Robust quasi-convex risk measures and applications

Robust quasi-con v ex risk measures and applications ∗ F rancesca Cen trone † Asmerilda Hita j ‡ Elisa Mastrogiacomo § Eman uela Rosazza Gianin ¶ This v ersion: March 19, 2026 Abstract This paper dev elops a unified framew ork for the robustification of risk measures b e- y ond the classical con vex and cash-additive setting. W e consider general risk measures on L p spaces ( p ∈ [1 , + ∞ ]) and construct their robust counterparts through families of uncertain ty sets that capture ambiguit y . Tw o complementary mechanisms generate ro- bust quasi-con vex measures: in the first, quasi-conv exit y is inherited from the initial risk measure under con vex uncertain ty sets; in the second it comes from the quasi-con v ex (or c-quasi-con vex) structure of the uncertain ty sets themselv es. Building on Cerreia-Vioglio et al. (2011); F rittelli and Maggis (2011), we derive dual (p enalt y-type) representations for robust quasi-conv ex and cash-subadditive risk measures, sho wing that the classical con vex cash-additive case arises as a special instance. W e further analyze acceptance fam- ilies and capital allo cation rules under robustification, highlighting how ambiguit y affects acceptabilit y and the distribution of capital. 1 In tro duction Quite recently , the construction of robust v ersions of risk measures has received increasing atten tion as a wa y to incorp orate mo del uncertain ty and am biguity 1 in to financial decision- making. V arious approaches to the robustness issue are p ossible, depending on differen t sources of uncertain ty that can arise, just to recall some, from partial kno wledge ab out the distributions of the risks or from the ambiguit y of a Decision Maker’s preferences (see, e.g., among the many , ∗ All the authors are members of Gruppo Nazionale per l’Analisi Matematica, la Probabilit` a e le loro Appli- cazioni (GNAMP A), Italy † Department of Economics and Business Studies, Universit y of Eastern Piedmont, 28100 Nov ara, Italy . francesca.centrone@uniupo.it ‡ Dipartimento di Economia, Universit` a dell’Insubria, via Monte Generoso 71, 21100 V arese, Italy . asmer- ilda.hita j@uninsubria.it § Dipartimento di Economia, Univ ersit` a dell’Insubria, via Monte Generoso 71, 21100 V arese, Italy . elisa.mastrogiacomo@uninsubria.it ¶ Department of Statistics and Quan titative Methods, University of Milano-Bico cca, Via Bicocca degli Arcim- boldi 8, 20126 Milan, Italy . emanuela.rosazza1@unimib.it 1 The terms am biguity and uncertaint y are used in terchangeably throughout the pap er. 1 W ang and Ziegel (2021), W ang and Xu (2023)). A central line of research also extends the classical framework of conv ex, cash-additive risk measures to robust settings. In particular, Righi (2024) has recen tly introduced uncertaint y at the lev el of an y financial p osition X (instead of considering a fixed uncertaint y set) and has defined robust con vex risk measures as: ˜ ρ ( X ) ≜ sup Z ∈U X ρ ( Z ) , for any X ∈ L p , (1) where ρ is a cash-additive and conv ex risk measure – hereafter referred to as the initial risk measure – and U X represen ts the uncertaint y set for X that is, a set of random v ariables represen ting p ossible p erturbations or alternative realizations of X . Righi (2025) has extended the robust approach to risk sharing, also providing concrete examples based on the p -norm and the W asserstein distance. Moresco et al. (2025) hav e considered the dynamic framework and clarified how the c hoice of uncertaint y sets affects time consistency . Ho wev er, as discussed in the literature, con vexit y and cash-additivity ma y b e to o restrictive in realistic settings. In markets with frictions, illiquidit y , or nonlinear pricing, conv ex combina- tions of positions need not preserv e linear costs, and the addition of cash ma y not reduce risk one-to-one. This motiv ates the use of cash-subadditiv e and quasi-con vex risk measures, which generalize the conv ex, cash-additive paradigm while preserving economic in terpretability . Cash- subadditivit y has b een in tro duced in El Karoui and Ra v anelli (2009) to address discoun ting am biguity due to uncertain in terest rates. Quasi-conv ex risk measures and their dual represen- tation ha ve been inv estigated from an axiomatic p oint of view in Cerreia-Vioglio et al. (2011), while their conditional counterpart has b een studied in F rittelli and Maggis (2011). Drap eau and Kupp er (2013) hav e studied risk functionals that satisfy only quasi-conv exity and mono- tonicit y and ha ve pro vided a one-to-one corresp ondence with risk-acceptance families and risk orders. Applications of quasi-conv ex risk measures confirm their flexibility . Mastrogiacomo and Rosazza Gianin (2015a,b), for instance, analyze p ortfolio optimization and optimal risk shar- ing under quasi-con vex preferences, showing that such measures preserve diversification while capturing nonlinear pricing and illiquidity . Their results highlight the practical relev ance of quasi-con vex risk measures and motiv ate the need for a systematic robustification theory b e- y ond the con vex, cash-additive setting. Despite significant adv ances in representation and applications of quasi-conv ex and cash- subadditiv e risk measures, the construction of robust counterparts for such functionals under am biguity has not yet b een systematically addressed. This work develops a unified theory of robust quasi-con vex (possibly cash-subadditive) risk measures and con tributes to the existing literature in several w ays, as summarized here below. First, we iden tify and connect tw o distinct approaches to construct a robust quasi-conv ex risk measure ˜ ρ of the form (1). The first approac h begins with a quasi-con vex risk measure ρ 2 and a family U ≜ ( U X ) X ∈ L p of uncertaint y sets satisfying the condition: U λX +(1 − λ ) Y ⊆ λ U X + (1 − λ ) U Y , for an y X , Y ∈ L p , λ ∈ [0 , 1] . This condition is commonly referred to as conv exity of the uncertain ty sets in the literature (see, e.g., Righi (2024); Moresco et al. (2025)). Alternativ ely , we show that ˜ ρ remains quasi-conv ex ev en without any assumption on ρ (other than monotonicit y), provided that the uncertaint y sets satisfy: U λX +(1 − λ ) Y ⊆ U X ∪ U Y , for an y X , Y ∈ L p , λ ∈ [0 , 1] . W e refer to this prop erty as quasi-con vexit y , b orro wing this terminology from the literature on set-v alued mappings (see, e.g., Seto et al. (2018)). Finally , we explore an alternative notion of quasi-conv exity – defined with resp ect to the cone of p ositiv e financial p ositions – and demonstrate that this framework also yields a robust quasi-conv ex risk measure. Second, w e c haracterize the largest family of uncertaint y sets that generates a given robust risk measure, extending the corresp ondence in Moresco et al. (2025) b eyond cash-additivity; w e also identify the conditions under which robustification preserv es monotonicit y , (quasi- )con vexit y , law in v ariance, and contin uity from ab o ve, clarifying ho w these properties are trans- ferred from the (initial) risk measure ρ and the family U of uncertain ty sets. Third, building on Cerreia-Vioglio et al. (2011); F rittelli and Maggis (2011), w e deriv e dual represen tations for robust quasi-conv ex (not necessarily cash-subadditive) risk measures, and sho w that the classical conv ex cash-additive case – including the dual forms in Righi (2024) – arises as a sp ecial case of our formulation. F ourth, adopting the acceptance-family p oin t of view of Drap eau and Kupp er (2013), we relate the acceptance families of ρ to those of ˜ ρ and pro vide tractable characterizations in a represen tative example. Finally , w e extend the robustification framework to the capital allocation problem and pro vide illustrativ e examples based on widely used uncertain ty sets, such as the W asserstein and the p -norm balls, and quasi-con v ex families commonly used in set-v alued analysis (see, e.g., Seto et al. (2018)). The pap er is organized as follows. Section 2 introduces the setting and recalls the main notions of risk measures, with an emphasis on quasi-conv exity , cash-subadditivity , and their dual representations. Section 3 develops the framework of (families of ) uncertaint y sets and robustification, presenting general prop erties and the correspondence with the largest family of uncertain ty sets inducing ˜ ρ . Section 4 establishes dual representations of robustified risk mea- sures under differen t assumptions on the initial risk measure and on the families of uncertain ty sets, including the quasi-conv ex, cash-subadditive, and conv ex cash-additive cases. Section 5 studies acceptance families of robust risk measures and their relationship with those of the initial risk measure. Applications to capital allocation principles and to the law in v ariant case with uncertain ty sets based on the W asserstein distance as well as some illustrative examples 3 are provided in Section 6. Finally , Section 7 concludes and outlines p ossible directions for future researc h. 2 Preliminaries and setting In this section, w e introduce the notations and setting used in the pap er, as well as the defini- tions and basic results on risk measures that will b e useful later. 2.1 Setting and notations W e fix a probabilit y space (Ω , F , P ) where, as usual, Ω is a set of p ossible states of the w orld, F is a σ -algebra of ev ents, and P is a reference probability measure. Throughout the pap er, random v ariables on (Ω , F , P ) will represen t profits and losses of financial p ositions where p ositive v alues indicate gains while negative ones losses. All equalities and inequalities betw een random v ariables should b e mean t to hold P -almost surely . The exp ectations under the reference probabilit y P will be denoted by E [ · ]. P stands for the set formed by all the probability measures that are absolutely contin uous with resp ect to P . The Radon–Nikodym deriv ativ e of Q ∈ P with resp ect to P is denoted by dQ dP . W e work with L p ≜ L p (Ω , F , P ) spaces, with p ∈ [1 , + ∞ ], formed b y all random v ariables that are p -in tegrable when p ∈ [1 , + ∞ ) or essen tially b ounded when p = + ∞ . L p is endo wed with the norm topology for p ∈ [1 , + ∞ ), while with the weak top ology σ ( L ∞ , L 1 ) for p = + ∞ . W e denote L p + ≜ { X ∈ L p : X ≥ 0 } . 2.2 Risk measures Financially speaking, we recall that risk measures can be interpreted as tools to assess the riski- ness of financial p ositions. See Artzner et al. (1999); Delbaen (2002); F¨ ollmer and Sc hied (2002); F rittelli and Rosazza Gianin (2002) (and many subsequen t pap ers) for a detailed treatmen t of the fundamentals of risk measures. In this paper, with risk measure we mean a functional ρ : L p → [ −∞ , + ∞ ] that is monotone de cr e asing , i.e., X ≤ Y implies that ρ ( X ) ≥ ρ ( Y ). Note that this con ven tion of decreasing monotonicit y is different from the “actuarial” conv en tion assuming increasing monotonicit y (see, e.g., Moresco et al. (2025)). The following additional axioms are sometimes imp osed to risk measures: • conv exity: ρ ( λX + (1 − λ ) Y ) ≤ λρ ( X ) + (1 − λ ) ρ ( Y ) for all X , Y ∈ L p , λ ∈ [0 , 1]; • cash-additivity: ρ ( X + m ) = ρ ( X ) − m for all X ∈ L p , m ∈ R ; • quasi-conv exity: ρ ( λX + (1 − λ ) Y ) ≤ max { ρ ( X ) , ρ ( Y ) } , for all X , Y ∈ L p , λ ∈ [0 , 1]; • cash-subadditivity: ρ ( X + m ) ≤ ρ ( X ) − m for all X ∈ L p , m ≥ 0; 4 • contin uity from ab ov e: for an y decreasing sequence ( X n ) n ∈ N in L p with X n ↓ X as n → + ∞ , then lim n → + ∞ ρ ( X n ) = ρ ( X ). See, among others, Artzner et al. (1999); Delbaen (2002); F¨ ollmer and Schied (2002); F rittelli and Rosazza Gianin (2002) for a discussion on coherent/con vex risk measures, El Karoui and Ra v anelli (2009) on cash-subadditiv e risk measures, and Cerreia-Vioglio et al. (2011); F rittelli and Maggis (2011) on quasi-conv ex risk measures. It is w orth recalling that con vexit y captures and incentiv ates risk diversification, while cash- additivit y reflects the assumption that adding m units of sure cash reduces risk exactly b y m . This allows to in terpret cash-additive risk measures as capital requirements (or margins). In man y applications, how ever, the axioms of con vexit y and cash-additivity are to o re- strictiv e. In mark ets with frictions, illiquidity or nonlinear pricing, conv ex com binations of p ositions ma y not preserve linear costs, and under ambiguit y on (sto c hastic) interest rates the effect of injecting cash is not necessarily linear. F or this reason, cash-subadditiv e risk measures w ere introduced by El Karoui and Ra v anelli (2009) to deal with discounting ambiguit y , while quasi-con vex risk measures w ere considered firstly b y Cerreia-Vioglio et al. (2011) to incentiv ate div ersification of risk without cash-additivit y . W e recall (see Cerreia-Vioglio et al. (2011); F rittelli and Maggis (2011)) that if ρ : L p → [ −∞ , + ∞ ], with p ∈ [1 , + ∞ ], is monotone, quasi-con vex, and con tinuous from ab ov e, then it admits the following dual represen tation ρ ( X ) = sup Q ∈P R ρ ( E Q [ − X ] , Q ) , for an y X ∈ L p , (2) with a “p enalt y-type” functional R ρ ( t, Q ) = R ( t, Q ) ≜ inf { ρ ( Y ) : E Q [ − Y ] = t } , for any t ∈ R , Q ∈ P , (3) b eing a map R ρ : R × P → [ −∞ , + ∞ ] that is monotone increasing and quasi-concav e in t , with inf t ∈ R R ( t, Q ) = inf t ∈ R R ( t, Q ′ ) for any Q, Q ′ ∈ P . See Theorem 2.9, Corollary 2.14, Lemma 3.2 of F rittelli and Maggis (2011), and Theorem 3.1 of Cerreia-Vioglio et al. (2011). In the sp ecial case of conv ex cash-additive risk measures ρ , R ρ reduces to R ρ ( t, Q ) = t − c ρ ( Q ) , for an y t ∈ R , Q ∈ P , where c ρ ( Q ) is the (minimal) p enalty functional in the dual representation of ρ , i.e. c ρ ( Q ) ≜ sup X ∈ L p { E Q [ − X ] − ρ ( X ) } , for any Q ∈ P . (4) See F rittelli and Rosazza Gianin (2002); F¨ ollmer and Schied (2002); F rittelli and Maggis (2011). Quite recently , in order to mo del amb iguity in the risk ev aluation Righi (2024, 2025) and Moresco et al. (2025) built r obust risk me asur es b y means of uncertain ty sets. F or eac h X ∈ L p , 5 an uncertain ty set U X ⊆ L p is a set of random v ariables represen ting p ossible perturbations or alternativ e realizations of X . W e recall from the aforemen tioned papers that, once the initial cash-additive risk measure ρ is fixed, • given a family of uncertaint y sets U = ( U X ) X ∈ L p (satisfying suitable prop erties), the induced U -robust risk measure ˜ ρ is defined as ˜ ρ ( X ) ≜ ˜ ρ U ( X ) ≜ sup Z ∈U X ρ ( Z ) , for any X ∈ L p . (5) In other words, ˜ ρ ( X ) represents the worst-case v alue of ρ when Z may v ary within the uncertain ty set U X . F urthermore, a one-to-one corresp ondence b et ween robust risk measures and families of un- certain ty sets can be found in Moresco et al. (2025) in terms of the largest family of uncertaint y sets. Indeed, • given a robust cash-additive risk measure ˜ ρ , the largest family of uncertain ty sets inducing ˜ ρ (called “consolidated” in Moresco et al. (2025)) is defined as ˜ U ≜ ( ˜ U X ) X ∈ L p , with ˜ U X ≜ [  U X uncertain ty set : ˜ ρ ( X ) = ˜ ρ U ( X )  , for an y X ∈ L p . (6) W e address to Righi (2024); Moresco et al. (2025) for precise statemen ts and assumptions. 3 Uncertain t y sets and robustification A central comp onen t in robustifying a risk measure is the c hoice of the structural prop erties of U – such as monotonicit y , conv exity , or (quasi-)con vexit y – whic h directly influence the regularit y of the robustified functional. Two complementary approaches can b e adopted to obtain a robust quasi-conv ex risk measure. In the first, quasi-conv exity is inherited from the risk measure ρ , under conv exity of the family U of the uncertaint y sets; in the second, the initial functional ρ is left general (i.e., quasi-con vexit y is not needed), and quasi-conv exity (or the stronger prop erty of c-quasi-conv exity) is instead imp osed on the uncertaint y sets of the family U . In this case, the source of quasi-conv exity shifts from the initial functional to the geometry of the uncertain ty sets, so that even a merely monotone ρ yields a robust quasi-con vex risk measure. In the following, w e formalize these tw o approaches. W e b egin by in tro ducing the structural assumptions on the families U of uncertaint y sets that will b e sometimes required later. Later on, w e will study the impact of such assumptions on the corresp onding robust risk measure. When not sp ecified otherwise, the elements ( U X ) of the family U are mean t to b elong to L p with general p ∈ [1 , + ∞ ]. 6 Definition 1. A family U ≜ ( U X ) X ∈ L p of unc ertainty sets is said to satisfy: - Monotonicity: if X ≤ Y then U X ⊇ U Y . - Or der pr eservation: if X ≤ Y then for any Y ′ ∈ U Y ther e exists X ′ ∈ U X such that X ′ ≤ Y ′ . - Convexity: U λX +(1 − λ ) Y ⊆ λ U X + (1 − λ ) U Y for any X , Y ∈ L p , λ ∈ [0 , 1] . - Quasi-c onvexity: U λX +(1 − λ ) Y ⊆ U X ∪ U Y for any X , Y ∈ L p , λ ∈ [0 , 1] . - c-quasi-c onvexity: for any X, Y ∈ L p , λ ∈ [0 , 1] U λX +(1 − λ ) Y ⊆ U X ∪ U Y + L p + . (7) - Continuity fr om ab ove: if ( X n ) n ∈ N is a de cr e asing se quenc e in L p with X n ↓ X as n → + ∞ , then ∪ n ∈ N U X n = U X . - Solidity: for any X ∈ L p it holds that: if Y ∈ U X and ¯ Y ≥ Y (with ¯ Y ∈ L p ), then also ¯ Y ∈ U X . - L aw invarianc e: if X ∼ X ′ , i.e. X and X ′ have the same distribution with r esp e ct to P , then U X = U X ′ . - Cash-invarianc e: for any X ∈ L p , c ∈ R , it holds that: U X + c = U X + c. F or what concerns the ab o ve prop erties of families of uncertain t y sets, con vexit y is the same as in Righi (2024), while monotonicity and order preserv ation corresp ond to the analo- gous axioms in Moresco et al. (2025). Ho wev er, we observe that order preserv ation is called monotonicit y in Righi (2024). Quasi-conv exity of the family of uncertaint y sets is the “natural” generalization of the classical quasi-conv exity notion to set-v alued maps (where, in our con text, w e consider the set-v alued map which asso ciates to each X an uncertaint y set U X ). In this framew ork, the order relation ≤ is replaced by the inclusion op erator and supremum corre- sp onds to union of sets. As for c-quasi-conv exity , it can b e found in the literature on set-v alued maps (see the notion of (u3)-type c-quasi-con vexit y in Seto et al. (2018) with the cone C b eing − L p + in our context). Remark 1. (a) We observe that c-quasi-c onvexity (define d as in (7) ) is not implie d in gener al by the c onvexity assumption of Righi (2024). Se e Example 1 b elow and Seto et al. (2018, Se ction 4, p ag.13) for further c omments. (b) Sever al other typ es of c onvexity and quasi-c onvexity for families of sets c ould b e also intr o duc e d (se e, e.g., Seto et al. (2018, pp.5–6)), and the pr op erties of the c orr esp onding r obust risk me asur es c ould then b e investigate d. The following example illustrates that in general conv exity do es imply neither quasi-conv exity nor c-quasi-con vexit y . Examples of (c-)quasi-con vex families of uncertain ty sets will be pro vided in Section 6.3. Example 1. Fix ε > 0 and c onsider U on L ∞ with U X = X + B ε , for any X ∈ L ∞ , 7 wher e B ε ≜ { Z ∈ L ∞ : ∥ Z ∥ ∞ ≤ ε } . It fol lows imme diately that the family U is c onvex. Nevertheless, U is neither quasi-c onvex nor c-quasi-c onvex. T o show this, let X = 0 , Y = 10 ε , and α = 1 2 , it holds U αX +(1 − α ) Y = 5 ε + B ε . F or ¯ Z = 1 2 ε , then, it fol lows that 5 ε + ¯ Z = 11 2 ε ∈ U αX +(1 − α ) Y but it do es not b elong to U X ∪ U Y = B ε ∪ (10 ε + B ε ) . This implies that U λX +(1 − λ ) Y ⊈ U X ∪ U Y , that is, quasi-c onvexity of U fails. With a slight mo dific ation of this example, it c an b e also shown that c onvexity of U do es not guar ante e c-quasi-c onvexity. The following Lemma clarifies the link b etw een the t wo concepts of quasi-conv exity and c-quasi-con vexit y for families of uncertain ty sets. Lemma 1. i) Solidity of U is e quivalent to U X = U X + L p + for any X ∈ L p . ii) If the family U is quasi-c onvex, then it is also c-quasi-c onvex. The c onverse implic ation holds under solidity of U . Pr o of. i) If U X = U X + L p + holds for an y X ∈ L p , then solidit y is straightforw ard. Conv ersely , if solidit y holds, then the inclusion ⊆ is immediate. Concerning the ⊇ inclusion, if Z ∈ U X + L p + then Z ≥ ˜ Z for some ˜ Z ∈ U X . By solidity of U , it follo ws that also Z ∈ U X . ii) The first statement is straigh tforward. The conv erse implication is due to U X ∪ U Y + L p + = ( U X + L p + ) ∪ ( U Y + L p + ) = U X ∪ U Y , where the former equality can be c heck ed easily , the latter follows from solidit y . In the sequel, w e will com bine the approaches of Righi (2024) and Moresco et al. (2025) (recalled at the end of Section 2, equations (5) and (6)) to robustify a risk measure ρ that is not necessarily cash-additiv e. Also, in the present setting where we consider L p spaces and w e do not necessarily require cash-additivity , it can b e prov ed (as in Moresco et al. (2025), Lemma 3) that the largest family of uncertain ty sets inducing ˜ ρ corresponds to: ˜ U X =  Z ∈ L ∞ : ρ ( Z ) ≤ ˜ ρ ( X )  , for any X ∈ L p . (8) In the following result, items a) and b) generalize items 2. and 7. of Theorem 2 of Moresco et al. (2025) to the case of robustification of risk measures ρ that are not necessarily cash- additiv e, for general families of uncertain ty sets in L p spaces, while items c), d) and e) are new and deal with quasi-conv exity of the robust risk measure. Prop osition 1. L et ρ b e a risk me asur e, U b e a family of unc ertainty sets and ˜ ρ the r obustifi- c ation of ρ via U . a) If U is monotone or or der pr eserving then ˜ ρ is monotone. 8 b) If ρ is c onvex and U is a c onvex family, then ˜ ρ is c onvex. c) If U is c-quasi-c onvex (or quasi-c onvex), then ˜ ρ is quasi-c onvex. d) If ρ is quasi-c onvex and U is c onvex, then ˜ ρ is quasi-c onvex. e) If U is monotone and c ontinuous fr om ab ove, then ˜ ρ is c ontinuous fr om ab ove. f ) If U is law invariant, then ˜ ρ is law invariant. Pr o of. a) Let X, Y ∈ L p with X ≤ Y b e fixed arbitrarily . If U is monotone, then U X ⊇ U Y , implying that ˜ ρ ( X ) = sup Z ∈U X ρ ( Z ) ≥ sup Z ∈U Y ρ ( Z ) = ˜ ρ ( Y ) , that is, monotonicity of ˜ ρ . If U is order preserving, then for an y Y ′ ∈ U Y there exists X ′ ∈ U X , such that X ′ ≤ Y ′ . Then, b y monotonicit y of ρ , w e ha ve ρ ( X ′ ) ≥ ρ ( Y ′ ). T aking in to account that suc h an inequalit y holds for all Y ′ ∈ U Y and by definition of ˜ ρ , we obtain ˜ ρ ( X ) ≥ ˜ ρ ( Y ). b) Suppose that ρ is a conv ex risk measure and that U is conv ex. By definition of ˜ ρ , it holds that for any X, Y ∈ L p and α ∈ [0 , 1] ˜ ρ ( αX + (1 − α ) Y ) = sup Z ∈U αX +(1 − α ) Y ρ ( Z ) ≤ sup Z ∈ ( α U X +(1 − α ) U Y ) ρ ( Z ) (9) ≤ sup Z 1 ∈U X ; Z 2 ∈U Y ρ ( αZ 1 + (1 − α ) Z 2 ) ≤ sup Z 1 ∈U X ; Z 2 ∈U Y  αρ ( Z 1 ) + (1 − α ) ρ ( Z 2 )  (10) = α sup Z 1 ∈U X ρ ( Z 1 ) + (1 − α ) sup Z 2 ∈U Y ρ ( Z 2 ) = α ˜ ρ ( X ) + (1 − α ) ˜ ρ ( Y ) , where (9) is due to conv exity of U and (10) to c on vexit y of ρ . c) By Lemma 1- ii), quasi-conv exity of U implies c-quasi-con vexit y of U . Therefore, it is enough to prov e that c-quasi-con vexit y of U implies quasi-conv exit y of ˜ ρ . F rom c-quasi-conv exity of U , it follo ws that, for an y X , Y ∈ L p and α ∈ [0 , 1], ˜ ρ ( αX + (1 − α ) Y ) = sup Z ∈U αX +(1 − α ) Y ρ ( Z ) ≤ sup Z ∈U X ∪U Y + L p + ρ ( Z ) . If Z ∈ U X ∪ U Y + L p + then Z = Z X,Y + K for some Z X,Y ∈ U X ∪ U Y and K ∈ L p + . Then, b y monotonicit y of ρ , ρ ( Z ) = ρ ( Z X,Y + K ) ≤ ρ ( Z X,Y ) . 9 The previous arguments imply that ˜ ρ ( αX + (1 − α ) Y ) ≤ sup Z ∈U X ∪U Y + L p + ρ ( Z ) ≤ sup Z ∈U X ∪U Y ρ ( Z ) = max { ˜ ρ ( X ); ˜ ρ ( Y ) } , i.e. quasi-con vexit y of ˜ ρ . d) Pro ceeding similarly as in item b), conv exity of U implies that, for any X , Y ∈ L p and α ∈ [0 , 1], ˜ ρ ( αX + (1 − α ) Y ) = sup Z ∈U αX +(1 − α ) Y ρ ( Z ) ≤ sup Z 1 ∈U X ; Z 2 ∈U Y ρ ( αZ 1 + (1 − α ) Z 2 ) ≤ sup Z 1 ∈U X ; Z 2 ∈U Y max { ρ ( Z 1 ); ρ ( Z 2 ) } (11) = max  sup Z 1 ∈U X ρ ( Z 1 ); sup Z 2 ∈U Y ρ ( Z 2 )  = max { ˜ ρ ( X ); ˜ ρ ( Y ) } , where (11) is due to quasi-conv exity of ρ . e) Let ( X n ) n ∈ N b e a decreasing sequence of L p with X n ↓ X as n → + ∞ . Then ˜ ρ ( X ) ≥ lim n → + ∞ ˜ ρ ( X n ) = sup n ∈ N ˜ ρ ( X n ) (12) = sup n ∈ N sup Z ∈U X n ρ ( Z ) = sup Z ∈U X ρ ( Z ) = ˜ ρ ( X ) , (13) where (12) is due to monotonicity of U –hence, b y item a), monotonicit y of ˜ ρ –, while the first equalit y in (13) is implied by contin uity from ab o ve of U . Contin uit y from ab ov e of ˜ ρ then follo ws. f ) is immediate. In other words, Prop osition 1 underlines that robust risk measures can b e built at least in t wo w ays: with a general risk measure ρ and a monotone quasi-conv ex (or c-quasi-conv ex) family U of uncertain ty sets; or with a quasi-conv ex risk measure ρ and a monotone con vex U . Notice that the tw o approac hes are different since, as already observed in Remark 1, con vexit y of U do es imply neither quasi-conv exity nor c-quasi-con v exity . Roughly sp eaking, the “sources” of quasi-conv exity of ˜ ρ can b e either at the level of the primal risk measure ρ or of the family U . Consider now the largest family ˜ U of uncertain ty sets inducing a giv en ˜ ρ as in (6). Sev eral prop erties of ˜ U hav e b een established in Moresco et al. (2025) for cash-additiv e risk measures. 10 Item a) of the follo wing result generalizes item 2. of Theorem 2 of Moresco et al. (2025) to the case of robustification of risk measures ρ that are not necessarily cash-additive, while item b) is new. Prop osition 2. L et ρ b e a risk me asur e, ˜ ρ the r obustific ation of ρ and ˜ U the lar gest family of unc ertainty sets define d in (6) asso ciate d to ˜ ρ . Then ˜ U is solid and the fol lowing implic ations hold. a) If ˜ ρ is monotone, then ˜ U is monotone. b) If ˜ ρ is quasi-c onvex, then ˜ U is quasi-c onvex (henc e, also c-quasi-c onvex). Pr o of. Solidity of ˜ U follows immediately from (8) and from the monotonicity of ρ . a) can b e prov ed similarly to Theorem 2, item 2., of Moresco et al. (2025). Assume, indeed, that ˜ ρ is monotone. If X ≤ Y , then ˜ ρ ( Y ) ≤ ˜ ρ ( X ), hence ˜ U Y = { Z ∈ L p : ρ ( Z ) ≤ ˜ ρ ( Y ) } ⊆ { Z ∈ L p : ρ ( Z ) ≤ ˜ ρ ( X ) } = ˜ U X . b) Assume that ˜ ρ is quasi-con v ex. Let X , Y ∈ L p and α ∈ [0 , 1] b e fixed arbitrarily . If Z ∈ ˜ U αX +(1 − α ) Y , then ρ ( Z ) ≤ ˜ ρ ( αX + (1 − α ) Y ) ≤ max { ˜ ρ ( X ); ˜ ρ ( Y ) } , where the former inequalit y comes from (8), while the latter from quasi-con vexit y of ˜ ρ . Again b y (8), it follows that Z ∈ ˜ U X ∪ ˜ U Y , so ˜ U is quasi-conv ex. 4 Dual represen tation of robust quasi-con v ex risk mea- sures In this section, we will provide the dual representation of robust risk measures in the t wo approac hes discussed in Section 3, b oth building robust quasi-con vex risk measures. As in Righi (2024), a key ingredient of the dual representation is the supp ort function of an uncertain ty set, defined as φ Q ( X ) ≜ sup Z ∈U X E Q [ − Z ] , for any X ∈ L p , (14) once a family U of uncertaint y sets is fixed. In the se quel, families U of unc ertainty sets wil l b e always supp ose d to satisfy monotonicity. 4.1 First approac h: quasi-con v ex ρ , conv ex U W e will now pro ve a dual representation of robust quasi-con vex risk measures in the first approac h of robustification, that is, for a quasi-con vex ρ and a conv ex family U of uncertaint y sets. 11 Prop osition 3. If U is a c onvex family of unc ertainty sets and ρ : L p → [ −∞ , + ∞ ] , with p ∈ [1 , + ∞ ] , is a quasi-c onvex and c ontinuous fr om ab ove risk me asur e, then the asso ciate d r obust risk me asur e ˜ ρ is quasi-c onvex. F urthermor e, it has the fol lowing dual r epr esentation: ˜ ρ ( X ) = sup Q ∈P sup Z ∈U X R ρ ( E Q [ − Z ] , Q ) , for any X ∈ L p , wher e R ρ is the p enalty-typ e functional in (3) of ρ . Pr o of. Monotonicity and quasi-conv exity of ˜ ρ follow, resp ectiv ely , from items a) and d) of Prop osition 1. F rom Theorem 2.9, Corollary 2.14, Lemma 3.2 of F rittelli and Maggis (2011) (see also Prop o- sitions 4.3-4.4 and Theorem 3.1 of Cerreia-Vioglio et al. (2011) on L ∞ ), ρ can b e represented as ρ ( X ) = sup Q ∈P R ρ ( E Q [ − X ] , Q ) , for any X ∈ L p , where R ρ ( t, Q ) is monotone increasing and quasi-concav e in t ∈ R , with inf t ∈ R R ( t, Q ) = inf t ∈ R R ( t, Q ′ ) for any Q, Q ′ ∈ P . By the previous arguments and b y definition of ˜ ρ , it follows that, for an y X ∈ L p , ˜ ρ ( X ) = sup Z ∈U X ρ ( Z ) = sup Z ∈U X sup Q ∈P R ρ ( E Q [ − Z ] , Q ) = sup Q ∈P sup Z ∈U X R ρ ( E Q [ − Z ] , Q ) . Under stronger assumptions on ρ , w e will provide a “more explicit” dual representation of the robust quasi-con vex risk measure. With this in mind, we will start with the particular case of risk measures ρ corresp onding to certaint y equiv alen ts on L ∞ , i.e. ρ ℓ ( X ) = ℓ − 1 ( E P [ ℓ ( − X )]) , for an y X ∈ L ∞ , where ℓ : R → R is a strictly increasing conv ex function (also called loss function) and ℓ − 1 denotes its inv erse function. By Prop osition 5.3 in Cerreia-Vioglio et al. (2011), ρ ℓ is quasi- con vex and admits the following robust represen tation: ρ ℓ ( X ) = sup Q ∈P R ℓ ( E Q [ − X ] , Q ) , for any X ∈ L ∞ , where R ℓ ( t, Q ) is a p enalt y-type function of the follo wing form: R ℓ ( t, Q ) ≜ ℓ − 1  max x ≥ 0  xt − E P  ℓ ∗  x dQ dP  , for an y ( t, Q ) ∈ ( R , P ) , (15) with ℓ ∗ denoting the conv ex conjugate of ℓ . 12 No w, consider a robustified version of the risk measure ρ ℓ , where uncertain ty affects the underlying outcomes. Namely , ˜ ρ ℓ ( X ) ≜ sup Z ∈U X ρ ℓ ( Z ) = sup Z ∈U X sup Q ∈P R ℓ ( E Q [ − Z ] , Q ) , for any X ∈ L ∞ . (16) Prop osition 4 (Robustified certain ty equiv alent) . L et U b e a c onvex family of unc ertainty sets on L ∞ . Then the r obustifie d c ertainty e quivalent ˜ ρ ℓ is quasi-c onvex and has the fol lowing dual r epr esentation: ˜ ρ ℓ ( X ) = sup Q ∈P R ℓ ( φ Q ( X ) , Q ) , for any X ∈ L ∞ . Pr o of. Monotonicity and quasi-conv exity of ˜ ρ ℓ follo w directly from quasi-conv exity of ρ ℓ and from Prop osition 1, items a) and d). W e are now going to prov e the dual representation. By interc hanging the suprema in (16) and applying the explicit formulation of R ℓ in (15), we obtain that, for an y X ∈ L ∞ , ˜ ρ ℓ ( X ) = sup Q ∈P sup Z ∈U X R ℓ ( E Q [ − Z ] , Q ) = sup Q ∈P sup Z ∈U X ℓ − 1  sup x ≥ 0  x E Q [ − Z ] − E P  ℓ ∗  x dQ dP  = sup Q ∈P ℓ − 1  sup x ≥ 0  x · sup Z ∈U X E Q [ − Z ] − E P  ℓ ∗  x dQ dP  (17) = sup Q ∈P ℓ − 1  sup x ≥ 0  x · φ Q ( X ) − E P  ℓ ∗  x dQ dP  = sup Q ∈P R ℓ ( φ Q ( X ) , Q ) , (18) where equality (17) is due to contin uity of the function ℓ − 1 (as a consequence of its concavit y on R ), while (18) follows from (15) and (14). In the same spirit of Prop osition 4, the follo wing result provides a dual representation of robust risk measures for a quasi-con vex cash-subadditive ρ and a conv ex family U in general spaces L p , with p ∈ [1 , + ∞ ]. Theorem 1. If U is a c onvex family of unc ertainty sets on L p and ρ : L p → [ −∞ , + ∞ ] is a quasi-c onvex, c ash-sub additive and c ontinuous fr om ab ove risk me asur e, then the asso ciate d r obust risk me asur e ˜ ρ is quasi-c onvex and has the fol lowing dual r epr esentation: ˜ ρ ( X ) = sup Q ∈P R ρ ( φ Q ( X ) , Q ) , for any X ∈ L p . Pr o of. Monotonicity and quasi-con vexit y of ˜ ρ follow from Prop osition 1, items a) and d). 13 As in the pro of of Prop osition 3, ρ can b e represented as ρ ( X ) = sup Q ∈P R ρ ( E Q [ − X ] , Q ) , for any X ∈ L p , (19) with R ρ b eing monotone increasing and quasi-concav e in t , and satisfying inf t ∈ R R ρ ( t, Q ) = inf t ∈ R R ρ ( t, Q ′ ) for any Q, Q ′ ∈ P . Moreo ver, cash-subadditivity of ρ implies that R ρ is also non-expansive, i.e., R ρ ( t ′ , Q ) ≤ R ρ ( t, Q ) + | t − t ′ | for any t, t ′ ∈ R , Q ∈ P . This can b e pro ved (on L p ) exactly as in the pro of of Theorem 3.1 of Cerreia-Vioglio et al. (2011). F rom the non-expansivit y of R ρ it follo ws that R ρ ( · , Q ) is left-contin uous. Indeed, for an y arbitrary t ∈ R and sequence ( t n ) n ∈ N with t n ↑ t as n → + ∞ , increasing monotonicity and non-expansivit y of R ρ ( · , Q ) guaran tee that 0 ≤ R ρ ( t, Q ) − R ρ ( t n , Q ) ≤ | t − t n | . It then follows that lim n → + ∞ R ρ ( t n , Q ) = R ρ ( t, Q ) for any Q ∈ P , hence the left-contin uity of R ρ ( · , Q ) holds. By (19), ˜ ρ can b e rewritten, for any X ∈ L p , as: ˜ ρ ( X ) = sup Z ∈U X ρ ( Z ) = sup Z ∈U X sup Q ∈P R ρ ( E Q [ − Z ] , Q ) = sup Q ∈P sup Z ∈U X R ρ ( E Q [ − Z ] , Q ) = sup Q ∈P R ρ  sup Z ∈U X E Q [ − Z ] , Q  (20) = sup Q ∈P R ρ ( φ Q ( X ) , Q ) , where equality in (20) is due to increasing monotonicity and left-con tinuit y of R ρ ( · , Q ). Here below, w e show that, when dealing with con vex and cash-additive risk measures and with cash-inv ariant uncertain ty sets, the dual representation provided in the previous result reduces to Theorem 1 of Righi (2024). Corollary 1. If U is a c onvex, c ash-invariant and c ontinuous fr om ab ove family of unc ertainty sets, and ρ : L p → [ −∞ , + ∞ ] is a c onvex, c ash-additive and c ontinuous fr om ab ove risk me a- sur e, then the asso ciate d ˜ ρ is c onvex and c ash-additive and has the fol lowing dual r epr esentation: ˜ ρ ( X ) = sup e Q ∈P n E e Q [ − X ] − inf Q ∈P { c φ Q ( e Q ) + c ρ ( Q ) } o , for any X ∈ L p , (21) wher e c ρ and c φ Q denote the minimal p enalty function of ρ and φ Q , r esp e ctively. Pr o of. It can be easily c heck ed that ˜ ρ is a con vex cash-additiv e risk measure. 14 It remains to prov e (21). It is well-kno wn (see Cerreia-Vioglio et al. (2011) and F rittelli and Maggis (2011)) that if ρ is a con vex cash-additiv e risk measure, then R ρ ( t, Q ) = t − c ρ ( Q ) , for any ( t, Q ) ∈ ( R , P ) . (22) By Theorem 1 and by (22), ˜ ρ b ecomes ˜ ρ ( X ) = sup Q ∈P { φ Q ( X ) − c ρ ( Q ) } , for any X ∈ L p . (23) W e prov e now that φ Q is a conv ex cash-additiv e and contin uous from ab ov e risk measure. Monotonicity. If X ≤ Y , then U X ⊇ U Y (b y monotonicity of U ). It is straightforw ard to c heck that φ Q ( X ) ≥ φ Q ( Y ), i.e. decreasing monotonicit y of φ Q . Continuity fr om ab ove. Let ( X n ) n ∈ N b e a decreasing sequence with X n ↓ X as n → + ∞ . Then φ Q ( X ) ≥ sup n ∈ N φ Q ( X n ) = sup n ∈ N sup Z ∈U X n E Q [ − Z ] = sup Z ∈∪ n U X n E Q [ − Z ] = sup Z ∈U X E Q [ − Z ] = φ Q ( X ) , where the first equalit y in the last line is due to contin uity from abov e of U . Contin uity from ab o ve of φ Q then follows. Convexity and c ash-additivity of φ Q follo w from Lemma 1 of Righi (2024). By the previous arguments, φ Q can b e represen ted as φ Q ( X ) = sup e Q ∈P  E e Q [ − X ] − c φ Q ( e Q )  , for any X ∈ L p . (24) F rom (23) and (24), it follo ws that, for an y X ∈ L p , ˜ ρ ( X ) = sup Q ∈P { φ Q ( X ) − c ρ ( Q ) } = sup Q ∈P  sup e Q ∈P  E e Q [ − X ] − c φ Q ( e Q )  − c ρ ( Q )  = sup Q ∈P sup e Q ∈P  E e Q [ − X ] − c φ Q ( e Q ) − c ρ ( Q )  = sup e Q ∈P n E e Q [ − X ] − inf Q ∈P { c φ Q ( e Q ) + c ρ ( Q ) } o . 15 4.2 Second approac h: con v ex ρ , quasi-con vex U In this section, we provide a dual representation of robust quasi-conv ex risk measures obtained with the second approach, that is, for a conv ex ρ and a quasi-conv ex U . Prop osition 5. L et ρ : L p → [ −∞ , + ∞ ] b e a c onvex, c ontinuous fr om ab ove, c ash-additive risk me asur e and let U b e a quasi-c onvex and c ontinuous fr om ab ove family of unc ertainty sets. Then: a) for any Q ∈ P , φ Q satisfies quasi-c onvexity, de cr e asing monotonicity and c ontinuity fr om ab ove; b) ˜ ρ is quasi-c onvex and has the fol lowing dual r epr esentat ion: ˜ ρ ( X ) = sup Q, ˜ Q ∈P  R φ Q ( E ˜ Q [ − X ] , ˜ Q ) − c ρ ( Q )  , for any X ∈ L p , wher e c ρ is the minimal p enalty function of ρ and R φ Q is the p enalty-typ e functional of φ Q . Pr o of. a) Quasi-c onvexity. By quasi-conv exit y of U , for any X , Y ∈ L p and α ∈ [0 , 1] it holds that φ Q ( αX + (1 − α ) Y ) = sup Z ∈U αX +(1 − α ) Y E Q [ − Z ] ≤ sup Z ∈U X ∪U Y E Q [ − Z ] = max { φ Q ( X ); φ Q ( Y ) } . Monotonicity and c ontinuity fr om ab ove of φ Q can b e pro ved as in Corollary 1. b) On the one hand, by Prop osition 2.5 and Theorem 2.9 of F rittelli and Maggis (2011) and b y item a), it follo ws that φ Q ( X ) = sup ˜ Q ∈P R φ Q ( E ˜ Q [ − X ] , ˜ Q ) , for any X ∈ L p . (25) On the other hand, b y the dual representation of ρ (see F¨ ollmer and Schied (2002) and F rittelli and Rosazza Gianin (2002)) and by definition of ˜ ρ , ˜ ρ ( X ) = sup Z ∈U X sup Q ∈P { E Q [ − Z ] − c ρ ( Q ) } = sup Q ∈P { sup Z ∈U X E Q [ − Z ] − c ρ ( Q ) } = sup Q ∈P { φ Q ( X ) − c ρ ( Q ) } . (26) 16 Then, (25) and (26) imply that, for an y X ∈ L p , ˜ ρ ( X ) = sup Q ∈P  sup ˜ Q ∈P R φ Q ( E ˜ Q [ − X ] , ˜ Q ) − c ρ ( Q )  = sup Q, ˜ Q ∈P  R φ Q ( E ˜ Q [ − X ] , ˜ Q ) − c ρ ( Q )  . 5 Acceptance families under robustification/uncertain t y It is well kno wn (see Delbaen (2002); F¨ ollmer and Sc hied (2002)) that conv ex cash-additive risk measures are in a one-to-one corresp ondence with acceptance sets A ρ ≜ { X ∈ L p : ρ ( X ) ≤ 0 } , formed by all p ositions that are acceptable and do not require any extra margin. F or risk measures that are not necessarily cash-additiv e, instead, a single acceptance set defined with resp ect to the target level 0 is no more enough but a family of acceptance sets at differen t target lev els is needed. Financially sp eaking, for general (e.g., quasi-conv ex or cash-subadditive) risk measures, the notion of acceptability dep ends on the sp ecified target level (see Drap eau and Kupp er (2013)). W e recall from Drapeau and Kupper (2013), Theorem 1, that any quasi-con vex risk measure ρ is in a one-to-one corresp ondence with the family ( A m ρ ) m ∈ R of acceptance sets given b y A m ρ ≜ { X ∈ L p : ρ ( X ) ≤ m } , for any m ∈ R , (27) via ρ ( X ) = inf { m ∈ R : X ∈ A m ρ } , for any X ∈ L p . (28) In other words, each set A m ρ collects the p ositions that are acceptable at the target level m , meaning that the p erceiv ed risk of X do es not exceed the threshold m , and ρ ( X ) represen ts the minimal lev el m for whic h X b ecomes acceptable. The family of acceptance sets thus pro vides a richer description of acceptability across different tolerance levels, capturing situations where the admissibility of a position dep ends on the benchmark or regulatory target. In the follo wing result, we provide a relationship betw een the family ( A m ρ ) m ∈ R of acceptance sets of the initial risk measure ρ and the family ( A m ˜ ρ ) m ∈ R of the robust risk measure ˜ ρ . Prop osition 6. L et U b e a c onvex family of unc ertainty sets, ρ b e a quasi-c onvex risk me asur e and ˜ ρ b e the c orr esp onding r obust risk me asur e. Then the fol lowing statements hold. 17 a) X ∈ A m ˜ ρ ⇐ ⇒ U X ⊆ A m ρ . b) ˜ ρ ( X ) = inf { m ∈ R : U X ⊆ A m ρ } for any X ∈ L p . c) If ρ is also c ash-additive (henc e, c onvex), then ˜ ρ ( X ) = inf { m ∈ R : U X + m ⊆ A 0 ρ } for any X ∈ L p . Pr o of. a) F rom Prop osition 1, item d), it follows that ˜ ρ is a quasi-conv ex risk measure. Hence, b y Theorem 1 of Drap eau and Kupp er (2013), it is in a one-to-one corresp ondence with the family ( A m ˜ ρ ) m ∈ R defined in (27). F urthermore, b y definition of ˜ ρ , it follo ws that A m ˜ ρ = { X ∈ L p : ˜ ρ ( X ) ≤ m } = { X ∈ L p : sup Z ∈U X ρ ( Z ) ≤ m } = { X ∈ L p : ρ ( Z ) ≤ m for any Z ∈ U X } . On the one hand, if X ∈ A m ˜ ρ then, by the argument ab o ve, ρ ( Z ) ≤ m for any Z ∈ U X . Hence, U X ⊆ A m ρ . On the other hand, if U X ⊆ A m ρ then ˜ ρ ( X ) = sup Z ∈U X ρ ( Z ) ≤ sup Z ∈A m ρ ρ ( Z ) ≤ m, implying that X ∈ A m ˜ ρ . b) follows immediately from item a) and (28). c) If ρ is quasi-conv ex and cash-additiv e, then it is also con vex (see Cerreia-Vioglio et al. (2011); Drap eau and Kupp er (2013); F rittelli and Maggis (2011)). Hence, by Prop osition 2 of Drapeau and Kupp er (2013), A m ρ = A 0 ρ − m for an y m ∈ R . This and item b) imply that ˜ ρ ( X ) = inf { m ∈ R : U X ⊆ A m ρ } = inf { m ∈ R : U X ⊆ A 0 ρ − m } = inf { m ∈ R : U X + m ⊆ A 0 ρ } for any X ∈ L p . The following example (on L ∞ ) illustrates the inclusion in item a) of the previous result where the acceptance sets of b oth ρ and ˜ ρ can b e explicitly computed. Example 2. Fix ε, K > 0 and c onsider, for any X ∈ L ∞ , U X = { Z ∈ L ∞ : − ε ≤ Z − X ≤ ε } ρ ( X ) = E [ − X ] ∨ K. It is imme diate to che ck that ρ is a quasi-c onvex risk me asur e. F urthermor e, the asso ciate d 18 r obust risk me asur e r e duc es to ˜ ρ ( X ) = sup X − ε ≤ Z ≤ X + ε  E [ − Z ] ∨ K  =  E [ − X ] + ε  ∨ K . It then fol lows that the family of ac c eptanc e sets of ρ is given by A m ρ = ( { X ∈ L ∞ : E [ − X ] ≤ m } ; m ≥ K ∅ ; m < K , while the family of ac c eptanc e sets of ˜ ρ is A m ˜ ρ = ( { X ∈ L ∞ : E [ − X ] ≤ m − ε } ; m ≥ K ∅ ; m < K The explicit formulation of the families ab ove al lows to verify dir e ctly the inclusion in a) in the pr evious r esult. F or any m < K , inde e d, we have that b oth A m ρ and A m ˜ ρ ar e empty, so that X / ∈ A m ˜ ρ is e quivalent to U X ⊆ A m ρ . Consider now the c ase of m ≥ K . F or any X ∈ A m ˜ ρ it holds that E [ − X ] ≤ m − ε . Thus, for any Z ∈ U X we have E [ − Z ] ≤ E [ − X + ε ] = E [ − X ] + ε ≤ m , so Z ∈ A m ρ . This shows that: X ∈ A m ˜ ρ ⇒ U X ⊆ A m ρ . On the other hand, if U X ⊆ A m ρ , then ρ ( Z ) ≤ m for any Z ∈ U X . In p articular, for Z = X − ε we de duc e ρ ( X − ε ) ≤ m , which implies E [ − X ] ≤ m − ε and thus X ∈ A m ˜ ρ . This shows that: U X ⊆ A m ρ ⇒ X ∈ A m ˜ ρ . 6 Applications and examples In this section, we provide tw o applications of the robustification: the former to capital al- lo cations, the latter to the law in v ariant case with uncertain ty sets based on the W asserstein distance. Some illustrativ e examples of families of uncertaint y sets and robust risk measures are also given. 6.1 Capital allo cation under robustification /ambiguit y W e start recalling the main features of capital allocation problems in order to pro vide an application to the robustness issue. Roughly speaking, a capital allocation rule replies to the question of ho w to share the capital requiremen t for an aggregate risky p osition (prescrib ed by a giv en risk measure ρ ) among its differen t sub-units. Here below, w e recall the classical definition of a capital allo cation rule. Definition 2 (see Kalkbrener (2005)) . Given a risk me asur e ρ : L p → [ −∞ , + ∞ ] , a c apital al lo c ation rule (CAR) for ρ is a map Λ : L p × L p → [ −∞ , + ∞ ] such that Λ( Y , Y ) = ρ ( Y ) for every Y ∈ L p . Notice that a CAR Λ is defined for an y pair ( X, Y ) ∈ L p × L p , where Y is alw a ys interpreted 19 as an aggregate p osition (or p ortfolio) and X as a sub-unit (or sub-p ortfolio). The condition Λ( Y , Y ) = ρ ( Y ) means that the capital allocated to Y when considered as a stand-alone port- folio is exactly the margin ρ ( Y ). Moreov er, Kalkbrener (2005) defines the following desirable prop erties for a capital allo cation rule Λ asso ciated to a risk measure ρ : - F ul l al lo c ation : for an y ( Y i ) i =1 ,...,n ⊆ L p , Y ∈ L p with P n i =1 Y i = Y , then P n i =1 Λ( Y i , Y ) = Λ( Y , Y ). - No-under cut : Λ( X, Y ) ≤ ρ ( X ) for every X , Y ∈ L p . F ull allo cation is satisfied by the most popular capital allo cation rules at least for the coheren t case, and it implies that the whole risk capital is entirely split among the sub-units of a risky p osition. Since Λ( X , Y ) represents the capital allo cated to the sub-p ortfolio X to hedge the total p ortfolio Y , the no-undercut axiom requires that the capital allo cated to X as a sub-unit of Y does not exceed the margin required for X . In the terminology of Tsanak as (2009), no-undercut corresp onds to the non-split requirement of X from Y , while from a game theory p erspective, it corresp onds to the core prop erty of Denault (2001). Although full allo cation and no-undercut are someho w incompatible when dealing with the non-coheren t case, capital allo cation can still b e p erformed for general risk measures without losing soundness (see, e.g., Centrone and Rosazza Gianin (2018); Canna et al. (2021)). When ambiguit y is introduced, b oth the ev aluation of total risk and the capital allo cation are affected. T o capture this effect, we extend the notion of capital allo cation to the robust framew ork by defining the “robustified” capital allo cation rule ˜ Λ asso ciated with the robust risk measure ˜ ρ . Definition 3. L et U b e a family of unc ertainty sets and Λ b e a CAR for a risk me asur e ρ . We define the r obustifie d version of Λ as a map ˜ Λ : L p × L p → [ −∞ ; + ∞ ] given by ˜ Λ( X, Y ) ≜ sup Z ∈U X Λ( Z, Y ) , for any X, Y ∈ L p . (29) The prop erties of ˜ Λ as a CAR for the robust risk measure ˜ ρ are inv estigated here b elow. As exp ected, the prop ert y of full allo cation is not preserved under robustification. No-undercut, instead, is fulfilled. Prop osition 7. L et Λ b e a CAR for a risk me asur e ρ and assume that the family U = ( U X ) X ∈ L p of unc ertainty sets is such that X ∈ U X for any X ∈ L p . a) If Λ satisfies no-under cut, then the r obustifie d ˜ Λ define d in (29) satisfies no-under cut and ρ ( Y ) ≤ ˜ Λ( Y , Y ) ≤ ˜ ρ ( Y ) , for any Y ∈ L p . (30) b) If ∪ n i =1 U Y i ⊇ U Y for any ( Y i ) i =1 ,...,n ⊆ L p and Y ∈ L p with P n i =1 Y i = Y , then ˜ Λ( Y , Y ) ≤ max i =1 ,...,n ˜ Λ( Y i , Y ) . If, in addition, 0 ∈ U Y i for any i = 1 , ..., n and Λ(0 , Y ) ≥ 0 , then ˜ Λ( Y , Y ) ≤ P n i =1 ˜ Λ( Y i , Y ) . 20 Pr o of. a) The no-undercut of Λ implies the no-undercut of ˜ Λ. Indeed, for any X , Y ∈ L p , ˜ Λ( X, Y ) = sup Z ∈U X Λ( Z, Y ) ≤ sup Z ∈U X ρ ( Z ) = ˜ ρ ( X ) . In particular, ˜ Λ( Y , Y ) ≤ ˜ ρ ( Y ). It remains to pro ve the first inequalit y in (30). Since Y ∈ U Y for an y Y ∈ L p , it holds that ˜ Λ( Y , Y ) = sup Z ∈U Y Λ( Z, Y ) ≥ Λ( Y , Y ) = ρ ( Y ) , where the last equality follows from Λ b eing a CAR for ρ . This completes the pro of of item a). b) Consider any ( Y i ) i =1 ,...,n ⊆ L p and Y ∈ L p with P n i =1 Y i = Y . By definition of ˜ Λ and by the prop erties on the uncertaint y sets, ˜ Λ( Y , Y ) = sup Z ∈U Y Λ( Z, Y ) ≤ sup Z ∈∪ n i =1 U Y i Λ( Z, Y ) (31) ≤ max i =1 ,...n sup Z ∈U Y i Λ( Z i , Y ) . Assume now that 0 ∈ U Y i for any i = 1 , ..., n and Λ(0 , Y ) ≥ 0. This implies that, for any i = 1 , .., n , sup Z ∈U Y i Λ( Z i , Y ) ≥ Λ(0 , Y ) ≥ 0 . (32) F rom (31) and (32), it then follows that ˜ Λ( Y , Y ) ≤ sup Z ∈∪ n i =1 U Y i Λ( Z, Y ) ≤ n X i =1 sup Z ∈U Y i Λ( Z, Y ) = n X i =1 ˜ Λ( Y i , Y ) . The previous result underlines that, in general, ˜ Λ is not necessarily a CAR but only a sub- CAR, meaning that ˜ Λ( Y , Y ) do es not necessarily coincide with ˜ ρ ( Y ) but ˜ Λ( Y , Y ) ≤ ˜ ρ ( Y ). See Cen trone and Rosazza Gianin (2018) for a detailed discussion on sub-CARs. In particular, for a stand-alone p ortfolio ( Y , Y ), the robustified ˜ Λ requires to allocate a greater amount than the margin ev aluated with the non-robust ρ ( Y ) but a low er amount than ˜ ρ ( Y ), that is, the margin prescrib ed by the robust ˜ ρ . Lo osely sp eaking, the prop osed ˜ Λ takes in to account ambiguit y (by “o verloading” ρ ( Y )) in a mo derate wa y , that is, without requiring the allo cation of the whole 21 robust capital ˜ ρ ( Y ). As exp ected, also full allo cation fails to b e satisfied for the robustified version of Λ. This fact is not surprising at all also reminding that no-undercut and full allo cation do not hold together for CARs of general risk measures (see the discussion in Centrone and Rosazza Gianin (2018)). How ever, under the additional assumptions that null p ositions b elong to all U Y i and that Λ(0 , Y ) ≥ 0, a w eaker prop ert y than full allo cation is fulfilled. This prop ert y means that the sum of the capital allocated to an y sub-unit Y i is greater than the total capital allo cated to the aggregate position Y . Financially speaking, P n i =1 ˜ Λ( Y i , Y ) − ˜ Λ( Y , Y ) ≥ 0 can b e in terpreted as a “security” extra capital to b e allo cated also in view of am biguity/robustification. Note that h yp othesis Λ(0 , Y ) = 0 (or, more generally , Λ(0 , Y ) ≥ 0) is reasonable and quite commonly assumed in the literature. Indeed, this means that the capital allo cated to a null p osition is zero (or, more generally , that even “doing nothing” ma y require capital). 6.2 La w in v ariant case under W asserstein distance W e apply now the robustification to the case of law inv ariant risk measures and uncertain ty sets based on the W asserstein distance. A notable example of a family of uncertaint y sets is, indeed, given by closed balls, centered at X ∈ L p with resp ect to a suitable metric and with a fixed radius ϵ > 0. Sp ecifically , one can consider the W asserstein distance of order p (see Villani (2021) for a comprehensive discussion of this metric), defined as d W p ( X, Y ) ≜  Z 1 0 | F X ( u ) − F Y ( u ) | p du  1 /p , for p ∈ [1 , + ∞ ) . F or p = + ∞ , one can set d W ∞ ( X, Y ) = lim p →∞ d W p ( X, Y ). Consider no w uncertain ty sets of the following form: U X = { Z ∈ L p : d W p ( X, Y ) ≤ ϵ } , for any X ∈ L p . (33) Lik e closed balls, these sets fit directly into our framew ork and hav e the additional prop erty of b eing conv ex. Moreo ver, it is straigh tforward to verify that they generate a law-in v ariant, closed, order preserving and conv ex family . In the following result, w e pro vide an estimate on ˜ ρ when uncertain ty sets are based on the W asserstein distance. This result is inspired b y Theorem 2 of Righi (2024). Compared to the result of this author (holding for conv ex cash-additive risk measures), ours is formulated for quasi-con vex and cash-subadditive risk measures. Due to this generalization, w e are only able to prov e an estimate of ˜ ρ . Prop osition 8. L et p ∈ [1 , + ∞ ] and ρ b e a quasi-c onvex and c ash-sub additive law invariant risk me asur e with dual r epr esentation (2) with the supr emum attaine d, i.e. ρ ( X ) = max Q ∈P R ρ ( E Q [ − X ] , Q ) . 22 If the unc ertainty sets U X ar e chosen as in (33) , then ˜ ρ ( X ) ≤ ρ ( X ) + ϵ      dQ ∗ X dP      q , for any X ∈ L p , wher e Q ∗ X ∈ arg max R ρ ( E Q [ − X ] , Q ) . Pr o of. Let X ∈ L p b e arbitrarily fixed and let Q ∗ X ∈ arg max R ρ ( E Q [ − X ] , Q ). F ollowing Righi (2024), w e denote b y f Q ∗ X ( Z ) = sup Z ′ ∼ Z E Q ∗ X [ Z ′ ] = Z 1 0 F − 1 Z ( u ) F − 1 d Q ∗ X / d P ( u )d u, for any Z ∈ L p . Then, by Righi (2024, Theorem 2), φ Q ∗ X ( X ) = sup Z ∈U X f Q ∗ X ( − Z ) ≤ f Q ∗ X ( − X ) + ϵ     dQ ∗ X dP     q . (34) F rom the definition of ˜ ρ and the dual representation of ρ , it holds: ˜ ρ ( X ) = sup Z ∈U X max Q ∈P R ρ ( E Q [ − X ] , Q ) = sup Z ∈U X R ρ  E Q ∗ X [ − X ] , Q ∗ X  = R ρ  sup Z ∈U X E Q ∗ X [ − Z ] , Q ∗ X  (35) = R ρ  φ Q ∗ X ( X ) , Q ∗ X  , (36) where (35) follo ws from the non-expansivity of R ρ (as in the pro of of Theorem 1). Applying (34) and increasing monotonicity and non-expansivit y of R ρ to (36), we obtain ˜ ρ ( X ) ≤ R ρ f Q ∗ X ( − X ) + ϵ     dQ ∗ X dP     q , Q ∗ X ! ≤ R ρ  f Q ∗ X ( − X ) , Q ∗ X  + ϵ     dQ ∗ X dP     q = sup X ′ ∼ X R ρ  E Q ∗ X [ − X ′ ] , Q ∗ X  + ϵ     dQ ∗ X dP     q (37) ≤ sup X ′ ∼ X ρ ( X ′ ) + ϵ     dQ ∗ X dP     q (38) = ρ ( X ) + ϵ     dQ ∗ X dP     q , (39) where (37) is due to non-expansivity of R ρ (see the pro of of Theorem 1), (38) to the dual represen tation of ρ , and (39) to law inv ariance of ρ . 23 6.3 Examples of families of uncertaint y sets Here b elow, we pro vide some examples of c-quasi-conv ex (and solid) families U of uncertaint y sets built on quasi-conv ex and cash-subadditive risk measures. Example 3. Given a quasi-c onvex and c ash-sub additive risk me asur e ρ 1 , c onsider U X = { Z ∈ L ∞ : | ρ 1 ( Z ) − ρ 1 ( X ) | ≤ ε } , for any X ∈ L ∞ . We ar e now going to show that the family U = ( U X ) X ∈ L ∞ is c-quasi-c onvex. L et X, Y ∈ L ∞ and λ ∈ [0 , 1] b e fixe d arbitr arily. Without loss of gener ality, we assume that ρ 1 ( X ) ≥ ρ 1 ( Y ) . F r om the definition of U and quasi-c onvexity of ρ 1 , it fol lows that, for any Z ∈ U λX +(1 − λ ) Y , ρ 1 ( Z ) ≤ ρ 1 ( λX + (1 − λ ) Y ) + ε ≤ max { ρ 1 ( X ) , ρ 1 ( Y ) } + ε = ρ 1 ( X ) + ε. (40) If ρ 1 ( Z ) ≥ ρ 1 ( X ) − ε , then (40) implies that Z ∈ U X , thus Z ∈ U X ⊆ U X ∪ U Y + L ∞ + . If ρ 1 ( Z ) < ρ 1 ( X ) − ε , inste ad, we ne e d to verify that ther e exist K ∗ ∈ L ∞ + and Z ∗ ∈ U X ∪ U Y such that Z = Z ∗ + K ∗ . In p articular, it would b e sufficient to pr ove that ther e exist k ∗ ∈ R + such that the r andom variable Z ∗ ≜ Z − k ∗ b elongs to U X . Inde e d, in this c ase, Z = Z ∗ + k ∗ ∈ U X + L ∞ + ⊆ U X ∪ U Y + L ∞ + . T o find a suitable k ∗ , define the function f : R + → R by: f ( k ) = ρ 1 ( Z − k ) , for any k ∈ R + . It fol lows imme diately that f is incr e asing (by monotonicity of ρ 1 ) and f (0) = ρ 1 ( Z ) . F urther- mor e, c ash-sub additivity of ρ 1 (henc e L ∞ -c ontinuity of ρ 1 by Pr op osition 2.1, b), of Cerr eia- Vio glio et al. (2011)) implies c ontinuity of f on R + . Set now: ¯ k ≜ ρ 1 ( X ) + ε − ρ 1 ( Z ) > 0 . By c ash-sub additivity of ρ 1 , it then fol lows that f ( ¯ k ) = ρ 1 ( Z − ¯ k ) ≥ ρ 1 ( Z ) + ¯ k = ρ 1 ( X ) + ε. Sinc e f is c ontinuous on [0 , ¯ k ] and its image c ontains the interval [ ρ 1 ( Z ) , ρ 1 ( X ) + ε ] , by the Interme diate V alue The or em ther e exists k ∗ ∈ [0 , ¯ k ] such that: f ( k ∗ ) ∈ [ ρ 1 ( X ) − ε, ρ 1 ( X ) + ε ] , i.e., ρ 1 ( Z − k ∗ ) ∈ [ ρ 1 ( X ) − ε, ρ 1 ( X ) + ε ] . So, Z ∗ = Z − k ∗ ∈ U X . 24 By the ab ove ar guments, the family U is c-quasi-c onvex (but it fails to satisfy monotonicity). Example 4. We now slightly mo dify Example 3 in or der to obtain a family U also satisfying solidity and monotonicity. Set U X = { Z ∈ L ∞ : ρ 1 ( Z ) ≤ ρ 1 ( X ) + ε } , for any X ∈ L ∞ . (41) c-quasi-c onvexity c an b e verifie d as in Example 3. Solidity is an imme diate c onse quenc e of monotonicity of ρ 1 . R e gar ding monotonicity, let X , Y ∈ L ∞ with X ≤ Y . If Z ∈ U Y , then, by monotonicity of ρ 1 , ρ 1 ( Z ) ≤ ρ 1 ( Y ) + ε ≤ ρ 1 ( X ) + ε, so Z ∈ U X , and thus U Y ⊆ U X . T o sum up, the family U in (41) satisfies c-quasi-c onvexity, solidity and monotonicity, henc e, by L emma 1, ii), also quasi-c onvexity. Remark 2. T o b e mor e c oncr ete, two r elevant c ases of risk me asur es ρ 1 fal ling within the setting of Examples 3 and 4 ar e, for instanc e, those taking into ac c ount ambiguity with r esp e ct to disc ount factors and those b ase d on gener alize d entr opy. Am biguity with respect to discount factors. Example 7.2 in El Kar oui and R avanel li (2009) c overs the c ase of a r e gulator who is evaluating the riskiness of financial p ositions under ambi- guity ab out the (sto chastic) disc ount factor D , which she/he b elieves lies within a known r ange [ d min , d max ] with d min , d max ∈ [0 , 1] b eing two c onstants. If the initial (sp ot) risk me asur e is ρ 0 , then the risk me asur e ρ 1 that takes into ac c ount ambiguity on the disc ount factor via worst-c ase sc enarios, would b e define d as ρ 1 ( X ) = sup 0 ≤ d min ≤ D ≤ d max ≤ 1 ρ 0 ( D X ) , for any X ∈ L ∞ . Se e El Kar oui and R avanel li (2009) for a detaile d tr e atment. Generalized entrop y . q-entr opic risk me asur es on losses ar e intr o duc e d in Di Nunno and R osazza Gianin (2024) to pr ovide a family of c ash non-additive (c onvex) risk me asur es gener al- izing entr opic ones and, in a dynamic setting and onc e mo difie d, fulfil ling the so-c al le d horizon longevity. Namely, a q-entr opic risk me asur e on losses is define d as ρ 1 ( X ) = ln q E [exp q (( X + β ) − )] , for q ∈ (0 , 1) , (42) wher e β > 0 is a fixe d tar get, ln q and exp q gener alize the the exp onential and lo garithmic 25 functions 2 , and ln q E [exp q ( · )] is the gener alize d entr opic risk me asur e (se e Tsal lis (1988, 2009)). As ar gue d in Ma and Tian (2021), c omp ar e d with the entr opic risk me asur e, the gener al- ize d entr opic risk me asur e do es not satisfy the c ash-additivity but satisfies c onvexity and c ash- sub additivity for any q ∈ (0 , 1) . The definition of q-entr opic risk me asur e as in (42) also guar ante es that it c an b e define d for al l risky p ositions and that is c onvex everywher e (se e Di Nunno and R osazza Gianin (2024)). 7 Conclusions This pap er dev elops a comprehensiv e framework for the robustification of risk measures beyond the classic al conv ex and cash-additive setting. W e in tro duced t wo complemen tary approaches for constructing robust quasi-con vex risk measures: one based on a quasi-conv ex base func- tional and con vex uncertaint y sets, and another relying on quasi-conv ex (or c-quasi-conv ex) uncertain ty families applied to a general risk measure. These tw o sc hemes clarify how the ge- ometry of the uncertaint y sets and the regularity of the base risk measure jointly determine the prop erties of the robust measure. Within this framework, we established general results on the preserv ation of monotonicit y , (quasi-)con vexit y , la w inv ariance, and contin uity from ab ov e under robustification. W e also c haracterized the largest, or consolidated, uncertaint y family asso ciated with a given robust risk measure, extending earlier results from the conv ex and cash-additive literature. Building on the dual represen tations of Cerreia-Vioglio et al. (2011) and F rittelli and Maggis (2011), w e deriv ed p enalt y-type dual forms for robust quasi-conv ex and cash-subadditiv e risk measures and show that the classical conv ex, cash-additive case emerges as a special instance of our general formulation. F urther, we analyzed acceptance families under robustification, establishing the relation b e- t ween the level sets of the base and robust risk measures and demonstrate ho w robustness can b e in terpreted as a systematic enlargement of acceptance sets to account for model ambiguit y . The theoretical results were illustrated through representativ e examples based on W asserstein distance and p -norm balls and c-quasi-conv ex uncertaint y families inspired by set-v alued analy- sis. Finally , w e extended the analysis to capital allocation rules, pro viding a consisten t approach under ambiguit y . The prop osed framew ork unifies and extends existing results on robust con vex risk measures, offering a general and flexible foundation for risk measuremen t and allo cation in the presence of 2 W e recall from Tsallis (1988, 2009) that exp q ( x ) ≜  1 + (1 − q ) x  1 1 − q      defined for an y x ≥ 1 q − 1 when q ∈ (0 , 1) defined for an y x < 1 q − 1 when q > 1 ln q ( x ) ≜ x 1 − q − 1 1 − q ( defined for an y x ≥ 0 when q ∈ (0 , 1) defined for an y x > 0 when q > 1 . 26 frictions, illiquidit y , or discounting am biguity . F uture researc h may address dynamic extensions, sto c hastic dominance consistency , or empirical applications to portfolio selection and stress testing under ambiguit y . References Artzner, P ., Delbaen, F., Eb er, J. M., Heath, D. (1999). Coherent measures of risk. Mathemat- ic al Financ e , 9(3), 203–228. Canna, E., Cen trone, A., Rosazza Gianin, E. (2021). Capital Allo cation Rules and the No- Undercut Prop ert y . Mathematics , 9(2), 175. 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