation results for SSD-consistent return risk measures.We finally provide three examples to illustrate that law-invariant star-shaped risk measures may arise naturally, also when cash-(sub)additivity is not preserved.
All the proof strategies share a common approach, relying on the approximation of starshaped functions by convex ones.Consequently, our proof strategies effectively encapsulate the concept of star-shapedness, providing flexibility and adaptability in a myriad of settings in which star-shapedness is assumed.
The remainder of this paper is organized as follows.In Section 2, we provide some preliminaries for law-invariant star-shaped functionals.In Section 3, we establish our first representation results for SSD-and CSD-consistent star-shaped functionals.Section 4 establishes representations à la Kusuoka [25] for SSD-and CSD-consistent star-shaped functionals.In Section 5, we derive representations as robustifications of Value-at-Risk for general lawinvariant star-shaped functionals.Finally, Section 6 provides three illustrative examples.

Preliminaries
Let (Ω, F , P) be a non-atomic probability space and let L 0 be the space of real-valued random variables.Equalities and inequalities between random variables are meant to hold almost surely (a.s.).Given a set X ⊆ L 0 , we denote by X + the subspace of X containing only positive random variables.The usual Lebesgue spaces of functions are denoted by We need the following assumptions on the spaces used in the sequel, which will be imposed throughout the paper unless explicitly mentioned otherwise.Let X , X * be two linear subspaces of L 0 and such that: • X contains the constants.
We consider the weakest topology σ(X , X * ) on X such that the linear functional f Y (X) := E[XY ] is continuous for every Y ∈ X * .Under these hypotheses, X is a locally convex topological space.Additionally, for any X ∈ X and c ∈ R, it holds that X + c ∈ X , because X is a linear space that includes constants.We will often use this fact without further mentioning.We recall that Orlicz spaces and L p spaces with p ∈ [1, +∞] are examples of spaces satisfying the previous assumptions.Definition 2. A functional f : X → R ∪ {+∞} with f (0) < +∞ is star-shaped if for any λ ∈ (0, 1) and X ∈ X it holds that: We define the proper domain of f as Dom(f ) := {X ∈ X : f (X) < +∞}, and analogously for ρ.
We note that, different from a substantial part of the literature on risk measures, Definition 2 adopts the sign convention that risk measures satisfy increasing rather than decreasing monotonicity.We recall that a functional f : X → R ∪ {+∞} is said to be lower semicontinuous if its level sets are σ(X , X * )-closed.In addition, if f is law-invariant and convex, and X is a rearrangement invariant space, σ(X , X * )-lower semicontinuity is equivalent to the Fatou property, i.e., for any sequence (X n ) n∈N ⊆ X and X ∈ X s.t.X n → X a.s. with these two properties are also equivalent to continuity from below, i.e., for any (X n ) n∈N ⊆ X and X ∈ X such that X n ↑ X a.s.we have f (X) = lim n→∞ f (X n ).In the subsequent definition, we present a non-exhaustive set of axioms that a risk measure (or a functional) may satisfy.Definition 3. Let ρ : X → R ∪ {+∞} be a risk measure.Then ρ is said to be • Positively homogeneous, if ρ(λX) = λρ(X) for any λ ∈ R + and X ∈ X .
Let us briefly revisit the core stochastic dominance concepts, particularly first-, secondand convex-order stochastic dominance.Under first-order stochastic dominance, we say that, for given ] for every increasing function g : R → R such that the expectations exist; it is denoted as X 1 Y .Similarly, under secondorder stochastic dominance, X dominates Y if E[g(X)] ≥ E[g(Y )] for every increasing and convex function g : R → R; it is denoted as X 2 Y .Finally, X dominates Y in the convex order sense if E[g(X)] ≥ E[g(Y )] for every (not necessarily increasing) convex function g : R → R; it is denoted as X c Y. We write ∼ • when both • and • apply.
Remark 5. We note that first-order stochastic dominance can be equivalently formulated as where V aR β (X) := inf{x ∈ R|P(X ≤ x) ≥ β}, for any X ∈ X , is the usual Value-at-Risk.Similarly, second-order stochastic dominance is equivalent to where, for any X ∈ X , 1), and ES 1 (X) := ess sup(X), is the usual Expected Shortfall. 4It is clear by this characterization that first-order stochastic dominance implies second-order stochastic dominance, i.e., if X 1 Y then the relationship X 2 Y also holds.Furthermore, convex-order stochastic dominance is equivalent to X 2 Y with the further constraint E(X) = E(Y ), i.e., ES β (X) ≥ ES β (Y ) ∀β ∈ [0, 1) and E(X) = E(Y ).Finally, we underline that if a functional f : X → R ∪ {+∞} is consistent with one of these three stochastic dominance conditions, then f is law-invariant.However, it is important to note that the converse implication does not hold, as we will explore in what follows.See [3,18] for further details on this subject.
It is well-known that a functional f : X → R∪{+∞} that adheres to either FSD-or SSDconsistency also displays monotonicity with respect to the standard pointwise order relation in L 0 .Therefore, FSD-or SSD-consistency are incompatible with functionals that lack monotonicity.In contrast to FSD-and SSD-consistency, the criterion of CSD-consistency does not entail monotonicity.

SSD-and CSD-Consistent Star-Shaped Functionals
In this section, we explore the implications of the properties of SSD-and CSD-consistency for star-shaped risk measures within the framework of min-max representations.It has been shown in the literature (see e.g., [8]) that not all law-invariant star-shaped risk measures can be expressed as the minimum of law-invariant convex risk measures.This is exemplified by VaR, which falls into the category of star-shaped risk measures that cannot be represented as the minimum of law-invariant convex risk measures.The following theorem establishes the connection between SSD-consistent star-shaped risk measures and the minimum of SSDconsistent convex risk measures.Theorem 6.A risk measure ρ : X → R ∪ {+∞} is SSD-consistent and star-shaped if and only if there exist a set of indexes Γ and a family of SSD-consistent convex risk measures (ρ γ ) γ∈Γ with ργ : X → R ∪ {+∞} such that ργ (0) = ρ(0) for all γ ∈ Γ and ρ(X) = min γ∈Γ ργ (X), X ∈ X . (3.1) In addition, ρ : X → R∪{+∞} is a SSD-consistent and positively homogeneous risk measure if and only there exists a family of SSD-consistent and sublinear risk measures (ρ γ ) γ∈Γ such that ρ can be represented as in Equation (3.1).
Proof.We provide a detailed proof for the star-shaped case, followed by a brief sketch of the proof for positively homogeneous risk measures, since it is similar to the star-shaped case.
We prove now that ρ Z is law-invariant.Indeed, when X ∼ Y , the law-invariance property of We are now prepared to define the family of SSD-consistent and convex risk measures . Hence, ρZ is also monotone and law-invariant (see Remark 5).By definition of the infimum, it follows that ρZ ≤ ρ Z .Moreover, for each fixed X ∈ X and Y 2 X it holds that ρ Z (Y ) ≥ ρ(Y ) ≥ ρ(X) (as shown in the first step of the proof), taking the infimum over Y 2 X we obtain ρZ (X) ≥ ρ(X).Summing up, ρ(X) ≤ ρZ (X) ≤ ρ Z (X), hence ρ(X) = min Z∈Γ ρZ (X).Now we prove convexity of ρZ .To see this, let us consider X 1 , X 2 ∈ X with X 1 = X 2 and λ ∈ (0, 1).If ρZ (X 1 ) = +∞ or ρZ (X 2 ) = +∞ there is nothing to prove.Moreover, if ρZ (λX 1 + (1 − 5 To prove this statement, let B a non-null measure set such that X < 0 on B. Setting Hence, there exists n ∈ N such that A := n n=1 Bn and P(A) > 0.Moreover, X ≤ − 1 n on A, taking c := − 1 n the thesis follows.
λ)X 2 ) = +∞ then either ρZ (X 1 ) = +∞ or ρZ (X 2 ) = +∞.Indeed, if ρZ (X i ) < +∞ for i = 1, 2, then there exist and Y i attains the infimum 6 in the definition of ρZ (X i ).This yields (λα , by positive homogeneity and subadditivity of Expected Shortfall.Thus by definition of ρZ we have ρZ (λX So, we only need to check the case ρZ (λX 1 + (1 − λ)X 2 ), ρZ (X 1 ), ρZ (X 2 ) < +∞.In this case we find at least one random variable for any β ∈ [0, 1].The first inequality follows from sublinearity of ES β , the second inequality is due to the condition Y i 2 X i for i = 1, 2, while the equality holds by positive homogeneity of ES β , SSD-consistency of ES β and the relation where the first inequality follows from SSD-consistency of ρZ , the second inequality follows from the relation ρZ ≤ ρ Z , the third equality is by definition of ρ Z , while the definition of ρ Z together with the relation ] and i = 1, 2 we get the thesis.'If': We proceed to demonstrate the converse implication.Given a collection of SSDconsistent convex risk measures (ρ γ ) γ∈Γ that share the same value at 0 for all γ ∈ Γ, it is established that their pointwise minimum forms a star-shaped risk measure (refer to Lemma 7 in [27]).Our focus now is to establish that ρ(X) := min γ∈Γ ρ γ (X) upholds SSDconsistency.To establish this, let us assume X 2 Y .By the SSD-consistency of the family (ρ γ ) γ∈Γ , it follows that ρ γ (X) ≥ ρ γ (Y ) for all γ ∈ Γ.This yields the result:

The positively homogeneous case
To prove the statement regarding positively homogeneous risk measures, we consider the family of functionals (ρ Z ) Z∈Γ defined as: By following the same reasoning as above, it can be proved that ρ Z is well-defined for all Z ∈ Γ.In addition, ρ Z (X) ≥ ρ(X) and ρ X (X) = ρ(X).Moreover, we observe that ρZ := inf{ρ Z (Y ) : Y 2 X} inherits the properties of ρ Z and it is SSD-consistent and sublinear.Thus, the family (ρ γ ) γ∈Γ with Γ := Dom(ρ) fulfills the second thesis of the theorem.The converse implication is straightforward by observing that the pointwise minimum of sublinear risk measures is positively homogeneous.✷ Corollary 7. f : X → R ∪ {+∞} is a CSD-consistent and star-shaped functional if and only if there exist a set of indexes Γ and a family ( fγ ) γ∈Γ of CSD-consistent and convex functionals fγ : 6 See Proposition 10 for a detailed proof of this statement.
In addition, f : X → R ∪ {+∞} is a CSD-consistent and positively homogeneous functional if and only there exists a family of CSD-consistent and sublinear functionals ( fγ ) γ∈Γ such that f can be represented as in Equation (3.2).
Proof.This can be proved similarly as in the proof of Theorem 6, recalling that the condition E(g(X)) ≥ E(g(Y )) for any convex (not necessarily increasing) function g : for any Z ∈ Γ := Dom(f ).Now, the proof follows verbatim from the proof of Theorem 6 considering fZ (X) : In the following proposition, we show that by requiring cash-additivity, we can obtain a result akin to Theorem 4 in [8], but within the broader space X .Proposition 8.A functional ρ : X → R ∪ {+∞} is an SSD-consistent, star-shaped and cash-additive risk measure if and only if there exist a set of indexes Γ and a family (ρ γ ) γ∈Γ of SSD-consistent, convex and cash-additive risk measures ργ : X → R ∪ {+∞} such that ργ (0) = ρ(0) for any γ ∈ Γ and Furthermore, ρ is SSD-consistent, positively homogeneous and cash-additive if and only if each element of the family (ρ γ ) γ∈Γ is SSD-consistent, sublinear and cash-additive.
Proof.We establish the implication that if ρ is SSD-consistent, star-shaped and cashadditive, then there exists a family of SSD-consistent, convex and cash-additive risk measures whose pointwise minimum is ρ.The converse implication follows analogously to the proof of Theorem 6.We assume ρ(0) = 0 for brevity.Let us define for any Z ∈ Γ := Dom(ρ) the functional: Let us observe that if Z is constant, then the initial case in the definition of ρ Z cannot occur.Indeed, when Z is constant, ES β (X) = ES β (Z + c) holds for all β ∈ [0, 1], and utilizing the cash-additivity of ES β , we deduce By invoking the first part of the proof of Theorem 6, we conclude that X = Z + c a.s., implying that X must be constant.Consequently, either X is constant and . By cash-additivity and positive homogeneity of . Hence, the previous considerations yield Z = c1−c2 α1−α2 a.s., i.e., Z is constant, which is a contradiction.A similar conclusion holds if we suppose by contradiction α 1 = α 2 and c 1 = c 2 or α 1 = α 2 and c 1 = c 2 .Furthermore, the cash-additivity of ρ Z becomes evident.In particular, for any This relation also holds when ρ Z (X) = +∞.Now, we proceed to demonstrate that ρ Z (X) ≥ ρ(X) for all X ∈ X , and ρ X (X) = ρ(X) for X ∈ Γ.Let us consider a non-constant X ∼ 2 αZ + c for some α ∈ (0, 1] and c ∈ R (the case where X = c is clear due to the cash-additivity of ρ).In this setting, ρ Z (X) = αρ(Z) + c = αρ(Z + c/α) ≥ ρ(αZ + c) = ρ(X), utilizing the cash-additivity and star-shapedness of ρ.Moreover, for X ∈ Γ, we can set Z = X, yielding ρ X (X) = ρ(X).Thus, for any X ∈ X , we have ρ(X) = min Z∈Γ ρ Z (X).We observe that ρ Z is law-invariant for any Z ∈ Γ.To see this, let us consider X ∼ X ′ , thus X ∼ 2 αZ + c if and only if We can build a family of SSD-consistent, star-shaped and cash-additive risk measures (ρ Z ) Z∈Γ by introducing ρZ (X) := inf{ρ Z (Y ) : Y 2 X}.In this case, the property of cash-additivity remains intact, as shown by where the last equality leverages the cash-additivity of ρ Z and the definition of ρZ .Additionally, similarly to Theorem 6, we establish ρ ≤ ρZ ≤ ρ Z , and also convexity of ρZ can be checked similarly.As a result, the thesis is obtained.The thesis concerning positively homogeneous risk measures is obtained by considering the family of sublinear functionals (ρ Z ) Z∈Γ defined by: and the corresponding family of SSD-consistent functionals (ρ Z ) Z∈Γ given by: ρZ (X) := inf{ρ Z (Y ) : Y 2 X}.

✷
The following corollary extends our finding to CSD-consistent and star-shaped functionals, which are not necessarily monotone.Corollary 9. f : X → R ∪ {+∞} is a CSD-consistent, star-shaped and cash-additive functional if and only if there exist a set of indexes Γ and a family (f γ ) γ∈Γ of CSD-consistent, convex and cash-additive functionals f γ : Furthermore, f is a CSD-consistent, positively homogeneous and cash-additive functional if and only if each element of the family (f γ ) γ∈Γ is CSD-consistent, sublinear and cash-additive.
Proof.The proof follows verbatim from the proofs of Corollary 7 and Proposition 8. ✷

Representations à la Kusuoka
In this section, we first present quantile-based representations for star-shaped functionals that satisfy SSD-or CSD-consistency.That is, we extend the representations established in [9,21,25] to our setting.We assume that X possesses a rearrangement-invariant structure, i.e., X is a solid lattice with a law-invariant and complete lattice norm (for further details on rearrangement invariant space cf.[11]).This assumption is satisfied by commonly used spaces such as Orlicz spaces, Orlicz hearts and Marcinkiewicz spaces.As emphasized in Section 2, within this framework, the concepts of σ(X , X * )-lower semicontinuity and the Fatou property are equivalent for any law-invariant convex functional f : X → R ∪ {+∞}.
Additionally, when f is also monotonic, continuity from below is equivalent to the previous two properties (see Proposition 2.5 in [3] for further details).We are ready to state the main result of this section.
Proposition 10.Let X be a rearrangement-invariant space.A functional f : X → R ∪ {+∞} is CSD-consistent and star-shaped if and only if it can be represented as: where ( fγ ) γ∈Γ is the family of functionals provided in Corollary 7, and f * γ is given by: In addition, if ρ : X → R ∪ {+∞} is an SSD-consistent and star-shaped risk measure, then the supremum in Equation (4.3) can be taken over L ∞ + , obtaining: Here, (ρ γ ) γ∈Γ is the family of risk measures provided in Theorem 6, and Finally, f : X → R ∪ {+∞} is a CSD-consistent and positively homogeneous functional if and only if the following representation holds: where Γ is a set of indexes and M γ ⊆ L ∞ for each γ ∈ Γ.If f is also monotone (thus, it is a risk measure), then the family of sets (M γ ) γ∈Γ can be chosen such that M γ ⊆ L ∞ + .
Proof.The 'if' part is straightforward.Let us prove the 'only if' implication.

The CSD-consistent and star-shaped case
According to Proposition 5.1 in [3], if fγ is law-invariant, convex and σ(X , X * )-lower semicontinuous, it admits the representation: valid for any γ ∈ Γ.Here, f * γ (Y ) is given by: Thus, it suffices to establish that fγ is σ(X , X * )-lower semicontinuous for any γ ∈ Γ, given that CSD-consistency and convexity of fγ have already been shown in Corollary 7, recalling that CSD-consistency implies law-invariance (see Remark 5).
Equivalent expression for fZ : Referring back to Corollary 7 and considering the explicit form of the law-invariant convex functional fγ , where γ := Z ∈ Dom(f ), involved in the representation of f (assuming for brevity that f (0) = 0), we can express fZ as follows: We also recall the definition of (f Z ) Z∈Γ : Note that for any fixed X ∈ X the infimum in the definition of fZ (X) is always attained as soon as A X = ∅.Indeed, by definition of the infimum, there exist a sequence of real numbers (α j ) j∈N ⊆ [0, 1] and a sequence of random variables (Y j ) j∈N verifying Y j ∼ c α j Z and Y j c X for all j ∈ N such that lim j→∞ α j f (Z) = inf α∈AX {αf (Z)}.We assume f (Z) = 0, otherwise the statement is trivial, once observed that any α ∈ A X is a minimizer in this case.In particular, we have that We want to prove that ᾱ := lim j→∞ α j is actually a minimizer.Let us consider the random variable Ȳ := ᾱZ.We need to check that Ȳ c X and Ȳ ∼ c ᾱZ.The second condition is true by construction while the first condition follows from positive homogeneity of Expected Shortfall: ES β (Y j ) = α j ES β (Z) ≥ ES β (X) for all j ∈ N, and letting j → ∞ we obtain thus the thesis follows.
Lebesgue property of ES β : We claim that ES β possesses the Lebesgue property for any β ∈ [0, 1).Based on Proposition 2.35 in [33], for any X, Y ∈ L 1 it follows that: Given the hypotheses, sup n∈N , and X n → X a.s., the dominated convergence theorem implies X n → X in L 1 .Consequently, σ(X , X * )-lower semicontinuity of fZ : As X is a rearrangement-invariant space, demonstrating the Fatou property of fγ for any γ ∈ Γ is sufficient.This property requires showing that for a sequence (X n ) n∈N ⊆ X and X ∈ X with X n → X a.s. and sup n∈N |X n | ∈ X , it holds that lim inf n∈N fγ (X n ) ≥ fγ (X).Based on the preceding argument and assuming that fZ (X) = +∞, it is evident that lim inf n∈N f (X n ) = +∞ only if there are infinitely many n ∈ N for which there exists This allows focusing on such sequences.We need to verify that lim inf n→∞ fZ (X n ) ≥ fZ (X).It holds that: where Ỹ := αZ with α := lim inf n→∞ α n .The last equality results from observing that α = lim inf n→∞ α n ∈ [0, 1] due to the fact that α n ∈ [0, 1] for all n ∈ N, and in accordance with the definition of f Z .If we show that Ỹ c X (thus, α ∈ A X ), then we can conclude f Z ( Ỹ ) ≥ inf α∈AX αf (Z) = fZ (X), leading to lim inf n→∞ fZ (X n ) ≥ fZ (X).To establish this, we need to verify ES β ( Ỹ ) ≥ ES β (X) and E( Ỹ ) = E(X).We have: where the inequality is due to Y n c X n and the last equality holds by the Lebesgue property of ES β .Thus, ES β ( Ỹ ) ≥ ES β (X) for any β ∈ [0, 1).Analogously, by the dominated convergence theorem, we have By a similar argument involving contradiction, it can be established that when fγ (X) = +∞, lim inf n→∞ f (X n ) = +∞, hence fZ verifies the Fatou property, yielding the σ(X , X * )-lower semicontinuity.

The SSD-consistent and star-shaped case
Let us consider an SSD-consistent and star-shaped risk measure ρ : X → R ∪ {+∞}.For brevity we suppose ρ(0) = 0. Our goal is to establish the σ(X , X * )-lower semicontinuity of ργ for any γ ∈ Γ. Importantly, the SSD-consistency of ρ inherently includes both monotonicity and law-invariance.Consequently, we can apply Proposition 5.1 as presented in [3] once more to derive the desired dual representation for ργ .
Equivalent expression for ρZ : By Theorem 6, we know that ργ takes the following form, with γ which can be written as: where Note that for any fixed X ∈ X the infimum in the definition of ρZ (X) is always attained as soon as A X = ∅.The proof of this statement is analogous to the case of CSD-consistency.σ(X , X * )-lower semicontinuity of ρZ : Under the monotonicity assumption, σ(X , X * )lower semicontinuity is equivalent to continuity from below.Let (X n ) n∈N ⊆ X be a sequence such that X n ↑ X ∈ X a.s.; then we want to show that ργ (X n ) → ργ (X) for any γ ∈ Γ.By monotonicity, it is enough to prove that lim inf n→∞ ργ (X n ) ≥ ργ (X) for any γ ∈ Γ.By hypotheses we have X n ↑ X, hence the continuity from below of Expected Shortfall ensures that ES β (X) = lim n→∞ ES β (X n ).We start assuming ρZ (X) < +∞.Once again, we can consider (X n ) n∈N such that for each n ∈ N there exists Y n 2 X n verifying Y n ∼ 2 α n Z with α n ∈ [0, 1] attaining the minimum of the set A Xn .Thus, for any β ∈ [0, 1), with α := lim inf n→∞ α n , where the inequality follows from the SSD-consistency of ES β and the relation Y n 2 X n for all n ∈ N. The above chain of inequalities, the definition of ρZ , the SSD-consistency of ρZ and the relation ρZ Similarly, if we assume ρ Z (X) = +∞ it can be proved that lim inf n→∞ ρ Z (X n ) = +∞, thus the thesis follows.
The positively homogeneous case: In the case of positive homogeneity, we can similarly deduce the σ(X , X * )-lower semicontinuity of ρZ and fZ for SSD-and CSD-consistent functionals, respectively.The key difference is that in this setting, the parameter α falls within the interval α ∈ [0, +∞) rather than the previously considered α ∈ [0, 1].✷ The following corollary shows that we are also able to characterize cash-additive and SSD-consistent star-shaped risk measures through a representation à la Kusuoka, as robustification of Expected Shortfall.Notably, these findings appear to be unprecedented, even when considering the setting in which X = L ∞ .
Corollary 11.Let X be a rearrangement-invariant space.A risk measure ρ : X → R ∪ {+∞} is SSD-consistent, star-shaped and cash-additive if and only if ρ admits the representation: where P((0, 1]) is the set of probability measures on ((0, 1], B((0, 1]), Γ is a set of indexes, (ρ γ ) γ∈Γ is the family of risk measures provided in Proposition 8 and α γ is a penalty function defined for any µ ∈ P((0, 1]) as: Proof.The 'if' part is trivial.Let us prove only the converse implication.The proof is similar to the proof of Proposition 10.For brevity we suppose ρ(0) = 0 and we use the same notation as in Proposition 10.We recall that for any γ := Z ∈ Dom(ρ) the functional ρZ is given by ρZ (X) := inf{ρ Z (Y ) : Y 2 X}, X ∈ X .
We know that ρZ is SSD-consistent, convex, monotone and cash-additive.If we can prove that ρZ is also continuous from below, then the thesis follows from Proposition 5.12 in [3], as law-invariance is implied by SSD-consistency and σ(X , X * )-lower semicontinuity is equivalent to continuity from below for a monotone functional defined on a rearrangement invariant space.
Equivalent definition of ρZ : Once again we can write an equivalent expression for ρZ .Fixing Z ∈ Dom(ρ), for each X ∈ X we define the set We observe that when B X is non-empty the infimum in the previous equation must be finite given that ρZ (X) ≥ ρ(X) > −∞.In addition, when B X is non-empty the infimum is in fact a minimum.Indeed, arguing as in the proof of Proposition 10, there exist a sequence of real numbers (α j , c j ) j∈N ⊆ [0, 1] × R and a sequence of random variables Y j such that α j ρ(Z) + c j → inf Continuity from below of ρZ : Given X n ↑ X we want to prove that lim n→∞ ρZ (X n ) = ρZ (X).Due to monotonicity, it suffices to demonstrate lim inf n→∞ ρZ (X n ) ≥ ρZ (X).We assume ρZ (X) < +∞, which, by monotonicity, implies lim inf n→∞ ρZ (X n ) < +∞.Thus, we can consider a sequence (X n ) n∈N such that ρZ ( . By monotonicity of ρZ , we infer that ρZ (X n ) must converge to some l ∈ R (otherwise, we would have lim inf n→∞ ρZ (X n ) = +∞, which contradicts the hypothesis).Thus, also ρZ (X n k ) → l and then 1), where the inequality follows from the SSD-consistency of ES β and the relation Y n k 2 X n k for all k ∈ N. The above chain of inequalities, the definition of ρZ and the SSD-consistency of ρZ yield: If ρZ (X) = +∞ we can proceed by contradiction, proving that also lim inf n→∞ ρZ (X n ) = +∞.The thesis follows.✷ 5 Law-Invariant Star-Shaped Functionals as Robustification of Value-at-Risk While we have proved that every SSD-consistent (resp.CSD-consistent) star-shaped functional arises as the minimum of SSD-consistent (resp.CSD-consistent) convex functionals, we aim to characterize the bigger family of law-invariant and star-shaped functionals in terms of Value-at-Risk.Indeed, as explained in the previous sections, not every law-invariant shar-shaped functional can be described as the minimum of a family of law-invariant convex functionals, having the Value-at-Risk as a classical counter-example to this statement.See also Section 7 in [8] and the references therein for a thorough discussion on this topic.More specifically, we are seeking to establish representations in a similar spirit to those provided in Theorem 5 of [8] and Proposition A.4 of [24].In [8], law-invariant star-shaped monetary risk measures appear as a robustification of Value-at-Risk, where the generalized scenarios are represented by a penalty function dependent on the β-level of V aR β .A similar representation is found in [24] for cash-subadditive risk measures, even when star-shapedness is not a requirement in this case.In the following proposition we show that this representation can be extended to the more general setting in which cash-additivity (or cash-subadditivity) and monotonicity are discarded, and X is a (much) more general space than L ∞ .The following results are presented under the assumption of normalization.It is important to note that the proof strategy remains applicable even for a non-normalized functional.
Proof.We prove the statement in the star-shapedness case, the positively homogeneous case follows similarly.We assume X ∈ Dom(f ), otherwise the statements are trivial. 7e start by proving the 'only if' part.It is worth noting that Γ X is well defined, given that if X ∼ 1 αZ, with α ∈ [0, 1] or α ∈ [0, +∞), then α is unique, as we have shown in the first part of the proof of Theorem 6.

Properties of γ X
Z : Let us fix λ ∈ (0, 1].Due to the star-shaped property of ρ and the set inclusion λΓ X ⊆ Γ λX for any λ ∈ (0, 1], it follows that for any Z ∈ Γ X and β ∈ (0, 1): In addition, we have: where the second equality follows from increasing monotonicity of V aR β w.r.t.β, while the last equality is due to normalization.Thus, max Z∈Γ0 γ 0 Z (0 + ) = 0. Now we are ready to prove the converse implication.We want to show that is normalized, star-shaped and law-invariant.
Normalization: We have that: Star-shapedness: Let us consider λ ∈ (0, 1] and X ∈ X .We have the following inequalities: where the inequalities follow from γ λX λZ ≥ λγ X Z and λΓ X ⊆ Γ λX .Law-invariance: This assertion follows from the law-invariance of VaR and the relation Γ X = Γ X ′ as long as X ∼ X ′ .Thus, we can express it as follows:

✷
Corollary 13.A functional f : X → R ∪ {+∞} is normalized, law-invariant, star-shaped and cash-additive if and only it admits the representation: where for each X ∈ X , Γ X is a set of random variables such that λΓ X ⊆ Γ λX for any λ ∈ (0, 1], Γ X = Γ X ′ whenever X ∼ X ′ and Γ X+m = Γ X for all m ∈ R. In addition, γ X Z : (0, 1) → R ∪ {−∞} is an increasing function such that max Z∈Γ0 γ 0 Z (0 + ) = 0.Moreover, for any Z ∈ Γ X , λ ∈ (0, 1] it holds that λγ X Z = γ λX λZ and for any m ∈ R it follows that γ X+m Z = γ X Z .Furthermore, for each X ∈ X , the set Γ X takes the following form: with that convention that if Γ X = ∅ then γ X Z = −∞.With the same notation as above, f : X → R ∪ {+∞} is law-invariant, positively homogeneous and cash-additive if and only f can be represented as in Equation (5.4), with λ ∈ [0, +∞) and λΓ X = Γ λX .In addition, the set Γ X can be defined for any α ∈ [0, +∞).
Proof.We prove the statement in the star-shapedness case, the positively homogeneous case follows similarly.We commence by establishing the 'only if' part.It is important to note that for any Z ∈ Γ X , the function γ X Z is well-defined.Indeed, if X is non-constant and X ∼ 1 αZ + c with (α, c) ∈ [0, 1] × R, then, as demonstrated in the first part of the proof of Proposition 8, (α, c) is a unique pair of values.Additionally, if X is constant, we have γ X Z ≡ 0, for any Z ∈ Γ X = {Z ∈ Dom(f ) : f (Z) ≤ 0}.Henceforth, we assume X ∈ Dom(f ) with X non-constant, otherwise the statements are trivial.
Properties of γ X Z : Let us fix λ ∈ (0, 1].By the inclusion λΓ X ⊆ Γ λX for any λ ∈ (0, 1], it follows for any Z ∈ Γ X and β ∈ (0, 1) that: where last equality follows from λZ ∈ λΓ X ⊆ Γ λX with λX = αλZ + λc.The equality max Z∈Γ0 γ 0 Z (0 + ) = 0 follows as in Proposition 12. Finally, consider m ∈ R. We have: where α ∈ [0, 1] is the same in the representation X ∼ 1 αZ + c, whether Z is regarded as an element of Γ X or as an element of Γ X+m , as demonstrated earlier in the proof.Now we are ready to prove the converse implication.We want to show that is normalized, star-shaped, law-invariant and cash-additive.Normalization and law-invariance follows as in the proof of Proposition 12.
Cash-additivity: Let m ∈ R. We have: The last equality follows from Γ X+m = Γ X and γ X+m Z = γ X Z .✷ In the subsequent lemma, we introduce a method for representing law-invariant risk measures using Value-at-Risk as a key component.Subsequently, we tailor our findings to derive a novel representation that adheres to the axioms of star-shapedness and cashadditivity, as expounded in Proposition 15.Lemma 14.Let ρ : X → R ∪ {+∞} be a normalized, law-invariant risk measure.Then it admits the representation: Here, and for each fixed X ∈ X and Z ∈ Γ X , we define and the real function γ X Z : (0, 1) → R ∪ {−∞} : where ᾱ = sup A X Z , with that convention that if Γ X = ∅ then γ X Z = −∞.Proof.We observe that γ X Z is well-defined for any X ∈ X and Z ∈ Dom(ρ), as we consider ᾱ = sup A X Z , which is clearly unique.Proceeding analogously as in the proof of Proposition A.4 in [24], it can be verified that any law-invariant risk measure can be represented as: We fix X ∈ Dom(ρ), otherwise the statements are trivial.If Z ∈ Γ X , then for any α ∈ A X Z it holds that αZ 1 X, implying that also Ȳ := ᾱZ verifies ᾱZ 1 X.Indeed, by the properties of the supremum, there exists a sequence (α n ) n∈N ⊆ A X Z such that α n → ᾱ.Thus, for any n ∈ N it results that Y n 1 X and Y n ∼ 1 α n Z. Letting n → ∞, we have that Ȳ := ᾱZ clearly satisfies Ȳ ∼ 1 ᾱZ and Ȳ 1 X, so the supremum is indeed a maximum.Hence, for any Z ∈ Γ X we have ρ(X) ≤ ρ(ᾱZ + sup β∈(0,1) (V aR β (X) − ᾱV aR β (Z)). (5.5) Taking the minimum over Z ∈ Γ X on both members of Equation (5.5) we have: where the last inequality follows by taking Z = X, given that ᾱ = 1 attains the supremum of the set A X X .✷ The following proposition demonstrates that our results genuinely extend those obtained in Theorem 5 of [8].Not only do we broaden the setting to encompass the general space X , but we also establish that the minimum can be taken over a set strictly smaller than the acceptance set of ρ, denoted as Bρ := {Z ∈ X : ρ(Z) ≤ 0}.More specifically, the corollary underscores that the minimum can be computed over a set Γ X ⊆ Bρ, which is contingent on the choice of X ∈ X .Proposition 15.A functional ρ : X → R ∪ {+∞} is a normalized, law-invariant, starshaped and cash-additive risk measure if and only it admits the representation: The properties of Γ X properties of γ X Z can be verified as in the proof of Corollary 13.Now we are ready to prove the converse implication.We want to show that is monotone, normalized, star-shaped, law-invariant and cash-additive.Normalization and star-shapedness can be established following the proof of Proposition 12, while the proof of cash-additivity closely mirrors the proof provided in Corollary 13.
Monotonicity: This property is obvious once observed that V aR β is monotone and Γ X1 ⊆ Γ X2 as soon as X 1 ≥ X 2 .
Law-invariance: Let us consider X, X ′ ∈ X such that X ∼ X ′ .We need to verify that Γ X = Γ X ′ .Given that X ∼ X ′ , both X 1 X ′ and X ′ 1 X apply, resulting in Γ X ⊆ Γ X ′ and Γ X ′ ⊆ Γ X , which in turns lead to Γ X = Γ X ′ .This equality, along with the law-invariance of V aR β , implies that ρ is law-invariant.✷

Illustrative Examples
While the prime motivation for this paper comes directly from [2,3,4,26,27], whence the aim of establishing novel characterization results for law-invariant return and star-shaped risk measures, we show in this section that such risk measures may also arise naturally from more classical settings.Indeed, we provide three examples to illustrate the inherent starshaped nature of certain risk measures, also when cash-additivity or cash-subadditivity are not preserved.Specifically, the examples show that during the recent extended period of negative interest rates induced by central banks' monetary policies, investors may naturally comply with non-cash-(sub)additive risk measures that remain star-shaped or even positively homogeneous.
Example 16.As shown in [16], in the absence of a zero coupon bond, the presence of ambiguity with respect to interest rates naturally leads to a relaxation of the axiom of cashadditivity, with cash-subadditivity being assumed instead.Over the last decade, the EONIA index, which tracks unsecured lending transactions in the interbank market, has assumed negative values.Consider a bank, or more generally, a financial institution, assessing the risk of an asset X T at the present time t = 0, where T > 0 denotes the asset's time to maturity.Suppose the institution evaluates the risk of X T using a spot risk measure ρ 0 defined on the discounted price D T X T , where D T represents a discount factor (for more details, see Section 2.4 in [16]).In the context of negative interest rates, the discount factor D T can take values greater than 1.Therefore, if the interest rate is subject to ambiguity, fluctuating between two constants 0 ≤ D b ≤ D u ≤ C, with C > 1, an ambiguity-adverse investor may select the risk measure as follows: ρ(X T ) = sup Even when ρ 0 is a monetary risk measure, ρ can be non-cash-(sub)additive, given that D T can take values greater than 1.
A classic industry measure of risk is the Value-at-Risk.Therefore, we can consider ρ 0 = V aR β .In this setting, the resulting risk measure is given by: ρ(X T ) = sup This risk measure is neither convex nor cash-(sub)additive, but it is positively homogeneous (thus star-shaped) and law-invariant, inheriting these properties from V aR β .In particular, the thesis of Proposition 12 concerning positively homogeneous functionals can be applied to represent ρ as in Equation (5.4).
Example 17.In the same context as in Example 16, we consider a generalization of V aR β that is closely connected to the concept of Λ-VaR, as shown in Theorem 3.1 of [24].For a thorough discussion of Λ-VaR risk measures, we refer to [19].Let us fix x ∈ R and define: Since the maximum operation preserves star-shapedness, and both V aR β and the trivial risk measure ρ(X) = x for any X ∈ X are star-shaped, the resulting risk measure ρ is lawinvariant and (genuinely) star-shaped.However, in this case, positive homogeneity does not hold in general.Once again, ρ is neither convex nor cash-(sub)additive.The parameter x can be interpreted as follows: it represents a barrier below which the risk evaluation corresponding to the asset X T cannot fall, possibly due to market frictions or other constraints.Consequently, the investor is required to retain the larger of two amounts -the assessment of the risk linked to the asset X T as determined by the Value-at-Risk and the minimum threshold x ∈ R. The fixed amount x remains constant and is unrelated to the value of X T , but instead may be influenced by external factors.In this case, it is also possible to derive a representation of ρ akin to the one presented in Equation (5.4).It should be noted that ρ is not normalized.However, the representation results remain valid, as stated at the outset of Section 5.
Example 18.Consider a firm that wishes to establish an insurance contract to cover the risk associated with an asset X T .Let A represent the set of insurance companies accessible to the firm.Suppose each insurer has its own spot monetary, convex, and normalized risk measure ρ a 0 with a ∈ A to assess the risk of X T .Furthermore, suppose the cost of each contract is equal to the risk assessment made by the insurer through its risk measure.Thus, for each asset X T , the firm will choose to pay the minimum amount: ρ(X T ) := min a∈A ρ a 0 (D T X T ).
Here, the resulting risk measure is star-shaped but not convex in general.Furthermore, as observed in the examples above, ρ lacks cash-(sub)additivity.If the risk measures employed by the insurance companies are SSD-consistent, such as in the case of Expected Shortfall, entropic risk measures, and risk measures generated from power or exponential utilities, the resulting risk measure ρ inherits this property.Under this circumstance, if X is a rearrangement-invariant space, we can represent ρ as in Equation (4.3).
DT ∈X {V aR β (D T X T ) : D b ≤ D T ≤ D u }.
ρ(X T ) = sup DT ∈X {V aR β (D T X T ) ∨ x : D b ≤ D T ≤ D u } = sup DT ∈X {V aR β (D T X T ) : D b ≤ D T ≤ D u } ∨ x.