Minkowski gauges and deviation measures
aa r X i v : . [ q -f i n . R M ] J u l Minkowski gauges and deviation measures
Marlon Morescoemail: [email protected] Marcelo Brutti Righiemail:[email protected] Hortaemail: [email protected] 6, 2020
Abstract
We propose to derive deviation measures through the Minkowski gauge of a given set of acceptablepositions. We show that given a suitable acceptance set, any positive homogeneous deviation mea-sure can be accommodated in our framework. In doing so, we provide a new interpretation for suchmeasures, namely, that they quantify how much one must shrink a position for it to become accept-able. In particular, the Minkowski gauge of a set which is convex, stable under scalar addition, andradially bounded at non-constants, is a generalized deviation measure. Furthermore, we explore therelations existing between mathematical and financial properties attributable to an acceptance set onthe one hand, and the corresponding properties of the induced measure on the other. In addition, weshow that any positive homogeneous monetary risk measure can be represented through Minkowskigauges. Dual characterizations in terms of polar sets and support functionals are provided.
Keywords : Risk measures, Deviation measures, Acceptance sets, Convex analysis, Minkowskigauges.
In modern financial theory — since the iconic paper of Markowitz (1952) — the standard deviation has been the measure most used to quantify the risk of a financial position. More recently, due to theincreasing necessity of paying attention to tail risks, monetary risk measures — which respect mono-tonicity and translation invariance (cash additivity) — came to light. Following the seminal paper ofArtzner et al. (1999), theoretical properties that are desirable for a risk measure have been widely stud-ied. Nonetheless, there is no consensus so far about which are the best properties (axioms) such a riskmeasure ought to satisfy, and even less regarding the best way to measure financial risk. The axiomaticapproach of Rockafellar et al. (2006a) represents a landmark in the literature, setting the tone for re-cent developments with the introduction of generalized deviation measures — generalizations of thestandard deviation and similar measures. A deviation measure is a functional D defined on a space X comprised of a suitable class of random variables, that captures the degree of “non-constancy” (disper-sion) of a financial position. Such measures have been proved useful in financial problems as can be seenin Rockafellar et al. (2006b), Pflug (2006), Grechuk et al. (2009), Rockafellar and Uryasev (2013) amongothers.Owing to the aforementioned lack of consensus regarding an appropriate way to measure risk, a hand-ful of coherent and convex risk measures have been proposed and, as a dénouement, many generalized andconvex deviation measures as well. Furthermore, due to the importance of variability, Righi and Ceretta(2016), Berkhouch et al. (2018) and Righi (2019) bring forward some novel convex risk measures, in thesense of Föllmer and Schied (2002), which explicitly take variability into account. Empirically, convexrisk measures in the latter class displayed consistently better performance for optimal portfolio strategies,as seen in the work of Righi and Borenstein (2018).In the present paper we propose a novel way to obtain deviation measures, by using the Minkowskigauge as a means to ascribe a deviation to an arbitrary set of acceptable positions — and, importantly,we show that every deviation measure can be thought of in this framework. One common interpretationfor deviation measures is that they quantify the distance between a random variable and constancy; ourapproach offers an alternative understanding: that they capture the amount that an agent must shrinka given position for it to be considered acceptable. Thus, in our quest to define a new class of deviationmeasures, we first have a glance into the framework of acceptance sets related to risk measures, whose1urpose is to define the range of positions which have an acceptable risk. Artzner et al. (1999) were thefirst to propose the concept, after which Delbaen (2002), Frittelli and Scandolo (2006), Artzner et al.(2009) among others, have deepened the literature.In the preceding framework, we consider a vector space X comprised of a suitable class of randomvariables (feasible outcomes). An element X ∈ X is to be interpreted as a real-valued, random resultof a given asset, corresponding to a certain position whose realized value depends on the outcome ω ofthe market; in this context, X ( ω ) > is a gain, whereas the reverse inequality corresponds to a loss.We would like to highlight the generality of our framework: we impose little restrictions on the space X , namely, we ask for it to be a topological vector space, which include the most used spaces in theliterature, such as the L p and Orlicz spaces. Given any functional f : X → R ∪ { + ∞} , one can interpretthe value f ( X ) as representing the financial risk of a position X — however, this interpretation is toogeneral and so it is customary, in the literature, to restrict attention to two broad classes of functionals,namely the class of monetary risk measures and the class of deviation measures . Regarding theformer it is well known that, under some weak assumptions on a set A ⊆ X of acceptable positions, amonetary risk measure can always be expressed in the form ρ ( X ) = inf { m ∈ R : X + m ∈ A } . The mainmessage of this paper is that, under possibly different assumptions on A , a deviation measure takes theform D ( X ) = inf { m > m − X ∈ A } . Thus, there is a straight connection between deviation measuresand the concept of a Minkowski functional (or gauge) which is of utmost importance in the theory oftopological vector spaces. This has the important consequence that such deviation measures are, in asense, a type of generalized seminorm. For precise definitions, see Sections 4 and 6.The underlying acceptance set A that we have in mind is, at least in principle, quite arbitrary —a convenient choice is to take A as an acceptance set (a sub-level set, that is) corresponding to somepre-specified deviation measure. Nonetheless, there is no all-encompassing approach so far by whichone can introduce such acceptance sets for deviation measures. The aforesaid lack in the literaturestems, mainly, from the fact that the classical notion of an acceptance set requires, indispensably, thatthe involved measures respect the axioms of translation invariance and monotonicity, whereas deviation measures are translation insensitive and, in general, not monotone. Besides, deviation measures are, bydefinition, non-negative, while the classical approach for monetary risk measures requires the positionto have a risk lower than . Therefore, simply replacing a monetary risk measure by a deviation riskmeasure D is of no use, as in this case a set of the form A = { X : D ( X ) ≤ } , is too restrictive — afterall, it would consider only constants to be acceptable! Secondly, we are specially interested in recoveringthe deviation measure corresponding to a given acceptance set, and the set { m : X + m ∈ A } , where A is induced by a deviation measure, would be empty for any X / ∈ A and equal to the whole real line if X ∈ A , even if is replaced by a positive constant in the definition of A . Hence, for the best of ourknowledge, there is no direct adaptation, from the preexisting notion of an acceptance set associated toa risk measure, that would allow one to encompass deviation measures.We are set out to fill this gap, by proposing an acceptance set for deviation measures, in the form of asub-level set A kD = { X : D ( X ) ≤ k } . At any rate, the highlight of our study is to provide a comprehensiveapproach for recovering the underlying deviation measure from any given set of acceptable positions.Drawing inspiration from the well known representation of a monetary risk measure as an infimum overthe set of acceptable cash additions on a given position, our approach consists in obtaining a deviationfunctional by taking the infimum over the set of acceptable positive expansions and contractions of such aposition. This change in perspective is required because deviation measures are commonly characterizedby at least two axioms, namely, non-negativity and translation insensitivity, but unfortunately thesecannot be used to reduce the dispersion of a financial position. Therefore, in order to extract a deviationmeasure from a given acceptance set, other axioms are likely going to be required, for example convexity,quasi-convexity, etc.Under convexity, one possibility is to adapt the approaches put forth by Frittelli and Scandolo (2006)and Artzner et al. (2009), where there are multiple eligible assets. These aim to recover the underlyingmeasure of a set A via ρ A ( X ) = inf { π ( Y ) : X + Y ∈ A } , where π : C → R is the cost to execute Y , and C is a set of feasible strategies. However, the preceding infimum yields a measure which is neither translationinsensitive nor non-negative — not a problem if one has risk measures in mind, but an impassable hurdleif the aim is to obtain measures of deviation . An alternative within reach is to assume that there existssome (constant) risk-free asset c , in which case — for a given position X and an acceptance set A — wecan use convexity to reduce the position’s risk, up to the point where it becomes acceptable; in other The tenured reader is probably familiar with the fact that the terminology monetary risk and deviation “measure”is misleading as the objects under study are not bona fide measures (as in “ σ -finite measure” for instance) but rather functionals (possibly non-linear) on a topological vector space. A via D A ( X ) = inf { λ ∈ [0 ,
1) : (1 − λ ) X + λc ∈ A } .In this setting, if A D is the sub-level set { X ∈ X : D ( X ) ≤ } corresponding to some previously givenconvex deviation measure D , then we have the equivalences D A D ( X ) = inf { λ ∈ [0 ,
1) : D (cid:0) (1 − λ ) X + λc (cid:1) ≤ } = inf { λ ∈ [0 ,
1) : D (cid:0) (1 − λ ) X (cid:1) ≤ } . The quantity D A D ( X ) can be understood as the amount by which we must shrink the position X untilit becomes acceptable. There is an important drawback in this approach, however — namely, that anytwo acceptable positions will always have the same deviation, whereas in general we want the “better”position to have a smaller deviation.In view of the above, we see that the idea of shrinking and expanding a position is closely relatedto positive homogeneity. Indeed, under positive homogeneity we can see the mapping λ D ( λX ) ,where λ > , as controlling simultaneously the size and the deviation of the position X . It appearsonly natural, then, to stipulate that a measure of ‘non-constancy’ be positive homogeneous, even moreso considering that most of the prominent deviation measures found in the literature do satisfy thisrequirement — and, besides, the relevant deviation measures that are not positive homogeneous, such asthe variance and the entropic deviation (Föllmer and Knispel, 2011), are only one transformation awayfrom positive homogeneity (for instance, the standard variation in relation to the variance, etc.). Insummary, positive homogeneity should translate into the following two properties for the correspondingacceptance set: in case the position X does not lie in A D , we should be able to shrink the position untilit “fits” in the set, and if X ∈ A D , then we should be able enlarge the position up to a limit where itstill “fits" in the set. Last but not least, we assume throughout that it is possible to invest the excesscapital resulting from shrinkage (similarly, to borrow the demanding capital for the enlargement) into a(constant) risk-free asset, i.e. we require our acceptance set to be stable with respect to translation bya constant. This can be interpreted as follows: adding a constant to a given position has no effect onwhether the latter is acceptable or not. This property is true, in particular, whenever our acceptanceset A is generated by a deviation measure, in which case (owing to translation insensitivity) allocationof capital in a risk-free manner leads to no change in the deviation of the position: in other terms, wehave that D ( λX + (1 − λ ) c ) = D ( λX ) , where λ is the amount to be shrunk and c is the risk-free asset.Monetary risk measures are representable as the minimum translation factor (corresponding to cashaddition/subtraction) which makes a given position acceptable. With deviation measures, on the otherhand, we propose to consider the least scaling factor (corresponding to expansion/shrinkage) which makessaid position acceptable. The function which describes the latter concept is the so called Minkowskigauge , which — for a given star-shaped acceptance set A — assigns a non-negative real number f A ( X ) = inf (cid:8) m ∈ R ∗ + : m − X ∈ A (cid:9) . (1)to each X ∈ X . See Figure 1. 3igure 1: Representation of the Minkowski gauge f A of a set A . XXf A ( X )0 A D at a certain level k (that is, sub-level sets ofthe form { X : D ( X ) ≤ k } ), may have the financial intuition that k represents an agent’s coefficientof aversion with respect to the deviation. Obviously, an agent with greater k has higher complianceregarding exposure to dispersion. Therefore, to compare positions of agents with different degrees ofaversion, we must bring the deviation measure to the same level for all agents — this is the case evenif the distinct agents agree about which deviation measure should be used. Since each set is uniquelydetermined by a positive number k , it is then possible, because of positive homogeneity, to normalizeeach set by multiplying it by the constant k − . In line with the the preceding heuristics, we propose torecover the normalized deviation measure through the identity D ( X ) = k · inf (cid:8) m ∈ R ∗ + : m − X ∈ A kD (cid:9) .Pflug and Romisch (2007) previously studied deviation measures generated by Minkowski gauges asin equation (1). The authors focused on sets of the form A = { X ∈ X : E ( h ◦ X ) ≤ h (1) } for aconvex, symmetric, non-negative real function h with h (0) = 0 and < h ( x ) < ∞ for x = 0 , thusestablishing a relation between financial risk and Orlicz norms. In particular, if h is invertible on [0 , + ∞ ) , then the set A is a sub-level set of the form A f , with the functional f constrained to be ofthe form f ( X ) = h − (cid:0) E ( h ◦ X ) (cid:1) . They in any event propose deviations of the form f A ( X − E X ) and f A (( X − E X ) − ) , and explore to exhaustion the different representations of this kind of functional. Ahomologous approach was studied in Bellini et al. (2018), who consider return risk measures ˜ ρ , whichare analogous to monetary risk measures but applied to the return of a position, not its profit/loss. Sucha functional is defined on the strictly positive returns ( { X ∈ L ∞ (Ω , F , P ) : X > } , where X is to beunderstood as a return) and maps into the strictly positive real line. Moreover, a return risk measure ˜ ρ ispositive homogeneous and satisfies ˜ ρ (1) = 1 , and stays in a one-to-one correspondence with a monetaryrisk measure ρ via the relation ˜ ρ ( X ) = exp( ρ (log( X ))) . Indeed, given a suitable acceptance set A = A ρ the return risk measure can be precisely recovered through the Minkowski gauge of A , i.e. ˜ ρ = f A ρ .When it comes to the interplay between Minkowski functionals and acceptance sets, two importantquestions arise: (i) given an arbitrary functional f : X → R + ∪ { + ∞} , under what conditions can wefind a set A ⊆ X such that f = f A ? (ii) if f is of the form f = f A for some A ⊆ X , what isthe relation between A and the sub-level set A f ? In particular, answering the preceding questions willtell us when it is the case that f = f A f . Fortunately, answers to both questions are readily available: f is of the form f = f A for some A if and only if it is positive homogeneous, and any A satisfying { X : f ( X ) < } ⊆ A ⊆ { X : f ( X ) ≤ } will do. Additionally, if A is closed and star-shaped, thenone has necessarily A = A f — see Lemma 3.8. More important, then, is to establish relations betweenproperties of a positive homogeneous functional f and properties of the set A f and, reciprocally, betweenproperties of a set A and properties of the corresponding Minkowski gauge f A . Of particular interestto us — as we focus our attention on f ’s that are risk functionals — are questions like: what does A have to be like to ensure that f A is a deviation measure? A convex deviation measure? (And so on).Similarly, if f is a deviation measure, is it the gauge of some set? Of A f ? What can we say about A f ?What if f is a convex deviation measure? (And so on). See Theorem 5.3.Our main goal, then, is to attain a generalized deviation measure f A ≡ D A that is generated by agiven acceptance set A , from which we could know how much we ought to shrink a position in orderto make it acceptable (i.e to make it “enter” A ). Here, by generated we mean that it is the Minkowskigauge of the acceptance set. In this context, it is of crucial importance to better understand the mannerwhereby each one of the desired properties to be satisfied by the underlying acceptance set impacts theassociated deviation measure. We may be willing to impose, for instance, that our acceptance set be star-shaped, because scaled down positions should have lower dispersion, and less dispersion should be “moreacceptable”. Or we could ask that A be stable under scalar addition, understanding in this case that wedo not care about the location of a random variable, being interested only in its dispersion, its asymmetry,etc. In other words, under stability for scalar addition we treat the positions X and X + c equally: oneis acceptable if and only if the other one is as well, and, as a consequence, the Minkowski gauge ofsuch an A will be a translation insensitive functional. Last but not least, if we are inclined towards therequirement that A be radially bounded at non-constants, then our inclination actually means that theconstant positions are the only ones we allow to be indefinitely scaled up. All in all, an acceptance set A which is star-shaped, stable under scalar addition, and radially bounded at non-constants turns outto generate a deviation measure D A ≡ f A . This is the content of our Proposition 4.2. If, in addition,we are interested in considering the effects of diversification, then we should be willing to assume that A is a convex set; in this case D A is a convex functional, which tells us that “diversification is good”. Wesee, then, that convexity, together with the previous conditions, hand us a generalized deviation measure— This is the message of Proposition 4.4. Furthermore, if we understand that the most conservativequantification of risk is provided by the Lower Range deviation measure LR( X ) := E [ X ] − ess inf X ,5hen our acceptance set should include the sub-level sets A (Proposition 4.13). Or, if we are onlyconcerned with distributional/statistical attributes of financial outcomes, then our acceptance set A should be required to be law invariant — in this case, the Minkowski functional does its magic againand delivers a law invariant deviation measure f A (Proposition 4.10). Last but not least, if we wantan acceptance set that does not reward (nor punishes) diversification with comonotone pairs, we shouldimpose an acceptance set which is comonotone, in which case our deviation would be comonotone additive(Corollary 4.7).The remainder of this paper is structured as follows: Section 2 introduces our notation and framework,and also provides the underlying financial intuition backing set properties that shall be used throughoutthis paper. We also state some relevant results concerning relations between attributes of sets in a(topological vector spaces). In Section 3 we focus on the Minkowski gauge as a general functional,recalling some definitions from the literature, as well as introducing new ones; we restate some importantresults from existing work, and, finally, we also provide new results. In section 4 we explore the Minkowskigauge as a deviation measure, developing the role of specific properties for the set and its impact on theproperties for the generated functional. In section 5 we develop the idea of an acceptance set generatedby a deviation measure by exploring the reverse implications from section 4. Section 6 is the icing onthe cake: we establish links existing between monetary risk measures and Minkowski gauges throughour proposed framework, and obtain a characterization of a class of positive homogeneous monetary riskmeasures, in terms of the notion of a risk system , which we introduce. In all that follows, (Ω , F , P ) is a fixed, underlying probability space. Every equality and inequality is tobe understood as holding P -almost surely. As usual, we write, for p ∈ (0 , ∞ ) , L p ≡ L p (Ω , F , P ) := “the setof all ( P -equivalence classes of) random variables X such that E | X | p < ∞ ”, whereas L ≡ L (Ω , F , P ) := “the set of all ( P -equivalence classes) of random variables on (Ω , F , P ) ”, and L ∞ ≡ L ∞ (Ω , F , P ) := “the setof all ( P -equivalence classes of) random variables X which are P -essentially bounded”. We work with aHausdorff topological vector space X , and assume beforehand that the inclusions L ⊇ X ⊇ L ∞ hold. The generic elements of X are denoted by X , Y , Z , etc, and are to be interpreted as the random resultof a financial position , which we assume throughout to be perfectly liquid and discounted by a risk-freerate. X ′ denotes the topological dual of X , and we shall write h X, X ′ i := X ′ ( X ) whenever X ∈ X and X ′ ∈ X ′ ; notice that this notation gives h X, Y i = E XY if X ∈ L p and Y ∈ L q , with ≤ p < ∞ and p − + q − = 1 , via the identification L q ≡ ( L p ) ′ . Furthermore, we write h X , X ′ i = X × X ′ , and callthis construct the dual pair . With this notation and terminology, the mapping ( X, X ′ )
7→ h
X, X ′ i givesa bilinear functional defined on the dual pair, one that separates points of both X and X ′ . The positiveand negative parts of an element X ∈ X are denoted by X + := max( X, and X − := min( − X, ,respectively. We define the cone X + of non-negative positions as X + := { X ∈ X : X ≥ } (this isthe range of X X + ), and similarly X − := { X ∈ X : X ≤ } . With a slight abuse of notation, weconsider the inclusion R ⊆ X by identifying each x ∈ R with the equivalence class of random variablesequal to x almost surely. A pair of random variables is said to be comonotone if the inequality ( X ( ω ) − X ( ω ′ ))( Y ( ω ) − Y ( ω ′ )) ≥ , ω, ω ′ ∈ Ω holds P ⊗ P -almost surely. As usual, F X represents the cumulative distribution function of a randomvariable X , while F − X denotes its left quantile function, that is to say, F − X ( α ) := inf { q ∈ R : F X ( q ) ≥ α } .We write X = d Y whenever X and Y are equal in distribution, a fact which we also express by writing Y ∈ L X (and this already defines L X implicitly). As mentioned, we denote the property of X beingalmost surely greater than Y by X ≥ Y , while for a generic partial order (cid:23) we write X (cid:23) Y , alsoadopting the obvious convention that the notation X (cid:22) Y means precisely that X (cid:23) Y . If not clearfrom context, we shall mention explicitly the partial order under consideration. We say that X isgreater than Y in the dispersive order of distributions , written Y (cid:22) D X , if the inequality F − X ( u ) − F − X ( v ) ≥ F − Y ( u ) − F − Y ( v ) holds for every < v < u < . In all that follows, R + denotes theset [0 , + ∞ ) , whereas R ∗ + := (0 , + ∞ ) .Given A, B ⊆ X we define the set A + B by saying that Z ∈ A + B if and only if Z = X + Y forsome X ∈ A and some Y ∈ B . Similarly, for a Λ ⊆ R , we write Z ∈ Λ A if and only if Z = λX forsome λ ∈ Λ and some X ∈ A . For simplicity, we write λA := { λ } A and Λ X := Λ { X } when one of the These inclusions are assumed to hold algebraically — no a priori assumption is made on the relation between thetopologies involved. X ∈ X as R X := R ∗ + X . In the samemanner, X + A := { X } + A , etc. We also denote by bd( A ) , int( A ) , cl( A ) , conv( A ) , cl-conv ( A ) , cone( A ) ,cl-cone ( A ) , and A c respectively the boundary, interior, closure, convex hull, closed convex hull, conichull, closed conic hull and the complement of A . Any A ⊆ X is called an acceptance set , and we saythat a given position X is acceptable (w.r.t. A ) if and only if is an element of A .We now focus on properties for sets that are considered alongside the text. We make an effort toclarify the financial intuition behind each of these attributes. Since not every attribute appearing in ouraxiom scheme is fundamental in functional and convex analysis — and thus it is likely that some of theseattributes are unknown to the reader —, we shall resort to figures as a means to illustrate them and helpto develop the intuition. In these figures, we are considering Ω as the binary market , i.e., Ω = { , } ; inthis setting, one can take X = L ≡ R , where the latter equivalence is given via the identification of arandom variable X with the ordered pair (cid:0) X (0) , X (1) (cid:1) in the Cartesian plane. Importantly, notice thatin this context the inclusion R ⊆ R corresponds to the diagonal { ( u, v ) : v = u, u ∈ R } , which may bebe different from what the reader has in mind at first thought. Definition 2.1.
Let A ⊆ X and { A ( k ) : k ∈ R } ⊆ X . We say that(i) ( Law invariance ) A is law invariant if X ∈ A and X = d Y implies Y ∈ A .This means that a financial position having the same distribution of a given, acceptable positionis also acceptable; that is, when deciding whether a position is to be deemed acceptable, we onlycare about its statistical properties.(ii) ( Monotonicity ) A is monotone with respect to a given partial order (cid:22) if the conditions X ∈ A and X (cid:22) Y imply Y ∈ A . A is said to be anti-monotone (w.r.t (cid:22) ) if the conditions Y ∈ A and X (cid:22) Y imply X ∈ A . For convenience, we say that A is (cid:22) -monotone whenever A is monotonewith respect to (cid:22) , and similarly for anti-monotonicity.Under monotonicity, a position is deemed acceptable whenever a “worse” (smaller) one is alsoacceptable (from a financial perspective, this is not very interesting). Anti-monotonicity, on theother hand, captures the notion that being “bigger” according to some partial order is actually worse,e.g. the dispersive order of distribution. Under anti-monotonicity, then, a position is regarded asacceptable whenever a “better” position is also acceptable. Note that if A is monotone then A c isanti-monotone: indeed, letting A be monotone and X (cid:22) Y , then X ∈ A implies that Y ∈ A , whichis equivalent to say that Y / ∈ A implies that X / ∈ A , thus yielding anti-monotonicity of A c .(iii) ( Conicity ) A is a cone with vertex at the origin , or simply a cone , if λX ∈ A for every λ ≥ and every X ∈ A . A is said to be a cone with vertex at V ∈ X if A is of the form A = V + C for some cone C . A cone with vertex at V is degenerate if it is a singleton; otherwise, it is saidto be a proper cone with vertex at V .Conicity means that if a position is acceptable, then every non-negative multiple of the positionis deemed acceptable as well. This is a reasonable assumption when we are concerned with losses,but not so much for dispersion, as it allows scaling any acceptable position up in a unboundedfashion.(iv) ( Radial boundedness ) A is radially bounded if, for every non-zero X ∈ A , there is some δ X ∈ (0 , ∞ ) , such that δX / ∈ A whenever δ ∈ [ δ X , ∞ ) . The set A is said to be radially boundedat non-constants if A \ R is radially bounded.Radial boundedness is, in a sense, the opposite of conicity: it says that there is always a bound onhow much it is possible to scale up a position while keeping it acceptable. It means precisely that A contains no cone (except for the trivial cone { } ) — see Figure 2 for an example. As constants haveno dispersion, financially it makes sense to always consider them acceptable; that is to say, whenwe are mainly concerned with positions that are acceptable with respect to their dispersion, it isfruitful to limit the scaling up of all positions except for constants. In this case we should requirethat A be radially bounded at non-constants. Figure 3 shows a set which is radially bounded atnon-constants but it is not radially bounded. 7igure 2: A set A which is absorbing, radially bounded and strongly star-shaped. The ray R X isrepresented by the dashed line in red, which clearly “leaves” the set (as any such ray). Xδ X XδX ∈ A / ∈ A (v) ( Stability under scalar addition ) A is stable under scalar addition if A + R = A , thatis, if X + c ∈ A , for all X ∈ A and c ∈ R .In our framework, as scalar addition does not affect the dispersion of a financial position, it is areasonable property to be imposed on acceptance sets — see Figure 3 for an example.It is important to note that stability under scalar addition is incompatible (from a financial perspec-tive) with monotonicity (or anti-monotonicity) with respect to some partial orders of interest, suchas the “almost surely ≥ ” order. To illustrate, assume A is ≤ -monotone, stable under scalar additionand that ∈ A . Then L ∞ ⊆ A : indeed, since ∈ A , stability under scalar addition immediatelyentails R ⊆ A . Then, for any Y ∈ L ∞ it follows that Y ≥ ess inf Y ∈ R ⊆ A , so monotonicitygives us Y ∈ A . Clearly such an A is way too large to be of any practical interest from a financialperspective. Also, stability under scalar addition is clearly incompatible with radial boundedness,as a non-empty acceptance set that respects stability under scalar addition contains at least thewhole real line, and hence it cannot be radially bounded. However, a set which is radially boundedat non-constants, such as the one in Figure 3, undoubtedly can accommodate stability under scalaraddition. 8igure 3: A set A which is absorbing, radially bounded at non-constants, stable under scalar addition andstar-shaped. The subspace R of constant random variables is represented by the thick black diagonal. X + 1 XX − (vi) ( Absorbency ) A is absorbing if, for every X ∈ X , there is some δ X > such [0 , δ X ] X ⊆ A ,that is, if ≤ λ ≤ δ X , then λX ∈ A . A being absorbing means that, for any random variable X ∈ X (not necessarily in A ), the linesegment joining to a suitable rescaling of X lies entirely in A . Absorbing sets are of interestin part because any positive homogeneous function is completely determined by its values on anyabsorbing set. Furthermore, when A is absorbing, it is possible to shrink any position until it “fits”in the set. In other words, any position may be scaled to a point where it becomes acceptable.Importantly, in a topological vector space, every neighborhood of zero is an absorbing set. Figure4 shows an example of an absorbing set.Figure 4: A set A which is absorbing and radially bounded. Notice that δ X is not uniquely defined. Xδ X X Convexity) A is convex if λX + (1 − λY ) ∈ A , for every pair X, Y ∈ A and every λ ∈ [0 , .Convexity is a fundamental property in the theory of vector spaces. In our context, it is closelyrelated to concept of diversification, in the following sense: if an acceptance set A is convex, thenone cannot obtain an unacceptable position via a convex combination of acceptable positions, i.e.we cannot get worse off when we diversify. Analogously, if the complement of an acceptance set A is convex, then we cannot get better off by taking convex combinations of non acceptable positions.(viii) ( Star-shapedness ) A is star-shaped if λX ∈ A , for every X ∈ A and λ ∈ [0 , . A is said to be costar-shaped if A c is star-shaped. A being star-shaped means that the line segment joining to X lies entirely in A , for every X already lying in A (thus, star-shapedness does not imply absorbency). For a star-shaped set A ,given any X ∈ X , there exists some non-negative number λ X (possibly with λ ∗ = ∞ ) such thatthat R + X ∩ A ⊇ (0 , λ X ) X and R + X ∩ A c ⊇ ( λ X , ∞ ) X ; note that if A is absorbing then we cantake λ X > , and if A is radially bounded then we can take λ X < ∞ . For sets containing zero,star-shapedness is a slightly weaker requirement than convexity: if ∈ A and A is convex, then A isstar-shaped. Figure 5 displays a star-shaped set which is not absorbing nor convex, while Figure 4shows a set that is not star-shaped, although absorbing. Notice that A = ∅ being costar-shapedimplies λX ∈ A , for every X ∈ A and λ ∈ (1 , ∞ ) .Star-shapedness captures the financial notion that any scaled down version of an acceptable positionshould also be deemed acceptable. This is clearly a desirable property, as it intuitively means thatif an agent accepts to invest a certain amount in a stock, then she also finds it acceptable to investa lesser amount in the same stock.Figure 5: A set A which is star-shaped set and radially bounded. XλX (ix) (
Strong star-shapedness ) A is strongly star-shaped if A is star-shaped and, for each X ∈ X ,the ray R X ≡ (0 , ∞ ) X intersects the boundary of A at most once, i.e. the set R X ∩ bd A is eitherempty or a singleton. For a similar concept, see Rubinov and Gasimov (2004). Figure 6 providesan example of a strongly star-shaped set having the origin as a boundary point.10igure 6: A set A which is strongly star-shaped, with ∈ bd( A ) . A X ∈ bd( A ) (x) ( Positive homogeneity ) { A ( k ) : k ∈ R } is positive homogeneous if it holds that λA ( k ) = A ( λk ) , for each λ > .A positive homogeneous family of sets { A ( k ) : k ∈ R } may be seen as arising from a group of agentswho take into account the same facts to quantify the risk, or dispersion, of a financial position. Insuch a family, each agent would have a particular tolerance k to dispersion; for example, if agent Ais twice as tolerant as agent B (who accepts any position in the set A ( k ) , say), then the acceptanceset of agent A would be A (2 k ) . For simplicity, we introduced the concept here letting the familyof sets be indexed by the whole real line, but the index set could be R + or R ∗ + as well without anymodification in the definition.(xi) ( Symmetry ) A is symmetric if X ∈ A implies − X ∈ A .While symmetry is useful, specially as — whenever X is a normed space — open balls centeredat the origin are symmetric, this attribute is not desirable from a financial perspective, when A represents a collection of acceptable positions. Indeed, there is no reason to require nor to expectthat, for a given portfolio X which is deemed acceptable, the corresponding short position − X should be considered acceptable as well.We now move to stating some results — involving the concepts introduced above — which will beused throughout the text. Lemma 2.2.
Let A ⊆ X be non-empty. If A is (cid:22) D -anti-monotone, then it is stable under scalaraddition, star-shaped and law invariant. Proof.
Let A be non-empty and assume it is (cid:22) D -anti-monotone. Let X ∈ A . Clearly F − c ( u ) − F − c ( v ) = 0 for any c ∈ R and < v < u < . Hence, it is clear that c (cid:22) D X for any c ∈ R , which entails R ⊆ A .Moreover, notice that F − X + c ( u ) − F − X + c ( v ) = F − X ( u ) + c − F − X ( v ) − c = F − X ( u ) − F − X ( v ) for any c ∈ R and < v < u < . Therefore, X + c (cid:22) D X for all c ∈ R , and due to anti-monotonicity of A , we get X + c ∈ A (this holds for all X ∈ A and c ∈ R ). Furthermore, A is star-shaped: indeed,given X ∈ A we have F − X ( u ) − F − X ( v ) ≥ λ (cid:0) F − X ( u ) − F − X ( v ) (cid:1) = F − λX ( u ) − F − λX ( v ) , for any λ ∈ [0 , and < v < u < . Hence, λX (cid:22) D X for any λ ∈ [0 , , from which star-shapedness of A follows.Additionally, anti-monotonicity w.r.t. (cid:22) D clearly implies that A is law invariant, as if Y and X followthe same distribution it is obvious that Y (cid:22) D X (cid:22) D Y . (cid:4) Lemma 2.3.
Let B ⊆ X . Then its law invariant hull L B := { X ∈ X : X = d Y, for some Y ∈ B } inherits from B the attributes of stability under scalar addition, star-shapedness, absorbency, conicity,symmetry and (cid:22) D -monotonicity. Proof. If B is stable under scalar addition, then taking any Y ∈ L B and c ∈ R we see — as, per definition,it holds that Y = d X for some X ∈ B — that Y + c = d X + c ∈ B , that is Y + c ∈ L B .11ssume now that B is a cone, and let Y ∈ L B and λ > . We have Y = d X for some X ∈ B , and,since B is a cone, λX ∈ B . But λY = d λX , and this is all we need to conclude that L B is also a cone.A similar argument yields that L B is star-shaped (resp., absorbing) whenever B is.For symmetry, just note that X = d Y if and only if − X = d − Y . Finally, (cid:22) D -monotonicity is clearas the dispersive order of distributions is defined in terms of distributions alone. (cid:4) Remark . Not every property that seems plausibly heritable turns out to be so: take, for instance,radial boundedness of B . It seems reasonable — since no random variable in B can be scaled upindeterminately while remaining acceptable — that the same should be true of L B . However, thefollowing counterexample shows that this is false: let Ω = { , } N be the Bernoulli space comprisedof all sequences of ’s and ’s, that is, the generic element ω ∈ Ω is of the form ω = ( ω , ω , . . . ) with ω n ∈ { , } for all n . The probability measure P is defined, for each n and each n tuple x , . . . , x n ∈ { , } ,via P { ω ∈ Ω : ω = x , . . . , ω n = x n , ω n +1 ∈ R , ω n +1 ∈ R , . . . } = 1 / n Now define X n ( ω ) = n × I ( ω n = 1) , and put B = { X , X , . . . } . Such B is radially bounded, sincefor any fixed element X n ∈ B , there is only one element of B in the direction X n . However, L B isnot radially bounded: indeed, since nX = d X n , we have that nX ∈ L B for all n , and thus L B is notradially bounded in the direction of X . Similarly, X / d X n / n and thus we have nX ∈ L B for all n , and so on. Lemma 2.5.
Let A ⊆ X . If A is closed, star-shaped, and contains a proper cone with vertex at someconstant x ∈ R , then A is not radially bounded. Hence, if A is closed, star-shaped, and radially bounded,then every proper cone with vertex at a constant intersects A c . Proof. As A contains a proper cone with vertex at some constant x ∈ R , there exists a non-zero X ∈ X such that { x + λX : λ ≥ } ⊆ A . As A is star-shaped, we have that k ( x + λX ) ∈ A for all k ∈ [0 , and all λ ≥ ; in particular, taking λ = 1 /k , we have kx + X ∈ A for all k ∈ (0 , and, as A is closed, X = lim k ↓ kx + X ∈ A . To conclude that A is not radially bounded, it is sufficient to show that thereis no δ X > such that δX / ∈ A for δ ≥ δ X . So, let us fix an arbitrary δ X > and put k n = 1 /n andlet λ n = δ X /k n . As A is closed, we have lim n →∞ ( k n x + k n λ n X ) ∈ A . Now, clearly the preceding limitequals δ X X and so, as δ X was chosen arbitrarily, we can conclude that A is not radially bounded. (cid:4) Remark . A quick inspection of the proof of Lemma 2.5 tells us that it remains true even when thevertex x is not assumed to be a constant. In any case, we opt to state it for constant vertices since thisis the case which will be used later on in the text. Lemma 2.7.
Let X ∈ X . Then the family C X := { Y ∈ X : Y is comonotone to X } is a convexcone which is closed with respect to the topology of convergence in probability. Furthermore, if ( X, Y ) is a comonotone pair, then any two elements of the convex cone C X,Y := conv(cone( { X } ∪ { Y } )) arecomonotone to one other. Proof.
In what follows all equalities and inequalities are in the P ⊗ P -almost sure sense, that is, theyhold for any pair ( ω, ω ′ ) lying in an event Ω ⊆ Ω × Ω having total P ⊗ P measure. To see that C X is a cone, note that for any Y ∈ C X we have, by definition, (cid:0) X ( ω ) − X ( ω ′ ) (cid:1) × ( Y ( ω ) − Y ( ω ′ )) ≥ , for any ( ω, ω ′ ) ∈ Ω . Hence, for any λ ≥ and ( ω, ω ′ ) ∈ Ω , (cid:0) X ( ω ) − X ( ω ′ ) (cid:1)(cid:0) λY ( ω ) − λY ( ω ′ ) (cid:1) = λ (cid:0) X ( ω ) − X ( ω ′ ) (cid:1)(cid:0) Y ( ω ) − Y ( ω ′ ) (cid:1) ≥ , yielding λY ∈ C X . For convexity, let Y, Z ∈ C X . Then, for λ ∈ [0 , we have that, h X ( ω ) − X ( ω ′ ) ih(cid:0) λY ( ω ) + (1 − λ ) Z ( ω ) (cid:1) − (cid:0) λY ( ω ′ ) + (1 − λ ) Z ( ω ′ ) (cid:1)i = λ [ X ( ω ) − X ( ω ′ )] [ Y ( ω ) − Y ( ω ′ )] + (1 − λ ) [ X ( ω ) − X ( ω ′ )] [ Z ( ω ) − Z ( ω ′ )] ≥ whenever ( ω, ω ′ ) ∈ Ω . To see that C X is closed in the asserted sense, consider a convergent sequence { Y n } ⊆ C X with Y n → Y in probability. By standard facts of measure theory, there is a subsequence { Y n ( k ) } such that Y n ( k ) → Y almost surely. Clearly this yields that Y is comonotone to X . Ω can be taken as the countable intersection of the events where the required inequalities (for any pairing of X , Y , Y n , Z and W ) hold. Z, W ∈ C X,Y . By definition we have Z = γ ( λ X ) + (1 − γ )( δ Y ) for sometriplet ( γ , λ , δ ) with ≤ γ ≤ and ≤ λ , δ , and similarly W = γ ( λ X ) + (1 − γ )( δ Y ) for sometriplet ( γ , λ , δ ) with ≤ γ ≤ and ≤ λ , δ . Then, for ( ω, ω ′ ) ∈ Ω , expanding the product (cid:0) Z ( ω ) − Z ( ω ′ ) (cid:1)(cid:0) W ( ω ) − W ( ω ′ ) (cid:1) yields a weighted sum whose terms are all non-negative. This completes the proof. (cid:4) Remark . Note that the set C := T Y ∈ C X C Y , where C X and C Y are defined as in the propositionabove, is a non-empty, closed, and convex set, such that all its elements are comonotone to one another.In particular, R ⊆ C .We now define and explore a very important concept regarding duality in convex analysis, namelythe polar of a set. Definition 2.9.
For a dual pair h X , X ′ i , the polar A ⊙ of a non-empty set A ⊆ X is defined through A ⊙ := { X ′ ∈ X ′ : sup X ∈ A h X, X ′ i ≤ } , and the bipolar of A is the set given by A ⊙⊙ := (cid:8) X ∈ X : sup X ′ ∈ A ⊙ h X, X ′ i ≤ (cid:9) . Remark . Notice that the bipolar is always defined with the dual pair h X , X ′ i in mind, which forcesthe inclusion A ⊙⊙ ⊆ X . If instead one had the bidual X ′′ in mind (or, which is the same, the dual pair h X ′ , X ′′ i ), it would then be natural to define ( A ⊙ ) ⊙ := (cid:8) X ′′ ∈ X ′′ : sup X ′ ∈ A ⊙ h X ′ , X ′′ i ≤ (cid:9) . In thiscase, however, unfortunately one may have A ⊙⊙ = ( A ⊙ ) ⊙ . This is a detail that is frequently overlookedin the literature, although it has important consequences: for instance, see the Bipolar Theorem (item(vi) in Lemma 2.11), and also example 2.13 below.
Lemma 2.11.
Given a dual pair h X , X ′ i , let A, B, { A i } i ∈ I be subsets of X :(i) If A ⊆ B , then B ⊙ ⊆ A ⊙ .(ii) ( λA ) ⊙ = λ − A ⊙ for each λ = 0 .(iii) ∩ A ⊙ i = ( ∪ A i ) ⊙ .(iv) A ⊙ is is nonempty, convex, weakly ∗ -closed and contains 0.(v) If A is absorbing, then A ⊙ is weakly*-bounded, i.e. the set {h XX ′ i : X ∈ A } is bounded in R , forevery X ′ ∈ X ′ .(vi) The bipolar A ⊙⊙ is the convex, weak-closed hull of A ∪ { } .(vii) If A is a cone, then A ⊙ = { X ′ ∈ X ′ : h X, X ′ i ≤ , ∀ X ∈ A } .(viii) If A is star-shaped and stable under scalar addition, then h , X ′ i = 0 for all X ′ ∈ A ⊙ . Proof.
For items (i) to (vi), see Lemma 5.102 and Theorem 5.103 of Aliprantis and Border (2006). Item(vii) follows from an argument similar to the proof that B = B ∗ in Proposition 3.19 below. For item(viii), let X ′ ∈ A ⊙ . Then — as R ⊆ A and A + R ⊆ A by assumption — we have, for any X ∈ A and c ∈ R , h X, X ′ i + c h , X ′ i = h X + c, X ′ i ≤ and, as c is arbitrary, it is necessarily true that h , X ′ i = 0 . (cid:4) Remark . Item (vi) above is the famous
Bipolar Theorem , which states, in other words, that if A isclosed, convex and contains zero, then A = A ⊙⊙ . It is important to have in mind that A ⊙⊙ ⊆ X bydefinition . The following (counter)example provides a reasoning for the bipolar to be defined in X andnot in X ′′ . 13 xample 2.13. Let X = L , so that X ′ = L ∞ and X ′′ = ba , where ba is the set of all finitelyadditive measures on (Ω , F ) that are absolutely continuous w.r.t. P . With the dual pair h L , L ∞ i inmind, if A is the unit ball in X , then clearly A ⊙ ⊇ ball( L ∞ ) . To see that the converse inclusion A ⊙ ⊆ ball( L ∞ ) also holds, notice that if X ′ ∈ X ′ is such that k X ′ k ∞ > then, since the randomvariable X = I [ X ′ >λ ] / P [ X ′ > λ ] belongs to ball( L ) for any conformable < λ < k X ′ k ∞ , we have forsuch an X h X, X ′ i = 1 P [ X ′ > λ ] Z [ X ′ >λ ] X ′ d P ≥ P [ X ′ > λ ] Z [ X ′ >λ ] λ d P > , hence X ′ / ∈ A ⊙ . Fix B := A ⊙ and, now with the dual pair h L ∞ , ba i in mind, notice that given any X ′′ ∈ ba with total variation less than 1, clearly one has h X ′′ , X ′ i ≤ for all X ′ ∈ B . That is, X ′′ ∈ B ⊙ .However, since L is not reflexive, not every such X ′′ is the image of an X ∈ L via the canonicalembedding. Therefore, ( A ⊙ ) ⊙ ) A ⊙⊙ . There is plethora of results concerning the Minkowski gauge to be found in the realms of functional andconvex analysis. In this section we recall and introduce important concepts, state some known resultsfrom the literature, and rediscover others that are of special interest from a financial perspective. Beforedefining the Minkowski gauge, we turn our focus to relevant properties — which regard functionals ingeneral, not only the Minkowski gauge — that are considered alongside the text.
Definition 3.1.
Let f : X → R ∪ {∞} be an arbitrary, extended real-valued functional on X . A sub-level set of a functional f (defined on X ) at level k ∈ R is denoted by A kf := { X ∈ X : f ( X ) ≤ k } .Moreover, we say that(i) ( Non-negativity ): f is non-negative if f ( X ) > for any non-constant X and f ( X ) = 0 forany constant X .If f is a deviation measure, non-negativity tells us that that the deviation can only assume strictlypositive values, except when evaluated at constants — which have no deviation.(ii) ( Translation insensitivity ) f is translation insensitive if f ( X + c ) = f ( X ) for any X ∈ X and c ∈ R .Whenever f is a deviation measure, translation insensitivity ensures that the deviation does notchange if a constant ammount is added to a given position.(iii) ( Translation invariance ) f is translation invariant if f ( X + c ) = f ( X ) − c for any X ∈ X and c ∈ R .Unlike translation insensitivity — which is typically a property imposed on deviation measures —translation invariance is one of the requirements defining a monetary risk measure; it says that the(monetary) risk of a position is reduced by the exact same amount of an invested sure gain on thatposition.(iv) ( Monotonicity ) f is monotone (w.r.t. a given partial order (cid:22) ) whenever Y (cid:22) X implies f ( Y ) ≤ f ( X ) . If − f is monotone, than f is said to be anti-monotone (w.r.t. (cid:22) ). For simplicity, wheneverthe partial order is not explicitly mentioned, we are assuming that it is the “almost surely ≤ ” partialorder.From a financial perspective, imposing anti-monotonicity on a risk functional f corresponds tothe requirement that, if a position yields better results than another in every possible state of theworld , then the former necessarily has lower risk than the latter.(v) ( Positive homogeneity ) f is positive homogeneous if f ( λX ) = λf ( X ) for all X ∈ X and λ ≥ .For a risk measure f , positive homogeneity has the financial interpretation that the risk of a positionincreases proportionally to its magnitude.(vi) ( Convexity ) f is convex if f ( λX + (1 − λ ) Y ) ≤ λf ( X ) + (1 − λ ) f ( Y ) , for every pair X, Y ∈ X and all λ ∈ [0 , . 14rom the financial viewpoint, convexity is a property which ensures that diversification reducesrisk. A mapping f : X → R ∪ { + ∞} with f (0) = 0 is said to be a sub-linear functional wheneverit satisfies any two of the following properties: (a) positive homogeneity; (b) convexity; (c) sub--additivity (the latter means that f ( X + Y ) ≤ f ( X ) + f ( Y ) for any X, Y ∈ X ).(vii) ( Lower range dominance ) f is lower-range dominated if domain( f ) ⊆ L and f ( X ) ≤ E X − ess inf X =: LR( X ) for all X .Lower range dominance is an essential property, as it reveals the interplay between coherent riskmeasures and generalized deviation measures — see in Rockafellar et al. (2006a) for instance.(viii) ( Law invariance ) f is law invariant if F X = F Y implies f ( Y ) = f ( X ) .If f is a risk functional, law invariance encapsulates the notion that, in appraising the risk ofa position, we should only care about its statistical properties — as these properties embody theuncertainty (w.r.t. the market outcome) faced by a given agent. Law invariance is also important inempirical implementations, as it allows the theoretical risk measure to be estimated from historicaldata.(ix) ( Lower-semicontinuity ) f is lower-semicontinuous if the set A kf is closed, for all real k .In the case when X is a metric space, lower-semicontinuity is equivalent to the following property:given any convergent sequence { X n } ⊆ X , it holds that f (lim X n ) ≤ lim inf f ( X n ) .The convex envelop of a mapping f : X → R is defined to be the extended real valued function conv f given by conv f ( X ) := sup g g ( X ) , X ∈ X , where the supremum runs through all afine,continuous g : X → R satisfying g ≤ f . Note that conv f is convex and lower-semicontinuous.(x) ( Upper-semicontinuity ) f is upper-semicontinuous if the set { X ∈ X : f ( X ) ≥ k } is closedfor all real k .In the case when X is a metric space, upper-semicontinuity is equivalent to the following property:given any convergent sequence { X n } ⊆ X , it holds that f (lim X n ) ≥ lim sup f ( X n ) . Note that afunctional f is continuous if and only if it is both upper- and lower-semicontinuous.(xi) ( Symmetry ) f is symmetric if f ( X ) = f ( − X ) for all X ∈ X .In the theory of topological vector spaces, symmetry is one of the sine qua non conditions indefining a seminorm; indeed, a seminorm is precisely the Minkowski gauge of a symmetric, convex,and absorbing set. Although some deviation measures — the standard deviation, for example —do enjoy this attribute, symmetry is not really something desirable in our framework. Indeed, weare interested in quantifying the downside risk or deviation of a position, and thus dispersion abovethe mean can even be considered as “good”.(xii) ( comonotone additivity ) f is comonotone additive if f ( X + Y ) = f ( X ) + f ( Y ) for everypair X, Y ∈ X such that X and Y are comonotone.Comonotone additivity implies that a comonotone pair does not yield a gain, nor a loss, in diver-sification. This property sums up the notion that, for such a pair, an agent should be indifferentabout how the two positions are kept, whether they are held in the same portfolio or separately.We are now in place to introduce the main tool used in this paper — the Minkowski gauge — as wellas some related functionals: Definition 3.2.
Let A ⊆ X . The Minkowski gauge of A is the functional f A : X → R + ∪ {∞} defined, for X ∈ X , by f A ( X ) := inf (cid:8) m ∈ R ∗ + : m − X ∈ A (cid:9) , (2)where inf ∅ = ∞ . The cogauge of A is the functional ϕ A : X → R + ∪ {∞} defined, for X ∈ X , by ϕ A ( X ) := sup (cid:8) m ∈ R ∗ + : m − X ∈ A (cid:9) , (3)where sup ∅ = 0 . Additionally, the support function h A ⊙ : X → R + ∪ {∞} on the polar A ⊙ isdefined, for X ∈ X , as h A ⊙ ( X ) := sup {h X, X ′ i : X ′ ∈ A ⊙ } . It is well known that, for such an f , any two of this three axioms imply the remaining one — see Aliprantis and Border(2006). In the present setting, the convention sup ∅ = 0 is a sensible one, as we are taking the supremum over some subset of (0 , ∞ ) .
15n words, the Minkowski gauge answers the following question: given a set A of acceptable positions,how much should we shrink (or “gauge”) a certain position X for it to become acceptable? The value f A ( X ) is the required amount of shrinkage. Notice that the following inclusions always hold: { X ∈ X : f A ( X ) < } ⊆ A ⊆ A f A . The cogauge, in turn, is a useful concept that is closely linked the the Minkowski gauge: if we take a set A comprised of non-acceptable positions, then the cogauge gives the most that we can shrink a positionwhile keeping it non-acceptable. Importantly, for a star-shaped set A , gauge and cogauge are linked— see Corollary 3.5 below. For more details on cogauges, we refer the reader to Rubinov and Yagubov(1986); Rubinov (2000); Zaffaroni (2008, 2013) and references therein.Before continuing with the Minkowski gauge, let us state a useful result for positive homogeneousfunctionals on topological vector spaces. Lemma 3.3.
Let f : X → R ∪ {∞} . If f is positive homogeneous, then the set E := { X ∈ X : f ( X ) =1 } has empty interior. Proof.
Let us proceed by contraposition by showing that if E has non-empty interior, then f is notpositive homogeneous. Assume, then, that X ∈ int E , and let V denote an open neighborhood of X with V ⊆ E . By continuity of scalar multiplication, for small enough u > we have (1 + u ) X ∈ V ⊆ E . Butthen f ((1 + u ) X ) = 1 < (1 + u ) f ( X ) , so f is not positive homogeneous. (cid:4) We begin with a result which we use many times in the remainder of the paper. It relates star-shapedness with the fact that the infimum in the definition of the Minkowski gauge is taken over aninterval.
Lemma 3.4.
Let A ⊆ X and X ∈ X . Then(i) f A ( X ) = ∞ if and only if { m ∈ R ∗ + : m − X ∈ A } = ∅ if and only if { m ∈ R ∗ + : m − X / ∈ A } = R ∗ + .Moreover, if A is star-shaped, then(ii) f A ( X ) = 0 if and only if { m ∈ R ∗ + : m − X ∈ A } = R ∗ + if and only if { m ∈ R ∗ + : m − X / ∈ A } = ∅ .If in addition < f A ( X ) < ∞ , then one of the following holds:(iii) { m ∈ R ∗ + : m − X ∈ A } = [ f A ( X ) , ∞ ) and { m ∈ R ∗ + : m − X / ∈ A } = (0 , f A ( X )) (this is true inparticular when A is closed).(iv) { m ∈ R ∗ + : m − X ∈ A } = ( f A ( X ) , ∞ ) and { m ∈ R ∗ + : m − X / ∈ A } = (0 , f A ( X )] (this is true inparticular when A is open). Proof.
The first item is immediate. For the remaining assertions, let T X ( m ) := m − X for m ∈ R ∗ + .Clearly T X is a continuous mapping from R ∗ + to X . We have T − X ( A ) = { m ∈ R ∗ + : m − X ∈ A } andsimilarly T − X ( A c ) = { m ∈ R ∗ + : m − X / ∈ A } . Assume now that A is star-shaped and m ∈ T − X ( A ) .Then, if m ′ > m , we have m ′ ∈ T − X ( A ) as well. This establishes that T − X ( A ) is always an interval with ∞ as its right endpoint, and by definition the left endpoint is f A ( X ) , thus establishing (ii), (iii) and (iv),where the topological assertions follow by continuity of T X . (cid:4) We then have the following direct corollary on the relation between gauge and co-gauge.
Corollary 3.5.
Let A ⊆ X be star-shaped. Then the equality f A ( X ) = ϕ A c ( X ) (4)holds for all X ∈ X . Remark . Let A ⊆ X and A ′ ⊆ X ′ . Then the Minkowski gauge f A × A ′ : X × X ′ → R + ∪ {∞} is, by definition, given by f A × A ′ ( X, X ′ ) = inf (cid:8) m ∈ R ∗ + : ( X, X ′ ) /m ∈ A × A ′ (cid:9) , for all X ∈ X and X ′ ∈ X ′ . Therefore, if we both A ⊆ X and A ′ ⊆ X ′ are star-shaped sets, we get that f A × A ′ ( X, X ′ ) =max( f A ( X ) , f A ′ ( X ′ )) , for all X ∈ X and X ′ ∈ X ′ . To see this, notice that the following chain holds: f A × A ′ ( X, X ′ ) = inf (cid:8) m ∈ R ∗ + : ( X, X ′ ) /m ∈ A × A ′ (cid:9) = inf (cid:8) m ∈ R ∗ + : X/m ∈ A and X ′ /m ∈ A ′ (cid:9) = max (cid:0) inf (cid:8) m ∈ R ∗ + : X/m ∈ A (cid:9) , inf (cid:8) m ∈ R ∗ + : , X ′ /m ∈ A ′ (cid:9)(cid:1) = max( f A ( X ) , f A ′ ( X ′ )) .
16 special case occurs when A is closed and convex, and A ′ = A ⊙ : in this scenario one has f A × A ⊙ ( X, X ′ ) =max( f A ( X ) , f A ⊙ ( X ′ )) . Importantly, the Minkowski gauge of a set A is positive homogeneous whenever ∈ A : Lemma 3.7.
Let A ⊆ X . Then the following holds, for each X ∈ X :(i) f λA ( X ) = λ − f A ( X ) , for every λ ∈ R ∗ + .(ii) If ∈ A , then f A is positive homogeneous. Proof.
Let X ∈ X . For the first item, given λ ∈ R ∗ + we have f λA ( X ) = inf (cid:8) m ∈ R ∗ + : m − X ∈ λA (cid:9) = inf (cid:8) m ∈ R ∗ + : ( λm ) − X ∈ A (cid:9) = inf (cid:8) mλ − ∈ R ∗ + : m − X ∈ A (cid:9) = λ − f A ( X ) as claimed.For the second item, clearly f A (0 X ) = f A (0) = inf { m ∈ R ∗ + : m − ∈ A } = inf R ∗ + = 0 = 0 f A ( X ) .Moreover, given λ > , we have f A ( λX ) = inf { m ∈ R ∗ + : λm − X ∈ A } = inf { m ∈ R ∗ + : m − X ∈ λ − A } = f λ − A ( X ) . Then, by item (i), we have f λ − A ( X ) = λf A ( X ) , so the claim holds. (cid:4) Item (i) in the above proposition tells us how a certain set operation on A (in this case, rescaling)modifies the corresponding gauge. This is further explored in Lemmata 3.8 and 3.11 below, which alsoestablish connections between properties of acceptance sets and properties of the associated Minkowskigauges. See Figure 7 to build up the intuition backing these results. Lemma 3.8. (Lemma 5.49 of Aliprantis and Border (2006)) Let
A, B ⊆ X be non-empty. Then thefollowing holds:(i) If A ⊆ B , then f A ( X ) ≥ f B ( X ) , for all X ∈ X .(ii) f A ( − X ) = f − A ( X ) for all X ∈ X ; in particular, if A is symmetric, so is f A .(iii) If A contains a cone M , then f A ( X ) = 0 , for all X ∈ M ; in particular as { } is a cone, if ∈ A then f A (0) = 0 .(iv) If A is closed and star-shaped, then A = A f A .(v) If A and B are star-shaped, then f A ∩ B ( X ) = max( f A ( X ) , f B ( X )) Remark . Note that A ∩ ( − A ) can be interpreted as a symmetrization of A , and whenever A isclosed, star-shaped, convex, stable under scalar addition and radially bounded at non constants, onehas f A ∩ ( − A ) ( X ) = max( f A ( X ) , f ( − A ) ( X )) , yielding a symmetric generalized deviation measure (i.e., aseminorm). The spaces generated by symmetrized sets as the ones just described were studied in Righi(2017). Remark . Let ∈ A , and let st( A ) be defined by the condition that Z ∈ st( A ) if and only if Z = λX for some λ ∈ [0 , and some X ∈ A (that is, st( A ) = [0 , A in our preceding notation). It is clear that st( A ) is the smallest star-shaped set that contains A . Also, as an arbitrary intersection of star-shapedsets is still star-shaped, we see that st( A ) is equal to the intersection of all star-shaped sets that contain A . Therefore, we have that B := { X ∈ X : f A ( X ) < } ⊆ st( A ) ⊆ A f A . In order to see this, note that A f A ≡ { X ∈ X : f A ( X ) ≤ } is clearly star-shaped, as f A is positive homogeneous, and that A ⊆ A f A because X ∈ A implies f A ( X ) ≤ . Thus, we know that st( A ) = [0 , A ⊆ [0 , A f A = A f A . Hence, weonly need to show that B ⊆ st( A ) which is also clear as f A ( X ) < implies that there is some m ∈ (0 , such that mX ∈ A . We also have the identities f B = f A = f A fA = f st( A ) , by item (i) in Lemma 3.8. Lemma 3.11.
Let
A, B ⊆ X . Then, for each X ∈ X , the following holds:(i) f A ∪ B ( X ) = min (cid:0) f A ( X ) , f B ( X ) (cid:1) .(ii) If B is a cone, then f A + B ( X ) = inf Z ∈ B f A ( X − Z ) .17 roof. Write y = min (cid:0) f A ( X ) , f B ( X ) (cid:1) .For item (i), if m < y then m − X / ∈ A and m − X / ∈ B . By contraposition, y is a lower bound forthe set { m ∈ R + : m − X ∈ A ∪ B } . Thus f A ∪ B ( X ) ≥ y . The reversed inequality is immediate: if m is such that m − X ∈ A , then obviously m − X ∈ A ∪ B and f A ∪ B ( X ) ≤ f A ( X ) . Similarly, we have f A ∪ B ( X ) ≤ f B ( X ) and, a fortiori , f A ∪ B ( X ) ≤ y .For item (ii), let m > . Then one has m − X ∈ A + B if and only if m − X = a + b for some a ∈ A andsome b ∈ B , if and only if m − X − b = a , for some b ∈ B and some a ∈ X such that f A ( a ) ≤ , if andonly if f A (cid:0) m − X − b (cid:1) ≤ for some b ∈ B . By positive homogeneity, the latter sentence is equivalentto the following: there exists a b ∈ B such that f A ( X − mb ) ≤ m . Additionally — as B is a cone —if there is an element b ∈ B that respects f A ( X − mb ) ≤ m , then by letting d = mb we see that thereis an element d ∈ B such that f A ( X − d ) ≤ m , and the reciprocal of the previous sentence is obviouslyalso true: that is, it holds that f A ( X − mb ) for some b ∈ B if and only if f A ( X − d ) ≤ m for some d ∈ B . In view of the above equivalences, by writing M b := { m ∈ R ∗ + : f A ( X − b ) ≤ m } and noticingthat f A ( X − b ) = inf M b , we finally have that f A + B ( X ) = inf [ b ∈ B M b = inf b ∈ B inf M b as asserted. (cid:4) Remark . In the context of item (ii) from last lemma, we have from Lemma 3.8, that f B ( X ) = 0 forany X ∈ B since B is a cone. Thus, f A + B = inf Z ∈ B { f A ( X − Z )+ f B ( Z ) } = inf Z ∈ X { f A ( X − Z )+ f B ( Z ) } .The last equality holds because, for any Z / ∈ B , as B is a cone, it follows by Lemma 3.4 that f B ( Z ) = ∞ .This concept is closely related to inf-convolution and optimal risk sharing. Inf-convolution is a wellknown operation for functionals in convex analysis — for details of the use of inf-convolution in riskshare we refer the reader to Barrieu and El Karoui (2005), Jouini et al. (2008) and Righi (2020a).Figure 7: This figure illustrates items (i) and (v) in Lemma 3.8, as well as item (i) in Lemma 3.11and item (ii) in Proposition 3.13. Here, we have A ∩ B = { ( x, y ) ∈ R : max(0 . x, y ) ≤ } and A ∪ B = { (1 . x, y ) ∈ R : min( x, y ) ∈ [0 , or max( x, y ) ∈ [0 , } Xf A ( X ) = Xf A ∪ B ( X ) = (1 , , . Xf B ( X ) = Xf A ∩ B ( X ) = (1 . , X = (1 . , A = { ( x, y ) ∈ R : x ∈ [0 , . } B = { ( x, y ) ∈ R : y ∈ [0 , } The next result deals with the solution of the minimization problem appearing in the definition ofthe Minkowski gauge.
Proposition 3.13.
Let A ⊆ X be non-empty. Then, we have the following:(i) If A is absorbing, then f A is finite-valued. 18ii) If A is star-shaped and y := f A ( X ) ∈ R ∗ + , then y − X ∈ bd( A ) , i.e. f A ( X ) = ϕ A c ( X ) = 1 implies X lies in the boundary of A .(iii) If A is strongly star-shaped, X ∈ bd( A ) and X = 0 then, f A ( X ) = 1 .(iv) If f A ( X ) ∈ R ∗ + , then f A ( X ) = inf { m ∈ R ∗ + : m − X ∈ A } = (sup { m ∈ R + : mX ∈ A } ) − . (v) If R X ∩ A = ∅ then f A ( X ) = ∞ . In particular if / ∈ A then f A (0) = ∞ .(vi) If A is closed, absorbing and radially bounded, then the infimum in equation (2) is attained forany X ∈ X \ { } , that is, X ∈ f A ( X ) A for any X ∈ X .(vii) If A is closed, then the infimum in equation (2) is attained for any X such that f A ( X ) ∈ R ∗ + . Proof.
Let X ∈ X and write y := f A ( X ) .For the first item, there exists — by the absorbing property — some δ X ∈ R ∗ + such that the inclusion [0 , δ X ] X ⊆ A holds. It is straightforward to see that in this case the set (cid:8) m ∈ R ∗ + : m − X ∈ A (cid:9) is neverempty. Therefore, f A ( X ) < ∞ .For the second item, it suffices to consider the case f A ( X ) = 1 , as the general case then easily followsfrom positive homogeneity. In order to verify that X ∈ bd( A ) , we only have to exhibit sequences { Y n } ⊆ A and { Z n } ⊆ A c such that lim Y n = lim Z n = X . Let, then, Y n be defined through Y n := (1 + 1 / n ) − X and, similarly, Z n := (1 − / n ) − X . Continuity of scalar multiplication immediatly yields the desiredequality of limits, so it only remains to show that Y n ∈ A and Z n ∈ A c for all n . For such, just noticethat — due to star-shapedness through Lemma 3.4 — if m > , then m − X ∈ A , so Y n ∈ A , and if < m < , then m − X / ∈ A , so Z n / ∈ A .For item (iii), as X = 0 by assumption, we see that whenever the ray R X ≡ { λX : λ > } has anon-empty intersection with bd( A ) , it necessarily also holds that f A ( X ) ∈ R ∗ + . Therefore, as X ∈ bd( A ) ,we also have that f A ( X ) − X ∈ bd( A ) , by item (ii). Thus, we have X ∈ R X ∩ bd A and f A ( X ) − X ∈ R X ∩ bd A , and hence strong star-shapedness of A tells us that f A ( X ) = 1 .For item (iv), notice that y = inf (cid:8) m ∈ R ∗ + : m − X ∈ A (cid:9) = inf (cid:8) m − ∈ R ∗ + : mX ∈ A (cid:9) . Now, if x − > is a lower bound for the set { m − ∈ R ∗ + : mX ∈ A } , then x is an upper bound forthe set { m ∈ R + : mX ∈ A } ; if x − is the largest such lower bound, then x is the smallest such upperbound. That is to say, one has y − = sup { m ∈ R + : mX ∈ A } .Item (v) is clear as if { λX : λ > } ∩ A = ∅ then the set { m ∈ R ∗ + : m − X ∈ A } is empty and theinfimum of such set is ∞ .For item (vi), notice that if A is radially bounded, then f A ( X ) > , for every non-zero X ∈ X .Indeed, if A is radially bounded then — by definition — for each X there is a m X > such that m − X / ∈ A , for all m < m X . Therefore, it holds that inf { m ∈ R ∗ + : m − X ∈ A } > . Now, let T X : R ∗ + → X be defined by T X ( m ) = m − X . Clearly, T X is continuous. Thus, if A is a closed subsetof X so is T − X ( A ) a closed subset of R ∗ + . Also, if A is absorbing, then T − X ( A ) is non-empty. Finally,since radial boundedness ensures f A ( X ) > , it follows that inf T − X ( A ) ∈ T − X ( A ) as stated.The proof of item the last item is identical to the previous one. (cid:4) We have already seen above that if ∈ A , then its Minkowski gauge f A is positive homogeneousand that, if the stronger requirement ∈ int A (i.e. A is absorbing) holds, then f A is also finite-valued.The next lemma shows that positive homogeneity is also a sufficient condition ensuring that an arbitraryfunctional f (which does not assume negative values) is the Minkowski gauge of some subset of X .We opt to state the result as it appears in Aliprantis and Border (2006), where it is assumed at theoutset that range( f ) ⊆ R + . This assumption can be easily dropped; if so, the set V appearing inProposition 3.14 is no longer (necessarily) absorbing. Instead, in this case the condition ∈ V musthold. Proposition 3.14. (Lemma 5.50 and Theorem 5.52 of Aliprantis and Border (2006) ) Let
A, B ⊆ X be non-empty, and let f : X → R + be an arbitrary function. Then the following holds:19i) f is positive homogeneous if and only if it is the Minkowski gauge of an absorbing set, in whichcase for every V ⊆ X satisfying { X ∈ X : f ( X ) < } ⊆ V ⊆ A f , we have f V = f .(ii) f is sub-linear (positive homogeneous and convex) if and only if it is the Minkowski gauge of aconvex absorbing set V , in which case we may take V = A f .(iii) f is sub-linear and symmetric if and only if it is the Minkowski gauge of a symmetric, convex,absorbing set V , in which case we may take V = A f .(iv) f is sub-linear and lower-semicontinuous if and only if it is the Minkowski gauge of an absorbing,closed convex set V , in which case we may take V = A f .(v) f is sub-linear and continuous if and only if it is the Minkowski gauge of a convex neighborhood V of zero, in which case we may take V = A f .(vi) f is sub-linear, symmetric and continuous if and only if it is the Minkowski gauge of a uniqueclosed, symmetric and convex neighborhood V of zero, namely V = A f . Remark . A locally convex topology is a topology generated by a family of seminorms. In particular,the neighborhood base at zero is given by the collection of all A kp , with k > and p belonging tosome collection of seminorms. Now, Lemma 3.8 item (vi) actually tells us that each p is the Minkowskigauge of some unique closed, symmetric, convex neighborhood A of zero, namely A = A p , with p = f A .Distinctively, Theorem 5.73 of Aliprantis and Border (2006) tell us that any locally convex topology isgenerated by the family of gauges of the convex symmetric closed neighborhoods of zero.Next, we show that one can establish a relation between the convex envelope of a positive homogeneousfunctional f defined on X and the Minkowski gauge of the closed convex hull of A f . Proposition 3.16.
Let A ⊆ X . If ∈ A , then the Minkowski gauge of the closed convex hull of A isequal to the convex envelope of the Minkowski gauge of A , i.e. one has f cl-conv A ( X ) = conv f A ( X ) for all X ∈ X . Proof.
First, notice that any lower-semicontinuous sub-linear function g ≥ that is dominated by f A can be written as g = f C , with C a closed convex set given by C = A g ⊇ A f A ⊇ A (see items (i) and (ii)in Lemma 3.8, item (iv) in Proposition 3.14, and also items (viii) and (xii) in Theorem 5.3 below, wherethe absorbing condition can be dropped by letting g assume + ∞ ). Reciprocally, if C is any closed convexset such that A ⊆ C , then the sub-linear function g := f C ≥ is dominated by f A . In summary, there isa one-to-one correspondence between the class S + ( f A ) comprised of all lower-semicontinuous sub-linearmappings g : X → R + ∪ { + ∞} dominated by f A and the class C comprised of all closed convex sets C ⊇ A . Therefore, since by definition cl-conv A = T C ∈C C , an easy generalization of item (v) in Lemma3.8 entails f cl-conv A ( X ) = sup C ∈C f C ( X ) = sup g ∈S + ( f A ) g ( X ) . Now, let S ( f ) be the set of all lower-semicontinuous sub-linear functions dominated by a mapping f ,and A ( f ) the set of all continuous affine functions dominated by f . The supremum over S + ( f A ) in theabove expression corresponds to the supremum over all lower-semicontinuous sub-linear functions withvalues in R + ∪ { + ∞} that are dominated by f A and it clearly coincides with the supremum over all (notnecessarily positive) lower-semicontinuous sub-linear functions that are dominated by f A . That is, wehave sup g ∈S + ( f A ) g ( X ) = sup g ∈S ( f A ) g ( X ) . As any lower-semicontinuous sub-linear function can be written as the supremum of the continuous affinefunctions that it dominates (by taking its convex envelope), we have that sup f ∈S ( f A ) sup g ∈A ( f ) g ( X ) = sup n g ( X ) : g ∈ [ f ∈S ( f A ) A ( f ) o = sup n g ( X ) : g ∈ A ( f A ) o . = conv f A ( X ) (cid:4) Remark . If the convex envelope of a function f is defined as the supremum over the (not necessarilycontinuous) affine functions that it dominates, then conv f is not necessarily lower-semicontinuous. Nev-ertheless, the proposition above can easily be adapted to yield the equality conv f = f conv A by changingconvex, closed sets for convex sets and dropping all the requirements of continuity over g, f and the affinefunctions appearing in the proof.Under strong star-shapedness we have the following result regarding to continuity. Proposition 3.18.
Let A be strongly star-shaped and closed. If ∈ bd( A ) , then f A is continuousexcept at . If ∈ int( A ) , then f A is continuous everywhere. Proof.
First, note that if A is closed and star-shaped, then due to Lemma 3.8, item (iv), we have A = A f A ,from which it follows that f A is lower-semicontinuous — see Theorem 5.3, item (i). Now, by definitionof strong star-shapedeness it is always true that ∈ A . We will show first the case ∈ bd( A ) , and thenconsider the case ∈ int( A ) .Assume then that ∈ bd( A ) . Note that if we let B := A \ { } , then it is an easy check to see that f B ( X ) = f A ( X ) for all X ∈ X \ { } and f B (0) = ∞ . Indeed, for X = 0 the conditions m − X ∈ A and m − X ∈ B are clearly equivalent, whereas for X = 0 the condition X ∈ mA is always truewhereas X ∈ mB is vacuous. Hence, we have that f B is lower-semicontinuous everywhere, except at . Furthermore, we have B = A f B . To see it, note that A f B = { X ∈ X : f B ( X ) ≤ } , and obviously / ∈ A f B as f B (0) = ∞ . Therefore, A f B = A f B \ { } = { X ∈ X : f B ( X ) ≤ } \ { } = { X ∈ X \ { } : f B ( X ) ≤ } = { X ∈ X \ { } : f A ( X ) ≤ } = { X ∈ X : f A ( X ) ≤ } \ { } = B. It remains to show that the set V := { X ∈ X : f B < } is open, from which we will know (again fromTheorem 5.3, item (i)) that f B is upper-semicontinuous. This will give us then that f B is continuouseverywhere except at , which in turn entails continuity of f A everywhere except at . To see that V is indeed an open set, note that — due to Proposition 3.13 items (ii), (iii) and (v) — if X = 0 then X ∈ bd( A ) if and only if f A ( X ) = 1 , hence bd( A ) = { X ∈ X : f A ( X ) ≡ f B ( X ) = 1 } ∪ { } . Now, thereader should realize that, again since f B (0) = ∞ , V = { X ∈ X : f B ( X ) < and X = 0 } = { X ∈ X : f A ( X ) < and X = 0 } = ( A \ bd( A )) \ { } = A \ bd( A )= int( A ) , and as int( A ) is by definition an open set, the claim that V is open holds.Finally, if ∈ int( A ) , as we already have that f A is lower-semicontinuous, it is enough to show thatit is also upper-semicontinuous. It suffices to show that the set U := { X ∈ X : f A < } is open. Clearly,again due to Proposition 3.13 items (ii), (iii) and (v), we have bd( A ) = { X ∈ X : f A ( X ) = 1 } . Hence, int( A ) = A \ bd( A ) = U and the claim follows. (cid:4) The following result characterizes polar sets through Minkowski gauges. Recall that, by definition,the polar of a set A ⊆ X is given by A ⊙ = { X ′ ∈ X ′ : h X, X ′ i ≤ for all X ∈ A } . Proposition 3.19.
Let A be star-shaped. Then it holds that A ⊙ = { X ′ ∈ X ′ : h X, X ′ i ≤ f A ( X ) for all X ∈ X } . (5) Proof.
Notice that we can write A ⊙ = { X ′ ∈ X ′ : h X, X ′ i ≤ for all X ∈ A } = B ∩ B ∩ B ∞ , B = { X ′ ∈ X ′ : h X, X ′ i ≤ for all X ∈ A such that f A ( X ) = 0 } ,B = { X ′ ∈ X ′ : h X, X ′ i ≤ for all X ∈ A such that < f A ( X ) < ∞} ,B ∞ = { X ′ ∈ X ′ : h X, X ′ i ≤ for all X ∈ A such that f A ( X ) = ∞} . Similarly, we can write the right hand side in (5) as { X ′ ∈ X ′ : h X, X ′ i ≤ f A ( X ) for all X ∈ X } = B ∗ ∩ B ∗ ∩ B ∗∞ , where B ∗ = { X ′ ∈ X ′ : h X, X ′ i ≤ for all X ∈ X such that f A ( X ) = 0 } ,B ∗ = { X ′ ∈ X ′ : h X, X ′ i ≤ f A ( X ) for all X ∈ X such that < f A ( X ) < ∞} ,B ∗∞ = { X ′ ∈ X ′ : h X, X ′ i ≤ ∞ for all X ∈ X such that f A ( X ) = ∞} . Clearly B ∞ = B ∗∞ = X ′ since B ∞ is defined by a vacuous sentence and the upper bound f A ( X ) = ∞ in B ∗∞ is non-binding. Thus, to establish the proposition it suffices to show that B = B ∗ and B = B ∗ .For the equality B = B ∗ , suppose X ′ ∈ B and let X ∈ X be such that f A ( X ) = 0 . If h X, X ′ i ≤ then there is nothing to show as in this case X ′ ∈ B ∗ . If h X, X ′ i ≥ , then — as f A is positivehomogeneous — we have f A ( λX ) = 0 for all λ > and, by assumption, h λX, X ′ i ≤ for all λ > ,which necessarily entails h X, X ′ i = 0 . Thus, B ⊆ B ∗ . That B ∗ ⊆ B is obvious. Hence, B = B ∗ For the equality B = B ∗ , suppose X ′ ∈ B and let X ∈ X be such that < f A ( X ) < ∞ . Writing Y = X/f A ( X ) , we have Y ∈ A by item (iii) in Proposition 3.13 and < f A ( Y ) < ∞ by positivehomogeneity. Thus h Y, X ′ i ≤ or, which is the same, h X, X ′ i ≤ f A ( X ) . The preceding argumentshows that, B ⊆ B ∗ . Reciprocally, suppose X ′ ∈ B ∗ and let X ∈ A be such that < f A ( X ) < ∞ .Writing Y = f A ( X ) X ∈ X , then again positive homogeneity entails < f A ( Y ) = f A ( X ) < ∞ . Thus, h Y, X ′ i ≤ f A ( Y ) or, equivalently, h f A ( X ) X, X ′ i ≤ f A ( X ) , from which we deduce that h X, X ′ i ≤ since f A ( X ) ≤ . Therefore, B ∗ ⊆ B , which establishes the equality in B = B ∗ . (cid:4) A well know result in convex analysis is the duality associating the Minkowski gauge of a set A with the support function of its polar h A ⊙ ( X ) := sup X ′ ∈ A ⊙ h X, X ′ i , X ∈ X . This is related to theconvex biconjugate of the Fenchel-Moreau Theorem (when X is a locally convex topological space) viathe conjugate and biconjugate functions (the latter is also called the penalty function in the jargon ofconvex risk measures). If the gauge is a proper, convex and weakly lower-semicontinous functional, thenthe penalty is precisely the characteristic function of the polar. Below we present this duality result fortopological vector spaces, without relying on the Frenchel-Moreau Theorem. Proposition 3.20 (Dual representation) . Let A be a closed, convex set such that ∈ A . Then we havethe identity f A ( X ) = h A ⊙ ( X ) (6)for all X ∈ X . Proof.
For simplicity, let us write h := h A ⊙ . First of all, notice that h : X → R + ∪ {∞} is a lower-semicontinuous, sub-linear function. Then, by (iv) in Proposition 3.14 (we drop the absorbing conditionas h may assume infinity), we have h = f A h . Therefore, it is enough to show that A h = A . Note that A h = { X ∈ X : h ( X ) ≤ } = (cid:8) X ∈ X : sup X ′ ∈ A ⊙ h X, X ′ i ≤ (cid:9) = (cid:8) X ∈ X : h X, X ′ i ≤ for all X ′ ∈ A ⊙ (cid:9) = A ⊙⊙ . The characteristic function of the polar assumes if X ∈ A ⊙ and ∞ otherwise. A = A ⊙⊙ . Hence, h = f A h = f A ⊙⊙ = f A as claimed. (cid:4) Remark . The equality in the proposition above holds even if A is empty, as in this case f A ≡ ∞ and A ⊙ = X ′ , so h A ⊙ ≡ ∞ . It is also interesting to remember that if A = X , then f A ≡ , A ⊙ = { } and h A ⊙ ≡ . Furthermore, note that A ⊙ = ∂f A (0) := “the set of sub-gradients of f A at ”. Remark . By the Bipolar Theorem, for a closed, star-shaped set A , we have that A ⊙⊙ = conv A .Additionally, Proposition 3.20 above tells us that h A ⊙ = f A ⊙⊙ = f conv A . However, by Proposition 3.16,as A is closed, f conv A = conv f A . Therefore, we have the following representation for the support function h A ⊙ in terms of the convex envelope of f A : h A ⊙ ( X ) = conv f A ( X ) , X ∈ X . (7) In this section we explore deviation measures induced by a Minkowski gauge, interpreting this functionalas measuring the amount of shrinkage on a financial position required to accommodate it in the base set A of acceptable positions. Specifically, we present results that link properties of A to financial propertiesof f A . Whenever possible — and due to its importance for duality —, we also establish a connectionwith the support function h A ⊙ .Before proceeding, let us introduce some further terminology. A non-negative and translation insen-sitive functional D : X → R + ∪ {∞} is called a deviation measure ; if moreover D is convex, then it issaid to be a convex deviation measure . Non-negativity and translation insensitivity are taken as ax-ioms in defining deviation measures because they capture, respectively, the intuitions that (i) a positionwhose payoff does not depend on the market outcome should display zero dispersion, and; (ii) addinga fixed amount of cash to a given position should not alter its “degree of non-constancy”. A positivehomogeneous, convex deviation measure is said to be a generalized deviation measure . Notice thatthe sub-level set A kD of a deviation measure D , for k > , is never empty — indeed, it contains at leastthe set of all constant positions. Of course, we say that D is law invariant , (cid:22) -monotone , comonotoneadditive , lower-range dominated , etc, if it fulfills the corresponding properties. Remark . Rockafellar and Uryasev (2013) proposed measures of error to quantify the “non-zeroness”of a random variable. By definition, a functional ε : L p → R + ∪ {∞} is called a measure of error if itis lower-semicontinuous, sub-linear, positive homogeneous and satisfies (i) ε ( X ) = 0 if and only if X = 0 almost surely; and (ii) if lim ε ( X n ) = 0 then lim E X n = 0 . By the authors’ Quadrangle Theorem, if ε isa measure of error, then the functional D defined, for X ∈ X , by D ( X ) := min c ∈ R ε ( X − c ) , is a convexdeviation measure. Furthermore, such D is a generalized deviation measure whenever the inequality ε ( X ) ≤ | E X | holds for every X ≤ . From this we can conclude that, given a functional ε satisfying allthose conditions, the identity D ( X ) = f ( A ε + R ) ( X ) holds — see Lemma 3.11. Indeed, if the minimum isattained, it holds that f ( A ε + R ) ( X ) = inf c ∈ R f A ε ( X − c ) = min c ∈ R ε ( X − c ) = D ( X ) .We begin, then, by characterizing deviation measures in general. It is specially compelling that thesupport functional inherits the defining attributes of deviation measures: Proposition 4.2.
Let A ⊆ X be star-shaped. Then:(i) If A is radially bounded, then f A ∪ R and h ( A ∪ R ) ⊙ are both non-negative. Thus, if A is radiallybounded at non-constants, then f A and h A ⊙ are both non-negative.(ii) If A is stable under scalar addition, then f A and h A ⊙ are both translation insensitive.In particular, if A is star-shaped, radially bounded at non-constants and stable under scalar addition,then f A is a deviation measure. Proof.
For the first item, recall from item (vi) in Proposition 3.13 that f A ( X ) > for every non-zero X ∈ X , whenever A is radially bounded. Observe that, as R is a subspace of X , one has f R ( X ) = 0 for every X ∈ R , whereas f R ( X ) = ∞ , for each (a.s.) non-constant X . Then, as A is radially bounded,we have f A ∪ R ( X ) = min( f A ( X ) , f R ( X )) = f A ( X ) > , for every X / ∈ R , and for c ∈ R we have that f A ∪ R ( c ) = min( f A ( X ) , f R ( c )) = min( f A ( X ) ,
0) = 0 . Now, let B := A ∪ R . Note that, by item (iii) of23emma 2.11 ( A ∪ R ) ⊙ = A ⊙ ∩ R ⊙ , and it should be clear that h A ⊙ ∩ R ⊙ = min( h A ⊙ , h R ⊙ ) . Additionally,by Proposition 3.20 h R ⊙ ( X ) = f R ( X ) = 0 for all X ∈ R and h R ⊙ ( X ) = ∞ otherwise (i.e. if X isnon-constant). Hence, it is clear that h B ⊙ ( X ) = 0 for constant X .For the case of non-constant X , in order to demonstrate that h B ⊙ ( X ) > , it is enough to showthat in this case h A ⊙ ( X ) > , since h R ⊙ ( X ) = ∞ and h B ⊙ = min( h A ⊙ , h R ⊙ ) . We shall do so byshowing that { X ∈ X : h A ⊙ ( X ) = 0 } = { } . Letting C denote the set in the lefthandside in the latterequation, first note that C is clearly a cone. Furthermore it is an easy check to show that A ⊙ ⊆ C ⊙ . Bydefinition, then, it holds that C ⊙⊙ ⊆ A ⊙⊙ . This, together with item (vi) in Theorem 2.11 yields that C ⊆ C ⊙⊙ ⊆ A ⊙⊙ = cl-conv A , and as the closed convex hull of a radially bounded set is again radiallybounded, it follows that cl-conv A is radially bounded. Now, by definition, the only non-empty cone thatis contained in a radially bounded set is the trivial cone { } , it follows that C = { } . Therefore, for X = 0 , it holds that h A ⊙ ( X ) > and the claim holds. The proof of item (i) is complete once we recallthat A is radially bounded at non-constants if and only if A \ R is radially bounded.For item (ii), since R ⊆ A clearly we have f A ( c ) = 0 , for every c ∈ R . It is also clear that for such a c one has X + c ∈ A if and only if X ∈ A . In particular the condition ( X + c ) /m ∈ A is equivalent to X/m ∈ A , hence f A ( X + c ) = inf { m > X + c ) /m ∈ A } = inf { m > m − X ∈ A } = f A ( X ) , for any c ∈ R . Lastly, recall from item (viii) in Lemma 2.11 that h , X ′ i = 0 for all X ′ ∈ A ⊙ . By thedefinition of h A ⊙ , together with linearity we have that h , X ′ i = 0 and h A ⊙ ( X + c ) = sup X ′ ∈ A ⊙ n h X, X ′ i + c h , X ′ i o = h A ⊙ ( X ) as claimed. (cid:4) Remark . For a given, non-empty A ⊆ X , it is always true that A + R is stable under scalar addition.Assuming further that A is star-shaped, closed and radially bounded yields that A + R is radially boundedat non-constants. Indeed, by Lemma 2.5, in this case A contains no proper cone with vertex at some x ∈ R . Hence, A + R contains no cones other than R and { } , that is, A + R is radially bounded at non-constants. In this case, f A + R is a deviation measure. An (apparent) sensible choice for the acceptance set A would be a sub-level set A ρ corresponding to some pre-specified coherent risk measure ρ (see Section6 for the definition). However, such a set is never radially bounded. Nevertheless, if we insist on taking B := A ρ + R in order to force translation insensibility, then we would have that B ≡ { X ∈ X : ρ < ∞} ,which again is of no interest as it is clearly a cone, with f B ( X ) = 0 for X ∈ B and f B ( X ) = ∞ otherwise.Said another way, in this case f B is the characteristic function of ρ .Diversification (or lack thereof) is a very important feature from the financial point of view, a conceptwhich mathematically is expressed in terms of convexity. The next result conveys sufficient conditionsto be imposed on the acceptance set in order to ensure that the corresponding Minkowski gauge bea generalized deviation measure. As the polar and its support are convex, irrespective of the chosenacceptance set, we refrain from presenting the analogous result for the support of the polar in theproposition bellow. Proposition 4.4.
Let A ⊆ X . The following assertions hold:(i) If A is convex and ∈ A , then f A is sub-linear.(ii) If A is star-shaped and A c is convex, then f A is super-linear (concave and positive homogeneous)on cone( A c ) , i.e. f A ( X + Y ) ≥ f A ( X ) + f A ( Y ) for any X, Y ∈ cone( A c ) .In particular if A is convex, radially bounded, stable under scalar addition and contains the origin, then f A is a generalized deviation measure. Proof.
We already have positive homogeneity for both items, by Lemma 3.7, as ∈ A in either case.For the first item, it remains to show that f A is convex, so fix λ ∈ [0 , and X, Y ∈ X . Define A := { α ∈ R ∗ + : λX ∈ αA } and B := { β ∈ R ∗ + : (1 − λ ) Y ∈ βA } . By definition and positive homogeneity we have inf A = f A ( λX ) = λf A ( X ) and inf B = f A ((1 − λ ) Y ) =(1 − λ ) f A ( Y ) . We only need to consider the case where both A and B are non-empty, as otherwise theupper bound f A ( λX + (1 − λ ) Y ) ≤ ∞ holds trivially. Take α ∈ A and β ∈ B . Then, convexity of A λX + (1 − λ ) Y ∈ ( α + β ) A , and hence f A ( λX + (1 − λ ) Y ) ≤ α + β . Therefore, f A ( λX + (1 − λ ) Y ) ≤ inf A + inf B = λf A ( X ) + (1 − λ ) f A ( Y ) .For the second item, star-shapedness of A and equation (4) tell us that f A = ϕ A c . Hence, as positivehomogeneity already holds, it suffices to show that ϕ A c is a concave functional on cone( A c ) whenever A c is convex. To see that this is the case, let B = A c and fix λ ∈ [0 , and X, Y ∈ cone( A c ) . Let usfirst consider the case where < λ < and where both X and Y are non-zero. In this scenario the sets A := { α ∈ R ∗ + : λX ∈ αB } and B := { β ∈ R ∗ + : (1 − λ ) Y ∈ βB } are both non-empty (for instance, X ∈ cone( B ) means precisely that X = aZ for some a > and some non-zero Z ∈ B , and in this case wehave λa ∈ A ). By definition and using positive homogeneity of f A together with the equality f A = ϕ B ,we have sup A = ϕ B ( λX ) = λϕ B ( X ) and sup B = ϕ B ((1 − λ ) Y ) = (1 − λ ) ϕ B ( Y ) . Taking α ∈ A and β ∈ B , convexity of B yields λX + (1 − λ ) Y ∈ ( α + β ) B , so ϕ B ( λX + (1 − λ ) Y ) ≥ α + β . Therefore, ϕ B ( λX + (1 − λ ) Y ) ≥ sup A + sup B = λϕ B ( X ) + (1 − λ ) ϕ B ( Y ) . The remaining cases are just amatter of adapting the following argument: if, say, λX = 0 , then A = ∅ and ϕ B ( λX + (1 − λY )) = ϕ B ((1 − λ ) Y ) = (1 − λ ) ϕ B ( Y ) = λϕ B ( X ) + (1 − λ ) ϕ B ( Y ) . This completes the proof. (cid:4) Remark . Unfortunately, item (ii) in Proposition 4.4 cannot be relaxed as to accommodate super-linearity of f A on the whole X . However, if we are willing to let go from the identity f A = ϕ A c , itis possible to define the cogauge in a slightly different manner, by assigning the value ϕ B ( X ) := −∞ whenever { m ∈ R + : m − X ∈ B } = ∅ ; in this case, an easy adaptation of the proof of item (i) inProposition 4.4 yields concavity of ϕ B for convex B . This alternative definition of the cogauge wasstudied in Barbara and Crouzeix (1994). To see that the assumptions in item (ii) of Proposition 4.4do not, in general, yield super-linearity of f A on the whole X , consider the following counterexample,illustrated in Figure 8: let Ω = { , } be the binary market and identify L ≡ R as usual. Let A := { ( x, y ) ∈ R : y − | x | ≤ } . In this case, the set C := A \ cone( A c ) is a cone and hence, forany X ∈ C , we have that f A ( X ) = 0 , whereas f A ( X ) > for X / ∈ C . Now let Y = (1 , / ) ∈ int B , Z = (1 , ∈ bd B and W = (1 , ∈ bd A . We have f A ( Z ) = 0 < f A ( W ) , but Z is a convex combinationof W and Y , so f A is not concave on the whole domain.Figure 8: A star-shaped set A (in gray) with convex complement for which f A is not concave. A c A c o n e ( A c ) ZWY
Consider a cone C comprised of positions that do not provide any benefit or detriment from diversifi-cation. By a benefit from diversifying a position X with an asset Y we mean that the risk, or dispersion,of the overall portfolio will not increase if we take a convex combination of X and Y when compared toany one of the individual positions. In the acceptance set, such reasoning is reflected by noting that ifboth X and Y are acceptable, then their convex combinations cannot be worse — that is to say, convexcombinations of acceptable positions are acceptable as well. This rationale says that the acceptance set A , or at least its positions also lying in C , should be a convex set, i.e we should require that C ∩ A beconvex. On the other hand, by a detriment from diversifying X with a position Y ∈ C , we mean theexact opposite: that the risk or dispersion of any convex combination of X and Y should not be lessthen the individual positions. With respect to an acceptance set A , this means that if both X and Y are not deemed acceptable ( X, Y / ∈ A ), then combining them in a convex fashion yields a unacceptableposition as well. Hence, the complement of A should be convex, at least when restricted to C : we should25lso require that A c ∩ C be a convex set. Importantly, when restricted to such a cone, the Minkowskigauge of a star-shaped set A is linear: Proposition 4.6.
Let A be a star-shaped set, and let C ⊆ cone( A c ) be a cone for which both A ∩ C and A c ∩ C are convex sets. Then f A respects f A ( X + Y ) = f A ( X ) + f A ( Y ) for every X, Y ∈ C . Proof.
Let g be the restriction of f A to the cone C , i.e. g : C → R + ∪ {∞} is such that g ( X ) = f A ( X ) =max (cid:0) f A ( X ) , f C ( X ) (cid:1) = f A ∩ C ( X ) for all X ∈ C . It suffices to show that g is additive; we shall proceedby showing that this function is concave and sub-linear. Sub-linearity of g is yielded by item (i) ofProposition 4.4, as A ∩ C is a convex set containing the origin by assumption, and thus f A ∩ C is sub-linear on the whole X , in particular when restricted to C . For concavity, we shall summon the cogauge tohelp us: as A is a star-shaped set, the gauge coincides with the cogauge of its complement, i.e. f A = ϕ A c — see eq. (4). It follows that, for X ∈ C , one has g ( X ) = ϕ A c ( X ) .We now show that, for X ∈ C , the identity ϕ A c ( X ) = ϕ A c ∩ C ( X ) holds. As A ∪ C c is star-shapedsince C c ∪ { } is a cone, we have ϕ A c ∩ C ( X ) = ϕ ( A ∪ C c ) c ( X ) = f A ∪ C c ( X ) = min( f A ( X ) , f C c ( X )) = min (cid:0) ϕ A c ( X ) , ϕ C ( X ) (cid:1) . In particular, g = ϕ A c ∩ C on C , as ϕ C ( X ) = ∞ = f C c ( X ) if X ∈ C and ϕ C ( X ) = 0 = f C c ( X ) if X / ∈ C .Now, the only thing that is left is to show is that the cogauge of a convex set is a concave functionon C , and this follows from Proposition 4.4 as it tells us that ϕ A c ∩ C is concave on cone( A c ) ⊇ C . (cid:4) The preceding reasoning and results yield comonotonic additivity of f A whenever A and A c are bothconvex for comonotone pairs; this is the content of Corollary 4.7. As an example of a set A satisfyingthe assumptions in the corolary, take Ω = { , } , identify L ≡ R , and let A be the set of those X = ( u, v ) ∈ R for which u ≥ , v ≥ and | u | + | v | ≤ . In this case, the set of comonotone pairs inthe 1st quadrant is precisely { ( u, v ) ∈ R : u ≥ v } . Corollary 4.7.
Let A ⊆ X be radially bounded with ∈ A . Suppose both A and A c are convex forcomonotone pairs, i.e. λX + (1 − λ ) Y ∈ A for all λ ∈ [0 , whenever X, Y ∈ A are comonotone, andsimilarly for A c . Then A is star-shaped and f A is comonotone additive. Proof.
First, notice that A is star-shaped. Indeed, any X ∈ A is comonotone to , and by assumption A is convex for this pair, i.e. λX ≡ λX + (1 − λ )0 ∈ A for any ≤ λ ≤ . Furthermore, as A is radiallybounded, it follows that cone( A c ) = X and so any cone that we may take is contained in cone( A c ) .Now let X and Y be any comonotone pair. Note that any two members of the set C X,Y =conv(cone( { X } ∪ { Y } )) are comonotone to one another (see Lemma 2.7). Now, if we take any Z, W ∈ C X,Y ∩ A , as they are a comonotone pair, by assumption we have that λZ + (1 − λ ) W ∈ C X,Y ∩ A .Hence, C X,Y ∩ A is a convex set. The same argument tells us that C X,Y ∩ A c is also convex. Thus, byProposition 4.6, we have that f A ( X + Y ) = f A ( X ) + f A ( Y ) . (cid:4) Remark . If the conditions in the corollary above and in Proposition 4.6 are imposed only on A (andnot necessarily on A c ), then we have in the proposition that f A is convex on C , and in the corollarythat f A is convex for comonotone pairs. Similarly, if we only impose those conditions on A c , then theresulting f A is concave. Remark . Note that the assumptions on Corollary 4.7 above — particularly radial boundedness —imply that f A ( X ) > for any X ∈ X \ { } . Hence, such a set A cannot yield a deviation measure,as it cannot fulfill the axiom of non-negativity. Notwithstanding, we can take A + R as the acceptanceset — which, due to item (ii) of Lemma 3.11, yields a Minkowski gauge that satisfies f A + R ( X ) =inf c ∈ R f A ( X − c ) . Now let X be non-constant, and notice that any constant is comonotone to X . Fromthe Corollary 4.7, we have that inf c ∈ R f A ( X − c ) = f A ( X ) + inf c ∈ R f A ( − c ) = f A ( X ) > , whereas forconstant X it is clear that inf c ∈ R f A ( X − c ) = inf c ∈ R f A (0) = 0 . Therefore, f A + R is non-negative, and A + R is clearly stable under scalar addition. Consequently, in view of Proposition 4.2, item (ii), it holdsthat f A + R is translation insensitive, i.e. it is a deviation measure.The next result concerns law invariance. Proposition 4.10. If A ⊆ X is law invariant then f A is law invariant. Furthermore, if (Ω , F , P ) is anatomless probability space, then for X = L p , p ∈ [1 , ∞ ) , it holds that A ⊙ and h A ⊙ are law invariant. Ifadditionally A ⊙ ⊆ L , then the preceding is also true for X = L ∞ .26 roof. First, let X = d Y ∈ X and m ∈ R ∗ + . Clearly one has m − X = d m − Y and thus, as A is lawinvariant by assumption, the condition m − X ∈ A holds if and only if it holds that m − Y ∈ A . Thisleads to f A ( X ) = inf (cid:8) m ∈ R ∗ + : m − X ∈ A (cid:9) = inf (cid:8) m ∈ R ∗ + : m − Y ∈ A (cid:9) = f A ( Y ) . We will establish the second claim for X ∈ L . The case < p < ∞ is analogous. If X ∈ A then by assumption the set { Y ∈ L : Y = d X } is contained in A . Two applications of Lemma 4.60 inFöllmer and Schied (2016) then give us sup Y ∈L X E [ Y X ′ ] = Z F − X ( t ) F − X ′ ( t ) d t = sup Y ′ ∈L X ′ E [ XY ′ ] for X ∈ L and X ′ ∈ L ∞ . Furthermore, it is well known that this fact can be generalized to X ∈ L p and X ′ ∈ L q , p ∈ [1 , ∞ ) — see Filipović and Svindland (2012) for instance. Now, for any X ′ ∈ A ⊙ , theabove yields ≥ sup Y ∈L X E [ Y X ′ ] = sup Y ′ ∈L X ′ E [ XY ′ ] , for all X ∈ A . Therefore, if X ′ ∈ A ⊙ and Y ′ = d X ′ , the above gives E [ XY ′ ] ≤ sup Z ′ ∈L X ′ E [ XZ ′ ] ≤ for all X ∈ A , that is to say, Y ′ ∈ A ⊙ . In summary, the polar A ⊙ is law invariant. For the supportfunction, since X ∈ L X = L Y , we have h A ⊙ ( X ) ≡ sup X ′ ∈ A ⊙ E [ XX ′ ] ≤ sup X ′ ∈ A ⊙ sup Z ∈L Y E [ ZX ′ ] = sup X ′ ∈ A ⊙ sup Z ′ ∈L X ′ E [ Y Z ′ ]= sup X ′ ∈ A ⊙ E [ Y X ′ ] (8) = h A ⊙ ( Y ) , where equality (8) follows from the fact that the collection {L X ′ : X ′ ∈ A ⊙ } is a partition of A ⊙ .Finally, by symmetry we also have the reversed inequality h A ⊙ ( Y ) ≤ h A ⊙ ( X ) , which establishes thestated result. (cid:4) Remark . It is a well known result in measure theory that a probability space (Ω , F , P ) has noatoms if and only if there is a random variable U defined on Ω having a uniform distribution on theunit interval — see for instance Proposition 6.9 in Delbaen (2002). At first sight this seems to excludeimportant examples such as the binary market, which is usually modelled via the sample space Ω = { , } equipped with the discrete σ -field F = 2 Ω and a probability measure characterized by a realnumber ≤ p ≤ for which P { } = p . Of course, there is no continuous uniform random variabledefined on this space. A possible workaround is as follows: extend the underlying probability space via (cid:0)b Ω , b F , b P (cid:1) = (cid:0) Ω × (0 , , Ω ⊗ Borel( R ) , P ⊗ Lebesgue (cid:1) and set U ( ω, t ) := t which clearly has a uniformdistribution. Now, for each random variable X on the original sample space there corresponds a randomvariable b X on the extended space which ‘only depends on the first coordinate’, b X ( ω, t ) := X ( ω ) . In thisframework, by letting b L denote the collection of all such random variables, it is clear that we have theidentification b L ≡ R and that b L ⊆ L ∞ ( b P ) . Thus, we can still apply Proposition 4.10 in the contextof the binary market whenever A ⊆ b L .Remember that the “law invariant hull” L B := { X ∈ X : X = d Y, for some Y ∈ B } of a set B ,inherits some interesting properties from B — see remark 2.3. We then have the following connection. Proposition 4.12.
Let B ⊆ X . Then the equality f L B ( X ) = inf Y ∈L X f B ( Y ) holds for all X ∈ X . Proof.
Let X ∈ X and m ∈ R ∗ + . Now, we have X/m ∈ L B if and only if there exists an Y ∈ X suchthat X/m = d Y /m and
Y /m ∈ B , if and only if there exists an Y ∈ L X such that Y ∈ mB . Therefore, { m ∈ R ∗ + : X/m ∈ L B } = S Y ∈L X { m ∈ R ∗ + : Y ∈ mB } and then f L B ( X ) = inf [ Y ∈L X { m ∈ R ∗ + : Y ∈ mB } = inf Y ∈L X inf { m ∈ R ∗ + : Y ∈ mB } = inf Y ∈L X f B ( Y ) , as stated. (cid:4)
27e now explore lower-range dominance of f A and h A ⊙ . It is not a surprise that lower-range dominanceof these functionals has a connection with the relation between A and A . Proposition 4.13.
Let X = L p , where p ∈ [1 , ∞ ) . If A ⊆ A , then f A and h A ⊙ are lower-rangedominated. Proof.
We shall prove the case X = L and X ′ = L ∞ , since if we show the claim for all X ∈ L thenit also holds for the L p spaces due to the inclusion L ⊇ L p , for p ∈ (1 , ∞ ] . Write B := A . As B ⊆ A and since LR is sub-linear, we have that f A ( X ) ≤ f B ( X ) = LR( X ) , for all X ∈ X , by Lemma 3.8 andProposition 3.14. Therefore, f A is lower-range dominated. For h A ⊙ , Lemma 2.11 tells us that A ⊙ ⊆ B ⊙ and since B is a closed, convex set containing the origin, the dual representation (6) holds. Hence, for X ∈ X , we have h A ⊙ ( X ) ≤ h B ⊙ ( X ) = f B ( X ) = LR( X ) as stated. (cid:4) Remark . A natural way to force lower-range dominance is by taking an acceptance set of the form A = B ∪ A , where B ⊆ X is a given set of acceptable positions. This yields f A = min( f B , f A ) . How-ever, while the union operation preserves properties like stability under scalar addition, star-shapedness,law invariance and radial boundedness at non-constants, it is possible that convexity may be lost. Remark . Under the conditions of Proposition 4.13 we have that X ′ ≤ for all X ′ ∈ A ⊙ . Tosee it, first note that as B := A is closed and star-shaped (by Theorem 5.3, since LR is lower-semicontinuous and positive homogeneous), Proposition 3.19 then tells us that the polar can be writtenas B ⊙ = { X ′ ∈ X ′ : E [ XX ′ ] ≤ LR( X ) for all X ∈ X } . Therefore, we have B ⊙ = { X ′ ∈ X ′ : E [ XX ′ ] ≤ LR( X ) for all X ∈ X } = { X ′ ∈ X ′ : E [ X ( X ′ − ≤ − ess inf X for all X ∈ X } = { X ′ ∈ X ′ : E [ X (1 − X ′ )] ≥ ess inf X for all X ∈ X } , hence, inf X ′ ∈ B ⊙ E [ X (1 − X ′ )] ≥ ess inf X, for every X ∈ X . Now, letting X ′ ∈ X ′ be such that ess sup X ′ > , and, for ω ∈ Ω , defining X ( ω ) := ( , − X ′ ( ω ) ≥ , − X ′ ( ω ) < , we clearly have that X ∈ L ∞ ⊆ L p and E [ X (1 − X ′ )] < whereas ess inf X = 0 . Therefore X ′ / ∈ B ⊙ .We now explore monotonicity with respect to a given partial order (cid:22) . Despite the fact that this kindof property is not studied much in the literature (both for gauges and deviations), it becomes crucial fordecision making. Proposition 4.16.
Let (cid:22) be a partial order that is stable under positive scalar multiplication, and let A ⊆ X . Then, we have the following:(i) If A is monotone with respect to (cid:22) , then f A is anti-monotone with respect to (cid:22) .(ii) If A is anti-monotone with respect to (cid:22) , then f A is monotone with respect (cid:22) . Proof.
For the first item, let X (cid:22) Y . If m ∈ R ∗ + is such that m − X ∈ A , then m − Y ∈ A , as A ismonotone. Thus, { m ∈ R ∗ + : m − X ∈ A } ⊆ { m ∈ R ∗ + : m − Y ∈ A } and hence f A ( X ) ≥ f A ( Y ) .Similarly, for item (ii) let Y (cid:22) X . If m ∈ R ∗ + is such that m − X ∈ A , then m − Y ∈ A , as A isanti-monotone. Thus, { m ∈ R ∗ + : m − X ∈ A } ⊆ { m ∈ R ∗ + : m − Y ∈ A } and hence f A ( X ) ≥ f A ( Y ) . (cid:4) For the next proposition, recall that (cid:22) D denotes the dispersive order of distributions , according towhich one has Y (cid:22) D X if and only if the inequality F − Y ( u ) − F − Y ( v ) ≤ F − X ( u ) − F − X ( v ) holds for every < v < u < . Proposition 4.17.
Let ∅ = A ⊆ X be (cid:22) D -anti-monotone. Then(i) If ( A, (cid:22) D ) has a maximal element X , then A is stable under convex combinations of comonotonepairs and radially bounded at non-constants. Furthermore, f A admits the following representations: f A ( Y ) = inf { m ∈ R ∗ + : F − Y ( u ) − F − Y ( v ) ≤ m (cid:0) F − X ( u ) − F − X ( v ) (cid:1) , ∀ < v < u < } = sup { m ∈ R ∗ + : F − Y ( u ) − F − Y ( v ) > m (cid:0) F − X ( u ) − F − X ( v ) (cid:1) , for some < v < u < } = inf { m ∈ R ∗ + : Y (cid:22) D mX } . That is to say, it holds that Y (cid:22) X if and only if λY (cid:22) λX for all λ ∈ R + . ( A c , (cid:22) D ) has a minimal element X , then A c is stable under convex combinations of comonotonepairs. Furthermore, f A admits the following representation: f A ( Y ) = inf { m ∈ R ∗ + : F − Y ( u ) − F − Y ( v ) < m (cid:0) F − X ( u ) − F − X ( v ) (cid:1) , for some < v < u < } = sup { m ∈ R ∗ + : F − Y ( u ) − F − Y ( v ) ≥ m (cid:0) F − X ( u ) − F − X ( v ) (cid:1) , ∀ < v < u < } . = sup { m ∈ R ∗ + : mX (cid:22) D Y } . Proof.
Before proceeding, notice that A is necessarily star-shaped.For item (i), let X be a maximal element of A . First, note that the quantile function is comonotoneadditive, in the sense that F − Y + Z = F − Y + F − Z whenever ( Y, Z ) is a comonotone pair — see Lemma 4.90in Föllmer and Schied (2002). Hence, for any W that is a convex combination of some comonotone pair Y, Z ∈ A , it follows that F − W ( u ) − F − W ( v ) = F − λZ +(1 − λ ) Y ( u ) − F − λZ +(1 − λ ) Y ( v )= λF − Z + (1 − λ ) F − Y ( u ) − λF − Z + (1 − λ ) F − Y ( v )= λ (cid:0) F − Z ( u ) − F − Z ( v ) (cid:1) + (1 − λ ) (cid:0) F − Y ( u ) − F − Y ( v ) (cid:1) ≤ max (cid:0) F − Z ( u ) − F − Z ( v ) , F − Y ( u ) − F − Y ( v ) (cid:1) ≤ F − X ( u ) − F − X ( v ) , for all < v < u < , which shows that W ∈ A .To see that A is radially bounded at non-constants, note that one has F − Y ( u ) − F − Y ( v ) = 0 for all < v < u < if and only if Y is constant. Hence, for a non-constant Y , there is some u and v with u > v such that c := F − Y ( u ) − F − Y ( v ) > . Also, we have that λc = F − λY ( u ) − F − λY ( v ) for any λ > .Therefore, as k := F − X ( u ) − F − X ( v ) ≥ F − Y ( u ) − F − Y ( v ) , it is obvious that one can find a γ such thatfor any λ ≥ γ the inequality λc > k holds. This implies that λY (cid:22) X never holds, and hence — as X isa maximal element of A — we must have λY / ∈ A . As Y ∈ A was arbitrary, it follows that A is radiallybounded at non-constants.For the stated representations, note that Y ∈ A if and only if Y (cid:22) X . Therefore, the following holds, f A ( Y ) = inf { m ∈ R ∗ + : m − Y ∈ A } = inf { m ∈ R ∗ + : Y (cid:22) mX } = inf { m ∈ R ∗ + : F − Y ( u ) − F − Y ( v ) ≤ m (cid:0) F − X ( u ) − F − X ( v ) (cid:1) , ∀ < v < u < } . Furthermore, remember that ∈ A and that, if A is stable under convex combinations of comonotonepairs, then A is star-shaped (see Corollary 4.7). Hence we have, by equation (4), that f A = ϕ A c and so f A ( Y ) = ϕ A c ( Y )= sup { m ∈ R ∗ + : m − Y ∈ A c } = sup { m ∈ R ∗ + : m − Y / ∈ A } = sup { m ∈ R ∗ + : Y (cid:22) mX does not holds } = sup { m ∈ R ∗ + : F − Y ( u ) − F − Y ( v ) > m (cid:0) F − X ( u ) − F − X ( v ) (cid:1) , for some < v < u < } . For the second item, let X be a minimal element of A c . First, we shall show that A c is monotone withrespect to the dispersive order of distributions: let Y ∈ A c and Y (cid:22) Z . Suppose, by contradiction, that Z ∈ A . Then, as A is anti-monotone, we should have Y ∈ A , an absurd. Hence, Z ∈ A c . Now, noticethat for any W that is a convex combination of some comonotone pair Y, Z ∈ A c , i.e. W = λZ + (1 − λ ) Y for some λ in the unit interval, the following holds for all < v < u < : F − W ( u ) − F − W ( v ) = F − λZ +(1 − λ ) Y ( u ) − F − λZ +(1 − λ ) Y ( v )= λF − Z + (1 − λ ) F − Y ( u ) − λF − Z + (1 − λ ) F − Y ( v )= λ ( F − Z ( u ) − F − Z ( v )) + (1 − λ )( F − Y ( u ) − F − Y ( v )) ≥ min (cid:0) F − Z ( u ) − F − Z ( v ) , F − Y ( u ) − F − Y ( v ) (cid:1) ≥ F − X ( u ) − F − X ( v ) . Therefore, as A c is monotone w.r.t. (cid:22) D , we have W ∈ A c .29inally, for the stated representations note that Y ∈ A c if and only if X (cid:22) Y . Therefore, the followingholds, f A ( Y ) = inf { m ∈ R ∗ + : m − Y ∈ A } = inf { m ∈ R ∗ + : m − Y / ∈ A c } = inf { m ∈ R ∗ + : mX (cid:22) Y does not holds } = inf { m ∈ R ∗ + : F − Y ( u ) − F − Y ( v ) < m (cid:0) F − X ( u ) − F − X ( v ) (cid:1) , for some < v < u < } . For the second representation, as A is star-shaped, we are once again allowed to summon the cogauge inorder to obtain f A ( Y ) = ϕ A c ( Y )= sup { m ∈ R ∗ + : m − Y ∈ A c } = sup { m ∈ R ∗ + : mX (cid:22) Y } = sup { m ∈ R ∗ + : F − Y ( u ) − F − Y ( v ) ≥ m ( F − X ( u ) − F − X ( v )) , ∀ < v < u < } . This completes the proof. (cid:4)
An important result in the literature of risk and deviation measures is the following dual representation for convex deviation measures:
Theorem 4.18 (Rockafellar et al. (2006a), Theorem 1) . A given functional D : L → R + ∪ { + ∞} is alower-semicontinuous generalized deviation measure if and only if it it has a representation of the form D ( X ) = E X − inf Q ∈Q E [ XQ ] , for all X ∈ L (9)in terms of a convex envelope Q ⊆ L satisfying the following:( Q1 ) Q is non-empty, closed and convex;( Q2 ) for each non-constant X there is a Q ∈ Q for which E ( XQ ) < E X ;( Q3 ) E Q = 1 for all Q ∈ Q .Additionally, the set Q above is uniquely determined by D through Q = { Q ∈ L : D ( X ) ≥ E X − E [ XQ ] for all X } , and the finiteness of D is equivalent to boundedness of Q . Furthermore, D is lower-range dominated ifand only if Q has the additional property that( Q4 ) Q ≥ for all Q ∈ Q .With regard to our framework, we have the following correspondences for the dual representation inthe generalized and law invariant cases. Corollary 4.19.
Let A ⊆ X . Suppose A is convex, radially bounded, stable under scalar addition andcontains the origin. Then D A ≡ f A is a generalized deviation measure, and admits the dual representation D A ( X ) = E X − inf Q ∈Q E [ XQ ] = sup X ′ ∈ A ⊙ h X, X ′ i = h A ⊙ ( X ) , X ∈ X , where Q = 1 − A ⊙ . Furthermore, if A ⊆ A , then X ′ ≤ for any X ′ ∈ A ⊙ . Proof.
For the Minkowski gauge of a closed convex set with ∈ A , the dual coincides with the supportfunction of the polar — see Proposition 3.20. Furthermore, we have a one-to-one correspondence of theproperties of the well know theorem of dual representation for deviation measures of Rockafellar et al.(2006a) (see Theorem 4.18 above) with the acceptance set A and its polar A ⊙ ≡ − Q . (Q1) is directfrom Lemma 2.11.A closed convex set A is a cone if and only if its polar can be written as A ⊙ = { X ′ ∈ X ′ : sup X ∈ A h X, X ′ i ≤ } , where f A := ∞ × I A c . Therefore, if A is radially bounded (i.e. A contains no cone with vertex at theorigin), there is some X ′ ∈ A ⊙ such that h X, X ′ i > , for all X ∈ X . This yields that, if A is radially30ounded at non-constants, the previous sentence holds for all non-constant X ∈ X , which, again under Q = 1 − A ⊙ is equivalent to (Q2).Stability under scalar addition of A implies that, for all X ′ ∈ A ⊙ , it holds that h , X ′ i = 0 , and itmakes h A ⊙ a translation insensitive functional (Proposition 4.2). This is equivalent to (Q3).Clearly, the boundedness of Q is equivalent to the boundedness of A ⊙ , which in turn is equivalentto A being absorbing (see Lemma 2.11) and f A finite (see Proposition 3.13). If A ⊆ A , then byProposition 4.13, one has X ′ < , for every X ′ ∈ A ⊙ and, consequently, (Q4). (cid:4) Proposition 4.20.
Assume (Ω , F , P ) is an atomless probability space, and put X := L p ( p ∈ [1 , ∞ ] ).Let moreover B denote a law invariant, closed, radially bounded, convex subset of X containing theorigin, and define A := B + R . Then f A is a law invariant, lower semicontinuous generalized deviationmeasure, and the following representation holds, for all X ∈ X : f A ( X ) = sup X ′ ∈ A ⊙ Z F − X ( t ) F − X ′ ( t )d t = sup ψ ∈ Λ Z ψ ( t ) F − X ( t )d t = sup g ∈ G Z g ( t ) F − X (d t ) , where Λ is a collection of nondecreasing functions ψ ∈ L q [0 , such that R ψ ( t ) d t = 0 , and where G is a collection of positive concave functions g : [0 , → R . If in addition B c is convex for comonotonepairs, then f A is also comonotone additive, then the supremum in the above representations is attainedfor some, respectively, X ′ ∈ A ⊙ , ψ ∈ Λ , g ∈ G , for any X ∈ X . Proof.
First of all, notice that if B is a law invariant, closed, radially bounded, convex set containing theorigin, then, by the reasoning in Remark 4.3, the set A := B + R is radially bounded at non constantsand stable under scalar addition. Furthermore, the operation of set addition preserves convexity, lawinvariance and closedness; hence, A is convex, law invariant and closed. This yields that f A is a lawinvariant, lower semi continuous generalized deviation measure. The stated representations follow fromPropositions 2.1 and 2.2 of Grechuk et al. (2009). Also, Proposition 2.4 of the same paper yields, undercomonotonic additivity — which is given by the convexity for comonotonic pairs of B c and remark 4.9— that f A ( X ) = R g ( t ) F − X (d t ) , for some positive concave function g : [0 , → R . (cid:4) Remark . By taking B = A ε for a law invariant measure of error ε , one obtains a set B that fulfillsthe requirements from the above proposition. To ensure comonotonicity, one can take B of the form B = A ε as above, with the additional requirement that the error measure ε be comonotone additive. So far we have (mostly) focused on the scenario where an acceptance set A is given, and studied therelations existing between properties of this set and associated properties of its Minkowski gauge, espe-cially how the former manifest on the latter. Now, a special case occurs when the acceptance set itselfis already induced by a given, specified a priori deviation measure. Remember that under the mildrequirement that A is closed and star-shaped we have A = A f A (as ensured by item (iv) in Lemma 3.8).Additionally, item (i) in Proposition 3.14 tells us that a positive homogeneous function f coincides withthe Minkowski gauge of A f (where the requirement that the underlying set be absorbing may be droppedwhen + ∞ ∈ range f ).The crucial fact explored in this section is that we actually have a two-way correspondence betweenproperties of the functional and the properties of the associated acceptance set. In particular, a lower-semicontinuous, convex deviation measure yields an acceptance set which is stable under scalar addition,convex, closed and radially bounded at non-constants. Proposition 5.1.
Let (cid:8) A ( k ) : k ∈ R ∗ + (cid:9) be a positive homogeneous family of subsets of X . Then, foreach real positive k and λ , we have the following: k f A ( k ) = λ f A ( λ ) = inf { m ∈ R ∗ + : X ∈ A ( m ) } . (10) Proof.
For the first equality, let k, λ ∈ R ∗ + . Due to positive homogeneity, we have k − λA ( k ) = A ( λ ) .Therefore, kf A ( k ) ( X ) = λ inf (cid:8) mkλ − ∈ R ∗ + : X ∈ mA ( k ) (cid:9) = λ inf (cid:8) m ∈ R ∗ + : X ∈ mk − λA ( k ) (cid:9) = λ inf (cid:8) m ∈ R ∗ + : X ∈ mA ( λ ) (cid:9) = λf A ( λ ) ( X ) . λ = 1 above and noticing that mA (1) = A ( m ) . (cid:4) Remark . The representation inf { m ∈ R + : X ∈ A ( m ) } appearing in equation (10) was studied inthe context of risk measures in Drapeau and Kupper (2013), under some extra conditions on the family { A ( k ) } .For the theorem below, recall that A kf = { X ∈ X : f ( X ) ≤ k } . The following theorem provides acharacterization for acceptance sets generated by deviation measures, i.e. sub-level sets corresponding tonon-negative, translation insensitive functionals on X . These results are new in the literature, and canbe seen as reciprocals for the results studied in the previous sections. Theorem 5.3.
Let f, f ′ : X → R ∪ {∞} be positive homogeneous functionals. Then we have thefollowing, for all positive real k ,(i) The collection { A λf : λ ∈ R + } is a positive homogeneous family with each of its members beinga star-shaped set. Moreover, if f does not assume negative values, then the following string ofequalities holds, for all X ∈ X : f ( X ) = f A f ( X ) = kf A kf ( X ) = inf (cid:8) λ ∈ R ∗ + : X ∈ A λf (cid:9) . (ii) If f is finite, then A kf is absorbing.(iii) If f is translation insensitive, then A kf + R = A kf .(iv) If f is non-negative, then A kf is radially bounded at non-constants and R ⊆ A kf .(v) If f is a convex functional, then A kf is a convex set.(vi) If f is a concave functional, then ( A kf ) c is a convex set.(vii) If f is law invariant, then so is A kf .(viii) If f ≤ f ′ , then A kf ′ ⊆ A kf . In particular, if f is lower-range dominated then A k LR ⊆ A kf .(ix) If f is symmetric, then so is A kf .(x) If f ( X ) > for all X ∈ X , then A kf is radially bounded.(xi) If f respects f ( X + Y ) = f ( X ) + f ( Y ) for X, Y in some convex cone C , then A kf ∩ C and ( A kf ) c ∩ C are convex sets. In particular, if f is comonotone additive, then both A kf and its complement arestable under convex combinations of comonotone pairs in X .(xii) If f is lower-semicontinuous, then A kf is closed.(xiii) If f is continuous, then A kf is strongly star-shaped.(xiv) If f is monotone, then A kf is anti-monotone and A k − f is monotone. Proof.
For item (i), star-shapedness of each A λf is clear as if f ( X ) ≤ k , then λf ( X ) ≤ k , for any λ ∈ [0 , .Also, note that (by positive homogeneity of f ) λA kf = { λX ∈ X : f ( X ) ≤ k } = { X ∈ X : f ( X ) ≤ λk } = A λkf , yielding the positive homogeneity for the generated family. It remains to prove that f = f A f , as theremaining equalities will follow from Proposition 5.1. Now, with A = A f , we have (again by positivehomogeneity of f ) f A ( X ) = inf n m ∈ R ∗ + : m − X ∈ { Z : f ( Z ) ≤ } o = inf n m ∈ R ∗ + : f ( X ) ≤ m o = f ( X ) . Item (ii) is clear, as if f is a positive homogeneous finite function and k > then, for any X ∈ X suchthat f ( X ) > one has f (cid:0) kX/f ( X ) (cid:1) = k . Therefore, we have tX ∈ A kf for any ≤ t ≤ δ X := k/f ( X ) .Of course, if f ( X ) ≤ then there is nothing to prove, as in this case we have f ( X ) ≤ k , that is, X ∈ A kf .32or item (iii), let Y ∈ A kf + R , that is, Y = X + c for some X ∈ A kf (meaning f ( X ) ≤ k ) and some c ∈ R . Then f ( Y ) = f ( X + c ) = f ( X ) ≤ k as f is translation insensitive. This yields A kf + R ⊆ A kf . Thereverse inclusion holds by definition.Item (iv) follows from the fact that, for any non-constant X , we have f ( X ) > (by assumption).Hence, by positive homogeneity of f , there is some δ X := k/f ( X ) > such that f ( mX ) > k for all m > δ X . Furthermore, as f ( c ) = 0 < k for any c ∈ R it follows that R ⊆ A kf .For item (v), let X, Y ∈ A kf and let Z be any convex combination of X and Y . It follows from theconvexity of f that f ( Z ) ≤ max( f ( X ) , f ( Y )) ≤ k , hence the claim holds.For item (vi), let B = ( A kf ) c , X, Y ∈ B and assume Z is any convex combination of X and Y . Itfollows from the concavity of f that f ( Z ) ≥ min( f ( X ) , f ( Y )) > k , hence the claim holds.Regarding item (vii), let X ∈ A kf and assume Y = d X . Then, due to law invariance of f , we have f ( Y ) = f ( X ) ≤ k, , that is Y ∈ A kf .For item (viii), let X ∈ A kf ′ . Clearly the claim holds, as f ( X ) ≤ f ′ ( X ) ≤ k. The particular case forwhen f is lower-range dominated is obvious from the definition.To prove item (ix) simply note that if X ∈ A kf , then — due to the symmetry of f — it holds that f ( − X ) = f ( X ) ≤ k, that is − X ∈ A kf .Item (x) follows the same reasoning as item (iv).For item (xi), note that the restriction of f to C is both convex and concave, hence the convexityof A kf ∩ C follows the same reasoning that item (v) and the convexity of ( A kf ) c ∩ C from item (vi). Forthe case when f is additive comonotone, let X, Y be a comonotone pair. Due to Lemma 2.7, the set C X,Y is a convex cone whose members are all comonotone to one another, and f is additive on C X,Y .By the preceding reasoning, the sets A kf ∩ C X,Y and ( A kf ) c ∩ C X,Y are both convex. In particular, if Z is any convex combination of X and Y , then Z ∈ A kf ∩ C X,Y ⊆ A kf whenever X, Y ∈ A kf , and similarly Z ∈ ( A kf ) c whenever X, Y ∈ ( A kf ) c .Item (xii) is just the definition of lower-semicontinuity.For item (xiii) we shall show only for the case A := A f . It holds for general A kf due to item (i). Bycontinuity of f , we have that A is closed whereas the set B := { X ∈ X : f ( X ) < } is open. Evidently, A c is open and B c is closed, and the inclusions B ⊆ int A and A c ⊆ int( B c ) hold; in particular this gives ∈ int A as f is positive homogeneous, so A is absorbing and f ( X ) = f A ( X ) < ∞ for all X . Therefore, B c ∩ A = { X ∈ X : f ( X ) = 1 } = bd( A ) , where the second equality is yielded by Lemma 3.3. We mustshow that, for each X , the ray R X := { λX : λ ∈ R ∗ + } intersects bd( A ) at most once. For all X such that f ( X ) ≤ it is clear that R X ⊆ B (so R X ∩ bd( A ) = ∅ ). It remains to consider the case < f ( X ) < ∞ .Clearly, f ( λX ) = 1 for λ − := f ( X ) , so R X ∩ bd( A ) is nonempty. Moreover, if γ > λ then clearly f ( γX ) > by positive homogeneity, and if < γ < λ then γX ∈ B ; in any case γX / ∈ bd A .Lastly, for item (xiv) again we shall show only for the case A := A f and B := A − f , as it holds forgeneral A kf and A k − f due to item (i). Let Y ∈ A and X (cid:22) Y . Now, remember that for any Z ∈ X , Z ∈ A if and only if f ( Z ) ≤ . Then we have, by monotoniticy f , that f ( X ) ≤ f ( Y ) ≤ , hence X ∈ A ,establishing the anti-monotonicity of A . By the same token, let X ∈ B and X (cid:22) Y . Again, we havethat for any Z ∈ X , Z ∈ B if and only if − f ( Z ) ≤ , and by anti-monotonicity of − f it follows that ≥ − f ( X ) ≥ − f ( Y ) . This completes the proof. (cid:4) Now, we analyze how some operations on a deviation measure are reflected on its correspondingacceptance set. For a comprehensive theory on combinations of monetary risk measures, see Righi(2020b).
Proposition 5.4.
Let f, f ′ : X → R + ∪{∞} be positive homogeneous functionals and k, λ ∈ R ∗ + . Then:(i) A k min( f,f ′ ) = A kf ∪ A kf ′ and A k max( f,f ′ ) = A kf ∩ A kf ′ .(ii) X ∈ A kf if and only if there are non-negative constants c and d , and positive homogeneous functions g and h such that k = c + d , f = g + h and X ∈ A cg ∩ A dh . In particular, one has A k + λf + f ′ ⊇ A kf ∩ A λf ′ .(iii) A kλf = λ − A kf . Proof.
For the first item, if X ∈ A k min( f,f ′ ) , then f ( X ) ≤ k or f ′ ( X ) ≤ k . That is, X ∈ A kf ∪ A kf ′ Reciprocally, if X ∈ A kf ∪ A kf ′ , then we must have f ( X ) ≤ k or f ′ ( X ) ≤ k , so min( f ( X ) , f ′ ( X )) ≤ k ,which is the same as X ∈ A k min( f,f ′ ) . The equality A k max( f,f ′ ) = A kf ∩ A kf ′ follows from a similar argument.33tem (ii) is established as follows: assume X ∈ A cg ∩ A dh , where k = c + d and f = g + h . Then,by definition, it holds that g ( X ) ≤ c and h ( X ) ≤ d . Hence, f ( X ) ≡ g ( X ) + h ( X ) ≤ c + d = k , whichis the same as X ∈ A kf . For the reverse inclusion, assume X ∈ A kf . Then, trivially, there are non-negative constants c := k and d := 0 , and positive homogeneous functions g := f and h := 0 such that X ∈ A cg ∩ A dh ≡ A kf . The last equivelence follows from the fact that A dh = { X ∈ X : 0( X ) ≤ } ≡ X .Finally, for the last item we have X ∈ A kλf if and only if f ( X ) ≤ k/λ if and only if X ∈ A k/λf . Thelatter set is equal to λ − A kf by item (i) in Theorem 5.3. (cid:4) In this section we discuss a few examples of well-known deviation measures and their respective accep-tance sets.
Example 5.5.
Variance ( σ ): One of the most widely used measures to quantify dispersion. It is defined,for X ∈ X ⊆ L (recall that we allow for deviations measures to assume + ∞ ), as σ ( X ) = E [( X − E X ) ] , and the associated acceptance sets are given by A kσ = (cid:8) X ∈ X : σ ( X ) ≤ k (cid:9) , k > . As the variance is not positive homogeneous, it does not coincide with the Minkowski gauge of A σ :indeed, for A = A kσ , we have f A ( X ) = σ ( X ) √ k . Also, notice that σ ( X ) < ∞ if and only if X ∈ L . Example 5.6.
Standard deviation ( σ ): The measure used to quantify risk in the seminal paper ofMarkowitz (1952). It has served as inspiration for the class of generalized deviation measures. It isdefined, for X ∈ X ⊆ L , as σ ( X ) = p σ ( X ) = k X − E X k , and the associated acceptance sets are given by A kσ = { X ∈ X : σ ( X ) ≤ k } , k > . (Note that A kσ = A k σ ). If X (cid:22) D Y then k X − E X k ≥ k Y − E Y k , for a detailed proof and more detailssee Shaked (1982). Furthermore, writing A k := A k k·k , we have that σ ( X ) = k f A kσ ( X ) = k f A k + R ( X ) , where the first equality above follows from Theorem 5.3, item (i), and the second one comes from item (ii)in Lemma 3.11, together with the identity k f A k = k · k yielded by item (i) in Theorem 5.3 and the wellknown fact that inf z ∈ R k X − z k = k X − E X k = σ ( X ) (indeed, k · k is the measure of error associatedwith the standard deviation ). Notice that σ ( X ) is finite if and only if X ∈ L . In Figure 9 bellow, wecan see the acceptance set A σ in blue, (note that A σ = A σ ) and the closed unit ball (on the norm k · k )in red. The figure also illustrates the relation A + R = A σ . Importantly, here k · k does not represent the Euclidian norm. A σ (in blue) and A ||·|| (in red) in the binary market Ω = { , } with P { } = / and P { } = / . A σ A k·k Example 5.7.
Standard lower-semi-deviation ( σ − ): It is a generalized deviation measure that considersonly the negative part of the deviation X − E X . This one is defined, for X ∈ X ⊆ L , as σ − ( X ) = k ( X − E X ) − k . The corresponding acceptance sets are given by A kσ − = { X ∈ X : k ( X − E X ) − k ≤ k } = (cid:8) X ∈ X : σ ( X | E X ≥ X ) ≤ k / P ( E X ≥ X ) (cid:9) , k > , where σ ( X | E X ≥ X ) := E (cid:8) ( X − E { X (cid:12)(cid:12) E X ≥ X } ) | E X ≥ X (cid:9) is the conditional variance of X giventhat X lies in the lower tail of its distribution. Importantly, the set A kσ − contains every random variablewhose standard deviation is bounded above by k , as k ( X − E X ) − k ≤ k X − E X k clearly yields A kσ ⊆ A kσ − .Such fact, can be seen in Figure 10, where the acceptance set A σ of the standard deviation is depictedin blue, and A σ − is represented in red. In particular, σ − is finite on a subspace which is larger than { X ∈ X : σ ( X ) < ∞} . 35igure 10: The sub-level sets A σ − (in red) and A σ (in blue) in the binary market Ω = { , } with P { } = / and P { } = / . A σ A σ − Example 5.8.
Lower range deviation ( LR ): It is the ‘most conservative’ among the class of lower-rangedominated generalized deviation measures, defined for X ∈ X ⊆ L as LR( X ) = E [ X − ess inf X ] , with acceptance set A k LR = { X ∈ X : E X − ess inf X ≤ k } = { X ∈ X : ess sup( − X ) ≤ E [ − X ] + k } . Thus, A k LR is comprised of all positions X whose penalized expected loss E ( − X ) + k is bounded belowby the maximum loss ess sup( − X ) . Furthermore writing A = ball k·k (0; k ) ∩ X + , we have that, LR( X ) = k f A k LR ( X ) = kf A + R ( X ) . The second equality follows from the fact that kf A ( X ) assumes ∞ for all X ≤ , and equals E | X | otherwise; thus it coincides with the error function associated to the lower-range deviation — see Lemma3.11. In Figure 11, we can see the acceptance set A in blue, and the closed unit ball (on the norm k · k ) restricted to R in red. The fact that A + R = A is clear from this figure.36igure 11: The sub-level sets A (in blue) and A = ball k·k (0; 1) ∩ X + (in red) in the binary market Ω = { , } with P { } = / and P { } = / . A A Example 5.9.
Upper range deviation ( UR ): Defined, for X ∈ X ⊆ L , as UR( X ) = ess sup X − E X = LR( − X ) , this measure is the symmetric opposite of LR. Its acceptance set is given by A k UR = { X ∈ X : ess sup X − E X ≤ k } = { X ∈ X : ess sup X ≤ E X + k } . Furthermore, writing A = ball k·k (0; k ) ∩ X − we have that UR( X ) = k f A k UR ( X ) = k f A + R ( X ) , where the second equality follows from the same reasoning as the one for LR . Example 5.10.
Full range deviation (
FRD ): Can be considered the most extreme generalized deviationmeasure, defined for X ∈ X = { X ∈ L : ess inf X < ∞ or ess sup X > −∞} as FRD( X ) = ess sup X − ess inf X, with acceptance set A k FRD = { X ∈ X : ess sup X ≤ k + ess inf X } . Furthermore, writing A = A k k·k ∞ we have that FRD( X ) = k f A k FRD ( X ) = 2 k f A + R ( X ) , where the second equality is due to Lemma 3.11 and the fact that k f A ( X ) = 2 k X k ∞ , which is theerror function associated to the full range deviation. Note that FRD( X ) < ∞ if and only if X ∈ L ∞ . InFigure 11, we can see the acceptance set A in blue, and the closed unit ball (on the norm k · k ∞ ),scaled down in half, in red. Clearly, A . k·k ∞ + R = A .37igure 12: The sub-level sets A (in blue) and A = A . k·k ∞ (in red) in the binary market Ω = { , } with P { } = / and P { } = / . A A Example 5.11.
Expected shortfall deviation (
ESD ): A generalized deviation measure derived fromthe (standard) expected shortfall. It is defined, for X ∈ X ⊆ L and < α ≤ , by ESD α ( X ) =ES α ( X − E X ) with, ES α ( X ) = − Z α α F − X ( t ) d t, and ESD α ( X ) = E X − ess inf X = LR( X ) for α = 0 . Note that if we take γ = 1 − α we have that ES α ( X ) = R γ − γ F − X ( t ) d t . Furthermore, if F X is continuous, then the following representation alsoholds. ESD α ( X ) = ES α ( X − E X ) ≡ − E (cid:0) X − E X | X ≤ F − X ( α ) (cid:1) = E ( X ) − E (cid:0) X | X ≤ F − X ( α ) (cid:1) with acceptance set A k ESD α = { X ∈ X : k − ES α ( X ) ≥ E X } If we let the
Koenker-Bassett error be defined as KB α ( X ) = E (cid:2) α − (1 − α ) X − + X + (cid:3) , which is the errorfunction associated with the ESD , then we have KB α = k f A , with A = A k KB α . Hence — by Lemma 3.11— it holds that ESD α ( X ) = k f A k ESD α ( X ) = k f A + R ( X ) . A α (in blue) and A = A α (in red), with α = 0 . , in the binary market Ω = { , } with P { } = / and P { } = / . A A α In this section we study the relation between Minkowski gauges and (positive homogeneous) monetaryrisk measures. First, we recall some terminology. A translation invariant and anti-monotone functionalthat does not attain −∞ is said to be a monetary risk measure , which we denote generically by ρ : X → R ∪ {∞} . If such a ρ is moreover convex, then we say that ρ is a convex risk measure .A positive homogeneous, convex risk measure is said to be a coherent risk measure . A sub-levelset of a monetary risk measure is never empty; yielding well defined acceptance sets of the form A kρ = { X ∈ X : ρ ( X ) ≤ k } for k ≥ .There is a compelling interplay between coherent risk measures and generalized deviation measures:Rockafellar et al. (2006a) show that any coherent risk measure ρ gives rise to a generalized deviationmeasure via D ρ ( X ) := ρ ( X − E X ) , whenever ρ is strict, i.e. ρ ( X ) > E X for non-constant X . Conversely,a generalized deviation measure which is lower-range dominated turns out to deliver a coherent riskmeasure through ρ ( X ) := − E X + D ( X ) .Any monetary risk measure — say, ρ — maps onto the real line; thus, such ρ clearly cannot beexpressed as a gauge, although it does admit a representation of the form ρ ( X ) = inf (cid:8) m ∈ R : X + m ∈ A ρ (cid:9) , X ∈ X . (11)Notwithstanding, if ρ is positive homogeneous, then its positive and negative parts (denoted respectivelyby ρ + and ρ − ) can easily be written as gauges. It is evident that A ρ ⊆ A ρ : the acceptance set A := A ρ contains the cone of all “riskless positions”, i.e. those X such that ρ ( X ) ≤ . Therefore, f A ( X ) canappraise the risk of a position X . By the same token, A − ρ contains the cone of the non-acceptablepositions, i.e. those X such that ρ ( X ) > , so f A − ρ ( X ) assesses the “risklessness” of a given position X .The next results formalizes the preceding line of thought, and also provides a general representation forpositive homogeneous monetary risk measures — and does so without resorting to convexity assumptions,thus characterizing a departure from the standard approach found in the literature, where convexity playsa major role. Theorem 6.1.
Let ρ : X → R ∪ {∞} be a monetary risk measure. If ρ is positive homogeneous, thenit holds that ρ + = f A ρ and ρ − = f A − ρ . In particular, one has ρ ( X ) = f A ρ ( X ) − f A − ρ ( X ) Recall that in this paper we use the term acceptance set to refer to sub-level sets of the form A kρ := { X ∈ X : ρ ( X ) ≤ k } whereas in the literature on monetary risk measures an acceptance set takes the form { X ∈ X : ρ ( X ) ≤ } . X ∈ X . Proof. As ρ does not attain −∞ and is positive homogeneous, we have that − ρ is also positive ho-mogeneous and does not attain + ∞ , so A − ρ is an absorbing set and f A − ρ is finite. This avoids theindeterminacy ∞ − ∞ . Now let B − := { X ∈ X : ρ ( X ) ≤ } and B + := { X ∈ X : ρ ( X ) ≥ } . Clearly,for X ∈ B + we have f A − ρ ( X ) = 0 = ρ − ( X ) , and similarly for X ∈ B − we have f A ρ ( X ) = 0 = ρ + ( X ) .Moreover, whenever X ∈ B − it holds that f A − ρ ( X ) = inf (cid:8) m ∈ R ∗ + : − ρ ( X ) ≤ m (cid:9) = − ρ ( X ) = ρ − ( X ) , as − ρ ( X ) ≥ , and for X ∈ B + we have f A ρ ( X ) = inf (cid:8) m ∈ R ∗ + : ρ ( X ) ≤ m (cid:9) = ρ ( X ) = ρ + ( X ) , as ρ ( X ) ≥ . This establishes the equalities ρ + = f A ρ and ρ − = f A − ρ . (cid:4) Remark . Note that the Theorem 6.1 holds even if ρ is not monetary. In fact any positive homogeneousfunctional f : X → R ∪ {∞} can be represented by f ( X ) = f A f ( X ) − f A − f ( X ) .Now, assume we are given four sets (let us call them C + , C − , A + and A − ) that compose a “systemof acceptable and non-acceptable positions”, or simply a risk system . We shall interpret C + as a coneof positions deemed “riskless”, i.e. with no positive risk (this set must contain the cone of non-negativerandom variables), and C − as a cone of risky assets, i.e. those with non-negative risk (a set whichcontains, at least, the set of non-positive random variables). Clearly, we want any position to lie in atleast one of those sets, so C + ∪ C − = X . These two sets give us information on whether a position haspositive or negative risk. However, such information is “not enough” if our objective is to differentiate— through gauges — how risky (or riskless) a position is. Therefore, we require two additional sets inour system: one (call it A + ) to gauge the riskiness of a position and the other (call it A − ) to gauge itsrisklessness. Fortunately, we have so far introduced and developed the tools that allow us to formalizethis reasoning.We shall make the requirement that the set A + contain the cone C + of riskless positions, but we areflexible in allowing that it also comprises some risky, but acceptable, positions. As convex combinationsshould not increase risk, A + should also be convex. Additionally, as X ≤ Y clearly implies that X hasgreater risk than Y , we demand that A + be monotone as well. Furthermore, to asses the “risklessness”of the position, we make the requirement that the complement of A − only contain positions that yieldat least as much as a risk free position. In other words, we should have ∈ bd (cid:0) A c − (cid:1) = bd( A − ) , and A c − should be monotone and convex (for the same reason that we want A + to be monotone and convex),yielding anti-monotonicity of A − . We require that A − contain C − . As an example of such a system ofsets, suppose we already have been handed a given continuous, coherent risk measure, say ρ . Takingthen C + := A ρ , C − := A − ρ , A + := A ρ and A − := A − ρ provides a system with the required properties(see Theorem 6.8). Figure 14 illustrates another possible choice for A + and A − , in the context of thebinary market, i.e. with X = R . Recall that X + is the set comprised of positions X ≥ whereas X − contains all positions X ≤ . For the next definition, recall that R X := (0 , + ∞ ) X . Definition 6.3.
Let A := ( A + , A − , C + , C − ) be a quadruple of non-empty subsets of X . We say that A is a risk system on X if the following conditions are satisfied:(A1) A + is strongly star-shaped, closed, and ≤ -monotone, with − ∈ bd( A + ) and ∈ int( A + ) (A2) C + is a conic, ≤ -monotone subset of X . Moreover, the inclusion C ⊆ C + ⊆ A + holds for any cone C contained in A + .(B1) A − is strongly star-shaped, closed, and ≤ -anti-monotone, with ∈ bd( A − ) and ∈ int( A − ) .(B2) C − is a conic, ≤ -anti-monotone subset of X . Moreover, the inclusion C ⊆ C − ⊆ A − holds for anycone C contained in A − .(C) C + ∪ C − = X , and C + ∩ C − has an empty interior.(D) For every X / ∈ C + , the sets A + ∩ ( R + X + R − ) and A c + ∩ ( R + X + R − ) are convex. Similarly, forevery X / ∈ C − , the sets A − ∩ ( R + X + R + ) and A c − ∩ ( R + X + R + ) are convex.40f the risk system further respects the requirement that(E) A + is convex and A − has convex complement,then A is said to be a coherent risk system on X . We say a risk measure is generated bythe risk system A if it takes the form ρ A := f A + − f A − . Similarly, we say that the quadruple A ρ = ( A ρ , A − ρ , A ρ , A − ρ ) is the risk system generated by ρ whenever ρ is a continuous, positivehomogeneous monetary risk measure. Remark . The functional ρ A appearing in the above definition, which we have boldly referred to as arisk measure, is indeed a monetary risk measure, as ensured by Theorem 6.9 below. In fact, this Theoremtells us that every risk system on X gives rise to a continuous, positive homogeneous, monetary riskmeasure. The latter result is complemented by Corollary 6.10, which tells us that if the generating risksystem is moreover required to be coherent, then the corresponding functionl is in fact a continuouscoherent risk measure. Theorem 6.8 provides the reciprocal to these assertions. Remark . It will be convenient, in order to make the proofs of the results below clearer, to introducethe set K X , defined for X ∈ ( C − ∩ C + ) c , as K X := ( R + X + R + , X / ∈ C − R + X + R − , X / ∈ C + (12)In this case, assumption (D) can be restated as(D’) If X / ∈ C + , then A + ∩ K X and A c + ∩ K X are convex. If X / ∈ C − , then A − ∩ K X and A c − ∩ K X areconvex. Remark . The requirements ∈ int( A + ) and ∈ int( A − ) in assumptions (A1) and (B1) can be relaxedif we consider in X a topology not coarser than the (metric) topology of convergence in probability.Indeed, if that requirements are dropped from those assumptions, it still holds that ∈ int A + and ∈ int A − . To see that this is the case, notice that, since A + is a strongly star-shaped set with − ∈ bd A + , one has that λ ∈ int A + for any λ ∈ ( − , . In particular, − / ∈ int A + . Now proceedby contradiction: assume ∈ bd A + , so that there is a sequence Z n / ∈ A + converging to the origin inprobability. For such a sequence, let X n := Z n I [ Z n ≤ . Note that monotonicity of A + forces X n / ∈ A + ,since X n ≤ Z n . Moreover, as it also holds that | X n | ≤ | Z n | , we have for any ε > that P { ω : | X n ( ω ) | ≥ ε } ≤ P { ω : | Z n ( ω ) | ≥ ε } → as n → ∞ . Therefore, X n → in probability. Let now W n := X n − / . We have W n → − / in probability, butmonotonicity of A + and the fact that X n / ∈ A + oblige W n / ∈ A + for all n . This contradicts the fact that − / ∈ int A + . A similar argument yields ∈ int( A − ) .We begin by obtaining some properties of the sets in a risk system. Lemma 6.7.
Let A = ( A + , A − , C + , C − ) be a risk system on X . Then the following holds:(i) C + and C − are closed cones, with X + ⊆ C + and X − ⊆ C − .(ii) ∈ int C + and − ∈ int C − .(iii) ρ A is a continuous function.(iv) bd C + = bd C − = C + ∩ C − is a cone and ρ A ( X ) = 0 if and only if X ∈ C + ∩ C − .(v) R + C + + R + ⊆ C + and R + C − + R − ⊆ C − . Proof.
We shall prove the statements concerning C + and A + , as the ones for C − and A − are analogous.For item (i), assumption (A2) requires that C + be the largest conic set contained in A + . This is welldefined since an arbitrary union of cones is again a cone, and we know by assumption (A1) that A + contains at least one cone, namely X + . Assumption (A2) then tells us that cl( C + ) ⊆ C + ⊆ A + since A + is closed and the closure of a cone is again a cone.For the second item, the assumption that ∈ bd A − tells us that it must be the case that / ∈ C − — otherwise, we would have λ ∈ bd A − for every λ > , violating the requirement that A − be alsostrongly star-shaped. Since C − is closed by item (i), we have that C c − is open, and assumption (C)entails C c − ⊆ C + . Therefore, ∈ int C + . 41or item (iii), we have that the functional ρ := ρ A is continuous, as both f A + and f A − are continuousdue to assumptions (A1) and (A2) and Proposition 3.18.For (iv), the fact that C + ∩ C − in item (v) is a (closed) cone is immediate. Furthermore, note that ρ ( X ) ≤ if and only if X ∈ C + (see Item (iii) in Lemma 3.8 ), i.e. { X ∈ X : ρ ( X ) ≤ } = C + ,additionally, continuity of ρ yields that B := { X ∈ X : ρ ( X ) < } is an open set, and clearly cl B = C + ,hence, bd C + = { X ∈ X : ρ ( X ) = 0 } the same reasoning with C − yields that bd C − = bd C + . For thelast equality, is enough to show that if ρ ( X ) < than X / ∈ C − , however it is clear as ρ ( X ) ≥ if andonly if X ∈ C − . This also yields that ρ ( X ) = 0 if and only if X ∈ C + ∩ C − .The fifth item is easily seen to hold: indeed, if X ∈ C + and y, z ≥ , then conicity of C + entails yX ∈ C + and monotonicity then yields yX + z ∈ C + since yX + z ≥ yX . (cid:4) First, we provide some sufficient conditions granting that a risk measure results in a risk system.
Theorem 6.8. If ρ is a continuous, positive homogeneous monetary risk measure, then the quadruple A ρ = ( A + , A − , C + , C − ) := ( A ρ , A − ρ , A ρ , A − ρ ) is a risk system. Additionally, if ρ is coherent, then A ρ isa coherent risk system. Proof.
For monotonicity in the assumptions , first note that due to the ≤ -anti-monotonicity of ρ , A ρ and A ρ are clearly ≤ -monotone, while due to the monotonicity of − ρ , A − ρ and A − ρ are ≤ -anti-monotone,see Theorem 5.3 item (xiv). For assumption (A1) and (B2), the continuity of ρ (and − ρ ) entail strongstar-shapedness of A ρ and A − ρ — see Theorem 5.3 item (xiii). Continuity also yields that both setsare closed, and that they contain in its interior. To see that the latter is true, take the pre-image B of the open unit ball on the real line, that is, B = { X : | f ( X ) | < } ; by definition of continuity, B is an open set in X , contained in A − ρ and containing the origin. To see that ∈ bd A − ρ , since wealready have that A − ρ is strongly star-shaped and closed, and since ρ (0 + 1) = ρ (0) − by translationinvariance and positive homogeneity of ρ , clearly then − ρ (1) = 1 and thus ∈ A − ρ . Finally, for any δ > we have − ρ (1 + δ ) = (1 + δ )( − ρ (1)) = 1 + δ > , so the constant position δ does not liein A − ρ , and similarly − ρ ((1 − δ )) < so − δ ∈ A − ρ . This establishes ∈ bd A − ρ . The argumentfor − ∈ bd A ρ is similar. To see that X + ⊆ A ρ , just notice that we have — by anti-monotonicity of ρ — that ρ ( X ) ≤ ρ (0) = 0 ≤ whenever ≤ X ( X ∈ X + ). Similarly, monotonicity of − ρ tells usthat − ρ ( X ) ≤ − ρ (0) = 0 ≤ whenever X ≤ ( X ∈ X − ). To demonstrate that A ρ is the largest cone(clearly it is a cone) contained in A ρ , it suffices to show that the set { } ∪ ( A ρ \ A ρ ) contains no propercone (otherwise, if C were such a cone, then C ∪ A ρ would be a cone contained in A ρ larger than A ρ ).But { } ∪ ( A ρ \ A ρ ) ≡ { } ∪ { X ∈ X : ρ ( X ) ∈ (0 , } , and positive homogeneity of ρ forbids the latterset from containing any proper cone. The same reasoning yields that A − ρ is the largest cone containedin A − ρ .For assumption (C), it is obvious that A ρ ∪ A − ρ = X , and since A ρ ∩ A − ρ = bd( A ρ ) ∩ bd( A − ρ ) clearly this set has an empty interior.For assumption (D), we have to show that, for X as above, the sets A ± ∩ K X and ( A ± ) c ∩ K X areconvex, where K X is defined as in equation (12). Let us start with X / ∈ C + ≡ A ρ . First of all, notice that K X is a convex cone: indeed, let Z, Y ∈ K X . By definition, Z = λX + z for some λ ≥ and some z ≤ ,and similarly Y = δX + y for some δ ≥ and some y ≤ . Then, if α ≥ , clearly αZ = ( αλ ) X + αz ∈ K X ,and if α ∈ [0 , , we have αZ + (1 − α ) Y = ( αλ + (1 − α ) δ ) X + αz + (1 − α ) y ∈ K X , thus establishingthat K X is a convex cone. Moreover, for Z, Y as above, we have ρ ( Z + Y ) = ρ (( λ + δ ) X + ( z + y )) =( λ + δ )( ρ ( X )) − ( z + y ) = ρ ( λX + z ) + ρ ( δX + y ) = ρ ( Z ) + ρ ( Y ) . Now it is just a matter of evoking item(xi) of Theorem 5.3 to conclude that both K X ∩ A ρ and K X ∩ ( A ρ ) c are convex sets. The same line ofthought will show that assumption (D) holds for X / ∈ C + .Lastly, if ρ is coherent (in particular, convex) then A ρ is convex by item (v) in Theorem 5.3, andsince − ρ is concave, item (vi) of the same theorem tells us that ( A − ρ ) c is a convex set. (cid:4) A + = { ( x, y ) ∈ R : y ≥ e − (1+ x ) − } (in blue) and A − = { ( x, y ) ∈ R : min( xy, x | y | ) ≤ } (in red). Here, one has C + = X + and C − = cl( X \ X + ) . The quadruple ( A + , A − , C + , C − ) in thisexample is not a risk system, since assumption (D) in the definition is not satisfied. For example, taking X / ∈ C − as depicted below, the set K X ∩ A c + is not convex. The problem here arises from the fact thatthe generating functional is not translation invariant. A + A − A + ∩ A − − X Theorem 6.9.
Let A be a risk system. Then the functional ρ A defined, for X ∈ X , via ρ A ( X ) := f A + ( X ) − f A − ( X ) ≡ f A + ( X ) − ϕ A c − ( X ) is a continuous, positive homogeneous monetary risk measure.Furthermore, ρ + A = f A + and ρ −A = f A − . Proof.
To ease notation, let ρ := ρ A . Clearly, ρ does not attain ∞ ( −∞ ) as A + ( A − ) contains an openneighborhood of , which makes it an absorbing set and, therefore, we have f A + < ∞ ( f A − < ∞ ). Thus, ρ = f A + − f A − ∈ R . Furthermore, as assumption (C) gives us C + ∪ C − = X , we have then by item (iii)of Lemma 3.8 and item (i) of Lemma 3.11, that min( f A + ( X ) , f A − ( X )) = f A + ∪ A − ( X ) = f X ( X ) = 0 forall X ∈ X , i.e. at most one of f A + ( X ) and f A − ( X ) is non-zero. This yields ρ + = f A + and ρ − = f A − .To see that ρ is anti-monotone, we have by Proposition 4.16, together with assumptions (A1) and(B1), that f A + and − f A − are anti-monotone functions, so their sum is as well.Positive homogeneity of ρ is inherited, directly from the definition, from f A + and f A − .Continuity of f A − and f A + follows from the fact that A − and A + are closed, strongly star-shapedand ∈ int A − (see Proposition 3.18). Hence, ρ as the sum of continuous functions, is continuous.We now proceed to establish translation invariance in each one of the following scenarios, for α ≥ :1. X + α with X / ∈ C − ;2. X − α with X / ∈ C + ;3. X ± α with X ∈ C − ∩ C + ;4. X − α with X / ∈ C − , broken in the respective subcases:(a) X − α / ∈ C − ;(b) X − α ∈ C − ∩ C + ;(c) X − α / ∈ C + ;5. X + α with X / ∈ C + .For 1 and 2, note that if X / ∈ C − then X + α / ∈ C − for all α ∈ R + , and similarly, if X / ∈ C + then X − α / ∈ C + for all α ∈ R + (by ≤ -anti-monotonicity of C − and ≤ -monotonicity of C + , respectively).43oreover, as C − is the largest conic set contained in A − , we have that A − \ C − is radially bounded.This yields that cone (cid:0) ( A − \ C − ) c (cid:1) = X . Moreover, we have that cone (cid:0) ( A − \ C − ) c (cid:1) = cone (cid:0) ( X \ A − ) ∪ C − (cid:1) = cone( X \ A − ) ∪ C − = cone( A c − ) ∪ C − . Then, since cone( A c − ) ∩ C − = { } , we obtain cone (cid:0) ( A − \ C − ) c (cid:1) \ C − = cone( A c − ) \ { } . Hence, fromassumption (C), we have that the condition X / ∈ C − implies X ∈ cone( A c − ) . From this we have, by ≤ -monotonicity of C + in assumption (A2), that for X / ∈ C − it holds that K X ⊆ cone( A c − ) (where K X is defined as in equation (12)).By Proposition 4.6 and assumption (D) — and recalling that f A − (1) = 1 because A − is a stronglystar-shaped set with ∈ bd( A ) —, we have, for X / ∈ C − and α ∈ R + , f A − ( X + α ) = f A − ( X ) + αf A − (1) = f A − ( X ) + α and f A + ( X ) = f A + ( X + α ) = 0 . Hence, ρ ( X + α ) = f A + ( X + α ) − f A − ( X + α ) = − f A − ( X ) − α = ρ ( X ) − α, fulfilling item 1. On the other hand, for item 2, letting X / ∈ C + and α ∈ R + we have f A − ( X ) = f A − ( X − α ) = 0 and, by the same reasoning as above (item 1), now using the fact that f A + ( −
1) = 1 , f A + ( X − α ) = f A + ( X + α · ( − f A + ( X ) + zf A + ( −
1) = f A + ( X ) + α, and then ρ ( X − α ) = f A + ( X − α ) − f A − ( X − α ) = f A + ( X ) + α = ρ ( X ) + α. Now, for item 3, let X ∈ C + ∩ C − , and let X n := X + 1 /n , so that { X n } ⊆ ( C + ∩ C c − ) = C c − is asequence converging to X . To see that X + 1 /n / ∈ C − whenever X ∈ C − ∩ C + notice that X/ ∈ bd( C + ) and / (2 n ) ∈ int( C + ) , the latter being true since otherwise we would have R + ⊆ bd( C − ) ; therefore, wehave X/ / (2 n )) ∈ int( C + ) . Note that, as A − is strongly star-shaped with in its interior, f A − iscontinuous, and so is ρ − . Therefore, using item 1 and given α ∈ R + , ρ ( X + α ) = − f A − ( X + α ) = − lim f A − ( X n + α ) = − (lim f A − ( X n ) + α ) = − f A − ( X ) − c, and so ρ ( X + c ) = ρ ( X ) − c . Similarly, ρ ( X ) = ρ ( X + α − α ) = ρ ( X − α ) − α, and then ρ ( X − α ) = α = α + ρ ( X ) . We now focus on the remaining cases, items 4 and 5, and in fact we shall consider item 4 only, sincethe other is quite similar. In what follows, then, we are taking
X / ∈ C − . For such X , we know that R X is not entirely contained A − (otherwise, the ray R + X is a coninc set contained in A − and, a fortiori, in C − ). Therefore, by star-shapedness of A − and item (ii) in Lemma 3.4, we get that f A − ( X ) > , whichin turn implies ρ ( X ) < . Now, let α ∈ R + and define Y = X − α . From assumption (C), we then havethree possibilities: either Y / ∈ C − (item 4.a), or Y ∈ C + ∩ C − (item 4.b), or else Y / ∈ C + (item 4.c).Items 4.a and 4.b follow from the identities ρ ( X ) = ρ ( Y + α ) ∗ = ρ ( Y ) − α = ρ ( X − α ) − α, which yield ρ ( X − α ) = ρ ( X ) + α , (the equality ∗ comes from case 1 for item 4.a, and from case 3 foritem 4.b).Lastly, for 4.c (that is, X / ∈ C − and Y = X − α / ∈ C + ), we will first show that there exists somescalar m such that X − m ∈ C + ∩ C − . Let s be a real, positive constant. We know that X ∈ int( C + ) since C c − is open, and also s ∈ int( C + ) . Therefore, X + s ∈ int( C + ) , as C c − is monotone — being thecomplement of an anti-monotone set —, and so X + s / ∈ C − . By the same token we have X − s / ∈ C − for small enough s . Furthermore, we know that there is some real k > such that X − k / ∈ C + (take k = α ), so let us define the sets S := { s ∈ R : X − s / ∈ C − } and K := { k ∈ R : X − k / ∈ C + } , and put moreover s ∗ := sup S and k ∗ := inf K . Henceforth, all k ∈ K and all s ∈ S . Due to monotonicity(and anti-monotonicity) of C − and C + , we have that s < k . Furthermore, we know that X − s ∗ ∈ C − X − k ∗ ∈ C + . Now let m be such that s ≤ s ∗ ≤ m ≤ k ∗ ≤ k , then due to monotonicity of C + it follows that X − m ∈ C + and due to anti-monotonicity of C − we have that X − m ∈ C − . Hence, X − m ∈ C + ∩ C − , which yields ρ ( X − m ) = 0 , and by case 3, ρ ( X − m ) = ρ ( X ) + m , i.e. ρ ( X ) = − m .Thus, X + ρ ( X ) ∈ C + ∩ C − . Indeed, we have Z ∈ C + ∩ C − if and only in ρ ( Z ) = 0 . Therefore, ρ ( X + ρ ( X ) − β ) = ρ ( X + ρ ( X )) + β for β ∈ R , by case 3. Furthermore, as X + ρ ( X ) ∈ C + and Y = X − α / ∈ C + , monotonicity of C + tells us that P ( X − α ≥ X + ρ ( X )) < and then ρ ( X ) ≥ − α . Byletting γ := ρ ( X ) + α ≥ , we finally have that ρ ( X − α ) = ρ ( X + ρ ( X ) − γ ) = ρ ( X + ρ ( X )) + γ = ρ ( X ) + γ − ρ ( X ) = ρ ( X ) + α. This completes the proof. (cid:4)
Corollary 6.10. If A is a coherent risk system, then the functional ρ A := f A + − f A − is a continuous,coherent risk measure. Proof.
Let ρ := ρ A . From Theorem 6.9, the only thing remaining to be established is convexity of ρ . We shall to separate it in four instances: 1) X, Y ∈ C − ; 2) X, Y / ∈ C − ; 3) X ∈ C − , Y / ∈ C − ; 4) Y ∈ C − , X / ∈ C − . Note that, as we are under translation invariance it is enough to show only quasi-convexity, i.e. ρ ( λX + (1 − λ ) Y ) ≤ max( ρ ( X ) , ρ ( Y )) for any X, Y ∈ X and any λ ∈ [0 . , from whichconvexity follows. In order to see it, note that ρ ( λX + (1 − λ ) Y ) − λρ ( X ) − (1 − λ ) ρ ( Y ) = ρ ( λ ( X + ρ ( X )) + (1 − λ )( Y + ρ ( Y ))) ≤ max( ρ ( X + ρ ( X )) , ρ ( Y + ρ ( Y ))= 0 . Hence, ρ ( λX + (1 − λ ) Y ) ≤ λρ ( X ) + (1 − λ ) ρ ( Y ) .For the first instance note that, due to Proposition 4.4 and assumption (E), f A + is convex. Fromthis, we have that ρ is convex for pairs X, Y ∈ C − (as in such case f A − ( X ) = f A − ( Y ) = 0 ),For the second instance, due to Proposition 4.4 and assumption (E), f A − is concave when restrictedto cone( A c − ) . Thus, we have that ρ is convex for X, Y ∈ cone( A c − ) (as f A + ( X ) = f A + ( Y ) = 0 and − f A − is convex). We shall show that C c − ⊆ cone( A c − ) . Remember that the complement of a cone (union with0) is also a cone. By assumptions (B2) and(C) we have the implications C − ⊆ A − ⇒ { } ∪ C c − ⊇ A c − ⇒ { } ∪ C c − ⊇ cone( A c − ) ⇒ C − ⊆ { } ∪ (cid:0) cone( A c − ) (cid:1) c . Now, by assumption (B2), C − is the largest conecontained in A − . Thus, C − = (cone( A c − )) c ∪ { } ⊇ (cone( A c − )) c , and taking the complement one lasttime implies C c − ⊆ cone( A c − ) , as desired.Lastly, note that the third and fourth cases are actually equivalents. Thus, let X ∈ C − and Y / ∈ C − ;this yields that f A − ( X ) = f A + ( Y ) = 0 . Hence, by assumption (E), we have for any λ ∈ [0 , , ρ ( λX + (1 − λ ) Y ) ≤ λf A + ( X ) + (1 − λ ) f A + ( Y ) − f A − ( λX − (1 − λ ) Y ) ≤ λf A + ( X ) + (1 − λ ) f A + ( Y ) ≤ f A + ( X ) − f A − ( X )= ρ ( X ) ≤ max( ρ ( X ) , ρ ( Y )) . Hence, ρ is quasi-convex, and, by Theorem 6.9 is also translation invariant. Therefore, ρ is convex, andthe claim holds. (cid:4) Corollary 6.11.
Let A = ( A + , A − , C + , C − ) be a risk system on X . Then the risk system generated by ρ A coincides with A , that is, A ρ A = A . Moreover, if ρ is a continuous, positive homogeneous monetaryrisk measure, then ρ = ρ A ρ . Proof.
Write ̺ := ρ A for simplicity. Note that, we have to show the following identities: A + = A ̺ , A − = A − ̺ , C + = A ̺ , and C − = A − ̺ . First note that by Theorems 6.1 and 6.9, we have that ̺ + = f A ̺ = f A + and ̺ − = f A − ̺ = f A − . Now, A ̺ = A ̺ + = A f A + , hence, as A + is closed and star-shaped, we get A f A + = A + , by item (iv) in Lemma3.8. For the equality A − = A − ̺ , note that A − ̺ = A ̺ − = A f A − , and by the same reasoning as above, A − = A − ̺ . That C + = A ̺ (respectively, C − = A − ̺ ) easily follows from noticing that A ̺ (respectively, A − ̺ ) is the largest cone contained in A ̺ (respectively A − ̺ ).For the remaining statement, let ρ be a continuous positive homogeneous monetary risk measure.That ρ = ρ A ρ follows directly from Theorem 6.1 and the preceding items, by noting that ρ A ρ = f A ρ − f A − ρ = ρ . (cid:4) emark . If one seeks additional protection against risk, it is possible to further penalize a givenmonetary acceptance set. For example, if ρ is a coherent risk measure and c > is the “amount ofadditional protection required”, one can obtain an acceptance set of the form A ρ − c = { X ∈ X : ρ ( X ) ≤ } − c = { X : ρ ( X ) ≤ − c } . However, this approach does not take variability in account; if the aim is to appraise variability, thenone can take the intersection A cρ ∩ A kD , where D is some generalized deviation measure. A possible wayto extract a risk measure from this set is to take inf (cid:8) m ∈ R : X + m ∈ λ ( A cρ ∩ A kD ) , for some λ ∈ R ∗ + (cid:9) = ρ ( X ) + ck D ( X ) . Moreover, by Proposition 4.7 of Righi (2019), if inf D ( X ) > (cid:18) ess sup( − X ) − ρ ( X ) D ( X ) (cid:19) ≥ ck , then ρ ( X ) + ck D ( X ) is coherent as well. Remark . Given a convex lower-semicontinuous function f : X → R ∪ {∞} , the space L f := { X ∈ X : f A f ( | X | ) < ∞} generated by the Minkowski gauge of its sub-level set, inspired on Orlicz spaces,is a Banach space. This approach was studied, for instance, in Kupper and Svindlandc (2011), Owari(2014), Svindland (2009) and Liebrich and Svindland (2017) on the context of risk measures and the socalled economic index of riskiness. In this section we discuss a few examples of well-known monetary risk measures and their respectiveacceptance sets in our framework.
Example 6.14.
Expected Loss ( E ): One of the simplest monetary measures. It is defined, for X ∈ X ⊆ L , as E ( X ) = E [ − X ] , and the associated acceptance sets are given by A E = { X ∈ X : E ( X ) ≥ − } ⊇ A E = { X ∈ X : E ( X ) ≥ } A − E = { X ∈ X : E ( X ) ≤ } ⊇ A − E = { X ∈ X : E ( X ) ≤ } , Clearly, taking A + = A E , C + = A E , A − = A − E and C − = A − E yields a risk system — see Figure 15,where the black line represents C + ∩ C − . Therefore, due to Theorem 6.1, the Expected Loss has thefollowing representation. E ( X ) = f A E ( X ) − f A − E ( X ) = f A E ( X ) − f A E ( X ) Ω = { , } , with P { } = / and P { } = / .The thick diagonal through the origin is C + ∩ C − . A + A − A + ∩ A − Example 6.15.
Mean plus standard deviation ( Eσ β ): It is a well known measure of risk, defined as Eσ β ( X ) = E ( X ) + β σ ( X ) , where β ∈ R is a prescribed constant. This risk measure may fail to be anti-monotone if β > inf σ ( X ) > (cid:18) ess sup( − X ) − E ( X ) σ ( X ) (cid:19) . (See remark 6.12). The associated acceptance sets are given by A Eσ β = { X ∈ X : E (1 + X ) ≥ βσ ( X ) } ⊇ A Eσ β = { X ∈ X : E ( X ) ≥ βσ ( X ) } A − Eσ β = { X ∈ X : E ( X − ≤ βσ ( X ) } ⊇ A − Eσ β = { X ∈ X : E ( X ) ≤ βσ ( X ) } , Again, by taking a suitable β , we have that A + = A Eσ β , C + = A Eσ β , A − = A − Eσ β and C − = A − Eσ β yield a risk system in X . Besides, the representation given by Theorem 6.1, due to Example 5.6, is Eσ β ( X ) = E ( X ) + f A β − σ ( X ) . Unfortunately, in R , the only suitable choice of β is β = 0 ; hence, in this setting the sets A Eσβ arealways of the form represented in Figure 15, and there is no risk system that generates
Eσβ .47igure 16: A Eσ and A Eσ in the binary market Ω = { , } , with P { } = / and P { } = / . A Eσ A Eσ Example 6.16.
Maximum loss ( ML ): Defined for X ∈ X as ML( X ) = − ess inf X = ess sup( − X ) , with acceptance sets A ML = { X ∈ X : P ( X < −
1) = 0 } ⊇ A ML = { X ∈ X : P ( X <
0) = 0 } = X + A − ML = { X ∈ X : P ( X ≤ > } ⊇ A − ML = { X ∈ X : P ( X ≤ > } . Figure 17 illustrates the maximum loss on X = R ; note that the sets appearing here are the same forall probability measure absolutely continuous with respect to P . Furthermore notice that C + = R .Figure 17: Risk system of Maximum Loss. A + A − emark . The above example does give rise to a coherent risk system, as ML is a continuouscoherent risk measure on R . However, if X = L p is infinite dimensional, where p ∈ [0 , ∞ ) , then MLmay fail to be lower semi-continuous, in which case neither A − ≡ A nor C − ≡ A are closed. Infact, the following counterexample shows that, if X = L (0 , , then we cannot have a risk systemwith C + = X + : let (Ω , P ) = (cid:0) (0 , , Leb (cid:1) be the unit interval equipped with Lebesgue measure. Let K + = X + = { X ∈ L : X ≥ } and K − = cl( K c + ) . Clearly K + is monotone and so K − is anti-monotone.We now show that thar / ∈ int( K + ) in the topology of convergence in probability. Let X n := I (1 /n, − I (0 , /n ] , n ∈ N . Clearly X n / ∈ K + , and X n → in probability, so ∈ bd( K + ) = bd( K − ) . In fact, an easy adaptationof the argument above shows that K + has empty interior! In particular, K + ⊆ K − . Thus, the sets K + and K − above never coincide with the maximal cones C + and C − from a risk system ( A + , A − , C + , C − ) ,since Lemma 6.7 ensures that ∈ int C + . Example 6.18.
Value at Risk (
VaR α ): A (non-convex) monetary risk measure defined, for X ∈ X ⊆ L and < α < , as VaR α ( X ) := − inf { x ∈ R : F X ( x ) ≥ α } = − F − X ( α ) with acceptance sets A α = { X ∈ X : P ( X < − ≤ α } ⊇ A α = { X ∈ X : P ( X < ≤ α } A − VaR α = { X ∈ X : P ( X ≤ ≤ α } ⊇ A − VaR α = { X ∈ X : P ( X ≤ ≥ α } . The representation of the risk system for the Value at Risk on the plane is actually equal to the one forthe Maximum Loss, whenever α ≤ min( P { } , P { } ) — see Figure 17. If α ≥ max( P { } , P { } ) then thegraph of VaR α ( − X ) coincides with the one in Figure 17. Example 6.19.
Expected Shortfall ( ES α ): A coherent risk measure defined for X ∈ X ⊆ L and ≤ α ≤ as ES α ( X ) := − Z α α F − X ( t ) d t ≡ ESD α ( X ) − E [ X ] , with acceptance sets A α = (cid:26) X ∈ X : Z α F − X ( t ) d t ≥ − α (cid:27) ⊇ A α = (cid:26) X ∈ X : Z α F − X ( t ) d t ≥ (cid:27) A − ES α = (cid:26) X ∈ X : Z α F − X ( t ) d t ≤ α (cid:27) ⊇ A − ES α = (cid:26) X ∈ X : Z α F − X ( t ) d t ≤ (cid:27) . Figure 18 illustrates the risk system of ES α in the context of the binary market.Figure 18: Risk system of Expected Shortfall in the binary market Ω = { , } , with P { } = 0 . and P { } = 0 . . A + A − eferences Aliprantis, D Charalambos, C. and Border, K. C. (2006).
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