Connection between higher order measures of risk and stochastic dominance

Pichler, Alois Article — Published Version Connection between higher order measures of risk and stochastic dominance Computational Management Science Provided in Cooperation with: Springer Nature Suggested Citation: Pichler, Alois (2024) : Connection between higher order measures of risk and stochastic dominance, Computational Management Science, ISSN 1619-6988, Springer, Berlin, Heidelberg, Vol. 21, Iss. 2, https://doi.org/10.1007/s10287-024-00523-0 This Version is available at: https://hdl.handle.net/10419/315084 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Vol.:(0123456789) Computational Management Science (2024) 21:41 https://doi.org/10.1007/s10287-024-00523-0 ORIGINAL PAPER Connection betweenhigher order measures ofrisk andstochastic dominance AloisPichler1 Received: 8 February 2024 / Accepted: 21 August 2024 / Published online: 5 September 2024 © The Author(s) 2024 Abstract Higher order risk measures are stochastic optimization problems by design, and for this reason they enjoy valuable properties in optimization under uncertainties. They nicely integrate with stochastic optimization problems, as has been observed by the intriguing concept of the risk quadrangles, for example. Stochastic dominance is a binary relation for random variables to compare random outcomes. It is demonstrated that the concepts of higher order risk measures and stochastic dominance are equivalent, they can be employed to characterize the other. The paper explores these relations and connects stochastic orders, higher order risk measures and the risk quadrangle. Expectiles are employed to exemplify the relations obtained. Keywords Higher order risk measure· Higher order stochastic dominance· Risk quadrangle Mathematics Subject Classification 62G05· 62G08· 62G20 1 Introduction Risk measures are considered in various disciplines to assess and quantify risk. Similarly to assigning a premium to an insurance contract with random losses after appraising its risk, risk measures assign a number to a random variable, which itself has stochastic outcomes. This paper focuses on higher order risk measures, as these risk measures naturally combine with stochastic optimization problems or in ‘learning’ objectives, as they are the result of optimization problems. In addition, these risk measures relate to the risk quadrangle. The paper derives explicit representations of higher order risk measures for general, elementary risk measures in a first main result. These characterizations are employed * Alois Pichler [email protected]hemnitz.de 1 Technische Universität Chemnitz, Faculty ofMathematics, 90126Chemnitz, Germany A.Pichler 41 Page 2 of 28 to characterize stochastic dominance relations, which are built on general norms. The second main result is a verification theorem. This is a characterization of higher order stochastic dominance relations, which is numerically tractable. For the norm in Lebesgue spaces, stochastic dominance relations have been considered for example in Dupačová and Kopa (2014), Kopa etal. (2016, 2023), Post and Kopa (2017) and Consigli etal. (2023), in portfolio optimization involving commodities (cf. Frydenberg etal. (2019)), and by Dentcheva and Martinez (2012) and Maggioni and Pflug (2016, 2019) in a multistage setting. The paper employs the characterizations obtained to establish relations for general norms. A comparison of these methods is given in Gutjahr and Pichler (2013). The paper illustrates these connections for expectiles (Bellini etal. 2016; Bellini and Caperdoni 2007) and adds a comparison with other risk measures. Outline of the paper The following Sect. 2 recalls the mathematical framework for higher order risk measures. Section3 addresses the higher order risk measure associated with the spectral risks, as these risk measures constitute an elementary building block for general risk measures. This section develops the first main result, which is an explicit representation of a spectral risk’s higher order risk measure. As a special case, the subsequent Sect. 4 links and relates stochastic dominance and higher order risk measures. This section presents the second main result, which allows verifying a stochastic dominance relation by involving only finitely many risk levels. The final Sect.5 addresses the expectile and establishes the relations of the preceding sections for this specific risk measure. Section6 concludes. 2 Mathematical framework Higher order risk measures are a special instance of risk measures, often also termed risk functionals. To introduce and recall their main properties we consider a space Y of ℝ -valued random variables on a probability space with measureP containing at least all bounded random variables, that is, L∞(P)⊆Y . A risk measure then satisfies the following axioms, originally introduced by Artzner etal. (1999). Definition 2.1 (Risk functional) Let Y be a space of ℝ -valued random variables on a probability space (Ω,Σ,P) . A mapping R∶Y→ℝ is (i) monotone, if R(X)≤R(Y) , provided that X≤Y almost everywhere; (ii) positively homogeneous if R(𝜆Y)=𝜆R(Y) for all 𝜆>0 ; (iii) translation equivariant, if R(c+Y)=c+R(Y) for all c∈ℝ ; (iv) subadditive, if R(X+Y)≤R(X)+R(Y) for all X and Y∈Y . A mapping satisfying(i)–(iv) is called a risk functional, or a risk measure. The risk quadrangle (cf. Rockafellar and Uryasev (2013)) relates risk measures with the measure of regret by Connection betweenhigher order measures ofrisk andstochastic… Page 3 of 28 41 where V is called regret function. Equation(2.1) was first introduced for the conditional value-at-risk in Rockafellar and Uryasev (2000). For the expectation type function, i.e., V(X)= 𝔼 v(X) , the relationship(2.1) is studied in Ben-Tal and Teboulle (2007), where V was called optimized certainty equivalent; also, Krokhmal (2007) study the relation(2.1). It follows from relation(2.1) that R —if given as in(2.1)—is translation equivariant, i.e, R satisfies R(Y+c)=c+R(Y) for any c∈ℝ (cf.(iii) above). In an economic interpretation, the amountc in(2.1) corresponds to an amount of cash spent today, while the remaining quantity Y−c is invested and consumed later, thus subject to V . The risk functional R is positively homogeneous, if the regret function V is positively homogeneous. If V is not positively homogeneous, then one may consider the positively homogeneous envelope where  𝛽≥0 is a risk aversion coefficient. The combined functional is positively homogeneous and translation equivariant (cf. (ii) and (iii)). The 𝜑 -divergence risk measure is an explicit example of a risk measure, which is defined exactly as(2.2), cf. Dommel and Pichler (2021). The paper suggests a regret for a higher-order risk starting from a given risk R . To this end consider a space Y⊂L1(P) endowed with norm ‖ ⋅ ‖ . We shall assume the norm to be monotone, that is, ‖X‖≤‖Y‖ provided that 0≤X≤Y almost everywhere. We associate the following family of risk measure with a given norm. Definition 2.2 (Higher order risk measure) Let ‖⋅‖ be a monotone norm on Y⊂L1(P) with ‖1‖=1 , where 1(⋅)=1 is the identically one function on Y . The higher order risk measure at risk level 𝛽∈[0, 1) associated with the norm ‖⋅‖ is where 𝛽∈[0, 1) is the risk aversion coefficient and x+∶= max(0, x) . We shall also omit the superscript and write R𝛽 instead of R‖⋅‖ 𝛽 in case the norm is unambiguous given the context. We shall demonstrate first that the higher order risk measure is well-defined for any 𝛽≥0 . (2.1) R(Y)=inf c∈ℝ c+V(Y−c), V  𝛽(Y)=inf t>0 t (  𝛽+V (Y t)), (2.2) R 𝛽 (Y)=inf c∈ℝ c+V 𝛽 (Y−c) =inf t>0 q∈ ℝ t( 𝛽+q+V(Y t−q )) (2.3) R‖ ⋅ ‖ 𝛽(Y)=inf t∈ℝ t+ 1 1−𝛽‖ (Y−t)+ ‖, A.Pichler 41 Page 4 of 28 Proposition 2.3 Let (Y,‖⋅‖) be a normed space of random variables. For the functional R𝛽 defined in(2.3) it holds that so that R𝛽(⋅) is indeed well-defined on (Y,‖⋅‖) for every 𝛽∈[0, 1) . Proof The upper bound follows trivially from the definition by choosing t=0 in the defining equation(2.3). For t≤0 , it holds that −t=−Y+(Y−t)≤−Y+(Y−t)+ . It follows from the triangle inequality that −t≤‖Y‖+‖(Y−t)+‖ and thus To establish the relation also for t≥0 , we start by observing the following monotonicity property of the objective in(2.3) in addition: for Δt≥0 , it follows from the reverse triangle inequality that where we have used that 0≤Y+−(Y−Δt)+≤Δt together with monotonicity of the norm. ReplacingY by Y−t in the latter expression gives that is, the function t↦t+‖(Y−t)+‖ is non-decreasing, which finally establishes that The lower bound in(2.4) thus follows from the latter inequality, as R0(Y)≤R𝛽(Y) for any 𝛽≥0 . ◻ Example 2.4 For Lebesgue spaces Lp(P) and norm ‖ Y ‖p ∶=( 𝔼 � Y �p ) 1∕p , p≥1 , the higher order risk measure has been introduced in Krokhmal (2007) and studied in Dentcheva etal. (2010). For the norm ‖⋅‖∞ , the higher order risk measure is indeed, it follows from(2.3) that the subgradient of the convex function in the latter expression at t= ess sup Y . The infimum in(2.3) is attained at t= ess sup Y , and thus(2.5). (2.4) −‖ Y ‖≤ R𝛽(Y) ≤1 1−𝛽‖ Y ‖, −‖Y‖≤t+‖(Y−t)+‖for all t≤0. ‖Y+‖−‖(Y−Δt)+‖≤‖Y+−(Y−Δt)+‖≤‖Δt1‖=Δt, t+‖(Y−t)+‖≤t+Δt+‖(Y−(t+Δt))+‖; −‖Y‖≤t+‖(Y−t)+‖for all t∈ℝ. (2.5) R‖⋅‖ ∞ 𝛽 (Y) = ess sup Y,𝛽> 0; 0 ∈ � 1−1 1−𝛽,1 � =𝜕t � t+1 1−𝛽 ‖ (Y−t)+ ‖ ∞ ������t=ess sup Y , Connection betweenhigher order measures ofrisk andstochastic… Page 5 of 28 41 Lemma 2.5 R𝛽(⋅) is a risk functional, provided that the norm is monotone. Further, R𝛽 is Lipschitz continuous with respect to the norm, the Lipschitz constant is 1 1−𝛽 . Proof The assertions(ii)–(iv) in Definition2.1 are straight forward to verify; to verify(i) it is indispensable to assume that the norm is monotone. As for continuity, it follows from subadditivity together with (2.4) that R 𝛽(Y)−R𝛽(Z) ≤ R𝛽(Y−Z) ≤1 1−𝛽‖ Y−Z ‖ , and � R𝛽(Y)−R𝛽(Z) �≤1 1−𝛽‖ Y−Z ‖ after interchanging the roles ofY andZ. Hence, the assertion. ◻ Note that the higher order risk measure as defined in(2.3) defines a risk functional based on a norm. In contrast to this construction, a risk functional R defines a norm via and a Banach space with Y = { Y∈L 1 ∶R( | Y | )<∞ } (cf. Pichler (2013)). Its natural dual norm for Z∈Z∶=Y∗ is The following relationship allows defining a regret functional to connect a risk functional R with the higher-order risk quadrangle. Proposition 2.6 (Duality) Let R be a risk functional with associated norm ‖⋅‖ and dual norm ‖⋅‖∗ . For the higher order risk functional it holds that where 𝛽∈[0, 1) . Remark 2.7 By the interconnecting formula(2.1), the higher order risk functional R‖⋅‖ 𝛽 associated with the norm ‖⋅‖ is the regret function V‖⋅‖ 𝛽(⋅)∶= 1 1−𝛽‖ (⋅)+ ‖ . Proof It holds by the Hahn–Banach theorem and as (Y−t)+≥0 that This establishes the first inequality ‘ ≤ ’ in(2.9) with t+(Y−t)+≥Y , as (2.6) ‖Y‖∶=R(�Y�) (2.7) ‖ Z ‖∗ ∶= sup {𝔼YZ ∶ ‖ Y ‖≤ 1} =sup { 𝔼 YZ ∶ R (�Y�)≤1}. (2.8) R 𝛽(Y)=sup � 𝔼YZ ∶Z ≥ 0, 𝔼Z=1 and ‖ Z ‖ ∗ ≤ 1 1−𝛽 � (2.9) = inf t∈ℝ t+ 1 1−𝛽‖ (Y−t)+ ‖, 1 1 −𝛽 ⋅‖(Y−t)+‖=sup ‖ Z ‖ ∗≤1 1−𝛽 𝔼Z(Y−t)+ ≥ sup 𝔼Z=1, Z≥0, ‖ Z ‖ ∗≤1 1−𝛽 𝔼Z(Y−t)+ . A.Pichler 41 Page 6 of 28 As for the converse inequality assume first that Y is bounded. Note, that so that it follows that Further, it holds that 𝔼YZ =t∗+ 𝔼 Z(Y−t∗)+ for t∗≤Y a.s. and thus thus the desired converse inequality, provided that Y is bounded; if Y is not bounded, then there is a bounded Y𝜀 with Y≤Y𝜀 ( 𝜀>0 ) and ‖Y𝜀 −Y‖<𝜀 , so that so that we may conclude that(2.9) holds for every Y∈Y . ◻ Example 2.8 (Lebesgue spaces) The dual norm of the genuine norm ‖ X ‖p ∶=( 𝔼 � X �p ) 1∕p in the Lebesgue space Lp(P) is ‖Z‖∗=( 𝔼 �Z�q)1∕q for the Hölder conjugate exponentq with 1 p + 1 q = 1 . With Proposition2.6 it follows that cf. also Pichler and Shapiro (2015) and Pichler (2017). In what follows, we shall elaborate the higher order risk measure and the associated regret function for specific risk measures, specifically the spectral risk measure. t + 1 1−𝛽 ⋅‖(Y−t)+‖ ≥ sup 𝔼Z=1 Z≥0, ‖Z‖∗≤1 1−𝛽 𝔼 � t+(Y−t)+ �Z ≥sup 𝔼Z=1 Z≥0, ‖ Z ‖ ∗≤1 1−𝛽 𝔼YZ. inf t ∈ℝ t+𝔼(Y−t)Z=𝔼YZ +inf t∈ℝ t⋅(1−𝔼Z)= { 𝔼YZ if 𝔼Z= 1, −∞ else, sup 𝔼Z=1 Z ≥0, ‖ Z ‖ ∗≤1 1−𝛽 𝔼YZ =sup Z ≥ 0, ‖ Z ‖ ∗≤1 1−𝛽 inf t∈ℝ t+𝔼(Y−t)Z. sup 𝔼Z=1, Z≥0, ‖ Z ‖ ∗≤1 1−𝛽 𝔼YZ = sup Z≥0, ‖ Z ‖ ∗≤1 1−𝛽 t∗+𝔼Z(Y−t∗)+=t∗+ 1 1−𝛽‖(Y−t∗)‖ ≥inf t∈ℝ t+ 1 1−𝛽‖(Y−t)+‖ , 𝔼Z(Y𝜀−t)+−𝜀 𝔼 Z≤ 𝔼 Z(Y−t)+≤ 𝔼 Z(Y𝜀−t)+, R‖ ⋅ ‖ p 𝛽(Y)=inf t∈ℝt+ 1 1−𝛽‖(Y−t)+‖p =sup � 𝔼YZ ∶ ‖ Z ‖ q≤1 1−𝛽,Z≥0 and 𝔼Z=1 �, Connection betweenhigher order measures ofrisk andstochastic… Page 7 of 28 41 3 Higher order spectral risk By Kusuoka’s theorem (cf. Kusuoka (2001)), every law invariant risk functional can be assembled by elementary risk functionals, each involving the average value-at-risk. The following section develops the explicit representations of the higher order risk measures associated with spectral risk measures first. The explicit representation then is extended to general risk functionals. Definition 3.1 (Spectral risk measures) The function 𝜎∶[0, 1)→ℝ is called a spectral function, if (i) 𝜎(⋅)≥0 , (ii) ∫1 0 𝜎(u)du= 1 and (iii) 𝜎(⋅) is non-decreasing. The spectral risk measure with spectral function 𝜎 is where is the value-at-risk, the generalized inverse or quantile function. The higher order risk measure of the spectral risk measure is a spectral risk measure itself. The following theorem presents the corresponding spectral function explicitly and generalizes (Pflug 2000). The result is central towards the main characterization presented in the next sections. Theorem3.2 (Higher order spectral risk) Let 𝛽∈[0, 1) be a risk level. The higher order risk functional of the risk functional R𝜎 with spectral function 𝜎(⋅) has the representation where 𝜎𝛽 is the spectral function here, u𝛽∈ℝ is the 𝛽 -quantile with respect to the density 𝜎 , that is, the solution of R 𝜎(Y)∶= ∫1 0 𝜎(u)F−1 Y(u)du , F−1 Y (u)∶= 𝖵@𝖱 u (Y)∶= inf {x∈ℝ∶P(Y ≤ x) ≥ u } (3.1) inf t ∈ℝ t+ 1 1−𝛽 R𝜎 ( (Y−t)+ ) =R𝜎𝛽(Y) , (3.2) 𝜎 𝛽(u)∶= { 0 if u<u𝛽 , 𝜎(u) 1−𝛽else; A.Pichler 41 Page 8 of 28 which is unique for 𝛽>0 . Proof We remark first that 𝜎𝛽 indeed is a spectral function, as ∫1 0𝜎𝛽(u)du=1 1−𝛽∫1 u 𝛽 𝜎(u)du=1 −𝛽 1−𝛽 = 1 by the defining property (3.3) and (ii) in Definition3.1. The quantile u𝛽 is uniquely defined for 𝛽>0 , as the function 𝜎 is non-decreasing by (iii). In what follows we shall demonstrate that the infimum in(3.1) is attained at t∗ ∶=F −1 Y (u 𝛽) . Note first that so that and Assume first that t≤t∗ . The inequality u≤FY(t) is equivalent to F−1 Y (u) ≤t (cf. vander Vaart (1998); this relation of functions FY and F−1 Y is occasionally called a Galois connection), and thus or equivalently Assume next that u𝛽≤FY(t∗) , then ∫1 FY(t ∗ ) 𝜎(u)du ≤ 1−𝛽 so that Combining the inequalities in the latter displays gives (3.3) ∫u 𝛽 0 𝜎(u)du=𝛽 , F −1 (Y−t)+ (u)= { 0 if u<FY(t) , F−1 Y (u)−telse, R 𝜎 ( (Y−t)+ ) =∫ 1 0 𝜎(u)F−1 (Y−t)+ (u)du=∫ 1 F Y (t) 𝜎(u) ( F−1 Y(u)−t ) d u (3.4) ( R𝜎)𝛽(Y)=inf t∈ℝ t+1 1−𝛽∫ 1 F Y (t) 𝜎(u) ( F−1 Y(u)−t ) du . �F Y (t∗) F Y (t) 𝜎(u) ( F−1 Y(u)−t ) du ≤0, �1 F Y (t) 𝜎(u) ( F−1 Y(u)−t ) du ≤ � 1 F Y (t∗) 𝜎(u) ( F−1 Y(u)−t ) du . t −t∗ 1 −𝛽� 1 FY(t ∗ ) 𝜎(u)du ≤ t−t∗ . (3.5) t ∗+1 1−𝛽� 1 FY(t ∗ ) 𝜎(u) ( F−1 Y(u)−t∗ ) du ≤ t+1 1−𝛽� 1 FY(t) 𝜎(u) ( F−1 Y(u)−t ) d u Connection betweenhigher order measures ofrisk andstochastic… Page 15 of 28 41 Definition 4.1 (Stochastic dominance) Let X, Y∈Y be ℝ -valued random variables in a Banach space (Y,‖⋅‖) . The random variableX is dominated by Y, denoted if If the norm is unambiguous from the context, we shall also simply write ≼ instead of ≼‖⋅‖ . The cone of random variables triggered by a single variable is convex. Lemma 4.2 (Convexity of the stochastic dominance cone) For X∈Y given, the set is convex. Proof The map y↦(t−y)+ is convex, as follows from reflecting and translating the convex function x↦x+ . Suppose that X≼Y0 and X≼Y1 . Then it follows for Y𝜆∶=(1−𝜆)Y0+𝜆Y1 , together with monotonicity of the norm and(4.1), that That is, it holds that X≼Y𝜆 and thus the assertion. ◻ 4.1 Characterization ofstochastic dominance relations Stochastic dominance relations can be fully characterized by higher order risk measures. The following theorem presents this main result, which integrates the details developed above for these risk functionals and stochastic dominance relations. Theorem4.3 (Characterization of stochastic dominance, cf. Gómez et al. (2022)) The following are equivalent: (i) X≼‖⋅‖Y , (ii) R𝛽(−X)≥R𝛽(−Y) for all 𝛽∈[0, 1) , and X≼‖⋅‖Y, (4.1) ‖(t−X)+‖≥‖(t−Y)+‖for all t∈ ℝ . {Y∈Y∶X≼Y} ‖ (t−Y𝜆)+‖ ≤� � � � (1−𝜆)(t−Y0)+𝜆(t−Y1) � + � � � ≤(1−𝜆)‖(t−Y0)+‖+𝜆‖(t−Y1)+ ‖ ≤(1−𝜆) ‖ (t−X)+ ‖ +𝜆 ‖ (t−X)+ ‖ = ‖ (t−X) +‖ . A.Pichler 41 Page 16 of 28 (iii) inf Z∈Z 𝛽 𝔼ZX ≤inf Z∈Z 𝛽 𝔼ZY for every 𝛽∈(0, 1) , where is the positive cone ( Z≥0 ) in the dual ball with radius 1 1−𝛽 ( ‖ Z ‖ ∗ ≤1 1−𝛽 ), intersected with the simplex ( 𝔼Z=1 ). Proof Suppose that X≼‖⋅‖Y , then, by definition, ‖(t−X)+‖≥‖(t−Y)+‖ for every t∈ℝ . It follows that t + 1 1−𝛽‖ (−X−t)+ ‖≥ t+ 1 1−𝛽‖ (−Y−t)+ ‖ for all t∈ℝ , and thus assertion(ii) after passing to the infimum. As for the contrary, assume that (ii) holds. To demonstrate (i) note first that q↦‖(q−X)+‖ is convex; indeed, with q𝜆∶=(1−𝜆)q0+𝜆q1 and (a+b)+≤a++b+ it holds that and thus by the triangle inequality of the norm. For q∈ℝ fixed, choose that is, the subdifferential (of the convex function 𝜂↦‖(𝜂−Y)+‖ ) evaluated at 𝜂=q , and note that 𝛼∈[0, 1] . Set 𝛽∶=1−𝛼 , and observe that so that by(2.3). Employing the definition(2.3) again and assumption(ii), it follows that or equivalently Z 𝛽∶= � Y∈Y∗∶ ‖ Z ‖ ∗ ≤ 1 1−𝛽,𝔼Z=1, Z ≥ 0 � ( q𝜆−X)+= ( (1−𝜆)(q0−X)+𝜆(q1−X) )+≤ (1−𝜆)(q0−X)+𝜆(q1−X) + ‖(q𝜆−X)‖≤(1−𝜆)⋅‖(q0−X)+‖+𝜆⋅‖(q1−X)+‖ 𝛼 ∈𝜕𝜂 ‖ (𝜂−Y)+ ‖� � �𝜂=q , 0 ∈𝜕q−q+ 1 1−𝛽‖ (q−Y)+ ‖ R 𝛽(−Y)=−q+ 1 1−𝛽‖ (q−Y)+ ‖ − q+ 1 1−𝛽‖(−X+q)+‖ ≥ R𝛽(−X) ≥R𝛽(−Y) =−q+1 1−𝛽‖ (q−Y)+ ‖, ‖(q−X)+‖≥‖(q−Y)+‖. Connection betweenhigher order measures ofrisk andstochastic… Page 17 of 28 41 The assertion(i) follows, as q∈ℝ was arbitrary; this establishes equivalence of(i) and(ii). Finally, let 𝛽∈(0, 1) . With(ii) and Proposition2.6 we have that where the infimum in both expressions is among Z ∈Z𝛽= � Z∈Z∶ ‖ Z ‖ ∗ ≤ 1 1−𝛽� , as the set Z𝛽 collects the constraints in (2.8). This establishes equivalence between(ii) and(iii). ◻ Remark 4.4 The quantity −R(− Y ) =∶ A( Y ) arising naturally in Theorem 4.3 (ii) above is often called an acceptability functional, cf. Pflug and Römisch (2007). Corollary 4.5 Suppose that then X is dominated by Y, X≼‖⋅‖Y . Further, the assertion(4.2) is equivalent to Proof Fix 𝛽∈(0, 1) , then inf Z∈Z 𝛽 𝔼ZX ≤inf Z∈Z 𝛽 𝔼ZY by (4.2). With (iii) in the preceding Theorem4.3 it follows that X≼Y . With(2.7), the statement(4.3) is equivalent with 𝔼Z(X−Y)≤0 for Z∈Z and hence the assertion. ◻ Remark 4.6 The assertion (4.3), however, is strictly stronger than (ii) in Theorem4.3. Indeed, it follows with convexity and(4.3) that and hence(ii), the assertion, although the reverse implication does not hold true. Example 4.7 (Uniform norm) For the uniform norm ‖⋅‖∞ , the defining relation(4.1) is equivalent to this relation derives from the characterization(i) in Theorem4.3 as well. inf Z ∈Z 𝛽 𝔼ZX ≤inf Z∈Z 𝛽 𝔼ZY, (4.2) 𝔼 ZX ≤ 𝔼ZY for all Z∈Z∶= ⋃ 𝛽∈(0,1) Z𝛽 , (4.3) R𝛽(X−Y)≤0 for all 𝛽∈(0, 1). R(−Y)≤R(X−Y)+R(−X)≤R(−X), X≼‖⋅‖ ∞ Y ⟺ ess inf X≤ess inf Y; A.Pichler 41 Page 18 of 28 4.2 Higher order stochastic dominance A traditional way of introducing stochastic dominance relations is by iterating integrals of the cumulative distribution function. This is a special case of the Lebesgue norm ‖⋅‖p , p∈[1, ∞) , with p∈ℕ . Definition 4.8 (Higher order stochastic dominance, cf. Müller and Stoyan (2002)) The random variableX is dominated byY in first order stochastic dominance, if where FX(x)∶=P(X≤x) is the cumulative distribution function. We shall write X≼(1)Y . For p∈[1, ∞] , the random variableX is stochastically dominated by Y in pth-stochastic order, if we write X≼(p)Y . By(4.1) in Definition4.1, where ‖⋅‖p is the usual norm in the Lebesgue space Lp . It is for historical—although unfortunate—reasons that the p-indici in the preceding display do not match. The higher order stochastic dominance of integral orders has been indtroduced and considered in earlier publications. Lemma 4.9 (Cf. Ogryczak and Ruszczyński (1999, 2001)) With F(1) X (⋅)∶=F X (⋅ ) , the kth ( k=2, 3, … ) repeated integral is F(k) X (x)∶= ∫x −∞ F (k−1) X (y)d y . The following two points are equivalent, they characterize stochastic dominance of integer orders ( k=1, 2, … ) by repeated integrals: (i) X≼(k)Y , (ii) F(k) Y (x) ≥ F (k) X (x ) for all x∈ℝ . Proof It holds with Cauchy’s formula for repeated integration that By integration by parts, the latter is so that FX(x)≥FY(x)for all x∈ ℝ , (4.4) 𝔼 (x−X) p−1 +≥ 𝔼(x−Y) p−1 + for all x∈ℝ ; X≼(p+1)Yis equivalent to X≼‖⋅‖ p Y,p≥1, F (k) X(x)= 1 (k−2)! ∫x −∞ (x−y)k−2FX(y)dy . F (k) X(x)= 1 (k−1)! ∫x −∞ (x−y)k−1dFX(y) , Connection betweenhigher order measures ofrisk andstochastic… Page 19 of 28 41 from which the assertion follows from the defining condition(4.1) in Definition4.1. ◻ Remark 4.10 It follows from the iterated integral and (ii) in Lemma 4.9 that X≼(k)Y⟹X≼(k+1)Y for all natural numbers k=1, 2, … . We notice next that To this end note first that the characterization(4.4) is equivalent to With ∫x z (x−y)𝛼 − 1(y−z)𝛽 − 1dy=B(𝛼,𝛽)(x−z)𝛽 + 𝛼 −1 (B is Euler’s integral of the first kind) and integration by parts it follows that where we have used the characterization (4.6) in (4.7), as x−y≥0 and that B(p,p�−p) is well-defined and positive for p′>p . The assertion again follows with(4.6). 4.3 Characterization ofstochastic dominance forspectral risk measures The following builds on the spectral risk measure R𝜎(⋅) introduced in Definition3.1 and considers the norm for the spectral function 𝜎 . Theorem 4.3 and the characterization of higher order spectral risk measures (Theorem3.2) give rise to the following result. Theorem4.11 The stochastic dominance relation F (k) X(x)= 1 (k−1)! ∫∞ −∞ (x−y)k−1 +dFX(y)= 1 (k−1)! 𝔼(x−X)k−1 + , (4.5) X≼( p )Y ⟹ X≼( p �)Yfor all real numbers 1 ≤p≤p�∈ ℝ . (4.6) �x −∞ (x−z)p−1dFX(z) ≥�x −∞ (x−z)p−1dFY(z)for all x∈ℝ . (4.7) �x −∞ (x−z)p�−1dFX(z)= 1 B(p,p�−p) �x −∞ �x z (x−y)p�−p−1(y−z)p−1dydFX(z) =1 B(p,p�−p)�x −∞ (x−y)p�−1−p�y −∞ (y−z)p−1dFX(z)d y ≥1 B(p,p�−p)�x −∞ (x−y)p�−1−p�x −∞ (y−z)p−1dFY(z)d y = � x −∞ (x−z)p�−1dFY(z), ‖⋅‖𝜎∶=R𝜎(�⋅�) A.Pichler 41 Page 20 of 28 with respect to the norm associated with the spectral risk measure R𝜎 is equivalent to where 𝜎 p∶= ∫1 1−p 𝜎(u)d u and SX(x)∶=1−FX(x)=P(X>x) is the survival function of the random variableX. Proof We argue with the norm ‖Y‖𝜎∶=R𝜎(�Y�) . Note, that (Y−t)+≥0 , hence the defining equation(2.3) is where we have used Theorem3.2 in(4.8). From(3.8) we have that where we have used that F−Y(y)=P(−Y≤y)=P(Y≥−y)=1−FY(−y)=SY(−y) and 𝖵@𝖱𝛼(−Y)=−𝖵@𝖱1−𝛼(Y) at points of continuity of FY(⋅) . Now set 1−u𝛽=∶p . Then, by employing the characterizing relation(3.3) for the 𝛽 -quantile of 𝜎 , it holds that so that X≼‖⋅‖ 𝜎 Y − 𝜎p⋅𝖵@𝖱p(Y)+� 𝖵@𝖱 p (Y) −∞ Σ ( SY(y) ) dy ≤ −𝜎p⋅𝖵@𝖱p(X)+ � 𝖵@𝖱p(X) −∞ Σ ( SX(x) ) dxfor all p∈(0, 1) , (4.8) R‖ ⋅ ‖ 𝜎 𝛽(Y)=inf t∈ℝ t+ 1 1−𝛽‖(Y−t)+‖𝜎 =inf t∈ℝ t+1 1−𝛽 R𝜎�(Y−t)+ � =R𝜎 𝛽 (Y), R 𝛽(−Y)= 𝖵@𝖱u𝛽(−Y)+ 1 1−𝛽∫ ∞ 𝖵@𝖱u𝛽(−Y) Σ ( F−Y(y) ) dy =−𝖵@𝖱1−u𝛽(Y)+ 1 1−𝛽∫∞ −𝖵@𝖱1−u𝛽(Y) Σ(SY(−y))d y =−𝖵@𝖱1−u𝛽(Y)+ 1 1−𝛽∫ 𝖵@𝖱1−u𝛽(Y) −∞ Σ ( SY(y) ) dy, 1 −𝛽=∫ 1 u 𝛽 𝜎(u)du=∫ 1 1−p 𝜎(u)du=𝜎p , R 𝛽(−Y)=−𝖵@𝖱p(Y)+ 1 𝜎 p ∫ 𝖵@𝖱 p (Y) −∞ Σ ( SY(y) ) dy . Connection betweenhigher order measures ofrisk andstochastic… Page 21 of 28 41 By Theorem4.3, the relation X≼‖⋅‖ 𝜎 Y is equivalent to R‖⋅‖ 𝜎 𝛽 (−Y) ≤ R ‖⋅‖ 𝜎 𝛽 (−X ) for all 𝛽∈(0, 1) . With that, the assertion follows. ◻ 4.4 Comparison ofstochastic order relations Different stochastic dominance relations may vary in strength (the implication(4.5) in the preceding Remark 4.10 is an example). In what follows, we provide an explicit relation to compare stochastic dominance relations, which are built on different spectral functions. Proposition 4.12 (Comparison of spectral stochastic orders) Suppose that for some probability measure 𝜇 , where u𝛽 is as defined in(3.3). Then the stochastic order associated with 𝜎𝜇 is weaker than the genuine stochastic order associated with 𝜎 . Specifically, for different spectral functions 𝜎 and 𝜎𝜇 , it holds that Remark 4.13 The function 𝜎𝜇 in(4.9) is indeed a spectral function. It is positive, as 𝜇 is a positive measure (thus(i) in Definition3.1). The function is non-decreasing, as u𝛽 is non-decreasing for 𝛽 increasing. Finally, the function 𝜎𝜇 is a density: indeed, it holds that by integration by parts, where we have used the definition of u𝛽 in(3.3). Proof of Proposition 4.12 Since x≼‖⋅‖ 𝜎 Y , it holds with Theorem 4.3 that R 𝜎 𝛽(−X)≥R 𝜎 𝛽(−Y) for all 𝛽∈(0, 1) , where 𝜎𝛽 is defined in(3.2). By the characterization(3.1), this is Integrating the latter expression with respect to 𝜇(d𝛽) establishes the inequality Interchanging the order of integration together with(3.17) gives that (4.9) 𝜎 𝜇(u)=𝜎(u)⋅ ∫u 𝛽 0 𝜇( d 𝛽) 1−𝛽 X≼‖ ⋅ ‖ 𝜎 Y ⟹ X≼‖⋅‖ 𝜎𝜇 Y. ∫1 0 𝜎𝜇(u)du=∫ 1 0 𝜎(u)⋅∫ u 𝛽 0 𝜇(d𝛽) 1−𝛽 du=∫ 1 0 ∫ 1 𝛽u 𝜎(u)du 𝜇(d𝛽) 1−𝛽=∫ 1 0 𝜇(d𝛽)= 1 �1 u 𝛽 𝜎(u) 1−𝛽 F−1 −X(u)du ≥ � 1 u 𝛽 𝜎(u) 1−𝛽 F−1 −Y(u)du,𝛽∈(0, 1) . �1 𝛽� 1 u 𝛽 � 𝜎(u) 1−𝛽�F−1 −X(u)du𝜇(d𝛽�) ≥ � 1 𝛽� 1 u 𝛽 � 𝜎(u) 1−𝛽�F−1 −Y(u)du𝜇(d𝛽�),𝛽∈(0, 1) . A.Pichler 41 Page 22 of 28 which in turn is This is the assertion. ◻ 5 Example: theexpectile The expectile risk measure, originally introduced by Newey and Powell (1987), has recently gained additional interest (cf. Malandii etal. (2024), Balbás etal. (2023) or Farooq and Steinwart (2018) for conditional regressions). A main reason for the additional interest in this risk measure is because it is the only elicitable risk functional (cf. Ziegel (2014)). As Proposition2.6 indicates, the higher order risk measure can be based on the dual norm. For this reason, the following section establishes the dual norm of expectiles first, as it is crucial in understanding its regret function in the risk quadrangle. Next, we provide an explicit characterization of the higher order expectiles, that is, the higher order risk measure based on the expectile risk measure. The expectile is defined as a minimizer. Its Kusuoka representation is central in elaborating the corresponding higher order risk functional. Definition 5.1 For 𝛼∈(0, 1) , the expectile is where is the asymmetric loss, or quadratic error function. The expectile satisfies the first order condition and e𝛼(⋅) is a risk measure for 𝛼∈[1∕2, 1] . We mention that condition(5.2) provides a definition for Y∈L1 , it is thus more general than(5.1), which requires Y∈L2 . The Kusuoka representation of the expectile (cf. Bellini etal. (2014,Proposition9)) is given by �1 u 𝛽 � 𝛽 u 𝛽 𝜎(u) 1−𝛽�𝜇(d𝛽�)F−1 −X(u)du ≥ � 1 u 𝛽 � 𝛽 u 𝛽 𝜎(u) 1−𝛽�𝜇(d𝛽)F−1 −Y(u)du,𝛽∈(0, 1) , �1 u 𝛽 𝜎𝜇(u)F−1 −X(u)du ≥ � 1 u 𝛽 𝜎𝜇(u)F−1 −Y(u)du,𝛽∈(0, 1) . (5.1) e 𝛼 (Y) = arg min x∈ℝ 𝔼 𝓁 𝛼 (Y−x), 𝓁 𝛼(x)= { 𝛼x 2 if x ≥0, (1−𝛼)x2else (5.2) (1−𝛼)𝔼(x−Y)+=𝛼𝔼(Y−x)+, Connection betweenhigher order measures ofrisk andstochastic… Page 23 of 28 41 where 𝜂 = 1−𝛼 𝛼 , so that the risk level in(5.3) is 1 − 𝛾 1−𝛾 𝜂 1−𝜂 = 𝛼(2−𝛾)−1 (2𝛼−1)(1−𝛾) . Involving spectral risk measures, the expectile can be recast as where S = { 𝜎 𝛾 ∶𝛾∈[0, 1 −𝜂] } collects the spectral functions The higher order expectile can be described by involving its dual norm (cf.(2.9)), as well as its Kusuoka representation (cf. Corollary3.6). The following two (sub)sections elaborate these possibilities for the expectile. 5.1 The dual norm ofexpectiles The higher order expectile can be described with the dual representation(2.8), for which the dual norm of the expectile is necessary. By the characterization of the loss function (5.2) it holds that e𝛼(Y) is welldefined for Y∈L1(P) . This is enough to conclude that 𝔼|Y|≤C𝛼 ⋅e𝛼(|Y|) for some constant C𝛼>0 (Lakshmanan and Pichler 2023,Corollary2.16) elaborate the tight bound C 𝛼 = 𝛼 1−𝛼 ). It follows that Y∗=L∞ , so that ‖Z‖∞ is well-defined for Z∈Y∗ . The following result provides the dual norm of the expectile explicitly. Proposition 5.2 (Dual norm of the expectile) For 𝛼≥1∕2 , the dual norm is (cf.(2.7)) . It holds that Notably, the norm ‖⋅‖∗ 𝛼 is not a risk measure itself, and(5.5) is not a Kusuoka representation; indeed, the total weight in the representation(5.5) is for 𝛼∈(1∕2, 1] . Proof of Proposition 5.2 We may assume that Z≥0 , as otherwise we may consider sign (Z)⋅Y instead of Y. For arbitrary sets B and G with B⊂G and P(G)<1 define the random variable (5.3) e 𝛼 (Y)= max 𝛾∈[0,1−𝜂](1−𝛾)⋅𝔼Y+𝛾⋅𝖠𝖵@𝖱 1−𝛾 1−𝛾 𝜂 1−𝜂 (Y), e 𝛼(Y)=sup { R𝜎𝛾 (Y)∶𝜎𝛾∈S }, s 𝛾(u)= { 1−𝛾if u ≤ 1− 𝛾 1−𝛾 𝜂 1−𝜂 , 1−𝛾 𝜂 else. (5.4) ‖ Z ‖∗ 𝛼 ∶= sup � 𝔼YZ ∶e 𝛼 ( � Y � ) ≤ 1 � (5.5) ‖ Z ‖ ∗ 𝛼=sup 𝛽∈(0,1) (1−𝛽)⋅𝖠𝖵@𝖱𝛽( � Z � )+𝛽 1−𝛼 𝛼 ‖ Z ‖ ∞ . ( 1−𝛽)+𝛽 1−𝛼 𝛼 < 1 A.Pichler 41 Page 24 of 28 Note, that and hence e𝛼 (  Y B,G )= 1 by the defining equation(5.2). It follows with(5.4) that As B⊂G are arbitrary, we conclude in particular that because the random variables satisfy all conditions from above for any uniform variable U. Now let P(G)→1 and by denoting 𝛽=P(B) it follows that as 𝖠𝖵@𝖱𝛾(Z)→ess sup Z for 𝛾→1 . As for the converse observe that we may assume e𝛼(Y)=1 for the optimal random variable in(5.4). Consider the Lagrangian where the Lagrangian multiplier 𝜆∈ℝ is associated with the equality constraint e𝛼(Y)=1 , i.e.,(5.2), and the measurable variable 𝜇∈L1 , 𝜇≥0 , is associated with the inequality constraint Y≥0 . Provided That the derivative exists, the first order conditions are or Now note that the left-hand side of(5.8) involves the variableZ, while the righthand side only involves constants, except on {Y=0} , where 𝜇 is not necessarily constant. The first order conditions (5.8) thus hold true on plateaus of Z, if they (5.6)  Y B,G(𝜔)∶= ⎧ ⎪ ⎨ ⎪ ⎩ 0 if 𝜔∈B, 1 if 𝜔∈G⧵B , and 1−𝛼 𝛼 ⋅ P(B) 1−P(G)+1 else. ( 1−𝛼)⋅P(B)(1−0)=𝛼⋅ ( 1−P(G) )( (1−𝛼)P(B) 𝛼(1−P(G)) +1−1 ), ‖ Z ‖∗ 𝛼≥ 𝔼ZY B,G. ‖ Z ‖ ∗ 𝛼 ≥� (1−P(B) � ⋅𝖠𝖵@𝖱P(B)(Z)+P(B) 1−𝛼 𝛼 ⋅𝖠𝖵@𝖱P(G)(Z) ,  Y B,G= ( 1−P(B) ) ⋅ 1 1−P(B) 1[P(B),1](U)+P(B) 1−𝛼 𝛼 ⋅ 1 1−P(G) 1[P(G),1](U ) ‖ Z ‖ ∗ 𝛼 ≥ sup 𝛽∈(0,1) (1−𝛽)⋅𝖠𝖵@𝖱𝛽(Z)+𝛽 1−𝛼 𝛼 ess sup Z , (5.7) L (Y;𝜆,𝜇)∶= 𝔼ZY −𝜆 ( (1−𝛼)𝔼(1−Y) + −𝛼𝔼(Y−1) +) +𝔼𝜇Y , 0 = 𝜕 𝜕Y L(Y;𝜆,𝜇) , (5.8) Z =𝜆 ( −(1−𝛼)1 {Y<1} −𝛼1 {Y>1}) −𝜇⋅1 {Y=0}.