A theory for combinations of risk measures
aa r X i v : . [ q -f i n . M F ] J a n A theory for combinations of risk measures
Marcelo Brutti Righi ∗ Federal University of Rio Grande do Sul [email protected]
Abstract
We study combinations of risk measures under no restrictive assumption on the set ofalternatives. The main result is the representation for resulting risk measures from theproperties of both alternative functionals and combination functions. To that, we developa representation for arbitrary mixture of convex risk measures. In this case, we obtain apenalty that recall the notion of inf-convolution under theoretical measure integration. As anapplication, we address the context of probability-based risk measurements for functionals onthe set of distribution functions. We develop results related to this specific context. We alsoexplore features of individual interest generated by our framework, such as the preservationof continuity properties, the representation of worst-case risk measures, stochastic dominanceand elicitability.
Keywords : risk measures, uncertainty, combination, representations, probability-based func-tionals.
The theory of risk measures in mathematical finance has become mainstream, especially sincethe landmark paper of Artzner et al. (1999). For a comprehensive review, see the books ofPflug and R¨omisch (2007), Delbaen (2012) and F¨ollmer and Schied (2016). Nonetheless, thereis still no consensus about the best set of theoretical properties to possess, and even less re-garding the best risk measure. See Emmer et al. (2015) for a comparison of risk measures.This phenomenon motivates the proposition of new approaches, such as in Righi and Ceretta(2016) and Righi et al. (2019) for instance. Under the lack of a universal choice of a proper riskmeasure from a set of alternatives, one can think into a combination from candidates.In this paper we study risk measures of the form ρ = f ( ρ I ), where ρ I = { ρ i , i ∈ I} is a set ofalternative risk measures and f is some combination function. We propose a framework wherebyno assumption is made on the index set I , apart from non-emptiness. Typically, this kind ofprocedure uses a finite set of candidates, leading the domain of f to be some Euclidean space. Inour case, the domain of f is taken by a subset of the random variables over a measurable space ∗ We would like to thank Professors Ruodu Wang and Martin Schweizer for their comments, which have helpedto improve our manuscript. We are grateful for the financial support of FAPERGS (Rio Grande do Sul StateResearch Council) project number 17/2551-0000862-6 and CNPq (Brazilian Research Council) projects number302369/2018-0 and 407556/2018-4. I . From that, our main goal is to develop dual representations for such composedrisk measures from the properties of both ρ I and f in a general sense. For this purpose, weexpose results for some featured special cases, which are also of particular interest, such as aworst case and mixtures of risk measures.There are studies regarding particular cases for f , I and ρ I , such as the worst case inF¨ollmer and Schied (2002), the sum of monetary and deviation measures in Righi (2019), fi-nite convex combinations in Ang et al. (2018), scenario-based aggregation in Wang and Ziegel(2018), model risk-based weighting over a non-additive measure in Jokhadze and Schmidt (2018)and more. Nonetheless, our main contribution is the generality of our framework and resultsbecause the mentioned papers of the literature are special cases in our approach. Moreover, wedo not impose a restriction on the set of alternative risk measures, allowing for a more generalstructure.We apply our framework to the special situation where I is a subset of probability mea-sures since we frequently do not know if there is a correct one, but we have instead a set ofcandidates. We consider the concept of probability-based risk measurement, which is a collec-tion of risk measures from a functional on the set of distributions generated by probabilities in I . The stream of robust risk measures, as in Cont et al. (2010), Kratschmer et al. (2014) andKiesel et al. (2016), explores such uncertainty in a more statistical fashion, while our approach isprobabilistic. Works such as those of Bartl et al. (2019), Bellini et al. (2018), and Guo and Xu(2018) focus on particular risk measures, instead of a general framework. Laeven and Stadje(2013), Frittelli and Maggis (2018), Jokhadze and Schmidt (2018), Wang and Ziegel (2018) andQian et al. (2019) explore frameworks for risk measures considering multiple probabilities. How-ever, they do not develop exaclty the same features we do. In these studies, restrictive assump-tions are made on the set I , such as it being finite and possessing a reference measure. We inour turn solely assume non-emptiness.We have structured the rest of this paper as follows: in section 2 we expose preliminariesregarding notation, a brief background on the theory of risk measures in order to support ourframework and our proposed approach with some examples; in section 3 we present results re-garding properties of combination functions and how they affect the resulting risk measures inboth financial and continuity properties; in section 4 we develop and prove our results on repre-sentations of resulting risk measures in terms of properties from both ρ I and f for the generaland law invariant cases, as well we address a representation for the worst-case risk measure;in section 5 we explore the special framework of probability-based risk measurement, exposingresults specific to this context such as representations, stochastic orders and elicitability. Consider the atom-less probability space (Ω , F , P ). All equalities and inequalities are in the P -a.s. sense. We have that L = L (Ω , F , P ) and L ∞ = L ∞ (Ω , F , P ) are, respectively, thespaces of (equivalent classes under P -a.s. equality of) finite and essentially bounded randomvariables. We define 1 A as the indicator function for an event A ∈ F . We identify con-2tant random variables with real numbers. We say that a pair X, Y ∈ L is co-monotone if( X ( w ) − X ( w ′ )) ( Y ( w ) − Y ( w ′ )) ≥ , ∀ w, w ′ ∈ Ω. We denote by X n → X convergence in the L ∞ essential supremum norm k·k ∞ , while lim n →∞ X n = X means P -a.s. convergence. Let P be theset of all probability measures on (Ω , F ). We denote E Q [ X ] = R Ω Xd Q , F X, Q ( x ) = Q ( X ≤ x )and F − X, Q ( α ) = inf { x : F X, Q ( x ) ≥ α } , respectively, the expected value, the (increasing and right-continuous) probability function and its inverse for X under Q ∈ P . We write X Q ∼ Y when F X, Q = F Y, Q . We drop subscripts regarding probability measures when Q = P . Furthermore,let Q ⊂ P be the set of probability measures that are absolutely continuous in relation to P with Radon-Nikodym derivatives d Q d P . Definition 2.1.
A functional ρ : L ∞ → R is a risk measure. It may possess the followingproperties: • Monotonicity: if X ≤ Y , then ρ ( X ) ≥ ρ ( Y ) , ∀ X, Y ∈ L ∞ . • Translation Invariance: ρ ( X + C ) = ρ ( X ) − C, ∀ X, Y ∈ L ∞ , ∀ C ∈ R . • Convexity: ρ ( λX + (1 − λ ) Y ) ≤ λρ ( X ) + (1 − λ ) ρ ( Y ) , ∀ X, Y ∈ L ∞ , ∀ λ ∈ [0 , . • Positive Homogeneity: ρ ( λX ) = λρ ( X ) , ∀ X, Y ∈ L ∞ , ∀ λ ≥ . • Law Invariance: if F X = F Y , then ρ ( X ) = ρ ( Y ) , ∀ X, Y ∈ L ∞ . • Co-monotonic Additivity: ρ ( X + Y ) = ρ ( X ) + ρ ( Y ) , ∀ X, Y ∈ L ∞ with X, Y co-monotone. • Fatou continuity: if lim n →∞ X n = X ∈ L ∞ and { X n } ∞ n =1 ⊂ L ∞ bounded, then ρ ( X ) ≤ lim inf n →∞ ρ ( X n ) .We have that ρ is called monetary if it fulfills Monotonicity and Translation Invariance,convex if it is monetary and respects Convexity, coherent if it is convex and fulfills PositiveHomogeneity, law invariant if it has Law Invariance, co-monotone if it attends Co-monotonicAdditivity, and Fatou continuous if it possesses Fatou continuity. In this paper, we are workingwith normalized risk measures in the sense of ρ (0) = 0 . The acceptance set of ρ is defined as A ρ = { X ∈ L ∞ : ρ ( X ) ≤ } . Beyond usual norm and Fatou-based continuities, (a.s.) point-wise are relevant for riskmeasures.
Definition 2.2.
A risk measure ρ : L ∞ → R is said to be: • Continuous from above: lim n →∞ X n = X and { X n } decreasing implies in ρ ( X ) = lim n →∞ ρ ( X n ) , ∀ { X n } ∞ n =1 , X ∈ L ∞ . • Continuous from below: lim n →∞ X n = X and { X n } increasing implies in ρ ( X ) = lim n →∞ ρ ( X n ) , ∀ { X n } ∞ n =1 , X ∈ L ∞ . • Lebesgue continuous: lim n →∞ X n = X implies in ρ ( X ) = lim n →∞ ρ ( X n ) ∀ { X n } ∞ n =1 , X ∈ L ∞ . Theorem 2.3 (Theorem 2.3 of Delbaen (2002), Theorem 4.33 of F¨ollmer and Schied (2016)) . Let ρ : L ∞ → R be a risk measure. Then:(i) ρ is a Fatou continuous convex risk measure if, and only if, it can be represented as: ρ ( X ) = sup Q ∈Q (cid:8) E Q [ − X ] − α minρ ( Q ) (cid:9) , ∀ X ∈ L ∞ , (2.1) where α minρ : Q → R + ∪ {∞} , defined as α minρ ( Q ) = sup X ∈A ρ E Q [ − X ] , is a lower semi-continuous convex function called penalty term.(ii) ρ is a Fatou continuous coherent risk measure if, and only if, it can be represented as: ρ ( X ) = sup Q ∈Q ρ E Q [ − X ] , ∀ X ∈ L ∞ , (2.2) where Q ρ ⊆ Q is non-empty, closed and convex called the dual set of ρ . Example 2.4.
Examples of risk measures: • Expected Loss (EL): This is a Fatou continuous law invariant co-monotone coherent riskmeasure defined conform EL ( X ) = − E [ X ] = − R F − X ( s ) ds . We have that A EL = { X ∈ L ∞ : E [ X ] ≥ } and Q EL = { P } . • Value at Risk (VaR): This is a Fatou continuous law invariant co-monotone monetaryrisk measure defined as
V aR α ( X ) = − F − X ( α ) , α ∈ [0 , A V aR α = { X ∈ L ∞ : P ( X < ≤ α } . • Expected Shortfall (ES): This is a Fatou continuous law invariant co-monotone coherentrisk measure defined conform ES α ( X ) = α R α V aR s ( X ) ds, α ∈ (0 ,
1] and ES ( X ) = V aR ( X ) = − ess inf X . We have A ES α = (cid:8) X ∈ L ∞ : R α V aR s ( X ) ds ≤ (cid:9) and Q ES α = n Q ∈ Q : d Q d P ≤ α o . • Maximum loss (ML): This is a Fatou continuous law invariant coherent risk measuredefined as
M L ( X ) = − ess inf X = F − X (0). We have A ML = { X ∈ L ∞ : X ≥ } and Q ML = Q .When there is Law Invariance, which is the case in most practical applications, interestingfeatures are present. Theorem 2.5 (Theorem 2.1 of Jouini et al. (2006) and Proposition 1.1 of Svindland (2010)) . Let ρ : L ∞ → R be a law invariant convex risk measure. Then ρ is Fatou continuous. Theorem 2.6 (Theorems 4 and 7 of Kusuoka (2001), Theorem 4.1 of Acerbi (2002), Theorem7 of Fritelli and Rosazza Gianin (2005)) . Let ρ : L ∞ → R be a risk measure. Then: i) ρ is a law invariant convex risk measure if, and only if, it can be represented as: ρ ( X ) = sup m ∈M (Z (0 , ES α ( X ) dm − β minρ ( m ) ) , ∀ X ∈ L ∞ , (2.3) where M is the set of probability measures on (0 , and β minρ : M → R + ∪ {∞} , definedas β minρ ( m ) = sup X ∈A ρ R (0 , ES α ( X ) dm .(ii) ρ is a law invariant coherent risk measure if, and only if, it can be represented as: ρ ( X ) = sup m ∈M ρ Z (0 , ES α ( X ) dm, ∀ X ∈ L ∞ , (2.4) where M ρ = (cid:26) m ∈ M : R ( u,
1] 1 v dm = F − d Q d P (1 − u ) , Q ∈ Q ρ (cid:27) .(iii) ρ is a law invariant co-monotone coherent risk measure if, and only if, it can be representedas: ρ ( X ) = Z (0 , ES α ( X ) dm, ∀ X ∈ L ∞ , (2.5) where m ∈ M ρ . Let ρ I = { ρ i : L ∞ → R , i ∈ I} be some (a priori specified) collection of risk measures, where I is a non-empty set. We write, for fixed X ∈ L ∞ , ρ I ( X ) = { ρ i ( X ) , i ∈ I} . We would liketo define risk measures conform ρ ( X ) = f ( ρ I ( X )), where f is some combination (aggregation)function. When I is finite with dimension n , we have that f : R n → R . This situation, whichis common in practical matters, brings simplification to the framework. However, when I is anarbitrary set we need a more complex setup.Consider the measurable space ( I , G ), where G is a sigma-algebra of sub-sets from I . Wedefine K = K ( I , G ) and K ∞ = K ∞ ( I , G ) as the spaces of point-wise finite and boundedrandom variables on ( I , G ), respectively. In these spaces we understand equalities, inequalitiesand limits in the point-wise sense. We define V as the set of probability measures in ( I , G ). Inorder to avoid measurability issues we make the following assumption. Assumption 2.7.
The maps R X : I → R , defined as R X ( i ) = ρ i ( X ), are G -measurable for any X ∈ L ∞ . Remark . A possible, but not unique, choice is when G = σ ( { i → ρ i ( X ) : X ∈ L ∞ } ) = σ ( { R − X ( B ) : B ∈ B ( R ) , X ∈ L ∞ } ), where B ( R ) is the Borel set of R . Other possibility is, ofcourse, the power set 2 I .Thus we can associate the domain of f with X = X ρ I = span ( { R ∈ K : ∃ X ∈ L ∞ s.t . R ( i ) = ρ i ( X ) , ∀ i ∈ I} ∪ { } ) = span ( { R ∈ K : R = R X , X ∈ L ∞ } ∪ { } ). The linear span is in orderto preserve vector space operations. We can identify ρ I ( X ) with R X by ρ I : L ∞ → X . Fromnormalization, we have R = 0. When ρ I ( X ) is bounded, which is the case for any monetaryrisk measure since ρ ( X ) ≤ ρ (ess inf X ) = − ess inf X < ∞ , we have that X ⊂ K ∞ . Under this5ramework, the composition is a functional f : X → R . We use, when necessary, the canonicalextension convention that f ( R ) = ∞ for R ∈ K \X . We consider normalized combinationfunctions conform f ( R ) = f (0) = 0. Example 2.9.
The worst-case risk measure is a functional ρ W C : L ∞ → R defined as ρ W C ( X ) = sup i ∈I ρ i ( X ) . (2.6)This risk measure is typically considered when the agent (investor, regulator etc.) seeks pro-tection. When I is finite the supremum is, of course, a maximum. This combination is thepoint-wise supremum f W C ( R ) = sup { R ( i ) : i ∈ I} . If X ⊂ K ∞ , then ρ W C < ∞ . It is straight-forwrd to verify that A ρ WC = T i ∈I A ρ i . When I = Q and ρ Q ( X ) = E Q [ − X ] − α ( Q ), with α : Q → R + ∪ {∞} such that inf { α ( Q ) : Q ∈ Q} = 0, we have that ρ W C becomes a Fatoucontinuous convex risk measure conform (2.1) in Theorem 2.3. Analogously, for a non-emptyclosed convex
I ⊆ Q and ρ Q ( X ) = E Q [ − X ], we have that ρ W C becomes a Fatou continuouscoherent risk measure conform (2.2) in this same Theorem 2.3. Analogous analysis can be madeto obtain law invariant convex and coherent risk measures conform (2.4) and (2.3), respectively,as in Theorem 2.6.
Example 2.10.
The weighted risk measure is a functional ρ µ : L ∞ → R defined as ρ µ ( X ) = Z I ρ i ( X ) dµ, (2.7)where µ is a probability on ( I , G ). This risk measure represents an expectation of R X regarding µ . Since i → ρ i ( X ) is G -measurable, the integral is well-defined. When I is finite, ρ µ is a convexmixture of the functionals which compose ρ I . The combination function is f µ ( R ) = R I Rdµ . Wehave that | ρ µ ( X ) − ρ µ ( X ) | ≤ | ρ W C ( X ) |k µ − µ k T V , where k·k
T V is the total variation norm.Hence it is somehow continuous (robust) regarding choice of the probability measure µ ∈ V .If I = (0 ,
1] and ρ i ( X ) = ES i ( X ), we have that ρ µ ( X ) defines a law invariant co-monotoneconvex risk measure conform (2.5), which is Fatou continuous due to Theorem 2.5. Example 2.11.
A spectral (distortion) risk measure is a functional ρ φ : L ∞ → R defined as ρ φ ( X ) = Z V aR α ( X ) φ ( α ) dα, (2.8)where φ : [0 , → R + is a decreasing functional such that R φ ( u ) du = 1. Any law invariant co-monotone convex risk measure can be expressed in this fashion. The relationship between thisrepresentation and the one in (2.5) is given by R ( u,
1] 1 v dm = φ ( u ), where m ∈ M ρ . When φ is notdecreasing, we have that the risk measure is not convex and the representation as combinationsof ES does not hold. Let I = [0 , λ the Lebesgue measure, and µ ≪ λ with φ ( i ) = F − dµdλ (1 − i ).Thus R I φdλ = 1 and ρ φ ( X ) = R I ρ i ( X ) φ ( i ) dλ . By choosing ρ i ( X ) = V aR i ( X ) we have thatany spectral risk measure is a special case of ρ µ .6 xample 2.12. Consider the risk measure ρ u ( X ) : L ∞ → R defined as ρ u ( X ) = u ( ρ I ( X )) , (2.9)where u : X → R is a monetary utility in the sense that if R ≥ S , then u ( R ) ≥ u ( S ) and u ( R + C ) = u ( R ) + C, C ∈ R . In this case the combination is f u = u . Note that u ( R ) can beidentified with π ( − R ), where π is a risk measure on X . For instance, one can pick π as EL,VaR, ES or ML. In these cases we would obtain for some base probability µ , respectively thefollowing combinations: f µ , F − R,µ (1 − α ), α R α F − R,µ (1 − s ) ds , and ess sup µ R . Definition 3.1.
A combination f : X → R may have the following properties: • Monotonicity: if R ≥ S , then f ( R ) ≥ f ( S ) , ∀ R, S ∈ X . • Translation Invariance: f ( R + C ) = f ( R ) + C, ∀ R, S ∈ X , ∀ C ∈ R . • Positive Homogeneity: f ( λS ) = λf ( S ) , ∀ R ∈ X , ∀ λ ≥ . • Convexity: f ( λR + (1 − λ ) S ) ≤ λf ( R ) + (1 − λ ) f ( S ) , ∀ λ ∈ [0 , , ∀ R, S ∈ X . • Additivity: f ( R + S ) = f ( R ) + f ( S ) , ∀ R, S ∈ X . • Fatou Continuity: If lim n →∞ R n = R ∈ K ∞ , with { R n } ∞ n =1 ⊂ X bounded, then f ( R ) ≤ lim inf n →∞ f ( R n ) .Remark . Such properties for the combination function f are in parallel to those of riskmeasures, exposed in Definition 2.1. Note the adjustment in signs from there. We use the sameterms indiscriminately for both f and ρ with reasoning to conform the context. We could haveimposed a determined set of properties for the combination. However, we choose to keep a moregeneral framework where it may or may not possess such properties. Proposition 3.3.
Let
X ⊂ K ∞ . We have that:(i) f W C defined as in Example 2.9 fulfills Monotonicity, Translation Invariance, PositiveHomogeneity, Convexity and Fatou continuity.(ii) f µ defined as in Example 2.10 fulfills Monotonicity, Translation Invariance, Positive Ho-mogeneity, Convexity, Additivity and Fatou continuity.Proof. (i) Monotonicity, Translation Invariance, Positive Homogeneity and Convexity areobtained directly from the definition of supremum. Regarding Fatou continuity, let { R n } ∞ n =1 ⊂ X bounded such that lim n →∞ R n = R ∈ K ∞ . Then we have that f W C ( R ) = sup lim n →∞ R n ≤ lim inf n →∞ sup R n = lim inf n →∞ f W C ( R n ) . f µ respects Monotonicity, TranslationInvariance, Positive Homogeneity, Convexity and Additivity. For Fatou continuity, let { R n } ∞ n =1 ⊂ X bounded such that lim n →∞ R n = R ∈ K ∞ . Then we have from DominatedConvergence that f µ ( R ) = Z I lim n →∞ R n dµ ≤ lim inf n →∞ Z I R n dµ = lim inf n →∞ f µ ( R n ) . Remark . Note that for any combination f with the property of Boundedness, i.e | f ( R ) | ≤ f W C ( R ) , ∀ R ∈ X , we have ρ ( X ) ≤ ρ W C ( X ). Consequently A ρ WC ⊆ A ρ . From Theorem 2.3applied to functionals over K ∞ , we have that { f µ } µ ∈V are the only combination functions thatfulfill all properties in Definition 3.1. Proposition 3.5.
Let ρ I = { ρ i : L ∞ → R , i ∈ I} be a collection of risk measures, f : X → R ,and ρ : L ∞ → R a risk measure defined as ρ ( X ) = f ( ρ I ( X )) = f ( R X ) . Then:(i) If ρ I is composed of risk measures with Monotonicity and f posses this same property,then also does ρ .(ii) If ρ I is composed of risk measures with Translation Invariance and f posses this sameproperty, then also does ρ .(iii) If ρ I is composed of risk measures with Convexity and f posses this same property in pairwith Monotonicity, then ρ fulfills Convexity.(iv) If ρ I is composed of risk measures with Positive Homogeneity and f posses this sameproperty, then also does ρ .(v) If ρ I is composed of law invariant risk measures, then ρ fulfills Law Invariance.(vi) If ρ I is composed of co-monotone risk measures and f fulfills Additivity, then ρ has Co-monotonic Additivity.(vii) If ρ I is composed of Fatou continuous uniformly bounded risk measures and f has Fatoucontinuity in pair with Monotonicity, then also does ρ .Proof. (i) Let X, Y ∈ L ∞ with X ≥ Y . Then ρ i ( X ) ≤ ρ i ( Y ) , ∀ i ∈ I . Thus, R X ≤ R Y and ρ ( X ) = f ( R X ) ≤ f ( R Y ) = ρ ( Y ).(ii) Let X ∈ L ∞ and C ∈ R . Then ρ i ( X + C ) = ρ i ( X ) − C, ∀ i ∈ I . Thus, ρ ( X + C ) = f ( R X + C ) = f ( R X − C ) = f ( R X ) − C = ρ ( X ) − C .(iii) Let X, Y ∈ L ∞ and λ ∈ [0 , ρ i ( λX + (1 − λ ) Y ) ≤ λρ i ( X ) + (1 − λ ) ρ i ( Y ) , ∀ i ∈ I .Thus, ρ ( λX + (1 − λY )) = f ( R λX +(1 − λY ) ) ≤ f ( λR X + (1 − λ ) R Y ) ≤ λρ ( X ) + (1 − λ ) ρ ( Y ).8iv) Let X ∈ L ∞ and λ ≥
0. Then ρ i ( λX = λρ i ( X ) , ∀ i ∈ I . Thus ρ ( λX ) = f ( R λX ) = f ( λR X ) = λf ( R X ) = λρ ( X ).(v) Let X, Y ∈ L ∞ such that F X = F Y . Then ρ i ( X ) = ρ i ( Y ) , ∀ i ∈ I . Thus R X = R Y point-wisely. Hence, R X and R Y belong to the same equivalence class on X and ρ ( X ) = f ( R X ) = f ( R Y ) = ρ ( Y ).(vi) Let X, Y ∈ L ∞ be a co-monotone pair. Then ρ i ( X + Y ) = ρ i ( X ) + ρ i ( Y ) , ∀ i ∈ I . Thus ρ ( X + Y ) = f ( R X + Y ) = f ( R X + R Y ) = f ( R X ) + f ( R Y ) = ρ ( X ) + ρ ( Y ).(vii) Let { X n } ∞ n =1 ⊂ L ∞ bounded with lim n →∞ X n = X ∈ L ∞ . Then ρ i ( X ) ≤ lim inf n →∞ ρ i ( X n ) , ∀ i ∈I . Thus ρ ( X ) = f ( R X ) ≤ f (lim inf n →∞ R X n ) ≤ lim inf n →∞ f ( R X n ) = lim inf n →∞ ρ ( X n ). Remark . Converse relations are not always guaranteed. For instance, spectral risk measuresin Example 2.11 are convex despite the collection { V aR α , α ∈ [0 , } is not in general. Proposition 3.7.
Let ρ I = { ρ i : L ∞ → R , i ∈ I} be a collection of risk measures, f : X → R ,and ρ : L ∞ → R a risk measure defined as ρ ( X ) = f ( ρ I ( X )) . Then:(i) If f fulfills Monotonicity, Sub-Additivity, i.e. f ( R + S ) ≤ f ( R ) + f ( S ) , ∀ R, S ∈ X , andBoundedness, and ρ i is Lipschitz continuous ∀ i ∈ I , then ρ is Lipschitz continuous.(ii) If f is Fatou continuous and ρ i fulfills, ∀ i ∈ I , continuity from above, below or Lebesgue,then ρ is Fatou continuous for decreasing sequences, increasing sequences, or any se-quences, respectively.(iii) If f is Lebesgue continuous, i.e. lim n →∞ R n = R implies f ( R ) = lim n →∞ f ( R n ) , ∀ { R n } ∞ n =1 , R ∈X ∩ K ∞ , and ρ i with any property among continuity from above, below or Lebesgue, ∀ i ∈ I ,then ρ also have.Proof. (i) From Monotonicity and Sub-Additivity of f and R X ≤ R Y + | R X − R Y | we havethat | f ( R X ) − f ( R Y ) | ≤ f ( | R X − R Y | ). Moreover, | ρ i ( X ) − ρ i ( Y ) | ≤ k X − Y k ∞ , ∀ X, Y ∈ L ∞ , ∀ i ∈ I . Then from boundedness of f we get | ρ ( X ) − ρ ( Y ) | ≤ f ( | R X − R Y | ) ≤ f W C ( | R X n − R X | ) ≤ k X − Y k ∞ , ∀ X, Y ∈ L ∞ . (ii) Let { X n } ∞ n =1 ⊂ L ∞ bounded such that lim n →∞ X n = X ∈ L ∞ and ρ i Lebesgue continuousfor any i ∈ I . Then we have lim n →∞ R X n = R X point-wise. Hence ρ ( X ) = f (cid:16) lim n →∞ R X n (cid:17) ≤ lim inf n →∞ f ( R X n ) = lim inf n →∞ ρ ( X n ) . When each ρ i is continuous from above or below, the same reasoning which is restrictedto decreasing or increasing sequences, respectively, is valid.9iii) Similar to (ii), but in this case f (cid:16) lim n →∞ R X n (cid:17) = lim n →∞ f ( R X n ). Remark . Let I = Q , ρ Q ( X ) = E Q [ − X ] and f = f W C . In this case we have ρ = M L ,which is not continuous from below even ρ Q possessing such property. However, ML is Fatoucontinuous. This example illustrates the item (ii) in the last Proposition. Moreover, f µ satisfiesLebesgue continuity as in (iii) when X ⊂ K ∞ , which is the case for monetary risk measures forinstance. In this section, we expose results regarding the representation of composed risk measures ρ = f ( ρ I ) based on the properties of both ρ I and f . The goal is to highlight the role of such terms.We begin the preparation with a Lemma for representation of f , without dependence on theproperties of ρ I . Lemma 4.1.
Let
X ⊂ K ∞ . A functional f : X → R , posses Monotonicity, Translation Invari-ance, Convexity and Fatou continuity if, and only if, it can be represented conform f ( R ) = sup µ ∈V (cid:26)Z I Rdµ − γ f ( µ ) (cid:27) , ∀ R ∈ X , (4.1) where γ f : V → R + ∪ {∞} is defined as γ f ( µ ) = sup R ∈X (cid:26)Z I Rdµ − f ( R ) (cid:27) . (4.2) Proof.
The fact that (4.1) possesses Monotonicty, Translation Invariance, Convexity and Fatoucontinuity is straightforward. For the only if direction, one can understand f ( R ) as π ( − R ),where π is a Fatou continuous convex risk measure on K ∞ . Note that it is finite. Thus, fromTheorem 4.22 of F¨ollmer and Schied (2016), which is similar to Theorem 2.3 but without a baseprobability, and f ( R ) = ∞ for any R ∈ K ∞ \X we have that f ( R ) = π ( − R ) = sup µ ∈V (cid:26)Z I Rdµ − sup R ∈X (cid:20)Z I Rdµ − f ( R ) (cid:21)(cid:27) = sup µ ∈V (cid:26)Z I Rdµ − γ f ( µ ) (cid:27) . Remark . When R = R X point-wisely for some X ∈ L ∞ , we have that the representationbecomes f ( R X ) = sup µ ∈V { ρ µ ( X ) − γ f ( µ ) } . If f posses Positive Homogeneity, then γ f assumes value 0 in V f = { µ ∈ V : f ( R ) ≥ R I Rdµ, ∀ R ∈X } and ∞ otherwise. For instance, V f µ = { µ } and V f WC = V . Note that inf µ ∈V γ f ( µ ) = 0 fromthe assumption of normalization for f . 10e need the following auxiliary result, which may be of individual interest, for interchangingthe supremum and integral in a specific case that it is useful in posterior results. The followingLemma is a generalization of the (countable) additive property of supremum over the sum ofsets. Lemma 4.3.
Let ( I , G , µ ) be a probability space, h i : Y → R , i ∈ I , a collection of boundedfunctionals over a non-empty space Y such that i → h i ( y i ) is G -measurable for any { y i ∈ Y , i ∈I} . Then Z I sup y ∈Y i h i ( y ) dµ = sup { y i ∈Y i , i ∈I} Z I h i ( y i ) dµ where, for any i ∈ I : Y i ⊆ Y , Y i = ∅ , and sup y ∈Y i h i ( y ) is G -measurable.Proof. Note that the axiom of choice guarantees the existence of { y i ∈ Y i , i ∈ I} . For every ǫ > i ∈ I , y ∗ i ∈ Y i such thatsup y ∈Y i h i ( y ) − ǫ ≤ h i ( y ∗ i ) ≤ sup y ∈Y i h i ( y ) . Then, by integrating over I in relation to µ we obtain Z I sup y ∈Y i h i ( y ) dµ − ǫ ≤ Z I h i ( y ∗ i ) dµ ≤ Z I sup y ∈Y i h i ( y ) dµ. Because y ∗ i ∈ Y i , ∀ i ∈ I , by taking the supremum over all sets { y i ∈ Y i , i ∈ I} , we get Z I sup y ∈Y i h i ( y ) dµ − ǫ ≤ sup { y i ∈Y i , i ∈I} Z I h i ( y i ) dµ ≤ Z I sup y ∈Y i h i ( y ) dµ. Since one can take ǫ arbitrarily, the result follows.We also need an assumption to circumvent some measurability issues in order to avoidindefiniteness of posterior measure related concepts, such as integration. Assumption 4.4.
When ρ I = { ρ i : L ∞ → R , i ∈ I} is a collection of Fatou continuousconvex risk measures we assume that the following maps are G -measurable for any collection { Q i ∈ P , i ∈ I} :(i) i → E Q i [ X ] , ∀ X ∈ L ∞ .(ii) i → α minρ i ( Q i ). Remark . Similarly to Assumption 2.7, as a single, but not unique, example for G one couldconsider the sigma-algebra generated by all such maps. Note that the maps i → Q i ( A ) , ∀ A ∈ F are also G -measurable since by choosing X = − A , A ∈ F we get E Q [ − ( − A )] = Q ( A ) forany Q ∈ P . Moreover, also are G -measurable the maps i → α minρ i ( Q ) , ∀ Q ∈ P by picking Q i = Q ∀ i ∈ I . Note that we can consider the extension α minρ ( Q ) = ∞ , ∀ Q ∈ P\Q for anyminimal penalty term. 11 emma 4.6. Let ρ I = { ρ i : L ∞ → R , i ∈ I} is a collection of Fatou continuous convex riskmeasures. Then R I Q i dµ defines a probability measure for any { Q i ∈ P , i ∈ I} . Moreover, E R I Q i dµ [ X ] = R I E Q i [ X ] dµ for any { Q i ∈ P , i ∈ I} and any X ∈ L ∞ .Proof. For some { Q i ∈ P , i ∈ I} let Q µ ( A ) = R I Q i ( A ) dµ, ∀ A ∈ F . It is direct that both Q µ ( ∅ ) = 0 and Q µ (Ω) = 1. For countably additivity, let { A n } n ∈ N be a collection of mutuallydisjoint sets. Then, since i → Q i ( A ) is bounded ∀ A ∈ F we have Q µ ( ∪ ∞ n =1 A n ) = Z I ∞ X n =1 Q i ( A n ) dµ = ∞ X n =1 Z I Q i ( A n ) dµ = ∞ X n =1 Q µ ( A n ) . Hence Q µ is a probability measure. Regarding expectation interchange we have for any X ∈ L ∞ that E R I Q i dµ [ X ] = Z ∞ Z I Q i ( X ≥ x ) dµdx − Z −∞ Z I Q i ( X ≤ x ) dµdx = Z I (cid:18)Z ∞ Q i ( X ≥ x ) dx − Z −∞ Q i ( X ≤ x ) dx (cid:19) dµ = Z I E Q i [ X ] dµ. The role played by ρ µ becomes clear since it can be understood as the expectation under µ of elements R X ∈ X . Thus, it is important to know how the properties of ρ I affect the repre-sentation of ρ µ . Proposition 2.1 of Ang et al. (2018) explores a case with finite number coherentrisk measures, while we address a situation with an arbitrary set of convex risk measures. Theorem 4.7.
Let ρ I = { ρ i : L ∞ → R , i ∈ I} be a collection of Fatou continuous convex riskmeasures and ρ µ : L ∞ → R defined as in (2.7) . Then:(i) ρ µ can be represented as: ρ µ ( X ) = sup Q ∈Q { E Q [ − X ] − α ρ µ ( Q ) } , ∀ X ∈ L ∞ , (4.3) with convex α ρ µ : Q → R + ∪ {∞} defined as α ρ µ ( Q ) = inf (cid:26)Z I α minρ i (cid:0) Q i (cid:1) dµ : Z I Q i dµ = Q , Q i ∈ Q ∀ i ∈ I (cid:27) . (4.4) (ii) If in addition ρ i fulfills, for every i ∈ I , Positive Homogeneity, then the representation isconform ρ µ ( X ) = sup Q ∈ cl ( Q ρµ ) E Q [ − X ] , ∀ X ∈ L ∞ , (4.5) with Q ρ µ = (cid:8) Q ∈ Q : Q = R I Q i dµ, Q i ∈ Q ρ i ∀ i ∈ I (cid:9) convex and non-empty.Proof. (i) From Propositions 3.3 and 3.5 we have that ρ µ is a Fatou continuous convex risk12easure. Thus we have ρ µ ( X ) = Z I sup Q ∈Q n E Q [ − X ] − α minρ i ( Q ) o! dµ = sup { Q i ∈Q , i ∈I} (cid:26)Z I (cid:16) E Q i [ − X ] − α minρ i (cid:0) Q i (cid:1)(cid:17) dµ (cid:27) = sup Q ∈Q (cid:26) E Q [ − X ] − inf (cid:26)Z I α minρ i (cid:0) Q i (cid:1) dµ : Z I Q i dµ = Q , Q i ∈ Q ∀ i ∈ I (cid:27)(cid:27) = sup Q ∈Q { E Q [ − X ] − α ρ µ ( Q ) } . We have used Lemma 4.3 for the interchange of supremum and integral with Lemma 4.6for expectations. Assumption 4.4 is used to guarantee that integrals make sense. Notethat α ρ µ is well-defined since the infimum is not altered for the distinct choices of possiblecombinations that lead to R I Q i dµ = Q . Non-negativity for α ρ µ is straightforward. Wealso have α ρ µ ( Q ) = ∞ for any Q ∈ P\Q . For convexity, let λ ∈ [0 , Q , Q ∈ Q , Q = λ Q + (1 − λ ) Q , Q j = (cid:8) { Q i ∈ Q , i ∈ I} : R I Q i dµ = Q j (cid:9) , j ∈ { , , } , and theauxiliary set Q λ = (cid:8) { ( Q i , Q i ) ∈ Q × Q , i ∈ I} : R I ( λ Q i + (1 − λ ) Q i ) dµ = Q (cid:9) . We thusobtain λα ρ µ ( Q ) + (1 − λ ) α ρ µ ( Q ) ≥ inf { Q i ∈Q , i ∈I}∈Q , { Q i ∈Q , i ∈I}∈Q Z I α minρ i ( λ Q i + (1 − λ ) Q i ) dµ ≥ inf { ( Q i , Q i ) ∈Q×Q , i ∈I}∈Q λ Z I α minρ i ( λ Q i + (1 − λ ) Q i ) dµ ≥ inf { Q i ∈Q , i ∈I}∈Q Z I α minρ i ( Q i ) dµ = α ρ µ ( Q ) . (ii) In this framework we obtain ρ µ ( X ) = Z I sup Q ∈Q ρi E Q [ − X ] dµ = sup n Q i ∈Q ρi , i ∈I o Z I E Q i [ − X ] dµ = sup Q ∈Q ρµ E Q [ − X ]= sup Q ∈ cl ( Q ρµ ) E Q [ − X ] . To demonstrate that taking closure does not affect the supremum, let { Q n } ∞ n =1 ∈ Q µρ suchthat Q n → Q in the total variation norm. Then we have E Q [ − X ] = lim n →∞ E Q n [ − X ] ≤ sup n E Q n [ − X ] ≤ sup Q ∈Q ρµ E Q [ − X ] . Since every Q ρ i is non-empty, we have that Q ρ µ posses at least one element Q ∈ Q such13hat Q = R I Q i dµ, Q i ∈ Q ρ i ∀ i ∈ I . Let Q , Q ∈ Q ρ µ . Then, we have for any λ ∈ [0 , λ Q + (1 − λ ) Q = R I (cid:0) λ Q i + (1 − λ ) Q i (cid:1) dµ . Since Q ρ i is convex for any i ∈ I we have that λ Q + (1 − λ ) Q ∈ Q ρ µ as desired. Still remains to show that α minρ µ isan indicator function on cl ( Q µρ ). Note that α minρ i ( Q i ) = 0 , ∀ Q i ∈ Q iρ . Thus, α ρ µ is welldefined on Q ρ µ and we have that 0 ≤ α minρ µ ( Q ) ≤ α ρ µ ( Q ) = 0 , ∀ Q ∈ Q ρ µ . Due to the lowersemi-continuity property, we have that α minρ µ ( Q ) = 0 for any limit point Q of sequences in Q ρ µ . If Q ∈ Q\ cl ( Q ρ µ ), then α minρ µ ( Q ) = ∞ , otherwise the dual representation would beviolated. Remark . We have that α ρ µ in this case can be understood as some extension of the conceptof inf-convolution for arbitrary terms represented by theoretical measure and integral concepts.In fact, the sum of finite risk measures leads to the inf-convolutions of their penalty functions.By extrapolating the argument, such a result is also useful regarding general conjugate for anarbitrary mixture of convex functionals. Remark . Note that we could consider the families { Q i ∈ P , i ∈ I} that define both α ρ µ and Q ρ µ by belonging to determined sets in therms of µ − a.s. instead of point-wise in I . Thisis because the criteria is Lebesgue integral with respect to each specified µ ∈ V . We chooseto keep it point-wise in order to keep pattern since we have not assumed fixed probability om( I , G ) alongside the text. Remark . We have that α ρ µ is lower semi-continuous if, and only if, it coincides with α minρ µ . This because when α ρ µ is lower semi-continuous, by bi-duality regarding Legendre-Fenchel conjugates and the fact that ρ µ = ( α ρ µ ) ∗ , we obtain α ρ µ = ( α ρ µ ) ∗∗ = ( ρ µ ) ∗ = α minρ µ .In the case of finite cardinality for I , Q ρ µ is closed as exposed in Proposition 2.1 of Ang et al.(2018), which makes possible to drop the closure on 4.5 in such situation.We now have the necessary conditions to enunciate the main result in this section, which isa representation for composed risk measures in the usual framework of Theorem 2.3. Theorem 4.11.
Let ρ I = { ρ i : L ∞ → R , i ∈ I} be a collection of Fatou continuous convex riskmeasures, f : X → R possessing Monotonicity, Translation Invariance, Convexity and Fatoucontinuity, and ρ : L ∞ → R defined as ρ ( X ) = f ( ρ I ( X )) . Then:(i) ρ can be represented conform ρ ( X ) = sup Q ∈Q { E Q [ − X ] − α ρ ( Q ) } , ∀ X ∈ L ∞ , (4.6) where α ρ ( Q ) = inf µ ∈V { α ρ µ ( Q ) + γ f ( µ ) } , with γ f and α ρ µ defined as in (4.2) and (4.4) ,respectively.(ii) If in addition to the initial hypotheses f possess Positive Homogeneity, then the penaltyterm becomes α ρ ( Q ) = inf µ ∈V f α ρ µ ( Q ) , where V f is conform Remark 4.2.(iii) If in addition to the initial hypotheses ρ i possess, for any i ∈ I , Positive Homogeneity,then α ρ ( Q ) = ∞ ∀ Q ∈ Q\ ∪ µ ∈V cl ( Q ρ µ ) . iv) If in addition to the initial hypotheses we have the situations in (ii) and (iii), then therepresentation of ρ becomes ρ ( X ) = sup Q ∈Q V fρ E Q [ − X ] , ∀ X ∈ L ∞ , (4.7) where Q V f ρ is the closed convex hull of ∪ µ ∈V f cl ( Q ρ µ ) .Proof. From the hypotheses and Proposition 3.5 we have that ρ is a Fatou continuous convexrisk measure.(i) From Lemma 4.1 and Theorem 4.7 we have that ρ ( X ) = sup µ ∈V ( sup Q ∈Q [ E Q [ − X ] − α ρ µ ( Q )] − γ f ( µ ) ) = sup Q ∈Q (cid:26) E Q [ − X ] − inf µ ∈V [ α ρ µ ( Q ) + γ f ( µ )] (cid:27) = sup Q ∈Q { E Q [ − X ] − α ρ ( Q ) } . (ii) If f possesses Positive Homogeneity, then γ f assumes value 0 in V f and ∞ otherwise.Thus, we get α ρ ( Q ) = inf µ ∈V { α ρ µ ( Q ) + γ f ( µ ) } = inf µ ∈V f α ρ µ ( Q ) . (iii) When each element of ρ I fulfills Positive Homogeneity, we have that α ρ µ ( Q ) = ∞ ∀ µ ∈ V for any Q ∈ Q\ ∪ µ ∈V cl ( Q ρ µ ). By adding the non-negative term γ f ( µ ) and taking theinfimum over V , one gets the claim.(iv) In this context, the generated ρ is coherent from Theorem 3.5. Moreover, in this casefrom Lemma 4.1 and Proposition 4.7 together to items (ii) and (iii) we have that ρ ( X ) = sup µ ∈V f sup Q ∈ cl ( Q ρµ ) E Q [ − X ]= sup Q ∈∪ µ ∈V f cl ( Q ρµ ) E Q [ − X ]= sup Q ∈Q V fρ E Q [ − X ] . In order to verify that the supremum is not altered by considering the closed convex hull,let Q , Q ∈ ∪ µ ∈V f cl ( Q ρ µ ) and Q = λ Q + (1 − λ ) Q , λ ∈ [0 , E Q [ − X ] ≤ max( E Q [ − X ] , E Q [ − X ]) ≤ sup Q ∈∪ µ ∈V f cl ( Q ρµ ) E Q [ − X ] , thus convex combinations do not alter the supremum. For closure, the deduction is quitesimilar to that used in the proof of Theorem 4.7. We also have that α minρ is an indicatoron Q V f ρ . This fact is true because in this case α ρ µ ( Q ) = 0 for at least one µ ∈ V f for any15 ∈ ∪ µ ∈V f cl ( Q ρ µ ). The same is valid for convex combinations and limit points is due toconvexity and lower semi-continuity of α minρ . Remark . The supremum in (4.6) could be over ∪ µ ∈V Q µ ⊆ Q since α ρ ( Q ) = ∞ ∀ Q ∈Q\ ∪ µ ∈V Q µ . Note that when f = f µ we recover the result in Theorem 4.7. Moreover, Q V f ρ ⊆Q ρ WC since f ≤ f W C for any bounded combination f . Remark . When α ρ µ is lower semi-continuous we have that α ρ coincides with the minimalpenalty term because α minρ ( Q ) = sup X ∈ L ∞ { E Q [ − X ] − f ( R X ) } = sup X ∈ L ∞ ( E Q [ − X ] − sup µ ∈V { ρ µ ( X ) − γ f ( µ ) } ) = inf µ ∈V (cid:26) γ f ( µ ) + sup X ∈ L ∞ { E Q [ − X ] − ρ µ ( X ) } (cid:27) = α ρ ( Q ) . Regarding the specific case of ρ W C , Proposition 9 in F¨ollmer and Schied (2002) states that α ρ WC ( Q ) = inf i ∈I α minρ i ( Q ) , ∀ Q ∈ Q . Under coherence, Theorem 2.1 of Ang et al. (2018) claimsthat Q ρ WC = conv ( ∪ ni = i Q ρ i ) when I is finite with cardinality n . We now expose a result thatstates the equivalence between these facts and our approach, and generalize them. Proposition 4.14.
Let { ρ i : L ∞ → R , i ∈ I} be a collection of Fatou continuous convex riskmeasures, and ρ W C : L ∞ → R defined as in (2.9) . Then:(i) α ρ WC ( Q ) = inf µ ∈V α ρ µ ( Q ) = inf i ∈I α minρ i ( Q ) , ∀ Q ∈ Q .(ii) If in addition to the initial hypotheses ρ i possess, for any i ∈ I , Positive Homogeneity,then Q ρ WC = Q V ρ is the closed convex hull of ∪ i ∈I Q ρ i .Proof. From Propositions 3.3 and 3.5 we have that ρ W C is a Fatou continuous convex riskmeasure when all ρ i also are. Moreover, from the fact that ρ W C (0) = sup i ∈I ρ i (0) = 0 we havethat ρ W C takes only finite values.(i) For fixed Q ∈ P , we have that for any ǫ >
0, there is j ∈ I such thatinf i ∈I α minρ i ( Q ) ≤ α minρ j ( Q ) ≤ inf i ∈I α minρ i ( Q ) + ǫ. Recall that inf i ∈I α minρ i ( Q ) ≤ α ρ µ ( Q ) ≤ R I α minρ i ( Q ) dµ for any µ ∈ V . Then it is true that forany ǫ >
0, there is µ ∈ V such thatinf i ∈I α minρ i ( Q ) ≤ α ρ µ ( Q ) ≤ inf i ∈I α minρ i ( Q ) + ǫ. By taking the infimum over V and since ǫ was taken arbitrarily, we get that α ρ WC ( Q ) =inf i ∈I α minρ i ( Q ). 16ii) From Propositions 3.3 and 3.5 we have that ρ W C is a Fatou continuous coherent riskmeasure when all ρ i also are. Thus, in light of Theorem 2.3, it has a dual representation.We then have ρ W C ( X ) = sup i ∈I sup Q ∈Q ρi E Q [ − X ] = sup Q ∈∪ i ∈I Q ρi E Q [ − X ] = sup Q ∈Q ρWC E Q [ − X ] . The fact that supremum is not altered by considering the closed convex hull follows similarsteps as those in the proof of Theorem 4.11. We have that Q ρ WC is non-empty becauseevery Q ρ i contains at least P . Moreover, α minρ WC is an indicator function over Q ρ WC sincefor Q ∈ ∪ i ∈I Q ρ i , then α minρ i ( Q ) = 0 for at least one i ∈ I . Consequently 0 ≤ α minρ WC ( Q ) ≤ α ρ WC ( Q ) = 0. If Q ∈ Q ρ WC \ ∪ i ∈I Q ρ i , then Q is a convex combination or a limitpoint. In both cases α minρ WC ( Q ) = 0 from the convexity and lower semi-continuity of α minρ WC ,respectively. Regarding the equivalence between dual sets, note that all we have to showis that ∪ i ∈I Q ρ i = ∪ µ ∈V cl ( Q ρ µ ). This is true since for any i ∈ I we have that Q ∈ Q ρ i if,and only if, Q ∈ cl Q ρ δi , where δ i ∈ V is defined as δ i ( A ) = 1 A ( i ) , ∀ A ∈ G . Under Law Invariance of the components in ρ I , the generated ρ is representable in light of thoseformulations in Theorem 2.6. We begin addressing the measurability issues. Lemma 4.15.
Let ρ I = { ρ i : L ∞ → R , i ∈ I} be a collection of law invariant convex riskmeasures. Then the following maps are G -measurable for any { m i ∈ M , i ∈ I} :(i) i → R (0 , ES α ( X ) dm i , ∀ X ∈ L ∞ .(ii) i → m i ( B ) , ∀ B ∈ B (0 , , where B (0 , is the Borel sigma-algebra on (0 , .(iii) i → β minρ i ( m i ) .Proof. We have for any fixed { m i ∈ M , i ∈ I} the following:(i) From the definition of ES and Hardy-Littlewood inequality, for any m i ∈ M there is Q i ∈ Q such that R ( u,
1] 1 v dm i = F − d Q id P (1 − u ). Then the following relation holds: Z (0 , ES α ( X ) dm i = − Z F − X ( α ) F − d Q id P (1 − α ) dα = sup Y ∼ X E Q i [ − Y ] . From Assumption 4.4 we have that i → E Q i [ − Y ] is G -measurable for any Y ∼ X . More-over, for any n ∈ N , there is Y n ∼ X such that g n ( i ) = E Q i [ − Y n ] + 1 n ≥ sup Y ∼ X E Q i [ − Y ] ≥ E Q i [ − Y n ] = h n ( i ) . Note that both g n and h n are G -measurable for all n ∈ N and µ -integrable for any µ ∈ V since E Q i [ − Y n ] is bounded. The same is true for g = lim inf n →∞ g n and h = lim sup n →∞ h n . In17his case we have that g ( i ) ≥ α minρ i ( Q ) ≥ h ( i ) , ∀ i ∈ I . Moreover, we have Z I ( g n − h n ) dµ ≤ n , ∀ n ∈ N , ∀ µ ∈ V . From Fatou lemma it is true that0 ≤ Z I ( g − h ) dµ ≤ lim inf n →∞ Z I ( g n − h n ) dµ = 0 . Thus, g = h µ -a.s. for any µ ∈ V . Since δ i ∈ V , ∀ i ∈ I , with δ i ( A ) = 1 A ( i ) , ∀ A ∈ G , wethen have that g = h point-wisely. Then i → R (0 , ES α ( X ) dm i = sup Y ∼ X E Q i [ − Y ] = g ( i ) = h ( i ) is G -measurable for any { m i ∈ M , i ∈ I} .(ii) Let B ∈ B (0 , B = ∪ n ( a n , b n ], i.e. a countable union of intervals( a n , b n ] ⊂ (0 , n , we define a pair X n , Y n ∈ L ∞ such that P ( X n = −
1) = b n , P ( X n = 0) = 1 − b n , P ( Y n = −
1) = a n , P ( Y n = 0) = 1 − a n . Thus, for any m i ∈ M we have that m i ( B ) = X n Z (0 , ( a n ,b n ] dm i = X n Z (0 , (0 ,b n ] dm i − Z (0 , (0 ,a n ] dm i ! = X n Z (0 , ES α ( X n ) dm i − Z (0 , ES α ( Y n ) dm i ! . From item (i), we have that i → m i ( B ) = P n (cid:16)R (0 , ES α ( X n ) dm i − R (0 , ES α ( Y n ) dm i (cid:17) is G -measurable for any { m i ∈ M , i ∈ I} .(iii) From item (i) jointly to Theorems 2.3 and 2.6, we have that β minρ i ( m i ) = sup ( α minρ i ( Q ) : d Q d P ∼ d Q i d P , Z ( u, v dm i = F − d Q id P (1 − u ) , Q i ∈ Q ) = α minρ i ( Q i ) . Hence, from Assumption 4.4 the maps i → β minρ i ( m i ) = α minρ i ( Q i ) are G -measurable forany { m i ∈ M , i ∈ I} .We now need an auxiliary result for the representation when ρ µ is law invariant. The nextProposition follows in this direction. Proposition 4.16.
Let ρ I = { ρ i : L ∞ → R , i ∈ I} be a collection of law invariant convex riskmeasures and ρ µ : L ∞ → R defined as in (2.7) . Then:(i) ρ µ can be represented as: ρ µ ( X ) = sup m ∈M (Z (0 , ES α ( X ) dm − β ρ µ ( m ) ) , ∀ X ∈ L ∞ , (4.8)18 ith convex β ρ µ : M → R + ∪ {∞} , defined as β ρ µ ( m ) = inf (cid:26)Z I β minρ i ( m i ) dµ : Z I m i dµ = m, m i ∈ M ∀ i ∈ I (cid:27) . (4.9) (ii) If in addition ρ i fulfills, for every i ∈ I , Positive Homogeneity, then the representation isconform ρ µ ( X ) = sup m ∈ cl ( M ρµ ) Z (0 , ES α ( X ) dm, ∀ X ∈ L ∞ , (4.10) with M ρ µ = (cid:8) m ∈ M : m = R I m i dµ, m i ∈ M ρ i ∀ i ∈ I (cid:9) non-empty and convex.(iii) If ρ i also is, for every i ∈ I , co-monotone, then the representation is ρ µ ( X ) = Z (0 , ES α ( X ) dm, (4.11) where m ∈ cl ( M ρ µ ) .Proof. From the hypotheses and Proposition 3.5 we have that ρ µ is a law invariant convex riskmeasure. The proof follows similar steps to those of Theorem 4.7 with m i → R (0 , ES α ( X ) dm i linear and playing the role of Q i → E Q i [ − X ]. Note that from Lemma 4.15, i → R (0 , ES α ( X ) dm i is G -measurable. For the co-monotonic case in (iii), the result is due to the supremum in (2.5)be attained for each ρ i . Remark . The representation in item (iii) on (4.11) is equivalent to the spectral one conform ρ µ ( X ) = Z V aR α ( X ) φ µ ( α ) dα, (4.12)where φ µ ( α ) = R I φ i ( α ) dµ , and φ i : [0 , → [0 ,
1] is as in Example 2.11 for any i ∈ I . The map i → φ i ( α ) is G -measurable to any α ∈ [0 , i ∈ I it is true that φ i ( α ) = R ( α,
1] 1 s dm i ( s ) , m i ∈ M , then dν i dµ i = s defines a finite measure on (0 , η i ∈ M such that ν i = βη i , β ∈ R + . Since, from Lemma 4.15, i → η i ( α,
1] is G -measurable for any α ∈ [0 , i → ν i ( α,
1] = φ i ( α ).We are now able to propose a result for the dual representation under Law Invariance. Thenext Corollary exposes such content. Corollary 4.18.
Let ρ I = { ρ i : L ∞ → R , i ∈ I} be a collection of law invariant convex riskmeasures, f : X → R possessing Monotonicity, Translation Invariance, Convexity and Fatoucontinuity, and ρ : L ∞ → R defined as ρ ( X ) = f ( ρ I ( X )) . Then:(i) ρ can be represented conform ρ ( X ) = sup m ∈M (Z (0 , ES α ( X ) dm − β ρ ( m ) ) , ∀ X ∈ L ∞ , (4.13) where β ρ ( m ) = inf µ ∈V { β ρ µ ( m ) + γ f ( µ ) } , with γ f and β ρ µ defined as in (4.2) and (4.9) ,respectively. ii) If in addition to the initial hypotheses f possess Positive Homogeneity, then the penaltyterm becomes β ρ ( m ) = inf µ ∈V f β ρ µ ( m ) , where V f is conform Remark 4.2.(iii) If in addition to the initial hypotheses ρ i possess, for any i ∈ I , Positive Homogeneity,then β ρ ( m ) = ∞ , ∀ m ∈ M\ ∪ µ ∈V cl ( M ρ µ ) .(iv) If in addition to the initial hypotheses we have the situations in (ii) and (iii), then therepresentation of ρ becomes ρ ( X ) = sup m ∈M V fρ Z (0 , ES α ( X ) dm, ∀ X ∈ L ∞ , (4.14) where M V f ρ is the closed convex hull of ∪ µ ∈V f cl ( M ρ µ ) .(v) If in addition to the initial hypotheses ρ i possess, for any i ∈ I , Co-monotonic Additivity,then β ρ ( m ) = ∞ , ∀ m ∈ M\ ∪ µ ∈V { m µc } , where m µc = arg max m ∈ cl ( M ρµ ) Z (0 , ES α ( X ) dm, ∀ µ ∈ V . (vi) If in addition to the initial hypotheses we have (ii) and (v), then the representation of ρ becomes ρ ( X ) = sup m ∈M V fρ,c Z (0 , ES α ( X ) dm, ∀ X ∈ L ∞ , (4.15) where M V f ρ,c is the closed convex hull of ∪ µ ∈V f { m µc } .Proof. Direct from Theorem 4.11 and Proposition 4.16.
Remark . Since the co-monotonicity of a pair
X, Y does not imply the same property forthe pair R X , R Y , the only situation where ρ is surely co-monotone occurs, from Proposition 3.5and Lemma 4.1, when { f µ } µ ∈V . In this case we have ρ ( X ) = ρ µ ( X ) = Z (0 , ES α ( X ) dm µc , ∀ X ∈ L ∞ . (4.16)From (4.12), we have that φ µ ( α ) = R ( α,
1] 1 s dm µc ( s ), where φ µ is conform Remark 4.17. Remark . It would be possible also to investigate a situation where f posses representationin terms of ES over X as in Example 2.12, but we do not consider any base probability on( I , G ). However, note that those would be special cases of our framework. Nonetheless, we leftsuch task for future research. From now on we work on L ∞ = L ∞ (Ω , F ) the space of point-wise bounded random variableson (Ω , F ) by replacing P -a.s. concepts from previous sections by their point-wise counterparts.20ote that L ∞ ⊆ L ∞ ( Q ) , ∀ Q ∈ P . We denote F = { F X, Q : X ∈ L ∞ , Q ∈ P} . In thissection, uncertainty is linked to probabilities in the sense that I ⊆ P . We assume that each L ∞ ( Q ) , Q ∈ I , is atom-less. Extreme choices for I are a singleton or the whole P . Otherpossible choices are closed balls around a reference probability measure based on distance,metric, divergence, or relation, as in Shapiro (2017), for instance. We do not pursue suchdetails in order to conduct our approach in a more general fashion.We define risk measurement under the intuitive idea that we obtain the same functionalfrom distinct probabilities that represent scenarios. These may have different interpretations,such as models, economic situations, heterogeneous beliefs etc. Definition 5.1.
A probability-based risk measurement is a family of risk measures ρ I = { ρ Q : L ∞ → R , Q ∈ I} such that ρ Q ( X ) = R ρ ( F X, Q ) , ∀ X ∈ L ∞ , ∀ Q ∈ I , where R ρ : F → R is calledrisk functional.Remark . Such risk functional approach is inspired in the two step risk measurement ofCont et al. (2010). This definition implies that each ρ Q has Q -Law Invariance or is Q -based inthe sense that if F X, Q = F Y, Q , then ρ Q ( X ) = ρ Q ( Y ) , ∀ X, Y ∈ L ∞ . In fact, we have the strongerCross Law Invariance in the sense that if F X, Q = F Y, Q , then ρ Q ( X ) = ρ Q ( Y ) , ∀ X, Y ∈ L ∞ , ∀ Q , Q ∈ I . It is direct that if X ∈ A ρ Q and F X, Q = F Y, Q , then Y ∈ A ρ Q . For more detailson Cross Law Invariance, see Laeven and Stadje (2013). Example 5.3.
Cross Law Invariance is respected in all cases exposed in Example 2.4. We haveindeed: • Expected Loss (EL): EL Q ( X ) = R EL ( F X, Q ) = − E Q [ X ] = − R F − X, Q ( s ) ds . • Value at Risk (VaR):
V aR Q α ( X ) = R V aR α ( F X, Q ) = − F − X, Q ( α ) , α ∈ [0 , • Expected Shortfall (ES): ES Q α ( X ) = R ES α ( F X, Q ) = α R α V aR Q s ( X ) ds, α ∈ (0 ,
1] and ES Q ( X ) = R ES ( F X, Q ) = V aR Q ( X ) = − ess inf Q X . • Maximum loss (ML):
M L Q ( X ) = R ML ( F X, Q ) = − ess inf Q X = − F − X, Q (0).We could consider risk measurement composed by Q -based risk measures ρ Q without acommon link R ρ . However, we would be very close to the standard combination theory ofprevious sections, but without the modeling intuition. We then take an interest in risk measureswhich consider the whole set I in the sense that they are a combination of probability-basedrisk measurements conform ρ ( X ) = f ( ρ I ( X )), where f : X → R is a combination function. If F X, Q = F Y, Q , ∀ Q ∈ I , then ρ ( X ) = f ( ρ I ( X )) = f ( ρ I ( Y )) = ρ ( Y ). This kind of risk measureis called I -based functional in Wang and Ziegel (2018). Remark . Let Q µ be defined as Q µ ( A ) = R I Q ( A ) dµ, ∀ A ∈ F , µ ∈ V . We do not necessarilyhave that Q µ ∈ I . However, Assumption 4.4 assures that the integral is well defined when each ρ Q is convex. Under this framework one has a temptation to write ρ Q µ ( X ) = ρ µ ( X ). However,this in general is not the case. From proposition 5 of Acciaio and Svindland (2013), we havefor finite I , that R ρ is concave in F when each ρ Q is convex, i.e. ρ Q µ ( X ) ≥ ρ µ ( X ). There isequality if, and only if ρ Q ( X ) = EL Q ( X ). Such result is a consequence of linearity for the map Q → E Q [ X ] , ∀ X ∈ L ∞ and Corollary 2 of Acciaio and Svindland (2013), which claims EL isthe only law invariant convex risk measure that is convex in F .21 .2 Representation It is not possible to obtain a representation like those in Theorem 2.6 for combinations over aprobability-based risk measurement since these risk measures are not determined by a singleprobability measure. However, it is nonetheless possible to adapt such representations. To that,we need a Lemma regarding measurability issues.
Lemma 5.5.
Let ρ I = { ρ Q : L ∞ → R , Q ∈ I} be a probability-based risk measurement composedof convex risk measures. Then the following maps are G -measurable:(i) Q → V aR Q α ( X ) , ∀ X ∈ L ∞ , ∀ α ∈ [0 , .(ii) Q → ES Q α ( X ) , ∀ X ∈ L ∞ , ∀ α ∈ [0 , .(iii) Q → R (0 , ES Q α ( X ) dm Q , ∀ X ∈ L ∞ , ∀ α ∈ [0 , and for any { m Q ∈ M : Q ∈ I} .Proof. (i) From Assumption 4.4, we have that Q → F X, Q ( x ) = Q ( X ≤ x ) is G -measurable ∀ X ∈ L ∞ , ∀ x ∈ R . We then have for a fixed pair ( X, α ) ∈ L ∞ × [0 ,
1] that for any k ∈ R n Q ∈ I : V aR Q α ( X ) ≥ k o = n Q ∈ I : F − X, Q ( α ) ≤ − k o = { Q ∈ I : F X, Q ( − k ) ≥ α } ∈ G . Thus, the maps Q → V aR Q α ( X ) are G -measurable for any X ∈ L ∞ and any α ∈ [0 , X, α ) ∈ L ∞ × [0 ,
1] that ES Q α ( X ) = 1 α Z α V aR Q s ( X ) ds = sup Y Q ∼ X E Q α [ − Y ] , where F − d Q αd Q , Q (1 − s ) = 1 s ≤ α α , ∀ s ∈ [0 , . It is straightforward to note that Q α is a probability measure absolutely continuous inrelation to Q . From Assumption 4.4 we have that Q → E Q α [ − Y ] is G -measurable for any Y Q ∼ X . By repeating the deduction in item (i) of Lemma 4.15, we conclude that themaps Q → ES Q α ( X ) are G -measurable for any X ∈ L ∞ and any α ∈ [0 , X, α ) ∈ L ∞ × [0 ,
1] and { m Q ∈ M : Q ∈I} . We have for each Q ∈ I that Z (0 , ES Q α ( X ) dm Q = sup Y Q ∼ X E Q ′ [ − Y ] , where F − d Q ′ d Q , Q (1 − s ) = Z ( s, v dm Q ( v ) , ∀ s ∈ [0 , . By replacing the argument in item (ii), which comes to be a special case of item (iii), with Q ′ instead of Q α we prove that the maps Q → R (0 , ES Q α ( X ) dm Q are G -measurable forany X ∈ L ∞ , any α ∈ [0 ,
1] and for any { m Q ∈ M : Q ∈ I} .22n the following Proposition, we establish a direct representation between ρ µ and ES µα , oreven V aR µα . This fact links ρ µ to spectral risk measures. Proposition 5.6.
Let ρ I = { ρ Q : L ∞ → R , Q ∈ I} be a probability-based risk measurementcomposed of convex risk measures and ρ µ : L ∞ → R defined as in (2.7) . Then:(i) ρ µ can be represented as: ρ µ ( X ) = sup m ∈M (Z (0 , ES µα ( X ) dm − β ρ µ ( m ) ) , ∀ X ∈ L ∞ , (5.1) with β ρ µ : M → R + ∪ {∞} defined as in (4.9) .(ii) If in addition ρ Q fulfills, for every Q ∈ I , Positive Homogeneity, then the representationis conform: ρ µ ( X ) = sup m ∈ cl ( M ρµ ) Z (0 , ES µα ( X ) dm, ∀ X ∈ L ∞ , (5.2) where M ρ µ is defined as in (4.10) .(iii) If ρ Q also is, for every Q ∈ I , co-monotone, then the representation is conform: ρ µ ( X ) = Z (0 , ES µα ( X ) dm, ∀ X ∈ L ∞ , (5.3) where m ∈ cl ( M ρ µ ) .Proof. Since L ∞ ⊂ L ∞ ( Q ) , ∀ Q ∈ I , we have that each ρ Q can be considered as the restrictionof a functional over L ∞ ( Q ). Thus, from Theorems 2.5 and 2.6, ρ Q can be represented ascombinations of ES Q α over probabilities m ∈ M . The proof follows similar steps of Theorem4.7 by noticing that Z I Z (0 , ES Q α ( X ) dm Q dµ = Z I Z (0 , Z I ES Q α ( X ) dµdm Q dµ = Z (0 , ES µα ( X ) dm, where m = R I m Q dµ ∈ M . Remark . The representation in item (iii) have a spectral analogous conform ρ µ ( X ) = Z V aR µα ( X ) φ ( α ) dα, where φ is conform Example 2.11. This is directly obtained by the definition of ES µα . Remark . The case of ρ µ is a special one, instead of a rule. For instance, this link for ρ W C is frustrated since ρ W C ( X ) ≤ sup m ∈M (Z (0 , ES W Cα ( X ) dm − β ρ WC ( m ) ) , where β ρ WC ( m ) = inf Q ∈I β minρ Q ( m ). Indeed, Propositions 3.3 and 3.5 shows that ρ W C is sub-additive for co-monotone random variables because the supremum in (2.6) is not attained in23eneral. Under the probability-based risk measurement framework with some conditions onthe set I , such as being tight, Bartl et al. (2019) provide situations whereby the supremumis attained. Moreover, Theorem 1 of Wang and Ziegel (2018) shows that V aR
W Cα possessesCo-monotonic Additivity in the case of finite cardinality for I .In Theorems 3 and 4 of Wang and Ziegel (2018) a representation for I -based risk mea-sures for the co-monotone and coherent cases is exposed when I is a finite set and restrictionsregarding assumptions over the probabilities Q ∈ I . In the next Corollary, we expose a repre-sentation that suits for convex cases and general I , despite being limited to those generated bya combination. Corollary 5.9.
Let ρ I = { ρ Q : L ∞ → R , Q ∈ I} be a probability-based risk measurementcomposed of convex risk measures, f : X → R possessing Monotonicity, Translation Invariance,Convexity and Fatou continuity, and ρ : L ∞ → R defined as ρ ( X ) = f ( ρ I ( X )) . Then:(i) ρ can be represented conform ρ ( X ) = sup µ ∈V ,m ∈M (Z (0 , ES µα ( X ) dm − β ρ ( m ) ) , ∀ X ∈ L ∞ , (5.4) where β ρ as in (4.13) .(ii) If in addition to the initial hypotheses f possess Positive Homogeneity, then the penaltyterm becomes β ρ ( m ) = inf µ ∈V f β ρ µ ( m ) , where V f is conform Remark 4.2.(iii) If in addition to the initial hypotheses ρ i possess, for any i ∈ I , Positive Homogeneity,then β ρ ( m ) = ∞ , ∀ m ∈ M\ ∪ µ ∈V cl ( M ρ µ ) .(iv) If in addition to the initial hypotheses we have the situations in (ii) and (iii), then therepresentation of ρ becomes ρ ( X ) = sup µ ∈V f ,m ∈M V fρ Z (0 , ES µα ( X ) dm, ∀ X ∈ L ∞ , (5.5) where M V f ρ is as (4.14) .(v) If in addition to the initial hypotheses ρ i possess, for any i ∈ I , Co-monotonic Additivity,then β ρ ( m ) = ∞ , ∀ m ∈ M\ ∪ µ ∈V { m µcc } , where m µcc = arg max m ∈ cl ( M ρµ ) Z (0 , ES µα ( X ) dm, ∀ µ ∈ V . (vi) If in addition to the initial hypotheses we have (ii) and (v), then the representation of ρ becomes ρ ( X ) = sup µ ∈V f ,m ∈M V fρ,cc Z (0 , ES µα ( X ) dm, ∀ X ∈ L ∞ , (5.6) where M V f ρ,cc is the closed convex hull of ∪ µ ∈V f { m µcc } . roof. The claims are directly obtained from Theorem 4.11, Corollary 4.18 and Proposition5.6.
It is reasonable to consider risk measures that are suitable for decision making. This is typicallyaddressed under monotonicity with respect to stochastic orders, see B¨auerle and M¨uller (2006)for instance. However, in the presence of uncertainty regarding choice of a probability measure,adaptations must be done.
Definition 5.10.
We consider the following orders for
X, Y ∈ L ∞ : • X (cid:23) , Q Y ( Q -based stochastic dominance of first order) if, and only if, F − X, Q ( α ) ≥ F − Y, Q ( α ) , ∀ α ∈ [0 , . • X (cid:23) , Q Y ( Q -based stochastic dominance of second order) if, and only if, R α F − X, Q ( s ) ds ≥ R α F − Y, Q ( s ) ds, ∀ α ∈ [0 , . • X (cid:23) , I Y ( I -based stochastic dominance of first order) if, and only if, F − X, Q ( α ) ≥ F − Y, Q ( α ) , ∀ α ∈ [0 , , ∀ Q ∈ I . • X (cid:23) , I Y ( I -based stochastic dominance of second order) if, and only if, R α F − X, Q ( s ) ds ≥ R α F − Y, Q ( s ) ds, ∀ α ∈ [0 , , ∀ Q ∈ I .A risk measure ρ : L ∞ → R is said to respect any of the previous orders if X (cid:23) Y implies ρ ( X ) ≤ ρ ( Y ) , ∀ X, Y ∈ L ∞ .Remark . Of course, if ρ Q respects (cid:23) , Q or (cid:23) , Q for any Q ∈ I and f is monotone, then ρ = f ( ρ I ) respects (cid:23) , I or (cid:23) , I , respectively. When I = { P } and risk measures are functionalsover L ∞ (Ω , F , P ), the standard case, some well known results arise: P -Law Invariance andMonotonicity are equivalent to respect regarding (cid:23) , P ; and any P -law invariant convex riskmeasure respects (cid:23) , P . See the reference books cited in the introduction for more details.The facts in Remark 5.11 are not always true under general I , as we show in the nextProposition. Such results highlight the discussion that considering a framework robust to thechoice of probability measures may lead to paradoxes regarding decision making based on riskmeasures. Nonetheless, such situation is avoided when one considers risk measures generated byadequate composition of probability-based risk measurements. To that, we need the followingauxiliary definition. Definition 5.12.
A risk measure ρ : L ∞ → R has a ( ES, I ) representation if ρ ( X ) = sup µ ∈V ,m ∈M (Z (0 , ES µα ( X ) dm − β ( m ) ) , ∀ X ∈ L ∞ , where β : M → R + ∪ {∞} is a penalty term.Remark . It is straightforward to note that any functional of this kind is a I -based Fatoucontinuous convex risk measure. From Proposition 5.6 and Corolary 5.9 we have that this25s the case for risk measures composed by probability-based risk measurements with propercombinations f . It is also the case for I -based coherent risk measures for finite and mutuallysingular I , as in Theorem 4 of Wang and Ziegel (2018). Proposition 5.14.
Let ρ : L ∞ → R be a risk measure. Then:(i) If ρ respects (cid:23) , I , then ρ is I -based and monotone. The converse is true when I = P .(ii) If ρ has a ( ES, I ) representation (and thus it is a I -based Fatou continuous convex riskmeasure), then it respects (cid:23) , I . The converse is true under the additional hypotheses of ρ monetary and convex for co-monotone pairs, plus I be a singleton.Proof. (i) Let F X, Q = F Y, Q ∀ Q ∈ I for arbitrary X, Y ∈ L ∞ . Then by hypothesis we have ρ ( X ) = ρ ( Y ). Let X ≥ Y . Then X (cid:23) , I Y and hence ρ ( X ) ≤ ρ ( Y ). For the converse,note that when I = P we have that ρ is always I -based since F X, Q = F Y, Q for any Q ∈ P implies X = Y . Let X (cid:23) , I Y . Then X ≥ Y and thus Monotonicity implies ρ ( X ) ≤ ρ ( Y ).(ii) Let X (cid:23) , I Y . Thus ES Q α ( X ) ≤ ES Q α ( Y ) , ∀ α ∈ [0 , , ∀ Q ∈ I . From monotonicity ofintegral and supremum, as well non negativity of penalty terms, we have that ρ ( X ) ≤ ρ ( Y )for any ρ with a ( ES, I ) representation. Regarding the converse claim, it is a known resultfrom literature since we have the standard case for a singleton I , see Proposition 5.1 inF¨ollmer and Knispel (2013) for instance. Remark . In (i), the difficulty in stating a converse claim is because X (cid:23) , I Y does notimply X ≥ Y point-wisely. Even when we assume that there is a reference measure P andwe work on L ∞ (Ω , F , P ) under P almost surely sense, the best we can get is to replace P by Q ⊂ P . Since not every I -based convex risk measure possess a ( ES, I ) representation, it isnot possible to obtain the result in (ii) by only assuming Convexity, as occurs in standard case.The converse statement in (ii) needs to be restrictive due to the impossibility of existence forany pair X, Y ∈ L ∞ , a co-monotone pair Z X , Z Y ∈ L ∞ such that for any Q ∈ I we have F X, Q = F Z X , Q and F Y, Q = F Z Y , Q , which essential to the proof when I is a singleton. A recently highlighted statistical property is Elicitability, which enables the comparison ofcompeting models in risk forecasting. See Ziegel (2016) and the references therein for moredetails.
Definition 5.16.
A map S : R → R + is called scoring function if it has the following proper-ties: • S ( x, y ) = 0 if, and only if, x = y ; • y → S ( x, y ) is increasing for y > x and decreasing for y < x , for any x ∈ R ; • S ( x, y ) is continuous in y , for any x ∈ R . probability-based risk measurement ρ I = { ρ Q : L ∞ → R , Q ∈ I} is elicitable if exists a scoringfunction S : R → R + such that ρ Q ( X ) = − arg min y ∈ R E Q [ S ( X, y )] , ∀ X ∈ L ∞ , ∀ Q ∈ I . (5.7) Remark . Elicitability can be restrictive, because depending on the demanded financialproperties at hand, we end up with only one example of risk functional which satisfies therequisites, see Theorem 4.9 of Bellini and Bignozzi (2015) and Theorem 1 in Kou and Peng(2016). For instance, EL and VaR are elicitable under scores ( x − y ) and α ( x − y ) + + (1 − α )( x − y ) − , respectively, while ES and ML are not.In our framework, when Q = Q ′ , ∀ Q ∈ I and f posses Translation Invariance it is easy toobserve that ρ = f ( ρ I ) inherits elicitability from ρ I . In general, we can express a non-elicitablerisk measure in the case it is a worst case combination ( f W C ) of some elicitable probability-based risk measurements in terms of a minimization argument for the same score function. Thisis very useful, for instance, to coherent and convex risk measures, since they are combinationsof EL.
Proposition 5.18.
Let ρ I = { ρ Q : L ∞ → R , Q ∈ I} be a probability-based risk measurementwhich is elicitable under the scoring function S : R → R + , I a convex set, and ρ W C : L ∞ → R defined as in (2.6) . Then we have that ρ W C ( X ) = − arg min y ∈ R inf Q ∈I E Q [ S ( X, y )] , ∀ X ∈ L ∞ . (5.8) Proof.
Let G = [inf X, sup X ]. We have that G ⊂ R is a compact set. Moreover, we have fromDefinition 5.16 that S ( X, y ) ≥ S ( X, inf X ) for y ≤ inf X and S ( X, y ) ≥ S ( X, sup X ) when y ≥ sup X . Thus, the minimization of E Q [ S ( X, y )] is not altered if we replace R by G . Thenwe have ρ W C ( X ) = sup Q ∈I (cid:26) − arg min y ∈ G E Q [ S ( X, y )] (cid:27) = − arg min y ∈ G inf Q ∈I E Q [ S ( X, y )] . The last step in this deduction is due to the minimax theorem, which is valid since both Q and G are convex, G is compact, plus both score function and expectation possess the necessarycontinuity properties in the demanded argument. Remark . For instance, let I = Q ES P α and ρ Q = EL Q . Thus ES P α ( X ) = − arg min y ∈ R E Q X [( X − y ) ] , where Q X = arg max { E Q [ − X ] : Q ∈ Q ES P α } , which we know to have relative density d Q X d P = α X ≤ F − X, P ( α ) . 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