Inf-convolution and optimal risk sharing with countable sets of risk measures
aa r X i v : . [ q -f i n . R M ] F e b Inf-convolution and optimal risk sharing with countable sets ofrisk measures
Marcelo Brutti Righi a, ∗ [email protected] Marlon Ruoso Moresco a [email protected] a Business School, Federal University of Rio Grande do Sul, Washington Luiz, 855, Porto Alegre,Brazil, zip 90010-460
Abstract
The inf-convolution of risk measures is directly related to risk sharing and general equi-librium, and it has attracted considerable attention in mathematical finance and insuranceproblems. However, the theory is restricted to finite sets of risk measures. In this study,we extend the inf-convolution of risk measures in its convex-combination form to a count-able (not necessarily finite) set of alternatives. The intuitive principle of this approach ageneralization of convex weights in the finite case. Subsequently, we extensively generalizeknown properties and results to this framework. Specifically, we investigate the preservationof properties, dual representations, optimal allocations, and self-convolution.
Keywords : Risk measures, Inf-convolution, Risk sharing, Representations, Optimal alloca-tions.
The theory of risk measures has attracted considerable attention in mathematical finance andinsurance since the seminal paper by Artzner et al. (1999). The books by Pflug and R¨omisch(2007), Delbaen (2012), R¨uschendorf (2013), and F¨ollmer and Schied (2016) are comprehensiveexpositions of this subject. In these studies, a key topic is the inf-convolution of risk measures,which is directly related to risk sharing and general equilibrium. These problems may beconnected with regulatory capital reduction, risk transfer in insurance–reinsurance contracts,and several other applications in classic studies such as Borch (1962), Arrow (1963), Gerber(1978), and Buhlmann (1982), as well as more recent research as in Landsberger and Meilijson(1994), Dana and Meilijson (2003), and Heath and Ku (2004).Formally, the inf-convolution of risk measures is defined as (cid:3) ni =1 ρ i ( X ) = inf ( n X i =1 ρ i ( X i ) : n X i =1 X i = X ) , ∗ Corresponding author. We would like to thank the editor, anonymous associate editor, and reviewer forconstructive comments and suggestions, which have been very useful to improve the technical quality of themanuscript. We are grateful for the financial support of FAPERGS (Rio Grande do Sul State Research Council)project number 17/2551-0000862-6 and CNPq (Brazilian Research Council) projects number 302369/2018-0 and407556/2018-4. X and X i , i = 1 , · · · , n , belong to some linear space of random variables over a probabilityspace, and ρ i , i = 1 , · · · , n , are risk measures, which are functionals on this linear space. Byusing a slightly modified version, convex combinations, which represent weighting schemes,may be considered as follows: µ = { µ , · · · , µ n } ∈ [0 , n , P ni =1 µ i = 1; this modified version isdefined as ρ µ,nconv ( X ) = inf ( n X i =1 µ i ρ i ( X i ) : n X i =1 µ i X i = X ) . Letting ˆ ρ i = µ i ρ i , i = 1 , · · · , n immediately implies that ρ µ,nconv ( X ) shares some properties asthe standard (cid:3) ni =1 ˆ ρ i . See Starr (2011) for details of the use of such formulation in generalequilibrium theory to obtain all Pareto-optimal allocations.Convex risk measures, as initially proposed by F¨ollmer and Schied (2002) and Frittelli and Rosazza Gianin(2002), have recently attracted considerable attention in the context of inf-convolutions, as inseveral other areas of risk management. This subject is explored in Barrieu and El Karoui(2005), Burgert and R¨uschendorf (2006), Burgert and R¨uschendorf (2008), Jouini et al. (2008),Filipovi´c and Svindland (2008), Ludkovski and R¨uschendorf (2008), Ludkovski and Young (2009),Acciaio and Svindland (2009), Acciaio (2009), Tsanakas (2009), Dana and Le Van (2010), Delbaen(2012), and Kazi-Tani (2017). These studies present a detailed investigation of the propertiesof inf-convolution as a risk measure per se, as well as optimality conditions for the resultingallocations.Beyond the usual approach of convex risk measures, some studies have been concerned withinf-convolution in relation to specific properties, as in Acciaio (2007), Grechuk et al. (2009),Grechuk and Zabarankin (2012), Carlier et al. (2012), Mastrogiacomo and Rosazza Gianin (2015),and Liu et al. (2020), particular risk measures, as the recent quantile risk sharing in Embrechts et al.(2018), Embrechts et al. (2020), Weber (2018), Wang and Ziegel (2018), and Liu et al. (2019),or even specific topics, as in Liebrich and Svindland (2019). However, these studies are re-stricted to finite sets of risk measures.In this study, we extend the convex combination-based inf-convolution of risk measures toa infinite countable set of alternatives. Specifically, we consider a collection of risk measures ρ I = { ρ i , i ∈ I} , where I is a nonempty countable set. Then, we obtain the generalized versionof the convex inf-convolution as follows: ρ µconv ( X ) = inf (X i ∈I ρ i ( X i ) µ i : X i ∈I X i µ i = X ) . The intuitive principle of this approach is to regard µ = { µ i } i ∈I ⊂ [0 ,
1] such that P i ∈I µ i = 1 asa generalization of convex weights in the finite case. We extensively generalize known propertiesand results to this framework. More specifically, we investigate the preservation of propertiesof ρ I , dual representations, optimal allocations, and self-convolution. Of course, owing to theextent of the related literature, we do not intend to be exhaustive. To the best of our knowledge,there is no study in this direction.Regarding to interpretation, it is the minimum amount of risk, which may represent capitalrequirement for instance, among all possible ways of dividing a risk X into countable fragmentsand distributing the capital into countable units under weighting scheme. Such units could be2usiness lines, agents, lotteries, liquidation times etc. Thus, cuch formulation arises naturallywhen the agent is allowed to pulverize its position in as many fragments as desired. Weightscould be the importance of each ρ i in the global decision. The key point to be noted, is that inthis case the split of the position X can be taken at any number of fragments as desired insteadof a fixed finite one as in the usual approach. Thus, it is expected that the value generated forthe resulting risk measure be smaller than the one resulting from any fixed finite inf-convolution.The work of Wang (2016) considers a countable allocation, but with ρ i = ρ for any i ∈ I ,i.e., a fixed risk measure. In his paper, the regulatory arbitrage, which occurs when dividing aposition into several fragments results in a reduced capital requirement in relation to the jointposition. In our framework, we generalize such reasoning by allowing distinct risk measures.In this sense, our approach identifies the limit case regarding all possibilities for the agentconcerning to the division of a position. We also allow for weights in order to allow for distinctdegrees of importance for each risk measure.The study by Righi (2019b) considers an arbitrary set of risk measures and investigates theproperties of combinations of the form ρ = f ( ρ I ), where f is a combination function over alinear space generated by the outcomes of ρ I ( X ) = { ρ i ( X ) , i ∈ I} . Under the lack of a universalchoice of a best risk measure from a set of alternatives, one can think into considering the jointuse of many candidates in the goal of benefit from distinct qualities. Clearly, the inf-convolutionis not suitable for such a framework, as X is fixed. Thus, under our approach for a countableset of candidates it is possible to split the position in order to obtain the best allocation.From a mathematical point of view, the countable I is a limiting case. The considerationof an arbitrary (not necessarily finite or countable) set of risk measures done by considering ameasure (probability) µ over a suitable sigma algebra G of I . The problem then becomes ρ µconv ( X ) = inf (cid:26)Z I ρ i ( X i ) dµ : Z I X i dµ = X (cid:27) . However, it would be necessary to impose assumptions on G in order to avoid measurabilityissues. Such assumptions, that the maps i → X i ( ω ) are measurable for any ω ∈ Ω and everyfamily { X i ∈ L ∞ , i ∈ I} , would imply that G is the power set, which would leave us withoutmeaningful choices for probability measures. In fact, as a consequence of Ulam’s theorem everyprobability measure on the power set of set with cardinality as the powerset of N is a discreteprobability measure.The remainder of this paper is organized as follows. In Section 2, we present preliminariesregarding notation, and briefly provide background material on the theory of risk measures. InSection 3, we present the proposed approach and results regarding the preservation of financialand continuity properties of the set of risk measures. In Section 4, we prove results regardingdual representations for the convex, coherent, and law-invariant cases. In Section 5, we ex-plore optimal allocations by considering general results regarding the existence, comonotonicimprovement, and law invariance of solutions, as well as the comonotonicity and flatness ofdistributions. In Section 6, we explore the special topic of self-convolution and its relation toregulatory arbitrage. 3 Preliminaries
We consider a probability space (Ω , F , P ). All equalities and inequalities are in the P − a.s. sense.Let L = L (Ω , F , P ) and L ∞ = L ∞ (Ω , F , P ) be the spaces of (equivalence classes under P − a.s. equality of) finite and essentially bounded random variables, respectively. When not explicit,we consider in L ∞ its strong topology. We define 1 A as the indicator function for an event A ∈ F . We identify constant random variables with real numbers. A pair X, Y ∈ L is calledcomonotone if ( X ( w ) − X ( w ′ )) ( Y ( w ) − Y ( w ′ )) ≥ , w, w ′ ∈ Ω holds P ⊗ P − a.s. We denoteby X n → X convergence in the L ∞ essential supremum norm k·k ∞ , whereas lim n →∞ X n = X indicates P − a.s. convergence. The notation X (cid:23) Y , for X, Y ∈ L ∞ , indicates second-orderstochastic dominance, that is, E [ f ( X )] ≤ E [ f ( Y )] for any increasing convex function f : R → R .In particular, E [ X |F ′ ] (cid:23) X for any σ -algebra F ′ ⊆ F .Let P be the set of all probability measures on (Ω , F ). We denote, by 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 } , the expected value, the (increasingand right-continuous) probability function, and its left quantile for X ∈ L ∞ with respect to Q ∈ P . We write X Q ∼ Y when F X, Q = F Y, Q . We drop subscripts indicating probabilitymeasures when Q = P . Furthermore, let Q ⊂ P be the set of probability measures Q that areabsolutely continuous with respect to P , with Radon–Nikodym derivative d Q d P . We denote thetopological dual ( L ∞ ) ∗ of L ∞ by ba , which is defined as the space of finitely additive signedmeasures (with finite total variation norm k·k T V ) that are absolutely continuous with respectto P ; moreover, we let ba , + = { m ∈ ba : m ≥ , m (Ω) = 1 } and by abuse of notation, we define E m [ X ] = R Ω Xdm as the bilinear-form integral of X ∈ L ∞ with respect to m ∈ ba , + . Definition 2.1.
A functional ρ : L ∞ → R is called a risk measure. It may have the followingproperties:(i) Monotonicity: If X ≤ Y , then ρ ( X ) ≥ ρ ( Y ) , ∀ X, Y ∈ L ∞ .(ii) Translation invariance: ρ ( X + C ) = ρ ( X ) − C, ∀ X, Y ∈ L ∞ , ∀ C ∈ R .(iii) Convexity: ρ ( λX + (1 − λ ) Y ) ≤ λρ ( X ) + (1 − λ ) ρ ( Y ) , ∀ X, Y ∈ L ∞ , ∀ λ ∈ [0 , .(iv) Positive homogeneity: ρ ( λX ) = λρ ( X ) , ∀ X, Y ∈ L ∞ , ∀ λ ≥ .(v) Law invariance: If F X = F Y , then ρ ( X ) = ρ ( Y ) , ∀ X, Y ∈ L ∞ .(vi) Comonotonic additivity: ρ ( X + Y ) = ρ ( X ) + ρ ( Y ) , ∀ X, Y ∈ L ∞ with X, Y comonotone.(vii) Loadedness: ρ ( X ) ≥ − E [ X ] , ∀ X ∈ L ∞ .(viii) Limitedness: ρ ( X ) ≤ − ess inf X, ∀ X ∈ L ∞ .A risk measure ρ is called monetary if it satisfies (i) and (ii), convex if it is monetaryand satisfies (iii), coherent if it is convex and satisfies (iv), law invariant if it satisfies (v),comonotone if it satisfies (vi), loaded if it satisfies (vii), and limited if it satisfies (viii). Unlessotherwise stated, we assume that risk measures are normalized in the sense that ρ (0) = 0 . Theacceptance set of ρ is defined as A ρ = { X ∈ L ∞ : ρ ( X ) ≤ } .
4n addition to the usual norm-based continuity notions, P − a.s. pointwise continuity notionsare relevant in the context of risk measures. Definition 2.2.
A risk measure ρ : L ∞ → R is called(i) Fatou continuous: If lim n →∞ X n = X implies that ρ ( X ) ≤ lim inf n →∞ ρ ( X n ) , ∀ { X n } ∞ n =1 boundedin L ∞ norm and for any X ∈ L ∞ .(ii) Continuous from above: If lim n →∞ X n = X , with { X n } being decreasing, implies that ρ ( X ) =lim n →∞ ρ ( X n ) , ∀ { X n } ∞ n =1 , X ∈ L ∞ .(iii) Continuous from below: If lim n →∞ X n = X , with { X n } being increasing, implies that ρ ( X ) =lim n →∞ ρ ( X n ) , ∀ { X n } ∞ n =1 , X ∈ L ∞ .(iv) Lebesgue continuous: If lim n →∞ X n = X implies that ρ ( X ) = lim n →∞ ρ ( X n ) , ∀{ X n } ∞ n =1 boundedin L ∞ norm and X ∈ L ∞ . For more details regarding these properties, we refer to the classic books mentioned in theintroduction. We also have the following dual representations.
Theorem 2.3 (Theorem 2.3 in Delbaen (2002b), Theorem 4.33 in F¨ollmer and Schied (2016)) . Let ρ : L ∞ → R be a risk measure. Then,(i) ρ is a convex risk measure if and only if it can be represented as ρ ( X ) = max m ∈ ba , + (cid:8) E m [ − X ] − α minρ ( m ) (cid:9) , ∀ X ∈ L ∞ , (2.1) where α minρ : ba , + → R + ∪ {∞} , defined as α minρ ( m ) = sup X ∈ L ∞ { E m [ − X ] − ρ ( X ) } =sup X ∈A ρ E m [ − X ] , is a lower semi-continuous (in the total-variation norm) convex functionthat is called penalty term.(ii) ρ is a Fatou-continuous coherent risk measure if and only if it can be represented as ρ ( X ) = max m ∈Q ρ E m [ − X ] , ∀ X ∈ L ∞ , (2.2) where Q ρ ⊆ ba , + is a nonempty, closed, and convex set that is called the dual set of ρ .Remark . With the assumption of Fatou continuity, the representations in the previoustheorem could be considered over Q instead of ba , + , but with the supremum not necessarilybeing attained. Moreover, for convex risk measures, we can define certain subgradients usingLegendre–Fenchel duality (i.e., convex conjugates), as follows: ∂ρ ( X ) = { m ∈ ba , + : ρ ( Y ) − ρ ( X ) ≥ E m [ − ( Y − X )] ∀ Y ∈ L ∞ } = (cid:8) m ∈ ba , + : E m [ − X ] − α minρ ( m ) ≥ ρ ( X ) (cid:9) ,∂α minρ ( m ) = (cid:8) X ∈ L ∞ : α minρ ( n ) − α minρ ( m ) ≥ E ( n − m ) [ − X ] ∀ n ∈ ba (cid:9) = (cid:8) X ∈ L ∞ : E m [ − X ] − ρ ( X ) ≥ α minρ ( m ) (cid:9) , Q instead of ba , + . Moreover, by Theorem 2.3, we could replace the inequalities in definitionof sub-gradients by equalities. Further, it is immediate that X ∈ ∂α minρ ( m ) if and only if m ∈ ∂ρ ( X ). Example 2.5.
Examples of risk measures:(i) Expected loss (EL): This is a Fatou-continuous, law-invariant, comonotone, coherentrisk measure defined as EL ( X ) = − E [ X ] = − R F − X ( s ) ds . We have that A EL = { X ∈ L ∞ : E [ X ] ≥ } and Q EL = { P } .(ii) Value at risk (VaR): This is a Fatou-continuous, law-invariant, comonotone, monetaryrisk measure defined as V aR α ( X ) = − F − X ( α ) , α ∈ [0 , A V aR α = { X ∈ L ∞ : P ( X < ≤ α } .(iii) Expected shortfall (ES): This is a Fatou-continuous, law-invariant, comonotone, coher-ent risk measure defined as 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 .(iv) Entropic risk measure (Ent): This is a Fatou-continuous, law-invariant, convex risk mea-sure defined as Ent γ ( X ) = γ log (cid:0) E (cid:2) e − γX (cid:3)(cid:1) , γ ≥
0. Its acceptance set is A Ent γ = (cid:8) X ∈ L ∞ : E [ e − γX ] ≤ (cid:9) , and the penalty term is α min Ent γ ( Q ) = γ E h d Q d P log (cid:16) d Q d P (cid:17)i .(v) 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 .If law invariance is satisfied, as is the case in most practical applications, interesting featuresare present. In this paper, when dealing with law invariance we always assume that our baseprobability space (Ω , F , P ) is atomless. Theorem 2.6 (Theorem 2.1 in Jouini et al. (2006) and Proposition 1.1 in Svindland (2010)) . Let ρ : L ∞ → R be a law-invariant, convex risk measure. Then, ρ is Fatou continuous. Theorem 2.7 (Theorem 4.3 in B¨auerle and M¨uller (2006), Corollary 4.65 in F¨ollmer and Schied(2016)) . Let ρ : L ∞ → R be a law-invariant, convex risk measure. Then, X (cid:23) Y implies that ρ ( X ) ≤ ρ ( Y ) . Theorem 2.8 (Theorems 4 and 7 in Kusuoka (2001), Theorem 4.1 in Acerbi (2002), Theorem7 in 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)6 here M is the set of probability measures on (0 , , and β minρ : M → R + ∪ {∞} is 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, comonotone, coherent risk measure if and only if it can be representedas ρ ( X ) = Z (0 , ES α ( X ) dm (2.5)= Z V aR α ( X ) φ ( α ) dα (2.6)= Z −∞ ( g ( P ( − X ≥ x ) − dx + Z ∞ g ( P ( − X ≥ x )) dx, ∀ X ∈ L ∞ , (2.7) where m ∈ M ρ , φ : [0 , → R + is decreasing and right-continuous, with φ (1) = 0 and R φ ( u ) du = 1 , and g : [0 , → [0 , , called distortion, is increasing and concave, with g (0) = 0 and g (1) = 1 . We have that R ( u,
1] 1 v dm = φ ( u ) = g ′ + ( u ) ∀ u ∈ [0 , .Remark . (i) Functionals with representation as in (iii) of the last theorem are calledspectral or distortion risk measures. This concept is related to capacity set functions andChoquet integrals. Note that Comonotonic Additivity implies coherence for convex riskmeasures, see Lemma 4.83 of F¨ollmer and Schied (2016) for instance. In this case, ρ canbe represented by Q ρ = { Q ∈ Q : Q ( A ) ≤ g ( P ( A )) , ∀ A ∈ F } , which is the core of g , ifand only if g is its distortion function. If φ is not decreasing (and thus g is not concave),then the risk measure is not convex and cannot be represented as combinations of ES.(ii) Without law invariance, we can (see, for instance, Theorem 4.94 and Corollary 4.95 inF¨ollmer and Schied (2016)) represent a convex, comonotone risk measure ρ by a Choquetintegral as follows: ρ µconv ( X ) = Z ( − X ) dc = Z −∞ ( c ( − X ≥ x ) − dx + Z ∞ c ( − X ≥ x ) dx = max m ∈ ba c , + E m [ − X ] , ∀ X ∈ L ∞ , where c : F → [0 ,
1] is a normalized ( c ( ∅ ) = 0 and c (Ω) = 1), monotone (if A ⊆ B then c ( A ) ≤ c ( B )), submodular ( c ( A ∪ B ) + c ( A ∩ B ) ≤ c ( A ) + c ( B )) set function that is calledcapacity and is defined as c ( A ) = ρ ( − A ) ∀ A ∈ F , and ba c , + = { m ∈ ba , + : m ( A ) ≤ c ( A ) ∀ A ∈ F } . 7 Proposed approach
Let ρ I = { ρ i : L ∞ → R , i ∈ I} be some (a priori specified) collection of normalized monetaryrisk measures, where I is a nonempty infinite countable set. We define the set of weightingschemes V = (cid:8) { µ i } i ∈I ⊂ [0 ,
1] : P i ∈I µ i = 1 (cid:9) . Otherwise stated we fix µ ∈ V and denote I µ = { i ∈ I : µ i > } . We use the notations { X i ∈ L ∞ , i ∈ I} = { X i , i ∈ I} = { X i } i ∈I = { X i } for families indexed over I ; these families should be understood as generalizations of n -tuples.For any X ∈ L ∞ , we define its allocations as A ( X ) = ( { X i } i ∈I : X i ∈I X i µ i = X, { X i } is bounded ) . Evidently, ω → P i ∈I X i ( ω ) µ i defines a random variable in L ∞ for any { X i } i ∈I ∈ A ( X ) , X ∈ L ∞ . We note that the identity P i ∈I X i µ i = X should then be understood in the P − a.s. sense. The restriction to bounded allocations { X i } is to circumvent technical issues, such asconvergence for instance. Note that this is always the case for finite I .The countable case we study can be regarded as I = N where the set of allocations A ( X )consists of all sequences { X i } i ∈ N ⊂ L ∞ such that the associated sequence P ni =1 µ i X i convergesto X in the P − a.s. sense. Note that if P ni =1 X i µ i = X for some n ∈ N , then P n + ki =1 X i µ i = X for any k ∈ N by taking X i = 0 for i > n . In particular { X , . . . , X n , , . . . } ∈ A ( X ). Wehave that A ( X ) = ∅ for any X because we can select X i = X, ∀ i ∈ I . We also note that { X i } i ∈I ∈ A ( X + Y ) is equivalent to { X i − Y } i ∈I ∈ A ( X ) for any X, Y ∈ L ∞ . Furthermore, if { X i } i ∈I ∈ A ( X ) and { Y i } i ∈I ∈ A ( Y ), then { aX i + bY i } ∈ A ( aX + bY ) for any a, b ∈ R and X, Y ∈ L ∞ . We now define the core functional in our study. Definition 3.1.
Let ρ I = { ρ i : L ∞ → R , i ∈ I} be a collection of monetary risk measures and µ ∈ V . The µ -weighted inf-convolution risk measure is a functional ρ µconv : L ∞ → R ∪ {−∞} defined as ρ µconv ( X ) = inf (X i ∈I ρ i ( X i ) µ i : { X i } i ∈I ∈ A ( X ) ) . (3.1) Remark . We defined risk measures as functionals that only assume finite values. By abuseof notation, we will also consider ρ µconv to be a risk measure, and we will provide conditionswhereby it is finite. Since ρ I consists of monetary risk measures, then ρ µconv ( X ) < ∞ becausefor any X ∈ L ∞ we have that ρ µconv ( X ) ≤ P i ∈I ρ i ( X ) µ i ≤ k X k ∞ < ∞ . Moreover, we note thatnormalization is not directly inherited from ρ I ; indeed, ρ µconv (0) ≤
0. When ρ µconv is convex itis finite if and only if ρ µconv (0) > −∞ , which is a well-known fact from convex analysis that aconvex function that does assume ∞ is either finite or −∞ point-wise (see Lemma 16 of Delbaen(2012) for instance).The following proposition provides useful results regarding well definiteness, representationsand properties of ρ µconv . Proposition 3.3.
We have that(i) ρ µconv is well defined. ii) For any X ∈ L ∞ holds ρ µconv ( X ) = inf (X i ∈I ρ i ( X − X i ) µ i : { X i } i ∈I ∈ A (0) ) = lim n →∞ inf ( n X i =1 ρ i ( X i ) µ i : n X i =1 X i µ i = X ) = inf ( n X i =1 ρ i ( X i ) µ i : n ∈ N , n X i =1 X i µ i = X ) . (iii) If ρ I consists of risk measures satisfying positive homogeneity, then ρ µconv ( X ) ≤ ρ i ( X ) ∀ i ∈I µ , ∀ X ∈ L ∞ . In particular, ρ µconv < ∞ .Proof. (i) We must to show that P i ∈I ρ i ( X i ) µ i converges for any { X i } ∈ A ( X ). Due totranslation invariance, is enough to show that P ∞ i ρ ( X i ) µ i converges for any { X i } suchthat ρ ( X i ) ≥
0. As { X i } is bounded, there is Y ∈ L ∞ such that y := ess inf Y ≤ Y ≤ X i , ∀ i ∈ I . By monotonicity it follows that 0 ≤ ρ ( X i ) ≤ ρ ( Y ) ≤ ρ ( y ) = − y . Thus,0 ≤ X i ∈I ρ ( X i ) µ i ≤ − X i ∈I yµ i = − y X i ∈I µ i = − y. Hence, the claim follows by monotone convergence.(ii) For the first relation, we note that P i ∈I X i µ i = X if and only if P i ∈I ( X − X i ) µ i = P i ∈I ( X i − X ) µ i = 0. Thus, by letting Y i = X − X i , ∀ i ∈ I , we have that ρ µconv ( X ) = inf (X i ∈I ρ i ( X − Y i ) µ i : { Y i } i ∈I ∈ A (0) ) . Regarding the second relation, first notice thatlim n →∞ inf ( n X i =1 ρ ( X i ) µ i : n X i =1 X i µ i = X ) = inf [ n ∈ N ( n X i =1 ρ ( X i ) µ i : n X i =1 X i µ i = X ) . Let B n := (cid:8)P ni =1 ρ ( X i ) µ i : P ni =1 X i µ i = X (cid:9) and B := (cid:8)P ∞ i =1 ρ ( X i ) µ i : { X i } ∈ A ( X ) (cid:9) .Such sets depend on X and B n ⊆ B n +1 ⊆ B ⊆ R , ∀ n ∈ N . And as any convergent infinitesum is the limit of finite sums, we obtain B ⊆ cl ( ∪ n ∈ N B n ). Thus, cl ( B ) = cl ( ∪ n ∈ N B n ).If both B and ∪ n ∈ N B n are unbounded from below then their infimum coincide to −∞ .If they both are bounded from below, we have that ρ µconv ( X ) = inf B = inf cl ( B ) =inf( cl ( ∪ n ∈ N B n )) = inf ∪ n ∈ N B n = lim n inf (cid:8)P ni ρ ( X i ) µ i : P ni X i µ i = X (cid:9) . Therefore, weonly need to show that B is unbounded if and only if ∪ n ∈ N B n is unbounded from below.Since ∪ n ∈ N B n ⊆ B we clearly have that if ∪ n ∈ N B n is unbounded from below so is B . Forthe converse, let B be unbounded from below. Then there is a sequence { b j } ⊆ B suchthat b j ↓ −∞ . As any b j is a limit point of a sequence (in n ) { a jn } ⊆ ∪ n ∈ N B n , we canfind another sequence (in j ) { a jn ( j ) } ⊆ ∪ n ∈ N B n , where n ( j ) is sufficiently large naturalnumber, such that a jn ( j ) → −∞ . This fact implies that ∪ n ∈ N B n is also unbounded from9elow.For the third relation, we show that n → inf (cid:8)P ni =1 ρ i ( X i ) µ i : P ni =1 X i µ i = X (cid:9) is de-creasing. We have thatinf ( n +1 X i =1 ρ i ( X i ) µ i : n +1 X i =1 X i µ i = X ) ≤ inf ( n +1 X i =1 ρ i ( X i ) µ i : n X i =1 X i µ i = X, X n +1 = 0 ) = inf ( n X i =1 ρ i ( X i ) µ i : n X i =1 X i µ i = X ) . Hence, the infimum with respect to n ∈ N can be replaced by a limit.(iii) We assume, toward a contradiction, that there is X ∈ L ∞ such that ρ µconv ( X ) > ρ j ( X )for some j ∈ I µ . Let { Y i } i ∈I be such that Y i = ( µ j ) − X , recalling that µ j >
0, for i = j ,and Y i = 0 otherwise. Then, { Y i } i ∈I ∈ A ( X ). Thus, by positive homogeneity and thedefinition of ρ µconv we have that ρ j ( X ) < ρ µconv ( X ) ≤ X i ∈I ρ i ( Y i ) µ i = ρ j ( X ) , which is a contradiction. In this case, for any X ∈ L ∞ , we have that ρ µconv ( X ) ≤ inf i ∈I µ ρ i ( X ) < ∞ .We now present a result regarding the preservation by ρ µconv of financial properties of ρ I . Proposition 3.4. ρ µconv is monetary. Moreover, If ρ I consists of risk measures with convexity,positive homogeneity, law invariance, loadedness, or limitedness, then each property is inheritedby ρ µconv .Proof. (i) Monotonicity: Let X ≥ Y . Then, there is Z ≥ X = Y + Z . Since A ( Y ) + A ( Z ) ⊆ A ( Y + Z ), we have ρ µconv ( X ) = inf (X i ∈I ρ i ( X i ) µ i : { X i } i ∈I ∈ A ( Y + Z ) ) ≤ inf (X i ∈I ρ i ( Y i + Z i ) µ i : { Y i } i ∈I ∈ A ( Y ) , { Z i } i ∈I ∈ A ( Z ) , Z i ≥ ∀ i ∈ I ) ≤ inf (X i ∈I ρ i ( Y i + Z ) µ i : { Y i } i ∈I ∈ A ( Y ) ) ≤ inf (X i ∈I ρ i ( Y i ) µ i : { Y i } i ∈I ∈ A ( Y ) ) = ρ µconv ( Y ) . C ∈ R , we have that ρ µconv ( X + C ) = inf (X i ∈I ρ i ( X i ) µ i : { X i − C } i ∈I ∈ A ( X ) ) = inf (X i ∈I ρ i ( Y i + C ) µ i : { Y i } i ∈I ∈ A ( X ) ) = ρ µconv ( X ) − C. (iii) Convexity: For any λ ∈ [0 , λρ µconv ( X ) + (1 − λ ) ρ µconv ( Y ) = inf { X i }∈ A ( X ) , { Y i }∈ A ( Y ) X i ∈I [ λρ i ( X i ) + (1 − λ ) ρ i ( Y i )] µ i ≥ inf { X i }∈ A ( X ) , { Y i }∈ A ( Y ) X i ∈I ρ i ( λX i + (1 − λ ) Y i ) µ i ≥ inf (X i ∈I ρ i ( Z i ) µ i : { Z i } i ∈I ∈ A ( λX + (1 − λ ) Y ) ) = ρ µconv ( λX + (1 − λ ) Y ) . (iv) Positive homogeneity: For any λ ≥
0, we have that λρ µconv ( X ) = inf (X i ∈I ρ i ( λY i ) µ i : { Y i } i ∈I ∈ A ( X ) ) = inf (X i ∈I ρ i ( λY i ) µ i : { λY i } i ∈I ∈ A ( λX ) ) = ρ µconv ( λX ) . (v) Law invariance: We begin by showing that ρ µconv inherits law invariance on the sub-domain L ∞⊥ := { X ∈ L ∞ : ∃ uniform on [0 ,
1] r.v. independent of X } . To that, let X, Y ∈ L ∞⊥ with X ∼ Y and take some arbitrary { X i } i ∈I ∈ A ( X ). Note that we can have a countableset { U i , i ∈ I} of i.i.d. uniform on [0 ,
1] random variables independent of Y becauseour probability space is atomless, see Theorem 1 of Delbaen (2002b) for instance. Nowtake X = X , Y = Y and let Y i = F − X i | X i − , ··· ,X ( U i | Y i − , · · · , Y ) ∀ i ∈ I , which is theconditional quantile function. We thus get that ( Y, Y , · · · , Y n ) ∼ ( X, X , · · · , X n ) ∀ n ∈ I and { Y i } i ∈I ∈ A ( Y ). In this sense we obtain ρ µconv ( Y ) ≤ P i ∈I ρ i ( Y i ) µ i = P i ∈I ρ i ( X i ) µ i .Taking the infimum over A ( X ) we get ρ µconv ( Y ) ≤ ρ µconv ( X ). By reversing roles of X and Y ,we obtain Law Invariance on L ∞⊥ . Now, let X, Y ∈ L ∞ with X ∼ Y and define { X n } ⊂ L ∞ as X n = n ⌊ nX ⌋ and { Y n } ⊂ L ∞ as Y n = n ⌊ nY ⌋ , where ⌊·⌋ is the floor function. Moreover,it is easy to show that X n ∼ Y n , ∀ n ∈ N . By Lemma 3 in Liu et al. (2020) we have thatif X ∈ L ∞ takes values in a countable set, then X ∈ L ∞⊥ . Thus { X n } ⊂ L ∞⊥ and ρ µconv ( X n ) = ρ µconv ( Y n ) , ∀ n ∈ N . From item (i) in Proposition 3.6 we have that ρ µconv isLipschitz continuous, in particular posses continuity in k·k ∞ norm. Note that X n → X .Thus | ρ µconv ( X ) − ρ µconv ( Y ) | ≤ lim n →∞ | ρ µconv ( X ) − ρ µconv ( X n ) | + lim n →∞ | ρ µconv ( Y ) − ρ µconv ( Y n ) | =0. Hence ρ µconv ( X ) = ρ µconv ( Y ). 11vi) Loadedness: We fix X ∈ L ∞ and note that for any { X i } i ∈I ∈ A ( X ) we have that X i ∈I ρ i ( X i ) µ i ≥ X i ∈I E [ − X i ] µ i = E [ − X ] . By taking the infimum over A ( X ), we obtain that ρ µconv ( X ) ≥ − E [ X ].(vii) Limitedness: We fix X ∈ L ∞ . By the monotonicity of the countable sum, we have that ρ µconv ( X ) ≤ X i ∈I ρ i ( X ) µ i ≤ − ess inf X. Remark . (i) Concerning the preservation of subadditivity, that is, ρ ( X + Y ) ≤ ρ ( X ) + ρ ( Y ), the result follows by an argument analogous to that for convexity, but with X + Y instead of λX + (1 − λ ) Y . We note that in this case, we have normalization because ρ µconv (0) ≤
0, whereas ρ µconv ( X ) ≤ ρ µconv ( X ) + ρ µconv (0), which implies ρ µconv (0) ≥
0. If therisk measures of ρ I are loaded, we also have normalization because 0 = ρ (0) ≥ ρ µconv (0) ≥ E [ −
0] = 0. Of course, in the case of positive homogeneity, we also obtain normalization.(ii) Regarding the preservation of comonotonic additivity, let
X, Y ∈ L ∞ be a comonotonepair. Then, λX ,(1 − λ ) Y is also comonotone for any λ ∈ [0 , ρ , comonotonic additivity implies positive homogeneity. Then, byarguing as in the proof of (iii) and (iv) of the last proposition with λ = , we have that ρ µconv ( X + Y ) = 2 ρ µconv (cid:18) X Y (cid:19) ≤ (cid:18) ρ µconv ( X ) + 12 ρ µconv ( Y ) (cid:19) = ρ µconv ( X ) + ρ µconv ( Y ) . Thus, we obtain subadditivity for comonotone pairs. If, additionally, convexity (and hencecoherence) for ρ I comonotonic additivity is preserved, as shown in Theorem 4.7.In the following, we focus on the preservation by ρ µconv of continuity properties of ρ I . Proposition 3.6.
We have that(i) ρ µconv is Lipschitz continuous.(ii) If ρ I consists of continuous from below risk measures, then ρ µconv is continuous from below.Proof. (i) For each i ∈ I , we have that | ρ i ( X ) − ρ i ( Y ) | ≤ k X − Y k ∞ . Thus, | ρ µconv ( X ) − ρ µconv ( Y ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) inf (X i ∈I ρ i ( X − X i ) µ i : { X i } i ∈I ∈ A (0) ) − inf (X i ∈I ρ i ( Y − X i ) µ i : { X i } i ∈I ∈ A (0) )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sup ((cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i ∈I (cid:2) ρ i ( X − X i ) − ρ i ( Y − X i ) (cid:3) µ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) : { X i } i ∈I ∈ A (0) ) ≤ k X − Y k ∞ . (ii) Let { X n } ∞ n =1 ⊂ L ∞ be increasing such that lim n →∞ X n = X ∈ L ∞ . By the monotonicityof ρ I we have that each ρ i ( X n − X i ) is decreasing in n . Moreover, i → ρ i ( X n − X i ) is12ounded above by sup i ∈I k X − X i k ∞ < ∞ . Thus, by the monotone convergence Theoremwe have thatlim n →∞ ρ µconv ( X n ) = inf n ( inf { X i }∈ A (0) X i ∈I ρ i ( X n − X i ) µ i ) = inf { X i }∈ A (0) ( inf n X i ∈I ρ i ( X n − X i ) µ i ) = inf { X i }∈ A (0) (X i ∈I h inf n ρ i ( X n − X i ) i µ i ) = ρ µconv ( X ) . Remark . (i) It is important to note that Fatou continuity is not preserved even when I is finite, as lim n →∞ X n = X does not imply the existence of { X in } i ∈I ∈ A ( X n ) ∀ n ∈ N and { X i } i ∈I ∈ A ( X ) such that lim n →∞ X in = X i , ∀ i ∈ I . See Example 9 in Delbaen (2002a),for instance. Accordingly, one should be careful when dual representations that dependsof such continuity property are considered.(ii) If ρ I consists of convex risk measures that are continuous from below, then ρ µconv is convexLebesgue continuous. This is true because continuity from below is equivalent to Lebesguecontinuity for convex risk measures, see Theorem 4.22 in F¨ollmer and Schied (2016) forinstance.Robustness is a key concept in the presence of model uncertainty. It implies small variationin the output functional when there is bad specification. See, for instance, Cont et al. (2010),Kratschmer et al. (2014), and Kiesel et al. (2016). We now present a formal definition. Definition 3.8.
Let d be a pseudo-metric on L ∞ . Then, a risk measure ρ : L ∞ → R is called d -robust if it is continuous with respect to d .Remark . Convergence in distribution in the set of bounded random variables (i.e., con-vergence with respect to the Levy metric) is pivotal in the presence of uncertainty regardingdistributions, as in the model risk framework. It is well known that convex risk measures arenot upper semicontinuous with respect to the Levy metric. By Proposition 3.4, if each memberof ρ I is a convex risk measure, then so is ρ µconv . Thus, robustness with respect to this metric isruled out.In light of Proposition 3.6, we have that the continuity of the risk measures in ρ I with re-spect to d is not generally preserved by ρ µconv . Consequently, the same is true for d -robustness.Nonetheless, under stronger assumptions, we have the following corollary regarding the preser-vation of robustness. Corollary 3.10. If ρ I consists of risk measures that are Lipschitz continuous (with constant C ) with respect to d , then ρ µconv is d -robust.Proof. The proof is analogous to that of (i) in Proposition 3.6.13
Dual representations
We now present the main results regarding the representation of ρ µconv for convex cases. Theorem 4.1.
Let ρ I be a collection of convex risk measures. We have that(i) The acceptance set of ρ µconv is A ρ µconv = cl ( A µ ) , (4.1) where A µ = (cid:8) X ∈ L ∞ : ∃ { X i } i ∈I ∈ A ( X ) s.t. X i ∈ A ρ i ∀ i ∈ I µ (cid:9) = P i ∈I A ρ i µ i . More-over, A µ is not dense in L ∞ if and only if A µ = L ∞ .(ii) The minimal penalty term of ρ µconv is α minρ µconv ( m ) = X i ∈I α minρ i ( m ) µ i , ∀ m ∈ ba , + . (4.2) Proof.
We have that ρ µconv is a convex risk measure that is either finite or identically −∞ .Moreover, it is well known for any n ∈ N that n X i =1 α minρ i µ i = sup X ∈ L ∞ ( E m [ − X ] − inf ( n X i =1 ρ i ( X i ) µ i : n X i =1 X i µ i = X )) , which generates the acceptance set cl (cid:0)P ni =1 A ρ i µ i (cid:1) with { µ , . . . , µ n } ⊂ [0 , n . We now demon-strate the claims.(i) Let X ∈ A µ . Then, there is { X i } i ∈I ∈ A ( X ) such that X i ∈ A ρ i ∀ i ∈ I µ . Thus, ρ µconv ( X ) ≤ P i ∈I ρ i ( X i ) µ i ≤
0. This implies X ∈ A ρ µconv . By taking closures we get cl ( A µ ) ⊆ A ρ µconv . We note that A µ is not necessarily closed (for reasons similar to thosefor which ρ µconv does not inherit Fatou continuity). For the converse relation, let X ∈ int ( A ρ µconv ). Then there is { X i } ∈ A ( X ) such that k = P i ∈I ρ i ( X i ) µ i <
0. Since ρ i ( X i + ρ i ( X i ) − k ) < i ∈ I , we have that Y i = X i + ρ i ( X i ) − k ∈ int ( A ρ i ) forany i ∈ I . Moreover, P i ∈I Y i µ i = X + k − k = X . Then { Y i } ∈ A ( X ), which implies X ∈ A µ ⊆ cl ( A µ ). Thus, int ( A ρ µconv ) ⊆ A µ . By taking closures we get cl ( int ( A ρ µconv )) = A ρ µconv ⊆ cl ( A µ ), which gives the required equality. Moreover, let A µ be norm-dense in L ∞ . Thus, for any X ∈ L ∞ and k >
0, there is Y ∈ A µ such that k X − Y k ∞ ≤ k .Thus, X + k ≥ Y and, as A µ is monotone, we obtain X + k ∈ A µ . As both X and k arearbitrary, A µ = L ∞ . The converse relation is trivial.(ii) We have for any m ∈ ba , + that α minρ µconv ( m ) = sup X ∈ L ∞ ( E m [ − X ] − lim n →∞ inf ( n X i =1 ρ i ( X i ) µ i : n X i =1 X i µ i = X )) ≤ lim n →∞ sup X ∈ L ∞ ( E m [ − X ] − inf ( n X i =1 ρ i ( X i ) µ i : n X i =1 X i µ i = X )) = lim n →∞ n X i =1 α minρ i ( m ) µ i = X i ∈I α minρ i ( m ) µ i . α minρ µconv ( m ) = sup X ∈A ρµconv E m [ − X ] , A ρ µconv = n X ∈ L ∞ : α minρ µconv ( X ) ≥ E m [ − X ] ∀ m ∈ ba , + o . Moreover, note that if X ∈ P ni =1 A ρ i µ i , then X = P ni =1 X i µ i with X i ∈ A ρ i with i = 1 , . . . , n . Thus, X ∈ A µ since { X , . . . , X n , , , . . . } ∈ A ( X ) and 0 is acceptable forany ρ i . Then for any m ∈ ba , + we have X i ∈I α minρ i ( m ) µ i = lim n →∞ n X i =1 sup (cid:8) E m [ − X ] µ i : X ∈ A ρ i (cid:9) = lim n →∞ sup ( n X i =1 E m [ − X i ] µ i : { X i ∈ A ρ i } i =1 ,...,n ) = lim n →∞ sup ( E m [ − X ] : X ∈ n X i =1 A ρ i µ i ) ≤ lim n →∞ sup { E m [ − X ] : X ∈ A µ } = sup (cid:8) E m [ − X ] : X ∈ A ρ µconv (cid:9) = α minρ µconv ( m ) . Hence, α minρ µconv = P i ∈I α minρ i µ i . Regarding the properties of m → P i ∈I α minρ i ( m ) µ i , non-negativity is straightforward, whereas convexity follows from the monotonicity of theintegral and the convexity of each α minρ i because for any λ ∈ [0 ,
1] and m , m ∈ ba , + , wehave that X i ∈I α minρ i ( λm + (1 − λ ) m ) µ i ≤ X i ∈I h λα minρ i ( m ) + (1 − λ ) α minρ i ( m ) i µ i = λ X i ∈I α minρ i ( m ) µ i + (1 − λ ) X i ∈I α minρ i ( m ) µ i . Furthermore, by Fatou’s lemma (which can be used because each α minρ i is bounded frombelow by 0) and by the lower semi-continuity of each α minρ i with respect to the totalvariation norm on ba , for any { m n } such that m n → m , we have that X i ∈I α minρ i ( m ) µ i ≤ X i ∈I lim inf n →∞ α minρ i ( m n ) µ i ≤ lim inf n →∞ X i ∈I α minρ i ( m n ) µ i . Remark . (i) Under the assumption of Fatou continuity for both the risk measures in ρ I and ρ µconv , the claims in Theorem 4.1 could be adapted by replacing the finitely additivemeasures m ∈ ba + , by probabilities Q ∈ Q . Moreover, weak ∗ topological concepts couldreplace the corresponding strong (norm) topological concepts.(ii) The weighted risk measure is a functional ρ µ : L ∞ → R defined as ρ µ ( X ) = P i ∈I ρ i ( X ) µ i .Theorem 4.6 in Righi (2019b) states that, assuming Fatou continuity, ρ µ can be repre-15ented using a convex (not necessarily minimal) penalty defined as α ρ µ ( Q ) = inf (X i ∈I α minρ i (cid:0) Q i (cid:1) µ i : X i ∈I Q i µ i = Q , Q i ∈ Q ∀ i ∈ I ) . By the duality of convex conjugates, α ρ µ = α minρ µ if and only if α ρ µ is lower semi-continuous.Nonetheless, this represents a connection between weighted and inf-convolution functionsfor countable I , as in the traditional finite case.The following corollary provides interesting properties regarding the normalization, finite-ness, preservation, dominance and sub-gradients of ρ µconv . Corollary 4.3.
Let ρ I be a collection of convex risk measures. Then(i) n m ∈ ba , + : P i ∈I α minρ i ( m ) µ i < ∞ o = ∅ if and only if then ρ µconv is finite. In this case A µ is not dense in L ∞ .(ii) ρ µconv is normalized if and only if n m ∈ ba , + : α minρ i ( m ) = 0 ∀ i ∈ I µ o = ∅ .(iii) If ρ : L ∞ → R is a convex risk measure with ρ ( X ) ≤ ( ≥ or =) ρ i ( X ) ∀ i ∈ I µ , ∀ X ∈ L ∞ ,then ρ ( X ) ≤ ( ≥ or =) ρ µconv ( X ) , ∀ X ∈ L ∞ .(iv) (cid:8) m ∈ ba , + : m ∈ ∂ρ i ( X i ) ∀ i ∈ I µ (cid:9) = T i ∈I µ ∂ρ i ( X i ) ⊆ ∂ρ µconv ( X ) for any { X i } i ∈I ∈ A ( X ) and X ∈ L ∞ .(v) n X ∈ L ∞ : ∃ { X i } i ∈I ∈ A ( X ) s.t. X i ∈ ∂α minρ i ( m ) ∀ i ∈ I µ o = P i ∈I ∂α minρ i ( m ) µ i ⊆ ∂α minρ µconv ( m ) for any m ∈ ba , + .Proof. (i) By Proposition 3.4, we have ρ µconv < ∞ . If n m ∈ ba , + : P i ∈I α minρ i ( m ) µ i < ∞ o = ∅ , then there exists m ∈ ba , + such that −∞ < E m [ − X ] − X i ∈I α minρ i ( m ) µ i ≤ E m [ − X ] − α minρ µconv ( m ) ≤ ρ µconv ( X ) . If n m ∈ ba , + : P i ∈I α minρ i ( m ) µ i < ∞ o = ∅ , then α minρ µconv ( m ) = P i ∈I α minρ i ( m ) µ i = ∞ , ∀ m ∈ ba , + . Hence, ρ µconv ( X ) = −∞ , ∀ X ∈ L ∞ . Furthermore, if A µ is dense in L ∞ , then by(i) in Theorem 4.1, we have A µ = L ∞ . Thus, we have ρ µconv ( X ) = inf { m ∈ R : X + m ∈ L ∞ } = −∞ , ∀ X ∈ L ∞ .(ii) Let n m ∈ ba , + : α minρ i ( m ) = 0 ∀ i ∈ I µ o = ∅ and m ′ in this set. Then, α minρ µconv ( m ′ ) = P i ∈I α minρ i ( m ′ ) µ i = 0. Hence, ρ µconv (0) = − min m ∈ ba , + α minρ µconv ( m ) = 0 . For the converse relation, let n m ∈ ba , + : α minρ i ( m ) = 0 ∀ i ∈ I µ o = ∅ . Thus, for any m ∈ ba , + , we have that n i ∈ I µ : α minρ i ( m ) > o = ∅ . Then, ρ µconv (0) = − min m ∈ ba , + α minρ µconv ( m ) < , ρ µconv is not normalized.(iii) We prove for the leq relation since the others are quite similar. By Theorem 2.3 and andRemark 2.4, we have that α minρ ( m ) ≥ α minρ i ( m ) ∀ i ∈ I µ for any m ∈ ba , + . Thus, byTheorem 4.1, we obtain that α minρ ( m ) ≥ X i ∈I α minρ i ( m ) µ i = α minρ µconv ( m ) . Hence, ρ ( X ) ≤ ρ µconv ( X ) , ∀ X ∈ L ∞ .(iv) Let m ′ ∈ (cid:8) m ∈ ba , + : m ∈ ∂ρ i ( X i ) ∀ i ∈ I µ (cid:9) . Then, E m ′ [ − X ] − α minρ µconv ( m ′ ) ≥ X i ∈I (cid:16) E m ′ [ − X i ] − α minρ i ( m ′ ) (cid:17) µ i ≥ X i ∈I ρ i ( X i ) µ i ≥ ρ µconv ( X ) , which implies m ′ ∈ ∂ρ µconv ( X ).(v) We fix m ∈ ba , + . If X ∈ n X ∈ L ∞ : ∃ { X i } i ∈I ∈ A ( X ) s.t. X i ∈ ∂α minρ i ( m ) ∀ i ∈ I µ o , let { X i } i ∈I ∈ A ( X ) such that X i ∈ ∂α minρ i ( m ) ∀ i ∈ I µ Then, E m [ − X ] − ρ µconv ( X ) ≥ X i ∈I (cid:0) E m [ − X i ] − ρ i ( X i ) (cid:1) µ i = X i ∈I α minρ i ( m ) µ i ≥ α minρ µconv ( m ) . Thus, X ∈ ∂α minρ µconv ( m ). Remark . In the context of item (iii) and under coherence of ρ I , by Proposition 3.3 we havethat ρ ( X ) ≤ ρ µconv ( X ) ≤ ρ i ( X ) ∀ i ∈ I µ , ∀ X ∈ L ∞ for any convex risk measure ρ such that ρ ( X ) ≤ ρ i ( X ) ∀ i ∈ I µ , ∀ X ∈ L ∞ . In this sense, we can understand ρ µconv as the “lower-convexification” of the non-convex risk measure inf i ∈I µ ρ i in the sense that the former is thelargest convex risk measure that is dominated by the latter.We now present the main results regarding the representation of ρ µconv for coherent cases. Theorem 4.5.
Let ρ I be a collection of coherent risk measures. Then,(i) ρ µconv is finite, and its dual set is Q ρ µconv = (cid:8) m ∈ ba , + : m ∈ Q ρ i ∀ i ∈ I µ (cid:9) = \ i ∈I µ Q ρ i . (4.3) In particular, Q ρ µconv is non empty.(ii) The acceptance set of ρ µconv is A ρ µconv = clconv ( A ∪ ) = cl ( A µ ) , A ∪ = [ i ∈I µ A ρ i , (4.4) where clconv denotes the closed convex hull. roof. (i) By Proposition 3.4, we have that ρ µconv is a coherent risk measure. Since ρ µconv (0) =0, it is finite. In this case, its dual set is composed by the measures m ∈ ba , + such that α minρ µconv ( m ) = 0. By Theorems 2.3 and 4.1, we obtain that Q ρ µconv = ( m ∈ ba , + : X i ∈I α minρ i ( m ) µ i = 0 ) = n m ∈ ba , + : α minρ i ( m ) = 0 ∀ i ∈ I µ o = (cid:8) m ∈ ba , + : m ∈ Q ρ i ∀ i ∈ I µ (cid:9) = \ i ∈I µ Q ρ i . The convexity and closedness of Q ρ µconv follow from the convexity and lower semicontinuityof α minρ µconv . Furthermore, if Q ρ µconv = n m ∈ ba , + : α minρ i ( m ) = 0 ∀ i ∈ I µ o = ∅ , then byCorollary 4.3, ρ µconv (0) <
0, which contradicts coherence.(ii) We recall that, by Theorem 2.3 and Remark 2.4, for any coherent risk measure ρ : L ∞ → R ,we have X ∈ A ρ if and only if E m [ − X ] ≤ ∀ m ∈ Q ρ . Thus, Q ρ µconv = \ i ∈I µ (cid:8) m ∈ ba , + : E m [ − X ] ≤ ∀ X ∈ A ρ i (cid:9) = (cid:8) m ∈ Q : E m [ − X ] ≤ , ∀ X ∈ ∪ i ∈I µ A ρ i (cid:9) = { m ∈ Q : E m [ − X ] ≤ , ∀ X ∈ clconv ( A ∪ ) } = Q ρ clconv ( A∪ ) . By considering the closed convex hull does not affect is because the map X → E m [ X ]is linear and continuous for any m ∈ Q . Thus, ρ µconv = ρ clconv ( A ∪ ) . Hence, A ρ µconv = A ρ clconv ( A∪ ) = clconv ( A ∪ ). We note that A ∪ is nonempty, monotone (in the sense that X ∈ A ∪ and Y ≥ X implies Y ∈ A ∪ ), and a cone, as this is true for any A ρ i . Moreover,it is evident that A ∪ ⊆ A µ for normalized risk measures in ρ I . Thus, by Theorem 4.1, wehave that A ρ µconv = clconv ( A ∪ ) ⊆ cl ( A µ ) ⊆ A ρ µconv . Remark . (i) Similarly to Remark 4.2, under the assumption of Fatou continuity for boththe risk measures in ρ I and ρ µconv , the claims in Theorem 4.1 could be adapted by replac-ing the finitely additive measures m ∈ ba + , by probabilities Q ∈ Q . Moreover, weak ∗ topological concepts could replace the corresponding strong (norm) topological concepts.(ii) In light of Corollary 4.3, we have, under the hypotheses of Theorem 4.5, that ρ µconv isfinite and normalized if and only if the condition n m ∈ ba , + : α minρ i ( m ) = 0 ∀ i ∈ I µ o = ∅ is satisfied. Moreover, both assertions are equivalent to A µ not being dense in L ∞ . Theintuition for A ∪ is that some position is acceptable if it is acceptable for any relevant (inthe µ sense) member of ρ I .We now focus on dual representations under the assumption of law invariance and comono-tonic additivity. Theorem 4.7.
Let ρ I be a collection of convex, law-invariant risk measures. Then, i) ρ µconv is finite, normalized and has penalty term β minρ µconv ( m ) = X i ∈I β minρ i ( m ) µ i , ∀ m ∈ M . (4.5) (ii) If, in addition, ρ I consists of comonotone risk measures, then ρ µconv is comonotone, andits distortion function is g = inf i ∈I µ g i , (4.6) where g i is the distortion of ρ i for each i ∈ I .Proof. (i) By Theorem 2.6 and Proposition 3.4, we have that ρ µconv is a Fatou continuousconvex risk measure with ρ µconv < ∞ . Moreover, ρ µconv is either finite or identically −∞ .Regarding finiteness and normalization, we note that ρ ( X ) ≥ − E [ X ] , ∀ X ∈ L ∞ fornormalized, convex, law-invariant risk measures by second-order stochastic dominance.Thus, α minρ i ( P ) = sup X ∈ L ∞ { E [ − X ] − ρ ( X ) } ≤ sup X ∈ L ∞ { ρ ( X ) − ρ ( X ) } = 0 , ∀ i ∈ I . By the non-negativity of the penalty terms, we have α minρ i ( P ) = 0 , ∀ i ∈ I . Hence, byitem (ii) of Corollary 4.3, we conclude that ρ µconv is normalized and, consequently, finite.Moreover, the penalty term can be obtained by an argument similar to that in (ii) ofTheorem 4.1 by considering the m → R (0 , ES α ( X ) dm linear and playing the role of m → E m [ − X ] and recalling that acceptance sets of law invariant risk measures are lawinvariant in the sense that X ∈ A ρ and X ∼ Y implies Y ∈ A ρ .(ii) By Theorems 2.8, 4.1, and 4.5, as well as Remark 2.9, we have, recalling that ρ µconv isfinite and Fatou continuous, that Q ρ µconv = { Q ∈ Q : Q ( A ) ≤ g i ( P ( A )) ∀ i ∈ I µ ∀ A ∈ F } = (cid:26) Q ∈ Q : Q ( A ) ≤ inf i ∈I µ g i ( P ( A )) ∀ A ∈ F (cid:27) . By the properties of the infimum and { g i } i ∈I , we obtain that g : [0 , → [0 ,
1] is increasingand concave, and it satisfies g (0) = 0 and g (1) = 1. Thus, ρ µconv can be represented as aChoquet integral using (2.7), which implies that it is comonotone. Remark . As a direct consequence of (ii) in the last theorem, if ρ i = ES α i , α i ∈ [0 , ∀ i ∈ I ,with α = sup i ∈I µ α i , then ρ µconv ( X ) = ES α ( X ) , ∀ X ∈ L ∞ . The financial intuition is that theinf convolution of countable many ES at distinct significance levels provides the same risk asthe less conservative option.Regarding comonotonic additivity, (ii) in Theorem 4.7 remains true if we drop the lawinvariance of ρ I , as shown in the following corollary.19 orollary 4.9. Let ρ I = { ρ i : L ∞ → R , i ∈ I} be a collection of convex, comonotone riskmeasures. Then, ρ µconv is finite, normalized, and comonotone, and its capacity function is c ( A ) = inf i ∈I µ c i ( A ) , ∀ A ∈ F , (4.7) where c i is the capacity of ρ i for each i ∈ I .Proof. We note that, by Proposition 3.4, we have ρ µconv < ∞ and ρ µconv (0) = 0 > −∞ . Let theset function c : F → [0 ,
1] be defined as c ( A ) = inf i ∈I µ c i ( A ), where c i is the capacity relatedto ρ i for each i ∈ I . Thus, by an argument similar to that in Theorem 4.7, if we consider ba c , + = { m ∈ ba , + : m ( A ) ≤ c ( A ) ∀ A ∈ F } , then the reasoning in Remark 2.9 implies that theclaim is true. An interesting feature of traditional finite inf-convolution is capital allocation. Highly relevantconcept is Pareto optimality, which is defined as follows.
Definition 5.1.
We call { X i } i ∈I ∈ A ( X ) (i) Optimal for X ∈ L ∞ if P i ∈I ρ i ( X i ) µ i = ρ µconv ( X ) .(ii) Pareto optimal for X ∈ L ∞ if for any { Y i } i ∈I ∈ A ( X ) such that ρ i ( Y i ) ≤ ρ i ( X i ) ∀ i ∈ I µ ,we have ρ i ( Y i ) = ρ i ( X i ) ∀ i ∈ I µ .Remark . (i) If ρ µconv is normalized, { X i = 0 } i ∈I is optimal for 0. This implies the con-dition that if { X i } i ∈I ∈ A (0) and ρ i ( X i ) ≤ ∀ i ∈ I µ , then ρ i ( X i ) = 0 ∀ i ∈ I µ , andtherefore { X i = 0 } i ∈I is also Pareto optimal for 0. This can be understood as a non-arbitrage condition. We note that any optimal allocation must be Pareto optimal, andthat a risk sharing rule is also a Pareto-optimal allocation.If ρ I consists of monetary risk measures and I is finite, Theorem 3.1 in Jouini et al. (2008)shows that optimal and Pareto-optimal allocations coincide. In the following proposition, weextend this result to the context of countable I . Proposition 5.3.
We have that(i) { X i } i ∈I ∈ A ( X ) is optimal for X ∈ L ∞ if and only if it is Pareto optimal for X ∈ L ∞ .(ii) if { X i } i ∈I is optimal for X ∈ L ∞ , then so is { X i + C i } i ∈I , where C i ∈ R ∀ i ∈ I µ , and P i ∈I C i µ i = 0 .(iii) Under sub-additivity of ρ I and normalization of ρ µconv we have that if { X i } i ∈I is optimalfor X ∈ L ∞ , then so is { X i + Y i } for any { Y i } that is optimal for .Proof. (i) The “only if” part is straightforward, as in Remark 5.2. For the “if” part, let { X i } i ∈I ∈ A ( X ) be not optimal for X ∈ L ∞ . Then, there is { Y i } i ∈I ∈ A ( X ) such that20 i ∈I ρ i ( Y i ) µ i < P i ∈I ρ i ( X i ) µ i . Let k i = ρ i ( X i ) − ρ i ( Y i ) ∀ i ∈ I µ and k = P i ∈I k i µ i > { Y i − k i + k } i ∈I ∈ A ( X ) and X i ∈I ρ i ( Y i − k i + k ) µ i < X i ∈I ρ i ( Y i − ρ i ( X i ) + ρ i ( Y i )) µ i = X i ∈I ρ i ( X i ) µ i . Hence, { X i } i ∈I is not Pareto optimal for X .(ii) We note that X i ∈I ρ i ( X i + C i ) µ i = X i ∈I ρ i ( X i ) µ i − X i ∈I C i µ i = ρ µconv ( X ) . Thus, { X i + C i } i ∈I is also Pareto optimal.(iii) This claim follows by X i ∈I ρ ( X i + Y i ) ≤ X i ∈I ρ ( X i ) + X i ∈I ρ ( Y i ) = ρ µconv ( X ) + ρ µconv (0) = ρ µconv ( X ) . Hence, { X i + Y i } ∈ A ( X ) is Pareto optimal for X .We now determine a necessary and sufficient condition for optimality in the case of convexrisk measures. Theorem 5.4.
Let ρ I be a family of convex risk measures. Then { X i } i ∈I ∈ A ( X ) is optimalfor X ∈ L ∞ if and only if T i ∈I µ ∂ρ i ( X i ) = ∂ρ µconv ( X ) = ∅ .Proof. By Corollary 4.3 we have T i ∈I µ ∂ρ i ( X i ) ⊆ ∂ρ µconv ( X ). We assume T i ∈I µ ∂ρ i ( X i ) = ∅ .Then, let m ′ ∈ (cid:8) m ∈ ba , + : m ∈ ∂ρ i ( X i ) ∀ i ∈ I µ (cid:9) ⊆ ∂ρ µconv ( X ). We have that X i ∈I ρ i ( X i ) µ i = X i ∈I (cid:16) E m ′ [ − X i ] − α minρ i ( m ′ ) (cid:17) µ i ≤ E m ′ [ − X ] − α minρ µconv ( m ′ ) = ρ µconv ( X ) . Hence, { X i } i ∈I is optimal for X ∈ L ∞ . Regarding the converse, for any m ′ ∈ ∂ρ µconv ( X ), weobtain by an argument similar to that in Theorem 4.1 the following: ρ µconv ( X ) = E m ′ [ − X ] − X i ∈I α minρ i ( m ′ ) µ i = lim n →∞ E m ′ [ − X ] + n X i =1 inf Y ∈ L ∞ (cid:8) E m ′ [ Y ] + ρ i ( Y ) (cid:9) µ i ! ≤ lim n →∞ E m ′ [ − X ] + inf { Y ,...,Y n }⊂ L ∞ n X i =1 (cid:8) E m ′ [ Y i ] + ρ i (cid:0) Y i (cid:1)(cid:9) µ i ! ≤ lim n →∞ inf ( n X i =1 (cid:8) E m ′ [ Y i − X ] + ρ i (cid:0) Y i (cid:1)(cid:9) µ i : n X i =1 Y i µ i = X ) = lim n →∞ inf ( n X i =1 (cid:8) ρ i (cid:0) Y i (cid:1)(cid:9) µ i : n X i =1 Y i µ i = X ) = ρ µconv ( X ) . ρ µconv ( X ) = P i ∈I µ ρ i ( X i ) µ i if and only if ∃ m ′ ∈ ba , + such that ρ i ( X i ) = E m ′ [ − X ] + α minρ i ( m ′ ) ∀ i ∈ I µ . Hence, m ′ ∈ (cid:8) m ∈ ba , + : m ∈ ∂ρ i ( X i ) ∀ i ∈ I µ (cid:9) .We have the following corollary regarding subdifferential and optimality conditions. Corollary 5.5.
Let ρ I be a collection convex risk measures. If for any X ∈ L ∞ there is anoptimal allocation, then ∂α minρ µconv ( m ) = P i ∈I µ ∂α minρ i ( m ) µ i for any m ∈ ba , + .Proof. From Corollary 4.3 we have P i ∈I µ ∂α minρ i ( m ) µ i ⊆ ∂α minρ µconv ( m ) for any m ∈ ba , + . Forthe converse relation, if ∂α minρ µconv ( m ) = ∅ , then the claim is immediately obtained. Let then X ∈ ∂α minρ µconv ( m ). By the definition of Legendre–Fenchel convex-conjugate duality, the optimalitycondition is equivalent to the existence of m ∈ ba , + such that X i ∈ ∂α minρ i ( m ) ∀ i ∈ I µ . ByTheorem 5.4, we have that X i ∈ ∂α minρ i ( m ) ∀ i ∈ I µ . Then, X ∈ P i ∈I µ ∂α minρ i ( m ) µ i .Under the assumption of law invariance, it is well known that, for finite I , the mini-mization problem has a solution under co-monotonic allocations (see, for instance, Theorem3.2 in Jouini et al. (2008), Proposition 5 in Dana and Meilijson (2003), or Theorem 10.46 inR¨uschendorf (2013)). For the extension to general I , we should extend some definitions andresults regarding comonotonicity. We note that if I is finite, these are equivalent to theirtraditional counterparts. Definition 5.6. { X i } i ∈I is called I -comonotone if ( X i , X j ) is comonotone ∀ ( i, j ) ∈ I µ × I µ . Lemma 5.7. { X i } i ∈I ∈ A ( X ) , X ∈ L ∞ , is I -comonotone if and only if there exists a classof functions { h i : R → R , i ∈ I} that are ( ∀ i ∈ I µ ) Lipschitz continuous and increasing, andthey satisfy X i = h i ( X ) and P i ∈I h i ( x ) µ i = x, ∀ x ∈ R . In particular, if { X i } i ∈I ∈ A ( X ) is I -comonotone, then F − X ( α ) = P i ∈I F − X i ( α ) µ i , ∀ α ∈ [0 , .Proof. For the “if” part, let X i , X j ∈ I µ such that X i = h i ( X ) and X j = h j ( X ) for { h i } i ∈ I satisfying the assumptions. Then X i , X j are comonotone. For the “only if” part, let { X i } i ∈I be I -comonotone and X (Ω) = { x ∈ R : ∃ ω ∈ Ω s.t. X ( w ) = x } . Then, for any fixed ω ∈ Ω, thereis a family { x i = X i ( ω ) ∈ R : i ∈ I} such that X ( ω ) = x = P i ∈I x i µ i ∈ X (Ω). Moreover, wedefine h i ( x ) = x i ∀ i ∈ I µ . If there are ω, ω ′ ∈ Ω such that P i ∈I X i ( ω ) µ i = x = P i ∈I X i ( ω ′ ) µ i ,we then obtain P i ∈I (cid:0) X i ( ω ) − X i ( ω ′ ) (cid:1) µ i = 0. Assuming I -comonotonicity, we have that X i ( ω ) = X i ( ω ′ ) ∀ i ∈ I µ . Consequently, the map x → P i ∈I h i ( x ) µ i = Id ( x ) is well defined.Regarding the increasing behavior of h i , let x, y ∈ X (Ω) with x ≤ y . Then, there are ω, ω ′ suchthat P i ∈I X i ( ω ) µ i = x ≤ y = P i ∈I X i ( ω ′ ) µ i , which implies P i ∈I (cid:0) X i ( ω ) − X i ( ω ′ ) (cid:1) µ i ≤ h i ( x ) = X i ( ω ) ≤ X i ( ω ′ ) = h i ( y ) ∀ i ∈I µ . Concerning Lipschitz continuity, for any x, x + δ ∈ X (Ω) with δ > ≤ h i ( x + δ ) − h i ( x ) ≤ ( µ i ) − δ, ∀ i ∈ I µ . The first inequality is due to the increasing behavior of h i . The second is because for any i ∈ I we have x + δ = P j ∈I\{ i } h j ( x + δ ) µ j + h i ( x + δ ) µ i ≥ P j ∈I\{ i } h j ( x ) µ j + h i ( x ) µ i = h i ( x + δ ) µ i + x − h i ( x ) µ i . It remains to extend { h i } from X (Ω)to R . We first extend it to cl ( X (Ω)). If x ∈ bd ( X (Ω)) is only a one-sided boundary point, thenthe continuous extension poses no problem, as increasing functions are involved. If x can beapproximated from both sides, then Lipschitz continuity implies that the left- and right-sidedcontinuous extensions coincide. The extension to R is performed linearly in each connected22omponent of R \ cl ( X (Ω)) so that the condition P i ∈I h i ( x ) = x is satisfied. Then, the mainclaim is proved. Moreover, let { X i } i ∈I ∈ A ( X ) be I -comonotone. Then, x → P i ∈I h i ( x ) µ i is also Lipschitz continuous and increasing. We recall that F − g ( X ) = g ( F − X ) for any increasingfunction g : R → R . Then, for any α ∈ [0 , F − X ( α ) = F − P i ∈I h i ( X ) µ i ( α ) = X i ∈I h i ( F − X ( α )) µ i = X i ∈I F − h i ( X ) ( α ) µ i = X i ∈I F − X i ( α ) µ i . We now prove the following comonotonic-improvement theorem for arbitrary I . Theorem 5.8.
Let X ∈ L ∞ . Then, for any { X i } i ∈I ∈ A ( X ) , there is an I -comonotone { Y i } i ∈I ∈ A ( X ) such that Y i (cid:23) X i ∀ i ∈ I µ Proof.
Let F n be the σ -algebra generated by { ω : k − n ≤ X ( ω ) ≤ k n } ⊂ Ω for k > X n = E [ X |F n ], and X in = E [ X i |F n ] ∀ i ∈ I . Then, lim n →∞ X n = X , lim n →∞ X in = X i for any i ∈ I ,and X in (cid:23) X i for each n each i . By the arguments in Proposition 1 in Landsberger and Meilijson(1994) or Proposition 10.46 in R¨uschendorf (2013), we can conclude that every allocation of X taking a countable number of values is dominated by a comonotone allocation. Thus, by Lemma5.7, for any n ∈ N , there are Lipschitz continuous, increasing functions { h in : R → R , i ∈ I} with P i ∈I h in µ i = Id such that Y in = h in ( X n ) (cid:23) X in ∀ i ∈ I µ We note that these functions constitutea bounded, closed, equicontinuous family. Then, by Ascoli’s theorem, there is a subsequence of { h in } that converges uniformly on [ess inf X, ess sup X ] to the Lipschitz-continuous and increasing h i in the ∀ i ∈ I µ sense. Thus, P i ∈I h i µ i = Id on [ess inf X, ess sup X ]. We then have Y i = h i ( X ) (cid:23) X i ∀ i ∈ I µ by considering uniform limits. Finally, by Lemma 5.7, we obtain that { Y i } i ∈I is I -comonotone. It remains to show that { Y i } i ∈I belongs to A ( X ). We have that P i ∈I Y i µ i = P i ∈I h i ( X ) µ i = X P − a.s. Hence, { Y i } i ∈I ∈ A ( X ).We are now in a position to extend the existence of optimal allocations to our framework oflaw-invariant, convex risk measures. Theorem 5.9.
Let ρ I be a collection of law-invariant, convex risk measures. Then,(i) For any X ∈ L ∞ , there is an I -comonotone optimal allocation.(ii) In addition to initial hypotheses, if ρ I consists of risk measures that are strictly monotonewith respect to (cid:23) , then every optimal allocation for any X ∈ L ∞ is I -comonotone.(iii) In addition to initial hypotheses, if ρ I consists of strictly convex functionals, we haveuniqueness of the optimal allocation up to scaling (if { X i } i ∈I is optimal for X ∈ L ∞ ,then so is { X i + C i } i ∈I , where C i ∈ R ∀ i ∈ I µ and P i ∈I C i µ i = 0 ).Proof. (i) By Theorem 5.8, we can restrict the minimization problem to I -comonotonic al-locations, as, by Theorem 2.7, law-invariant risk measures preserve second-order stochas-tic dominance. Let { Y in = h in ( X ) ∈ L ∞ , i ∈ I} n be an optimal sequence for X ,i.e. lim n →∞ P i ∈I ρ i ( Y in ) µ i = ρ µconv ( X ), where h in : [ess inf X, ess sup X ] → R are increas-ing, bounded, and Lipschitz-continuous functions. Such sequence always exist because23 I is monetary and we can take h in = − x n for any i ∈ I , where x n → ρ µconv ( X ). Thus,each h in is increasing, bounded and Lipschitz continuous while lim n →∞ P i ∈I ρ i ( h in ( X )) µ i =lim n →∞ P i ∈I ρ i ( − x n ) µ i = lim n →∞ x n = x . By an argument similar to that in Theorem 5.8, wehave that h i is the uniform limit (after passing to a subsequence if necessary) of { h in } .Thus, Y in = h in ( X ) → h i ( X ) = Y i . By continuity in the essential supremum norm, wehave that lim n →∞ (cid:12)(cid:12) ρ i ( Y in ) − ρ i ( Y i ) (cid:12)(cid:12) = 0. As h i is the uniform limit of { h in } , we have bydominated convergence, since | ρ i ( Y in ) | ≤ k X k ∞ < ∞ for any n ∈ N , that ρ µconv ( X ) = lim n →∞ X i ∈I ρ i ( Y in ) µ i = X i ∈I lim n →∞ ρ i ( Y in ) µ i = X i ∈I ρ i ( Y i ) µ i . Hence, { Y i } i ∈I is the desired optimal allocation.(ii) We recall that strict monotonicity implies that if X (cid:23) Y and X Y , then ρ i ( X ) <ρ i ( Y ) ∀ i ∈ I µ for any X, Y ∈ L ∞ . Let { X i } i ∈I be an optimal allocation for X ∈ L ∞ .Then, by Theorem 5.8, there is an I -comonotone allocation { Y i } i ∈I ∈ A ( X ) such that ρ µconv ( X ) = X i ∈I ρ i ( X i ) µ i ≥ X i ∈I ρ i ( Y i ) µ i . Thus, { Y i } i ∈I is also optimal. If X i = Y i ∀ i ∈ I µ , then we have the claim. If thereis i ∈ I µ such that X i = Y i , then we have by strictly monotonicity regarding (cid:23) that ρ i (cid:0) Y i (cid:1) < ρ i ( X i ), contradicting the optimality of { X i } i ∈I . Hence, every optimal allocationfor X is I -comonotone.(iii) We assume, toward a contradiction, that both { X i } i ∈I and { Y i } i ∈I are optimal allocationsfor X ∈ L ∞ such that there is i ∈ I µ with X i = Y i and there is no C i ∈ R ∀ i ∈ I µ suchthat P i ∈I C i µ i = 0 and { X i + C i } i ∈I (otherwise, item (ii) in Proposition 5.3 assuresoptimality). We note that for any λ ∈ [0 , { Z i = λX i + (1 − λ ) Y i } i ∈I is in A ( X ). However, we would have X i ∈I ρ i ( Z i ) µ i < λ X i ∈I ρ i ( X i ) µ i + (1 − λ ) X i ∈I ρ i ( Y i ) µ i = ρ µconv ( X ) , which contradicts the optimality of both { X i } i ∈I and { Y i } i ∈I for X . Remark . The examples in Jouini et al. (2008) and Delbaen (2006) show that law invari-ance is essential to ensure the existence of an I -comonotone solution as above. However, theuniqueness of this optimal allocation is not ensured outside the scope of strict convexity as initem (iii) of the last Theorem.If ρ I consists of comonotone, law-invariant, convex risk measures, then we can prove anadditional result regarding the connection between optimal allocations and the notion of flatnessfor quantile functions. To this end, the following definitions and lemma are required. We recallthat dF − X is the differential of F − X . 24 efinition 5.11. Let g , g be two distortions with g ≤ g . A quantile function F − X , X ∈ L ∞ ,is called flat on { x ∈ [0 ,
1] : g ( x ) < g ( x ) } if dF − X = 0 almost everywhere on { g < g } and ( F − X (0 + ) − F − X (0))( g (0 + ) − g (0 + )) = 0 . Lemma 5.12 (Lemmas 4.1 and 4.2 in Jouini et al. (2008)) . Let ρ : L ∞ → R be a law-invariant,comonotone, convex risk measure with distortion g ; moreover, let m ∈ ba , + has a Lebesguedecomposition m = Z m P + m s into a regular part with density Z m and a singular part m s .Then,(i) g m : [0 , → R defined as g m (0) = 0 and g m ( t ) = k m s k T V + R t F − Z m (1 − s ) ds, < t ≤ ,is a concave distortion.(ii) For any m ∈ ∂ρ ( X ) , we have that X and − Z m are comonotone. Moreover, the measure m ′ such that Z m ′ = E [ Z m | X ] belongs to ∂ρ ( X ) .(iii) ∂ρ ( X ) = (cid:8) m ∈ ba , + : g m ≤ g, F − X is flat on { g m < g } (cid:9) . Theorem 5.13.
Let ρ I consist of law-invariant, comonotone, convex risk measures with dis-tortions { g i } i ∈I , g = inf i ∈I µ g i , and { X i } i ∈I ∈ A ( X ) be I -comonotone. If F − X i is flat on { g < g i } ∩ { dF − X > } ∀ i ∈ I µ , then { X i } i ∈I is an optimal allocation for X ∈ L ∞ . Theconverse is true if ( i, α ) → V aR α ( X i ) g ′ m ( α ) is bounded.Proof. By Theorem 4.7, g is the distortion of ρ µconv . Let U be a [0 , X = F − X ( U ). Wedefine m ∈ ba , + by m = g (0 + ) δ ( U ) + g ′ ( U )1 (0 , ( U ), where δ is the Dirac measure at 0. It iseasily verified using (i) in Lemma 5.12 that g m = g . Moreover, let { X i } i ∈I be an I -comonotoneoptimal allocation for X (the existence of which is ensured by Theorem 5.9). Thus, in the ∀ i ∈ I µ sense, g m ≤ g i , − Z m is comonotone with X i , and, by hypothesis, F − X i is flat on { g m < g i } ∩ { dF − X > } . By Lemma 5.7, we have { dF − X = 0 } = { α ∈ [0 ,
1] : dF − X i ( α ) =0 ∀ i ∈ I µ } . Thus, { g m < g i } ∩ { dF − X = 0 } = ∅ ∀ i ∈ I µ By (iii) of Lemma 5.12, we have that m ∈ ∂ρ i ( X i ) ∀ i ∈ I µ Then, by Theorem 5.4, we obtain that { X i } i ∈I is an optimal allocation.For the converse, by Theorem 5.4 and Corollary 5.5, we have that X i ∈ ∂α minρ i ( m ′ ) ∀ i ∈ I µ for some m ′ ∈ ba , + . By Corollary 5.5, we have that X ∈ ∂α minρ µconv ( m ′ ), and by convex-conjugateduality, m ′ ∈ ∂ρ µconv ( X ). Thus, by (ii) in Lemma 5.12, we obtain that m ∈ ba , + such that Z m = E [ Z m ′ | X ] belongs to ∂ρ µconv ( X ) = (cid:8) m ∈ ba , + : m ∈ ∂ρ i ( X i ) ∀ i ∈ I µ (cid:9) . By Theorem 2.8and (iii) of Lemma 5.12, we have that ρ i ( X i ) = R V aR α ( X i ) g ′ m ( α ) dα ∀ i ∈ I µ . As { X i } i ∈I isan optimal allocation, Theorem 4.7 and Lemma 5.7 imply that Z V aR α ( X ) g ′ ( α ) dα = ρ µconv ( X )= X i ∈I Z V aR α ( X i ) g ′ m ( α ) dαµ i = Z X i ∈I V aR α ( X i ) µ i g ′ m ( α ) dα = Z V aR α ( X ) g ′ m ( α ) dα.
25e can make the interchange of sum and integral for dominated convergence because of theboudedness assumption. By continuity,
V aR α ( X ) dα = − dF − X ( α ). Then, we have R ( g m ( α ) − g ( α )) dF − X ( α ) = 0. As m ∈ ∂ρ µconv ( X ), (iii) in Lemma 5.12 implies that g m ≤ g , and therefore g m = g in { dF − X > } . Hence, F − X i is flat on { g < g i } ∩ { dF − X > } ∀ i ∈ I µ . Remark . We also have in this context that for any optimal allocation { X i } i ∈I , F − X i is flaton { g i = g j } ∩ { dF − X = 0 } for any j = i in I µ sense. To see this, let t ∈ { g j < g i } ∩ { dF − X = 0 } .We note that, by Lemma 5.7, { dF − X = 0 } = { α ∈ [0 ,
1] : dF − X i ( α ) = 0 ∀ i ∈ I µ } . Then, byLemma 5.12, g m ( t ) < g i ( t ), and thus dF − X i is flat at { g m < g i } . By comonotonicity and Lemma5.7, the same is true for dF − X j . By repeating the argument for t ∈ { g j > g i } ∩ { dF − X = 0 } , weprove the claim.A relevant concept in the present context is the dilated risk measure, which is stable underinf-convolution and has a dilatation property with respect to the size of a position. In thisparticular situation, we can provide explicit solutions for optimal allocations even without lawinvariance. We now define this concept and extend some interesting related results to ourframework. Definition 5.15.
Let ρ : L ∞ → R be a risk measure, and γ > be a real parameter. The dilatedrisk measure with respect to ρ and γ is a functional ρ γ : L ∞ → R defined as ρ γ ( X ) = γρ (cid:18) γ X (cid:19) . (5.1) Remark . A typical example of dilated measure is the entropic
Ent γ with Ent as basis.It is evident that, for convex risk measures, α minρ γ = γα minρ . Moreover, a convex risk measureis coherent if and only if ρ = ρ γ pointwise for any γ >
0. Under normalization, lim γ →∞ ρ γ definesthe smallest coherent risk measure that dominates ρ . Proposition 5.17.
We have that(i) ( ρ µconv ) γ = inf (cid:8)P i ∈I ( ρ i ) γ ( X i ) µ i : { X i } i ∈I ∈ A ( X ) (cid:9) for any γ > .(ii) Let ρ be a convex risk measure and { γ i > } i ∈I , such that P i ∈I γ i µ i = γ . If ρ i = ρ γ i ∀ i ∈I µ , then n γ i γ X o i ∈I is optimal for X ∈ L ∞ and ρ µconv = ρ γ .Proof. (i) For any γ > X ∈ L ∞ , we have thatinf (X i ∈I ( ρ i ) γ ( X i ) µ i : { X i } i ∈I ∈ A ( X ) ) = γ inf (X i ∈I ρ i (cid:18) γ X i (cid:19) µ i : { X i } i ∈I ∈ A ( X ) ) = γ inf (X i ∈I ρ i (cid:0) Y i (cid:1) µ i : { Y i } i ∈I ∈ A (cid:18) γ X (cid:19)) = γρ µconv (cid:18) γ X (cid:19) = ( ρ µconv ) γ ( X ) . (ii) We obtain from Theorem 4.1 that α minρ µconv ( m ) = X i ∈I α minρ γi ( m ) µ i = X i ∈I γ i α minρ ( m ) µ i = γα minρ ( m ) = α minρ γ ( m ) , ∀ m ∈ ba. ρ µconv = ρ γ . It is straightforward to note that n γ i γ X o i ∈I ∈ A ( X ). We then obtainthat ρ µconv ( X ) ≤ X i ∈I ρ γ i (cid:18) γ i γ X (cid:19) µ i = X i ∈I γ i ρ (cid:18) γ i γ i γ X (cid:19) µ i = γρ (cid:18) γ X (cid:19) = ρ γ ( X ) ≤ ρ µconv ( X ) . Hence, n γ i γ X o i ∈I is optimal for X ∈ L ∞ . Remark . We note that for any X ∈ L ∞ , (cid:8) γ i γ − X (cid:9) i ∈I is I -comonotone, which is inconsonance with Theorem 5.9 when the risk measures in ρ I are law invariant. For instance, wehave that if ρ i = Ent γ i , γ i > , ∀ i ∈ I µ , then ρ µconv = Ent γ .We now follow the approach of Embrechts et al. (2018) by focusing in robustness of optimalallocations instead of the one for ρ µconv . Intuitively, if an optimal allocation is robust then undera small model misspecification, the true aggregate risk value would be close from the obtainedone. We now formally define such concept and prove a result that relate robustness and uppersemi continuity. To that, we need the concept of allocation principle, which is defined in thefollowing. Definition 5.19.
We define the following:(i) { h i : R → R , i ∈ I} is an allocation principle if ∀ i ∈ I µ : h i ( X ) ∈ L ∞ ∀ X ∈ L ∞ , h i hasat most finitely points of discontinuity, and P i ∈I h i µ i is the identity function. We denoteby H the set of allocation principles.(ii) let d be a pseudo-metric on L ∞ and { h i } i ∈I ∈ H . Then { h i ( X ) } i ∈I is d -robust if the mapon L ∞ defined as Y → P i ∈I ρ i ( h i ( Y )) µ i is continuous at X in respect to d .Remark . Note that from Lemma 5.7, I -comonotonicity implies the existence of allocationprinciples. Proposition 5.21.
Let d be a pseudo-metric on L ∞ . If { h i ( X ) } i ∈I is a d -robust optimalallocation of X ∈ L ∞ , then ρ µconv is upper semi continuous at X with respect to d .Proof. Let { h i ( X ) } i ∈I be a d -robust optimal allocation for X ∈ L ∞ , and { X n } ⊂ L ∞ be suchthat X n → X in d . Since ρ µconv ( X n ) ≤ P i ∈I ρ i ( h i ( X n )) µ i , by convergence regarding d we getthat lim sup n →∞ ρ µconv ( X n ) ≤ lim sup n →∞ X i ∈I ρ i ( h i ( X n )) µ i = ρ µconv ( X ) . Remark . Regarding the converse statement for proposition 5.21, Theorem 5 of Embrechts et al.(2018), which is focused in quantiles, asserts that even continuity in respect to d is not sufficientfor the existence of a robust optimal allocation. Furthermore, from Remark 3.9, when eachmember of ρ I is a convex risk measure there is not robust optimal allocations regarding theLevy metric. In fact, this is a relatively new concept and even in the finite I case there is stillnot a general sufficient condition. We left for future research a more complete study of generalsufficient conditions for robust allocations. 27 Self-convolution and regulatory arbitrage
In this section, we consider the special case ρ i = ρ, ∀ i ∈ I . In this situation, we have that ρ µconv is a self-convolution. This concept is highly important in the context of regulatory arbitrage (asin Wang (2016)), where the goal is to reduce the regulatory capital of a position by splitting it.The difference between ρ ( X ) and ρ µconv ( X ) is then obtained by a simple rearrangement (sharing)of risk. We now adjust this concept to our framework. Definition 6.1.
The regulatory arbitrage of a risk measure ρ is a functional τ ρ : L ∞ → R + ∪{∞} defined as τ ρ ( X ) = ρ ( X ) − ρ µconv ( X ) , ∀ X ∈ L ∞ . (6.1) Moreover, ρ is called(i) free of regulatory arbitrage if τ ρ ( X ) = 0 , ∀ X ∈ L ∞ ;(ii) of finite regulatory arbitrage if τ ρ ( X ) < ∞ , ∀ X ∈ L ∞ ;(iii) of partially infinite regulatory arbitrage if τ ρ ( X ) = ∞ for some X ∈ L ∞ ;(iv) of infinite regulatory arbitrage if τ ρ ( X ) = ∞ , ∀ X ∈ L ∞ .Remark . As ρ µconv ≤ ρ < ∞ , we have that τ ρ is well defined. Our approach is different fromthat in Wang (2016) because we consider “convex” inf-convolutions instead of direct sums. Morespecifically, the approach by Wang (2016) is R ( X ) = inf ( n X i =1 ρ ( X i ) , n ∈ N , X i ∈ L ∞ , i = 1 , · · · , n, n X i =1 X i = X ) = lim n →∞ inf ( n X i =1 ρ ( X i ) , X i ∈ L ∞ , i = 1 , · · · , n, n X i =1 X i = X ) , while in our case we have the role for { µ i } . This distinction leads to differences specially, as isexplored in Theorem 6.6 below, because in our approach convexity rules out regulatory aritrage,while in Wang (2016) sub-additivity plays this role. For instance, in that approach, Ent γ is oflimited regulatory arbitrage, whereas in ours (see Proposition 6.6 below), we have τ Ent γ = 0,that is, Ent γ is free of regulatory arbitrage. Moreover, it is evident that τ EL = 0 because P i ∈I EL ( X i ) µ i = EL ( X ) , ∀{ X i } i ∈I ∈ A ( X ).It was proved in Wang (2016) that V aR α is of infinite regulatory arbitrage. The followingproposition adapts this to our framework. Proposition 6.3.
Let α ∈ (0 , . If cardinality of I µ is at least k + 1 such that k < α , then V aR α is of infinite regulatory arbitrage.Proof. Let { i j } k +1 j =1 be members of I µ such that k < α . Then k >
1. Moreover, let { B j , j =1 , · · · , k } be a partition of Ω such that P ( B j ) = k for any j = 1 , · · · , k . We note that as(Ω , F , P ) is atomless, such a partition always exists. For fixed X ∈ L ∞ and some arbitrary real28umber m >
0, let { X i } i ∈I be defined as X i ( ω ) = m (1 − k B j ( ω ))( k − µ i , for i = i j , j = 1 , · · · , k,X ( ω ) µ i k +1 , for i = i k +1 , , otherwise . for any ω ∈ Ω. Thus, { X i } i ∈I ∈ A ( X ) because for any ω ∈ Ω, the following is true: X i ∈I X i ( ω ) µ i = k X j =1 (cid:20) m (1 − k B j ( ω ))( k − µ i j (cid:21) µ i j + (cid:20) X ( ω ) µ i k +1 (cid:21) µ i k +1 = mk − k X j =1 (1 − k B j ( ω )) + X ( ω )= mk − k − k ) + X ( ω ) = X ( ω ) . Furthermore, we note that for i = i j , j = 1 , · · · , k , we have that P ( X i <
0) = P (cid:18) B j > k (cid:19) = P (cid:0) B j = 1 (cid:1) = P ( B j ) = 1 k < α. Thus,
V aR α ( X i ) <
0. In fact, we have that
V aR α ( − B j ) = 0 and thus V aR α ( X i ) = m ( k − µ i j (cid:0) kV aR α ( − B j ) − (cid:1) = − m ( k − µ i j < . As X i ∈I V aR α ( X i ) µ i = k X j =1 V aR α ( X i ) µ i j + V aR α ( X i k +1 ) µ i k +1 = V aR α ( X ) − mkk − ,V aR α ( X ) < ∞ , and m > ρ µconv ( X ) ≤ V aR α ( X ) − lim m →∞ mkk − −∞ . Hence, we conclude that τ ρ ( X ) = ∞ for any X ∈ L ∞ , which implies that V aR α is of infiniteregulatory arbitrage. Remark . (i) The idea is that if any position X could be split into k + 1 random variableswith k < α , then we would obtain an arbitrarily smaller weighted V aR α . In the frameworkin Wang (2016), it is always possible to obtain a countable division of any position owingto the nature of the functional R in Remark 6.2.(ii) We note that smaller values of α require a richer structure on I to allow the regulatoryarbitrage strategy. This is in fact desired, as such values represent riskier scenarios. Forexample, for α = 0 .
01, which is the demanded level for regulatory capital in Basel accords,a cardinality of at least k + 1 = 102 would be necessary for the required partition.29iii) This result can be extended to the more general framework of ρ I = { V aR α i , i ∈ I} when α ∗ = inf i ∈I µ α i >
0. As
V aR α i ≤ V aR α ∗ ∀ i ∈ I µ , if cardinality of I µ is at least k + 1with k < α ∗ , then ρ µconv ( X ) = inf { X i }∈ A ( X ) (X i ∈I V aR α i ( X i ) µ i ) ≤ inf { X i }∈ A ( X ) (X i ∈I V aR α ∗ ( X i ) µ i ) = −∞ . In fact, analogous reasoning is valid for any choice of risk measures ρ I dominated by V aR α ∗ .We now state more general results regarding τ ρ in our framework. To this end, the followingproperty of risk measures is required. Definition 6.5.
A risk measure ρ : L ∞ → R is called I -convex if ρ (cid:0)P i ∈I X i µ i (cid:1) ≤ P i ∈I ρ (cid:0) X i (cid:1) µ i for any { X i } i ∈I ∈ ∪ X ∈ L ∞ A ( X ) . Theorem 6.6.
We have the following for a risk measure ρ :(i) ρ is I -convex if and only if it is free of regulatory arbitrage.(ii) If ρ is a convex risk measure, then it is free of regulatory arbitrage.(iii) If ρ is subadditive, then it is at most of finite regulatory arbitrage.(iv) Let ρ , ρ : L ∞ → R be risk measures such that ρ ≤ ρ . If ρ is of finite regulatoryarbitrage, then ρ is not of infinite regulatory arbitrage. Moreover, if ρ is of infinite (orpartially infinite) regulatory arbitrage, then so is ρ .(v) If ρ satisfies the positive homogeneity condition, then τ ρ (0) > if and only if τ ρ (0) = ∞ .(vi) If ρ is loaded, then it is not of infinite regulatory arbitrage. If, in addition, it has thelimitedness property, then it is of finite regulatory arbitrage.Proof. We note that as we consider finite risk measures, it holds that τ ρ ( X ) = ∞ if and only if ρ µconv ( X ) = −∞ . Then,(i) We assume that ρ is I -convex, and let X ∈ L ∞ . Then, ρ ( X ) ≤ P i ∈I ρ ( X i ) µ i for any { X i } i ∈I ∈ A ( X ). By taking the infimum over A ( X ), we obtain ρ µconv ( X ) ≤ ρ ( X ) ≤ ρ µconv ( X ). For the converse, we obtain ρ ( X ) = ρ µconv ( X ) ≤ P i ∈I ρ ( X i ) µ i for any { X i } i ∈I ∈ A ( X ), which is I -convexity.(ii) By Corollary 4.3, we have that ρ µconv ( X ) = ρ ( X ) , ∀ X ∈ L ∞ . As a direct consequence, weobtain that ρ is free of regulatory arbitrage.(iii) We begin with the claim that if ρ is subadditive, then it is of partially infinite regulatoryarbitrage if and only if it is of infinite regulatory arbitrage. By Proposition 3.4 and Remark3.5, we have that ρ µconv is also subadditive and normalized. We need only show thatpartially infinite regulatory arbitrage implies infinite regulatory arbitrage. Let X ∈ L ∞ be such that τ ρ ( X ) = ∞ . As ρ is finite, it holds that ρ µconv ( X ) = −∞ . Let now Y ∈ L ∞ .30e have that ρ µconv ( Y ) ≤ ρ µconv ( X ) + ρ µconv ( Y − X ) = −∞ . Thus, ρ is of infinite regulatoryarbitrage. However, we have that τ ρ (0) = ρ (0) − ρ µconv (0) = 0 < ∞ . Then, ρ is notof infinite regulatory arbitrage and, by the previous claim, it is not of partially infiniteregulatory arbitrage either. Thus, ρ is at most of finite regulatory arbitrage.(iv) It is evident that, in this case, we have, by abuse of notation, ( ρ ) µconv ≤ ( ρ ) µconv . If ρ isof finite regulatory arbitrage, then −∞ < ( ρ ) µconv ( X ) ≤ ( ρ ) µconv ( X ) , ∀ X ∈ L ∞ . Thus, ρ is also of infinite regulatory arbitrage. If now ρ is of infinite regulatory arbitrage,then ( ρ ) µconv ( X ) ≤ ( ρ ) µconv ( X ) = −∞ , ∀ X ∈ L ∞ . Thus, ρ is also of finite regulatoryarbitrage. For partially infinite regulatory arbitrage, the reasoning is analogous.(v) We need only prove the “only if” part because the converse is automatically obtained.As ρ (0) = 0, τ ρ (0) > ρ µconv (0) <
0. Then, there is { X i } i ∈I ∈ A (0) such that ρ µconv (0) ≤ P i ∈I ρ ( X i ) µ i <
0. As { λX i } i ∈I ∈ A (0) ∀ λ ∈ R + , by the positive homogeneityof ρ , we obtain that ρ µconv (0) ≤ lim λ →∞ X i ∈I ρ ( λX i ) µ i = lim λ →∞ λ X i ∈I ρ ( X i ) µ i = −∞ . Hence, τ ρ (0) = ρ (0) − ρ µconv (0) = ∞ .(vi) By Proposition 3.4, we have that ρ µconv inherits loadedness and limitedness from ρ . Theloadedness of ρ implies normalization of ρ µconv , and therefore τ ρ (0) = 0. Thus, ρ is not ofinfinite regulatory arbitrage. If, in addition, ρ is limited, then for any X ∈ L ∞ , we havethat τ ρ ( X ) ≤ E [ X ] − ess inf X < ∞ . Hence, we obtain finite regulatory arbitrage for ρ . Remark . (i) In the approach in Wang (2016), τ ρ is always subadditive, whereas in ourcase, this is not ensured. This fact alters most results and arguments, as it is crucial in hisstudy. Moreover, (iii), (iv), (v), and (vi) would remain true if we considered the generalframework of arbitrary ρ I and made the adaption τ ρ I = P i ∈I ρ i µ i − ρ µconv = ρ µ − ρ µconv ≥ ρ µ is as Remark 4.2.(ii) A remarkable feature is that is possible to identify τ ρ as a deviation measure in the senseof Rockafellar et al. (2006), Rockafellar and Uryasev (2013), Righi and Ceretta (2016),Righi (2019a), and Righi et al. (2019). For instance, the bound for τ ρ in the proof of (vi)is known as lower-range dominance for deviation measures. References
Acciaio, B., 2007. Optimal risk sharing with non-monotone monetary functionals. Finance andStochastics 11, 267–289.Acciaio, B., 2009. Short note on inf-convolution preserving the fatou property. Annals ofFinance 5, 281–287. 31cciaio, B., Svindland, G., 2009. Optimal risk sharing with different reference probabilities.Insurance: Mathematics and Economics 44, 426 – 433.Acerbi, C., 2002. Spectral measures of risk: A coherent representation of subjective risk aversion.Journal of Banking & Finance 26, 1505–1518.Arrow, K., 1963. Uncertainty and welfare economics of medica care. The American EconomicRewview 53, 941 – 973.Artzner, P., Delbaen, F., Eber, J., Heath, D., 1999. Coherent measures of risk. MathematicalFinance 9, 203–228.Barrieu, P., El Karoui, N., 2005. Inf-convolution of risk measures and optimal risk transfer.Finance and Stochastics 9, 269–298.B¨auerle, N., M¨uller, A., 2006. Stochastic orders and risk measures: Consistency and bounds.Insurance: Mathematics and Economics 38, 132–148.Borch, K., 1962. Equilibrium in a reinsurance market. Econometrica 30, 424 – 444.Buhlmann, H., 1982. The general economic premium principle. ASTIN Bulletin 14, 13 – 21.Burgert, C., R¨uschendorf, L., 2006. On the optimal risk allocation problem. Statistics &Decisions 24, 153 – 171.Burgert, C., R¨uschendorf, L., 2008. Allocation of risks and equilibrium in markets with finitelymany traders. Insurance: Mathematics and Economics 42, 177 – 188.Carlier, G., Dana, R.A., Galichon, A., 2012. Pareto efficiency for the concave order and multi-variate comonotonicity. Journal of Economic Theory 147, 207 – 229.Cont, R., Deguest, R., Scandolo, G., 2010. Robustness and sensitivity analysis of risk measure-ment procedures. Quantitative Finance 10, 593–606.Dana, R., Meilijson, I., 2003. Modelling agents’ preferences in complete markets by secondorder stochastic dominance. Working Paper .Dana, R.A., Le Van, C., 2010. Overlapping sets of priors and the existence of efficient allocationsand equilibria for risk measures. Mathematical Finance 20, 327–339.Delbaen, F., 2002a. Coherent risk measures. Lectures given at the Cattedra Galileiana at theScuola Normale di Pisa, March 2000, Published by the Scuola Normale di Pisa .Delbaen, F., 2002b. Coherent risk measures on general probability spaces, in: Sandmann, K.,Sch¨onbucher, P.J. (Eds.), Advances in Finance and Stochastics: Essays in Honour of DieterSondermann. Springer Berlin Heidelberg, pp. 1–37.Delbaen, F., 2006. Hedging bounded claims with bounded outcomes, in: Kusuoka, S., Yamazaki,A. (Eds.), Advances in Mathematical Economics. Springer, Tokyo, pp. 75–86.Delbaen, F., 2012. Monetary Utility Functions. Lecture Notes: University of Osaka.32ennerberg, D., 1994. Non-additivie Measure and Intgral. Springer.Embrechts, P., Liu, H., Mao, T., Wang, R., 2020. Quantile-based risk sharing with heteroge-neous beliefs. Mathematical Programming 181, 319 – 347.Embrechts, P., Liu, H., Wang, R., 2018. Quantile-based risk sharing. Operations Research 66,936–949.Filipovi´c, D., Svindland, G., 2008. Optimal capital and risk allocations for law- and cash-invariant convex functions. Finance and Stochastics 12, 423–439.F¨ollmer, H., Schied, A., 2002. Convex measures of risk and trading constraints. Finance andstochastics 6, 429–447.F¨ollmer, H., Schied, A., 2016. Stochastic Finance: An Introduction in Discrete Time. 4 ed., deGruyter.Fritelli, M., Rosazza Gianin, E., 2005. Law invariant convex risk measures. Advances inmathematical economics 7, 33–46.Frittelli, M., Rosazza Gianin, E., 2002. Putting order in risk measures. Journal of Banking &Finance 26, 1473–1486.Gerber, H., 1978. Pareto-optimal risk exchanges and related decision problems. ASTIN Bulletin10, 25 – 33.Grechuk, B., Molyboha, A., Zabarankin, M., 2009. Maximum Entropy Principle with GeneralDeviation Measures. Mathematics of Operations Research 34, 445–467.Grechuk, B., Zabarankin, M., 2012. Optimal risk sharing with general deviation measures.Annals of Operations Research 200, 9–21.Heath, D., Ku, H., 2004. Pareto equilibria with coherent measures of risk. MathematicalFinance 14, 163–172.Jouini, E., Schachermayer, W., Touzi, N., 2006. Law invariant risk measures have the Fatouproperty. Advances in Mathematical Economics 9, 49–71.Jouini, E., Schachermayer, W., Touzi, N., 2008. Optimal risk sharing for law invariant monetaryutility functions. Mathematical Finance 18, 269–292.Kazi-Tani, N., 2017. Inf-convolution of choquet integrals and applications in optimal risk trans-fer. Working Paper .Kiesel, R., R¨uhlicke, R., Stahl, G., Zheng, J., 2016. The wasserstein metric and robustness inrisk management. Risks 4, 32.Kratschmer, V., Schied, A., Zahle, H., 2014. Comparative and qualitative robustness for law-invariant risk measures. Finance and Stochastics 18, 271–295.33usuoka, S., 2001. On law invariant coherent risk measures. Advances in mathematical eco-nomics 3, 158–168.Landsberger, M., Meilijson, I., 1994. Co-monotone allocations, bickel-lehmann dispersion andthe arrow-pratt measure of risk aversion. Annals of Operations Research 52, 97–106.Liebrich, F.B., Svindland, G., 2019. Risk sharing for capital requirements with multidimensionalsecurity markets. Finance and Stochastics 23, 925–973.Liu, F., Wang, R., Wei, L., 2019. Inf-convolution and optimal allocations for tail risk measures.Working Paper .Liu, P., Wang, R., Wei, L., 2020. Is the inf-convolution of law-invariant preferences law-invariant? Insurance: Mathematics and Economics 91, 144 – 154.Ludkovski, M., R¨uschendorf, L., 2008. On comonotonicity of pareto optimal risk sharing.Statistics & Probability Letters 78, 1181 – 1188.Ludkovski, M., Young, V.R., 2009. Optimal risk sharing under distorted probabilities. Mathe-matics and Financial Economics 2, 87–105.Mastrogiacomo, E., Rosazza Gianin, E., 2015. Pareto optimal allocations and optimal risksharing for quasiconvex risk measures. Mathematics and Financial Economics 9, 149–167.Pflug, G., R¨omisch, W., 2007. Modeling, Measuring and Managing Risk. 1 ed., World Scientific.Righi, M., 2019a. A composition between risk and deviation measures. Annals of OperationsResearch 282, 299–313.Righi, M., 2019b. A theory for combinations of risk measures. Working Paper .Righi, M., Ceretta, P., 2016. Shortfall Deviation Risk: an alternative to risk measurement.Journal of Risk 19, 81–116.Righi, M.B., M¨uller, F.M., Moresco, M.R., 2019. On a robust risk measure-ment approach for capital determination errors minimization. Working Paper URL: https://arxiv.org/abs/1707.09829 .Rockafellar, R., Uryasev, S., 2013. The fundamental risk quadrangle in risk management,optimization and statistical estimation. Surveys in Operations Research and ManagementScience 18, 33–53.Rockafellar, R., Uryasev, S., Zabarankin, M., 2006. Generalized deviations in risk analysis.Finance and Stochastics 10, 51–74.R¨uschendorf, L., 2013. Mathematical Risk Analysis. Springer.Starr, R., 2011. General Equilibrium Theory: An Introduction. 2 ed., Cambridge UniversityPress. 34vindland, G., 2010. Continuity properties of law-invariant (quasi-)convex risk functions on L ∞∞