An optimal three-way stable and monotonic spectrum of bounds on quantiles: a spectrum of coherent measures of financial risk and economic inequality

A certain spectrum, indexed by a\in[0,\infty], of upper bounds P_a(X;x) on the tail probability P(X\geq x), with P_0(X;x)=P(X\geq x) and P_\infty(X;x) being the best possible exponential upper bound on P(X\geq x), is shown to be stable and monotonic in a, x, and X, where x is a real number and X is a random variable. The bounds P_a(X;x) are optimal values in certain minimization problems. The corresponding spectrum, also indexed by a\in[0,\infty], of upper bounds Q_a(X;p) on the (1-p)-quantile of X is stable and monotonic in a, p, and X, with Q_0(X;p) equal the largest (1-p)-quantile of X. In certain sense, the quantile bounds Q_a(X;p) are usually close enough to the true quantiles Q_0(X;p). Moreover, Q_a(X;p) is subadditive in X if a\geq 1, as well as positive-homogeneous and translation-invariant, and thus is a so-called coherent measure of risk. A number of other useful properties of the bounds P_a(X;x) and Q_a(X;p) are established. In particular, quite similarly to the bounds P_a(X;x) on the tail probabilities, the quantile bounds Q_a(X;p) are the optimal values in certain minimization problems. This allows for a comparatively easy incorporation of the bounds P_a(X;x) and Q_a(X;p) into more specialized optimization problems. It is shown that the minimization problems for which P_a(X;x) and Q_a(X;p) are the optimal values are in a certain sense dual to each other; in the case a=\infty this corresponds to the bilinear Legendre--Fenchel duality. In finance, the (1-p)-quantile Q_0(X;p) is known as the value-at-risk (VaR), whereas the value of Q_1(X;p) is known as the conditional value-at-risk (CVaR) and also as the expected shortfall (ES), average value-at-risk (AVaR), and expected tail loss (ETL). It is shown that the quantile bounds Q_a(X;p) can be used as measures of economic inequality. The spectrum parameter, a, may be considered an index of sensitivity to risk/inequality.

1. An optimal three-way stable and three-way monotonic spectrum of upper bounds on tail probabilities Consider the family ph α q αPr0,8s of functions h α : R Ñ R given by the formula for all u P R. Here, as usual, It¨u denotes the indicator function, u`:" 0 _ u and u ὰ :" pu`q α for all real u.
(1. 10) In fact (cf. e.g. [43,Proposition 3.3]), the bound P α pX; xq remains optimal given the larger class of generalized moments E gpXq for all functions g P H α , where H α :" g P R R : gpuq " ş R g α;t puq µpdtq for some µ P M α and all u P R This provides the other way to come to the mentioned conclusion that P α pX; xq is nondecreasing in α P r0, 8s.
(1. 13) By [40,Proposition 1.1], the class H α of generalized moment functions can be characterized as follows in the case when α is a natural number: for any g P R R , one has g P H α if and only if g has finite derivatives g p0q :" g, g p1q :" g 1 , . . . , g pα´1q on R such that g pα´1q is convex on R and lim xÑ´8 g pjq pxq " 0 for j " 0, 1, . . . , α´1. Also, by [43,Proposition 3.4], g P H 8 if and only if g is infinitely differentiable on R, and g pjq ě 0 on R and lim xÑ´8 g pjq pxq " 0 for all j " 0, 1, . . . . Thus, the greater the value of α, the narrower and easier to deal with is the class H α and the smoother are the functions comprising H α . However, the greater the value of α, the farther away is the bound P α pX; xq from the true tail probability PpX ě xq.
Of the bounds P α pX; xq, the loosest and easiest one to get is P 8 pX; xq, the so-called exponential upper bound on the tail probability PpX ě xq. It is used very widely, in particular when X is the sum of independent r.v.'s X i , in which case one can rely on the factorization A α pX; xqpλq " e´λ x ś i E e λXi . A bound very similar to P 3 pX; xq was introduced in [16] in the case when X the sum of independent bounded r.v.'s; see also [36,15,37]. For any α P p0, 8q, the bound P α pX; xq is a special case of a more general bound given in [38,Corollary 2.3]; see also [38,Theorem 2.5]. For some of the further developments in this direction see [39,7,8,41,42,9,43]. The papers mentioned in this paragraph used the representation (1.7) of P α pX; xq, rather than the new representation (1.5).
The new representation appears, not only of more unifying form, but also more convenient as far as such properties of P α pX; xq as the monotonicity in α and the continuity in α and in X are concerned; cf. (1.2) and the proofs of Propositions 1.4 and 1.5; those proofs, as well as proofs of most of the other statements in this paper, are given in Appendix A. Yet another advantage of the representation (1.5) is that, for α P r1, 8q, the function A α pX; xqp¨q inherits the convexity property of h α , which facilitates the minimization of A α pX; xqpλq in λ, as needed to find P α pX; xq by (1.5); relevant details on the remaining "difficult case" α P p0, 1q can be found in Section 3.1.
On the other hand, the "old" representation (1.7) of P α pX; xq is more instrumental in establishing the mentioned connection with the classes H α of generalized moment functions; in proving part (iii) of Proposition 1.2; and in discovering and proving Theorem 2.3. *** Some of the more elementary properties of P α pX; xq are presented in Proposition 1.1.
(i) P α pX; xq is nonincreasing in x P R.
In view of Proposition 1.1, it will be henceforth assumed by default that the tail bounds P α pX; xq -as well as the quantile bounds Q α pX; pq, to be introduced in Section 2, and also the corresponding expressions A α pX; xqpλq,Ã α pX; xqptq, B α pX; pqptq, and α´1 QpX; pq as in (1.3), (1.9), (2.9), and (3.15) -are defined and considered only for r.v.'s X P X α (unless indicated otherwise), where if α " 0, X P X : E X ὰ ă 8 ( if α P p0, 8q, X is the set of all real-valued r.v.'s on a given probability space (implicit in this paper), and Λ X :" λ P p0, 8q : E e λX ă 8 ( . (1.14) Observe that the set X α is a convex cone: for any θ P r0, 8q and any X and Y in X α , the r.v.'s θX and X`Y are in X α . Indeed, the conclusion that θX P X α for any θ P r0, 8q and X P X α is obvious. Concerning the conclusion that X`Y P X α for any X and Y in X α , use the inequalities EpX`Y q ὰ ď EpX``Y`q α ď E X ὰ`E Y ὰ if α P p0, 1s, Minkowski's inequality }X``Y`} α ď }X`} α`} Y`} α if α P r1, 8q, and Hölder's inequality E e λpX`Y q ď`E e pλX˘1 {p`E e qλX˘1 {q for any positive p and q such that 1 p`1 q " 1. Here, as usual, }Z} α :" pE |Z| α q 1{α , the L α -norm of a r.v. Z -which is actually a norm if and only if α ě 1. Also, it is obvious that the cone X α contains all real constants.
It follows from Proposition 1.1 and (1.6) that P α pX; xq is nonincreasing in x P R, with P α pX; p´8q`q " 1 and P α pX; 8´q " 0. (1.15) Here, as usual, f pa`q an f pa´q denote the right and left limits of f at a.
One can say more in this respect. To do that, introduce x˚:" x˚, X :" sup supp X and p˚:" p˚, X :" PpX " x˚q.
(1. 16) Here, as usual, supp X denotes the support set of (the distribution of the r.v.) X; speaking somewhat loosely, x˚is the maximum value taken by the r.v. X, and p˚is the probability with which this value is taken. It is of course possible that x˚" 8, in which case necessarily p˚" 0, since the r.v. X was assumed to be real-valued.
(ii) For all x P p´8, x˚q one has P α pX; xq ą 0.
(iii) The function p´8, x˚s X R Q x Þ Ñ P α pX; xq´1 {α is continuous and convex if α P p0, 8q; we use the conventions 0´a :" 8 and 8´a :" 0 for all real a ą 0; concerning the continuity of functions with values in the set r0, 8s, we use the natural topology on this set. Also, the function p´8, x˚s X R Q x Þ Ñ´ln P 8 pX; xq is continuous and convex, with the convention ln 0 :"´8. (iv) If α P p0, 8s then the function p´8, x˚s X R Q x Þ Ñ P α pX; xq is continuous.
(v) The function R Q x Þ Ñ P α pX; xq is left-continuous. (vi) x α is nondecreasing in α P r0, 8s, and x α ă 8 for all α P r0, 8s. (vii) If α P r1, 8s then x α " E X; even for X P X α , it is of course possible that E X "´8, in which case P α pX; xq ă 1 for all real x. (viii) x α ď x˚, and x α " x˚if and only if p˚" 1.
One may also note here that, by (1.15) and part (v) of Proposition 1.2, the function P α pX;¨q may be regarded as the tail function of some r.v. Z α : P α pX; uq " PpZ α ě uq for all real u. Example 1.3. Some parts of Propositions 1.1 and 1.2 are illustrated in the following picture with graphs of the function P α pX;¨q for various values of α in the important case when the r.v. X takes only two values. Then, by the translation invariance property stated below in Theorem 1.6, without loss of generality (w.l.o.g.) E X " 0. Thus, X " X a,b , where a and b are positive real numbers and X a,b is a r.v. with the uniquely determined zero-mean distribution on the set t´a, bu.
Explicit expressions for P α pX a,b ; xq were given in [43,Section 3,Example] in the cases α P p1, 8q and α " 8. In the case α P p0, 1s, one can see that P α pX a,b ; xq equals a a`b p a`b a`x q α for x P rx α , x˚s, 1 for x P p´8, x α s, and 0 for x P px˚, 8q, with x α " a`p a a`b q 1 α´1´1˘a nd x˚" b. Graphs t`x, P α pX a,b ; xq˘:´2 ď x ď 4u are shown here on the left, with a " 1 and b " 3, for values of α equal 0 (black), 1 2 (blue), 1 (green), 2 (orange), and 8 (red). In particular, here x 1{2 "´3 4 . Proposition 1.4. P α pX; xq is continuous in α P r0, 8s in the following sense: Suppose that pα n q is any sequence in r0, 8q converging to α P r0, 8s, with β :" sup n α n and X P X β ; then P αn pX; xq Ñ P α pX; xq.
In view of parts (ii) and (iii) of Proposition 1.1, the condition X P X β in Proposition 1.4 is essential.
Let us now turn to the question of stability of P α pX; xq with respect to (the distribution of) X. First here, recall that one of a number of mutually equivalent definitions of the convergence in distribution, X n D ÝÑ nÑ8 X, of a sequence of r.v.'s X n to a r.v. X is the following: PpX n ě xq ÝÑ nÑ8 PpX ě xq for all real x such that PpX " xq " 0.
We shall also need the following uniform integrability condition: PpX " x˚q.
Note that in the case α " 0 the convergence (1.21) may fail to hold, not only for x " x˚, but for all real x such that PpX " xq ą 0. *** Let us now discuss matters of monotonicity of P α pX; xq in X, with respect to various orders on the mentioned set X of all real-valued r.v.'s X. Using the family of function classes H α , defined by (1.11), one can introduce a family of stochastic orders, say α`1 ď , on the set X by the formula where α P r0, 8s and X and Y are in X . To avoid using the term "order" with two different meanings in one phrase, let us refer to the relation α`1 ď as the stochastic dominance of order α`1, rather than the stochastic order of order α`1. In view of (1.11), it is clear that (1.22) so that, in the case when α " m´1 for some natural number m, the order α`1 ď coincides with the "m-increasing-convex" order ď m´icx as defined e.g. in [56, page 206]. In particular, A stochastic order, which is a "mirror image" of the order α`1 ď , but only for nonnegative r.v.'s, was presented by Fishburn in [20]; note [20, Theorem 2] on the relation with a "bounded" version of this order, previously introduced and studied in [18]. Denoting the corresponding Fishburn [20] order by ď α`1 , one has for nonnegative r.v.'s X and Y . However, as shown in this paper (recall Proposition 1.1) the condition of the nonnegativity of the r.v.'s is not essential; without it, one can either deal with infinite expected values or, alternatively, require that they be finite. The case when α is an integer was considered, in a different form, in [5]. One may also consider the order ď´1 α defined by the condition that X ď´1 α Y if and only if X and Y are nonnegative r.v.'s and F p´αq X with Qp¨;¨q as in (2.3), and the integral in (1.28) is understood as the Lebesgue integral with respect to the nonnegative Borel measure µ´1 X on r0, 1q defined by the condition that µ´1 X`r 0, pq˘" F´1 X ppq for all p P p0, 1q; cf. [29,31]. Note that F p´1q X ppq " F´1 X ppq. For nonnegative r.v.'s, the order ď´1 α`1 coincides with the order ď α`1 if α P t0, 1u; again see [29,31]. Even for nonnegative r.v.'s, it seems unclear how the orders ď α`1 and ď´1 α`1 relate to each other for positive real α ‰ 1; see e.g. the discussion following Proposition 1 in [29] and Note 1 on page 100 in [33].
The following theorem summarizes some of the properties of the tail probability bounds P α pX; xq established above and also adds a few simple properties of these bounds. Theorem 1.6. The following properties of the tail probability bounds P α pX; xq are valid.
Model-independence: P α pX; xq depends on the r.v. X only through the distribution of X. Monotonicity in X: P α p¨; xq is nondecreasing with respect to the stochastic dominance of order α`1: for any r.v. Y such that X α`1 ď Y , one has P α pX; xq ď P α pY ; xq. Therefore, P α p¨; xq is nondecreasing with respect to the stochastic dominance of any order γ P r1, α`1s; in particular, for any r.v. Y such that X ď Y , one has P α pX; xq ď P α pY ; xq. Monotonicity in α: P α pX; xq is nondecreasing in α P r0, 8s. Monotonicity in x: P α pX; xq is nonincreasing in x P R. Values: P α pX; xq takes only values in the interval r0, 1s. α-concavity in x: P α pX; xq´1 {α is convex in x if α P p0, 8q, and ln P α pX; xq is concave in x if α " 8. Stability in x: P α pX; xq is continuous in x at any point x P R -except the point x " x˚when p˚ą 0. Stability in α: Suppose that a sequence pα n q is as in Proposition 1.4. Then P αn pX; xq Ñ P α pX; xq. Stability in X: Suppose that α P p0, 8s and a sequence pX n q is as in Proposition 1.5. Then P α pX n ; xq Ñ P α pX; xq. Translation invariance: P α pX`c; x`cq " P α pX; xq for all real c. Consistency: P α pc; xq " P 0 pc; xq " Itc ě xu for all real c; that is, if the r.v. X is the constant c, then all the tail probability bounds P α pX; xq precisely equal the true tail probability PpX ě xq. Positive homogeneity: P α pκX; κxq " P α pX; xq for all real κ ą 0.
A property similar to the model-independence was called "neutrality" in [57, page 97].
2. An optimal three-way stable and three-way monotonic spectrum of upper bounds on quantiles Take any p P p0, 1q (2.1) and introduce the generalized inverse (with respect to x) of the bound P α pX; xq by the formula Q α pX; pq :" inf E α,X ppq " inf x P R : P α pX; xq ă p ( ; (2.2) where E α,X ppq is as in (1.18). In particular, in view of the equality in (1.6), which is a p1´pq-quantile of (the distribution of) the r.v. X; actually, QpX; pq is the largest one in the set of all p1´pq-quantiles of X. Thus, one has a monotonic spectrum of upper bounds, Q α pX; pq, on the quantile QpX; pq, ranging from the tightest bound, Q 0 pX; pq " QpX; pq, to the loosest one, Q 8 pX; pq, which latter is based on the exponential bound P 8 pX; xq " inf λą0 E e λpX´xq on PpX ě xq. Also, it is obvious from (2.2) that Q α pX; pq is nonincreasing in p P p0, 1q. (i) Q α pX; pq P R.
(v) If α P p0, 8s, then the function is the unique inverse to the continuous strictly decreasing function So, the function (2.6) too is continuous and strictly decreasing. (vi) If α P p0, 8s, then for any y P`´8, Q α pX; pq˘one has P α pX; yq ą p. (vii) If α P r1, 8s, then Q α pX; pq ą E X.  Some parts of Proposition 2.1 are illustrated in the picture here on the left, with graphs t`p, Q α pX; pq˘: 0 ă p ă 1u for a r.v. X " X a,b as in Example 1.3, with the same a " 1 and b " 3, and the same values of α, equal 0 (black), 1 2 (blue), 1 (green), 2 (orange), and 8 (red). One may compare this picture with the one in Example 1.3, having in mind that the function Q α pX;¨q is a generalized inverse to the function P α pX;¨q.
The definition (2.2) of Q α pX; pq is rather complicated, in view of the definition (1.5) of P α pX; xq. So, the following theorem will be useful, as it provides a more direct expression of Q α pX; pq; at that, one may again recall (2.3), concerning the case α " 0. where T α is as in (1.8) and B α pX; pqptq :" t ln E e X{t p for α " 8. (2.9) Proof of Theorem 2.3. The proof is based on the simple observation, following immediately from the definitions (1.9) and (2.9), that the dual level sets for the functionsÃ α pX; xq and B α pX; pq are the same: TÃ αpX ;xq ppq " T BαpX;pq pxq (2.10) for all α P p0, 8s, x P R, and p P p0, 1q, where TÃ α pX;xq ppq :" tt P T α :Ã α pX; xqptq ă pu and T BαpX;pq pxq :" tt P T α : B α pX; pqptq ă xu. Indeed, by (1.7) and (2.10),  [46,Proposition 1.5]. Correspondingly, the duality presented in the above proof of Theorem 2.3 is a generalization of the bilinear Legendre-Fenchel duality considered in [46]. Theorem 2.4. The following properties of the quantile bounds Q α pX; pq are valid.
Model-independence: Q α pX; pq depends on the r.v. X only through the distribution of X. Monotonicity in X: Q α p¨; pq is nondecreasing with respect to the stochastic dominance of order α`1: for any r.v. Y such that X α`1 ď Y , one has Q α pX; pq ď Q α pY ; pq. Therefore, Q α p¨; pq is nondecreasing with respect to the stochastic dominance of any order γ P r1, α`1s; in particular, for any r.v. Y such that X ď Y , one has Q α pX; pq ď Q α pY ; pq. Monotonicity in α: Q α pX; pq is nondecreasing in α P r0, 8s. Monotonicity in p: Q α pX; pq is nonincreasing in p P p0, 1q, and Q α pX; pq is strictly decreasing in p P rp˚, 1q X p0, 1q if α P p0, 8s. Finiteness: Q α pX; pq takes only (finite) real values. Concavity in p´1 {α or in ln 1 p : Q α pX; pq is concave in p´1 {α if α P p0, 8q, and Q 8 pX; pq is concave in ln 1 p . Stability in p: Q α pX; pq is continuous in p P p0, 1q if α P p0, 8s. Stability in X: Suppose that α P p0, 8s and a sequence pX n q is as in Proposition 1.5. Then Q α pX n ; pq Ñ Q α pX; pq. Stability in α: Suppose that α P p0, 8s and a sequence pα n q is as in Proposition 1.4. Then Q αn pX; pq Ñ Q α pX; pq. Translation invariance: Q α pX`c; pq " Q α pX; pq`c for all real c. Consistency: Q α pc; pq " c for all real c; that is, if the r.v. X is the constant c, then all the quantile bounds Q α pX; pq equal c. Sensitivity: Suppose here that X ě 0. If at that PpX ą 0q ą 0, then Q α pX; pq ą 0 for all α P p0, 8s; if, moreover, PpX ą 0q ą p, then Q 0 pX; pq ą 0. Positive homogeneity: Q α pκX; pq " κQ α pX; pq for all real κ ě 0. Subadditivity: Q α pX; pq is subadditive in X if α P r1, 8s; that is, for any other r.v. Y (defined on the same probability space as X) one has Q α pX`Y ; pq ď Q α pX; pq`Q α pY ; pq.
Convexity: Q α pX; pq is convex in X if α P r1, 8s; that is, for any other r.v. Y (defined on the same probability space as X) and any t P p0, 1q one has Q α`p 1´tqX`tY ; p˘ď p1´tqQ α pX; pq`tQ α pY ; pq.
The inequality Q 1 pX; pq ď Q 8 pX; pq, in other notations, was mentioned (without proof) in [50]; of course, this inequality is a particular, and important, case of the monotonicity of Q α pX; pq in α P r0, 8s. That Q α p¨; pq is nondecreasing with respect to the stochastic dominance of order α`1 was shown (using other notations) in [13] in the case α " 1.
The following strict monotonicity property complements the monotonicity property of Q α pX; pq in X stated in Theorem 2.4. Proposition 2.5. Suppose that a r.v. Y is stochastically strictly greater than X`which may be written as X st ă Y ; cf. (1.23)˘in the sense that X st ď Y and for any v P R there is some u P pv, 8q such that PpX ě uq ă PpY ě uq. Then Q α pX; pq ă Q α pY ; pq if α P p0, 8s.
This proposition will be useful in the proof of Proposition 2.6 below. Given the positive homogeneity, it is clear that the subadditivity and convexity properties of Q α pX; pq easily follow from each other. In the statements in Theorem 2.4 on these two mutually equivalent properties, it was assumed that α P r1, 8s. One may ask whether this restriction is essential. The answer to this question is "yes": Proposition 2.6. There are r.v.'s X and Y such that for all α P r0, 1q and all p P p0, 1q one has Q α pX`Y ; pq ą Q α pX; pq`Q α pY ; pq, so that Q α pX; pq is not subadditive (and, equivalently, not convex) in X.
It is well known (see e.g. [3,35,53]) that QpX; pq " Q 0 pX; pq is not subadditive in X; it could therefore have been expected that Q α pX; pq will not be subadditive in X if α is close enough to 0. In a quite strong and specific sense, Proposition 2.6 justifies such expectations. *** In Figure 1, the graphs of the quantile bounds Q α pX; pq as functions of α P p0, 20s are given for (a) p " 0.05 (left panel) and (b) p " 0.01 (right panel), for the case when X has the Gamma distribution with the scale parameter equal 1 and values 0.5, 1, 2, and 5 of the shape parameter (say a) -shown respectively in colors red, green, blue, and black. These graphs illustrate the general monotonicity properties of Q α pX; pq in α, X, and p stated in Theorem 2.4; recall here that the Gamma distribution is (i) stochastically increasing with respect to the shape parameter a and (ii) close to normality for large values of a. It is also seen that Q α pX; pq varies rather little in α, so that the quantile bounds Q α pX; pq are not too far from the corresponding true quantiles QpX; pq " Q 0 pX; pq; cf. the somewhat similar observation made in [37,Theorem 2.8.].
In fact, it can be shown that for small values of p the quantile bounds Q α pX; pq are relatively close to the true quantiles Q 0 pX; pq whenever the right tail of the distribution of X is light enough (depending on α) and regular enough. One possible formalization of this general thesis is provided by Proposition 2.7 below, which is based on [38,Theorem 4.2 and Remark 4.3]. We shall need pertinent definitions from that paper.
Take any r P p0, 8s. Given a positive function q on R, let us say that qpxq is like x´r if there is a positive twice differentiable function q 0 on R such that qpxq " xÑ8 q 0 pxq and lim xÑ8 q 0 pxqq 2 0 pxq q 1 0 pxq 2 " 1`1 r ; (2.11) as usual, we write f " g if f {g Ñ 1. For any real r ‰ 0, the second relation in (2.11) can be rewritten as lim xÑ8´1 pln q 0 q 1 pxq¯1 "´1 r , (2.11a) which successively implies 1 pln q0q 1 pxq "´x r , pln q 0 q 1 pxq "´r x , ln q 0 pxq "´r ln x, and hence qpxq " x´r`o p1q as x Ñ 8. (2.11b) In particular, given any r P p0, 8q, s P R, and C P p0, 8q, Cx´r ln s x, then qpxq is like x´r.
Moreover, if qpxq is like x´8, then for all r P p0, 8q one has qpxq " opx´rq as x Ñ 8.
(i) If the r.v. X is bounded from above -that is, x˚ă 8, then (ii) If PpX ě xq is like x´r for some r P pα, 8s, then x˚" 8, Q α pX; pq ÝÑ where Kpr, αq :" (2.14) c r,α :" Γpα`1qΓpr´αq Γprq r r α α pr´αq r´α , and Γ is the Gamma function, given by the formula Γpβq " ş 8 0 u β´1 e´u du for β ą 0. Kr,Α Graphs `α , Kpr, αq˘: 0 ă α ď minp10, 0.99rq, Kpr, αq ď 7.5 ( are shown here on the left: for r " 1 2 (red), r " 1 (orange), r " 2 (green), r " 5 (blue), r " 10 (purple), and r " 8 (black). It is seen that, if the (right) tail of the distribution of X is very heavy -that is, if r is comparatively small, then even for α " 1 the quantile bound Q α pX; pq is much greater (for small p) than the true quantile Q 0 pX; pq. However, if the tail is not very heavy -that is, if r is not so small, then the graph of Kpr, αq is very flat -that is, Q α pX; pq varies little with α (if p is small).
Concerning the relevance of the condition p Ó 0 in the asymptotic relations (2.12) and (2.13) in Proposition 2.7, note that small values of p are of particular importance in financial practice. Indeed, values of p commonly used with the so-called value-at-risk (VaR) measure -equal to the quantile Q 0 pX; pq, where X is the amount of financial loss -are 1% and 5% for one-day and two-week horizons, respectively [34].
As for the condition that X be bounded from above in part (i) of Proposition 2.7, it is obviously fulfilled, in particular, whenever the r.v. X takes only finitely many values, as is assumed e.g. in [3].
By (2.11b), if PpX ě xq is like x´r for some r.v. X and some real r ‰ 0, then necessarily r ą 0which is in accordance with the assumption r P p0, 8s made above concerning (2.11).
Usual statistical families of continuous distributions, including the normal, log-normal, gamma, beta, Student, and Pareto families, are covered by Proposition 2.7. More specifically, part (i) of Proposition 2.7 applies to the beta family of distributions, including the uniform distribution. Part (ii) of Proposition 2.7 applies (with r " 8) to the normal, log-normal, and gamma families, including the exponential family; the Student family (with r " d, where d is the number of degrees of freedom); and Pareto family`with r " β, assuming that PpX ě xq " pa{xq β for some β and a in p0, 8q and all real x ě a˘. However, note that under the condition that PpX ě xq is like x´r, as in part (ii) of Proposition 2.7, (2.13) is guaranteed to hold only for α P r0, rq. Also, in the case when PpX ě xq is like x´r for some r P p0, 8q, (2.13) cannot possibly hold for any α P pr, 8s -because then, by part (ii) of Proposition 1.1, (1.13), and (2.2), Q α pX; pq " 8. So, the general tendency is that, the lighter the right tail of the distribution of X, the wider is the range of values of α for which (2.13) holds. Moreover, it appears that, the lighter the right tail of the distribution of X, the closer is the constant Kpr, αq in (2.13) to 1 and, more generally, the closer is the quantile bound Q α pX; pq to the true quantile Q 0 pX; pq.
One can also show, using [38,Remark 3.13] or [40,Remark 1.4], that (2.13) will hold -with r " 8 and hence Kpr, αq " 1 -for all α P r0, 8s and usual statistical families of discrete distributions, including the Poisson, geometric, and, more generally negative binomial families -because the right tails of those distributions are light enough. On the other hand, (2.12) will hold for any distributions with a bounded support, including the binomial and hypergeometric distributions.
More examples of distributions to which Proposition 2.7 is applicable can be found in [39]. ***

Computation of P α pX; xq
The computation of P α pX; xq in the case α " 0 is straightforward, in view of the equality in (1.6). If x P rx˚, 8q, then the value of P α pX; xq is easily found by part (i) of Proposition 1.2. So, in the rest of this subsection it may be assumed that α P p0, 8s and x P p´8, x˚q.

(3.4)
Therefore and because λ max,α ă 8, the minimization of A α pX; xqpλq in λ in (3.4) in order to compute the value of P α pX; xq can be done effectively if α P r1, 8s, because in this case A α pX; xqpλq is convex in λ. At that, the positive-part moments E`1`λpX´xq{αq ὰ , which express A α pX; xqpλq for α P p0, 8q in accordance with (1.3), can be efficiently computed using formulas in [44]; cf. e.g. [43, Section 3.2.3]. Of course, for specific kinds of distributions of the r.v. X, more explicit expressions for the positive-part moments can be used.
In the remaining case, when α P p0, 1q, the function λ Þ Ñ A α pX; xqpλq cannot in general be "convexified" by any monotonic transformations in the domain and/or range of this function, and the set of minimizing values of λ does not even have to be connected, in the following rather strong sense: Proposition 3.1. For any α P p0, 1q, p P p0, 1q, and x P R, there is a r.v. X (taking three distinct values) such that P α pX; xq " p and the infimum inf λPp0,8q in (1.5) is attained at precisely two distinct values of λ P p0, 8q.  Let X be a r.v. taking values´2 7 11 ,´1, 2 with probabilities 1 4 , 1 4 , 1 2 ; then x˚" 2. Also let α " 1 2 and x " 0, so that x P p´8, x˚q, and then let λ max be as in (3.3) with y " x˚" 2, so that here λ max " 3 4 . Then the minimum of A α pX; 0qpλq over all real λ ě 0 equals ? 3 2 and is attained at each of the two points, λ " 11 54 and λ " 1 2 , and only at these two points. The graph `λ , A 1{2 pX; 0qpλq˘: 0 ď λ ď λ max ( is shown here on the left.

Computation of Q α pX; pq
(3.5) moreover, for any s P R andp P pp, 1q, where x˚and p˚are as in (1.16). On the other hand, if conditions (3.9) hold then B 8 pX; pqptq is strictly increasing in t ą 0 and hence inf tPTα " inf tPp0,8q in (2.8) is not attained; rather, For instance, in the case when α " 0.5, p " 0.05, and X has the Gamma distribution with the shape and scale parameters equal to 2.5 and 1, respectively, Proposition 3.3 yields t min ą 4.01 (using p " 0.095) and t max ă 6.45.
When α " 0, the quantile bound Q α pX; pq is simply the quantile QpX; pq, which can be effectively computed by formula (2.3), since the tail probability PpX ą xq is monotone in x. Next, as was noted in the proof of Theorem 2.4, B α pX; pqptq is convex in t when α P r1, 8s, which provides for an effective computation of Q α pX; pq by formula (2.8).
Therefore, it remains to consider the computation -again by formula (2.8) -of Q α pX; pq for α P p0, 1q. In such a case, as in Subsection 3.1, one can use an interval method. As soon as the minimizing values of t are bracketed as in (3.6), one can partition the finite interval rt min , t max s into a large number of small subintervals ra, bs with t min ď a ă b ď t max . For each such subinterval, as b´a Ñ 0, uniformly over all subintervals ra, bs of the interval rt min , t max s. Thus, one can effectively bracket the value Q α pX; pq " inf tPR B α pX; pqptq; this same approach will work, and perhaps may be useful, for α P r1, 8q as well. *** If α P p1, 8q then, by part (ii) of Proposition 3.6 and part (i) of Proposition 3.3, the set argmin tPR B α pX; pqptq is a singleton one; that is, there is exactly one minimizer t P R of B α pX; pqptq.
If α " 1 then B α pX; pqptq " B 1 pX; pqptq is convex, but not strictly convex, in t, and the set argmin tPR B α pX; pqptq of all minimizers of B α pX; pqptq in t coincides with the set of all p1´pq-quantiles of X, as mentioned at the conclusion of the derivation of the identity (3.10). Thus, if α " 1, then the set argmin tPR B α pX; pqptq may in general be, depending on p and the distribution of X, a nonzerolength closed interval. Finally, if α P p0, 1q then, in general, the set argmin tPR B α pX; pqptq does not have to be connected: For any α P p0, 1q, p P p0, 1q, and x P R, there is a r.v. X (taking three distinct values) such that Q α pX; pq " x and the infimum inf tPTα " inf tPR in (2.8) is attained at precisely two distinct values of t.
Proposition 3.1 is illustrated by the following example, which is obtained from Example 3.2 by the same duality (2.10).
Example 3.5. As in Example 3.2, let α " 1 2 and let X be a r.v. taking valueś 27 11 ,´1, 2 with probabilities 1 4 , 1 4 , 1 2 . Also let p " Then the minimum of B α pX; pqptq over all real t equals 0 and is attained at each of the two points, t "´2 7 11 and t "´1, and only at these two points. The graph `t , B 1{2`X ; ? 3 2˘p tq˘: ( is shown here on the left. The minimizing values of t here,´2 7 11 and´1, are related with the minimizing values of λ in Example 3.2, 11 54 and 1 2 , by the mentioned formula t " x´α{λ (here with x " 0 and α " 1 2 ).
It follows that the right derivative of the convex function t Þ Ñ t`}pX´tq`} 1 {p at any point t P R is 1´PpX ą tq{p, which, by (2.3), is ď 0 if t ă QpX; pq and ą 0 if t ą QpX; pq. Hence, QpX; pq is a minimizer in t P R of t`}pX´tq`} 1 {p, and thus (3.10) follows by (2.8). It is also seen now that any p1´pq-quantile of X is a minimizer in t P R of t`}pX´tq`} 1 {p as well, and QpX; pq is the largest of these minimizers.
As was shown in [53], the expression for Q 1 pX; pq in (3.10) can be rewritten as a conditional expectation: Q 1 pX; pq " QpX; pq`E`X´QpX; pqˇˇX ě QpX; pq, U ě δ˘" E`X|X ě QpX; pq, U ě δ˘, (3.11) where U is any r.v. which independent of X and uniformly distributed on the interval r0, 1s, δ :" δpX; pq :" d ItX " QpX; pqu, and d is any real number in the interval r0, 1s such that P`X ě QpX; pq˘´p " P`X " QpX; pq˘d; such a number d always exists. Thus, the r.v. U is used to split the possible atom of the distribution of X at the quantile point QpX; pq in order to make the randomized tail probability P`X ě QpX; pq, U ě δ˘exactly equal to p. Of course, in the absence of such an atom, one can simply write Q 1 pX; pq " QpX; pq`E`X´QpX; pqˇˇX ě QpX; pq˘" E`X|X ě QpX; pq˘. (3.12) However, as pointed out in [52,53], a variational formula such as (2.8) has a distinct advantage over such ostensibly explicit formulas as (3.10) and (3.11), since (2.8) allows for incorporation into specialized optimization problems, with additional restrictions, say on the distribution of X; cf. e.g. [52,Theorem 2].
(i) If α P r1, 8s then B α pX; pqptq is convex in t P T α .
(ii) If α P p1, 8q then B α pX; pqptq is strictly convex in t P p´8, x˚˚s X R.
(iii) B 8 pX; pqptq is strictly convex in t P ts P p0, 8q : E e X{s ă 8u unless PpX " cq " 1 for some c P R.
Suppose at this point that α P r1, 8q. By part (i) of Proposition 3.3 (stated in Section 3), the minimum-attainment set argmin tPR B α pX; pqptq :" tt P R : B α pX; pqptq " Q α pX; pqu (3.14) is nonempty and bounded. Also, this set is closed, by the continuity of B α pX; pqptq in t P R. Therefore, the definition the largest p1´pq-quantile of X, as stated at the end of the paragraph containing (3.10) and its proof. Thus, indeed the representation (3.10) is extended to all α P r1, 8q: Further properties of α´1 QpX; pq are presented in Proposition 3.7. Suppose that α P r1, 8q. Then the following statements are true.
(b) Suppose here that α P p1, 8q and p P pp˚, 1q. Then α´1 QpX; pq " t α,p , where t α,p is the only root t P p´8, x˚q of the equation α´1 P pX; tq " p and, again for t P p´8, x˚q, Furthermore, for all t P p´8, x˚q one has α´1 P pX; tq ď PpX ą tq ď P 0 pX; tq, and also α´1 P pX; tq Ñ PpX ą tq as α Ó 1`assuming that X P X α for some α P p1, 8q˘; in addition, α´1 P pX; tq ă PpX ą tq for all t P p´8, x˚˚q.
By (3.21), α´1 QpX; pq is a lower bound on the true p1´pq-quantile QpX; pq of X. Therefore and in view of part (i) of Proposition 3.7, α´1 QpX; pq may be referred to as the lower pα´1, 1´pq-quantile of the r.v. X.  ( for a r.v. X " X a,b as in Examples 1.3 and 2.2, with the same a " 1 and b " 3, and the values of α equal 1 (green), 2 (orange), and 5 (red). Note that here x˚" b " 3 and x˚˚"´a "´1.
*** One may conclude this section by an obvious but oftentimes rather useful observation that -even when a minimizing value of λ or t in formulas (1.5), (1.7), or (2.8) is not identified quite perfectly -one still obtains, by those formulas, an upper bound on P α pX; xq or Q α pX; pq and hence on the true tail probability PpX ě xq or the true quantile QpX; pq, respectively.

Implications for risk control/inequality modeling in finance/economics
In financial literature -see e.g. [35,53,24], the quantile bounds Q 0 pX; pq and Q 1 pX; pq are known as the value-at-risk and conditional value-at-risk, denoted as VaR p pXq and CVaR p pXq, respectively: Q 0 pX; pq " VaR p pXq and Q 1 pX; pq " CVaR p pXq; (4.1) here, X is interpreted as a priori uncertain potential loss. The value of Q 1 pX; pq is also known as the expected shortfall (ES) [2], average value-at-risk (AVaR) [47], and expected tail loss (ETL) [26]. As indicated in [53], at least in the case when there is no atom at the quantile point QpX; pq, the quantile bound Q 1 pX; pq is also called the "mean shortfall" [28], whereas the difference Q 1 pX; pq´QpX; pq is referred to as "mean excess loss" [17,6]. Greater values of α correspond to greater sensitivity to risk; cf. e.g. [19]. For instance, let X and Y denote the potential losses corresponding to two different investments portfolios. Suppose that there are mutually exclusive events E 1 and E 2 and real numbers p˚P p0, 1q and δ P p0, 1q such that (i) PpE 1 q " PpE 2 q " p˚{2, (ii) the loss of either portfolio is 0 if the event E 1 Y E 2 does not occur, (iii) the loss of the X-portfolio is 1 if the event E 1 Y E 2 occurs, and (iv) the loss of the Y -portfolio is 1´δ if the event E 1 occurs, and it is 1`δ if the event E 2 occurs. Thus, the r.v. X takes values 0 and 1 with probabilities 1´p˚and p˚, and the r.v. Y takes values 0, 1´δ, and 1`δ with probabilities 1´p˚, p˚{2, and p˚{2, respectively. So, E X " E Y , that is, the expected losses of the two portfolios are the same. Clearly, the distribution of X is less dispersed than that of Y , both intuitively and also in the formal sense that X α`1 ď Y for all α P r1, 8s. Therefore, everyone will probably say that the Y -portfolio is riskier than the X-portfolio. However, for any p P pp˚, 1q it is easy to see, by (2.3), that Q 0 pX; pq " 0 " Q 0 pY ; pq and hence, in view of (3.10), Using also the continuity of Q α p¨; pq in p, as stated in Theorem 2.4, one concludes that the Q 1 p¨; pq " CVaR p p¨q risk value of the riskier Y -portfolio is the same as that of the less risky X-portfolio for all p P rp˚, 1q. Such indifference (which may also be referred to as insensitivity to risk) may generally be considered "an unwanted characteristic" [23, pages 36, 48].
The above example is illustrated in Figure 2, for p˚" 0.1 and δ " 0.6. It is seen that the sensitivity of the measure Q α p¨; pq to risk`reflected especially by the gap between the red and blue lines for p P rp˚, 1q " r0.1, 1q˘increases from the zero sensitivity when α " 1 to an everywhere positive sensitivity when α " 2 to an everywhere greater positive sensitivity when α " 5. *** Based on an extensive and penetrating discussion of methods of measurement of market and nonmarket risks, Artzner et al [3] concluded that, for a risk measure to be effective in risk regulation and management, it has to be coherent, in the sense that it possess the translation invariance, subadditivity, positive homogeneity, and monotonicity properties. In general, a risk measure, sayρ, is a mapping of a linear space of real-valued r.v.'s on a given probability space into R. The probability space (say Ω) was assumed to be finite in [3]. More generally, one could allow Ω to be infinite, and then it is natural to allowρ to take values˘8 as well. In [3], the r.v.'s (say Y ) in the argument of the risk measure were called risks but at the same time interpreted as "the investor's future net worth". Then the translation invariance was defined in [3] as the identityρpY`rtq "ρpY q´t for all r.v.'s Y and real numbers t, where r is a positive real number, interpreted as the rate of return. We shall, however, follow Pflug [35], who considers a risk measure (say ρ) as a function of the potential cost/loss, say X, and then defines the translation invariance of ρ, quite conventionally, as the identity ρpX`cq " ρpXq`c for all r.v.'s X and real numbers c. The approaches in [3] and [35] are equivalent to each other, and the correspondence between them can be given by the formulas ρpXq " rρpY q " rρp´Xq, X "´Y , and c "´rt. The positive homogeneity as defined in [3] can be stated as the identity ρpλXq " λρpXq for all r.v.'s X and real numbers λ ě 0.
Corollary 4.1. For each α P r1, 8s, the quantile bound Q α p¨; pq is a coherent risk measure.
This follows immediately from Theorem 2.4. The usually least trivial of the four properties characterizing the coherence is the subadditivity of a risk measure -which, in the presence of the positive homogeneity, is equivalent to the convexity, as was pointed out earlier in this paper. As is well known and also discussed above, the value-atrisk measure VaR p pXq is translation invariant, positive homogeneous, and monotone (in X), but it fails to be subadditive. Quoting [53, page 1458]: "The coherence of [ CVaR p pXq] is a formidable advantage not shared by any other widely applicable measure of risk yet proposed." Corollary 4.1 above addresses this problem by providing an entire infinite family of coherent risk measures, indexed by α P r1, 8s, including CVaR p pXq " Q 1 pX; pq just as one member of the family.
Theorem 2.4 also provides additional monotonicity and other useful properties of the spectrum of risk measures Q α p¨; pq. The terminology we use to name some of these properties differs from the corresponding terminology used in [3]. Namely, what we referred to as the "sensitivity" in Theorem 2.4 corresponds to the "relevance" in [3]. Also, in the present paper the "model-independence" means that the risk measure depends on the potential loss only through the distribution of the loss, rather than on the way to model the "states of nature", on which the loss may depend. In contrast, in [3] a measure of risk is considered "model-free" if it does not depend, not only on modeling the "states of nature", but, to a possibly large extent, on the distribution of the loss. The "model-independence" property is called "law-invariance" in [21, Section 12.1.2], where the consistency property was referred to as "constancy". An example of such a "model-free" risk measure is given by the Securities and Exchange Commission (SEC) rules, described e.g. in [3,Subsection 3.2]; this measure of risk depends only on the set of all possible representations of the investment portfolio in question as a portfolio of long call spreads, that is, pairs of the form (a long call, a short call). If a measure of risk is not "model-free", then it is called "model-dependent" in [3]. *** Yitzhaki [58] utilized the Gini mean difference -which had prior to that been mainly used as a measure of economic inequality -to construct, somewhat implicitly, a measure of risk; this approach was further developed in [14,11]. If (say) a r.v. X is thought of as the income of randomly selected person in a certain state, then the Gini mean difference can be defined by the formula whereX is an independent copy of X and H : r0, 8q Ñ R is a measurable function, usually assumed to be nonnegative and such that Hp0q " 0; clearly, given the function H, the Gini mean difference G H pXq depends only on the distribution of the r.v. X. So, if Hpuq is considered, for any u P r0, 8q, as the measure of inequality between two individuals with incomes x and y such that |x´y| " u, then the Gini mean difference E Hp|X´X|q is the mean H-inequality in income between two individuals selected at random (and with replacement, thus independently of each other). The most standard choice for H is the identity function id, so that Hpuq " idpuq " u for all u P r0, 8q. Based on the measure-of-inequality G H , one can define the risk measure where now the r.v. X is interpreted as the uncertain loss on a given investment, with the term G H pXq " E Hp|X´X|q then possibly interpreted as a measure of the uncertainty. Clearly, when there is no uncertainty, so that the loss X is in fact a nonrandom real constant, then the measure G H pXq of the uncertainty is 0, assuming that Hp0q " 0. If X " N pµ, σ 2 q (that is, X is normally distributed with mean µ and standard deviation σ ą 0) and H " κ id for some positive constant κ, then R H pXq " µ`2 κ ? π σ, a linear combination of the mean and the standard deviation, so that in such a case we find ourselves in the realm of the Markowitz mean-variance risk-assessment framework.
It is assumed that R H pXq is defined when both expected values in the last expression in (4.2) are defined and are not infinite values of opposite signs -so that these two expected values could be added, as needed in (4.2).
It is clear that R H pXq is translation-invariant. Moreover, R H pXq is convex in X if the function H is convex and nondecreasing. Further, if H " κ id for some positive constant κ, then R H pXq is also positive-homogeneous.
It was shown in [58], under an additional technical condition, that R H pXq is nondecreasing in X with respect to the stochastic dominance of order 1 if H " 1 2 id. Namely, the result obtained in [58] is that if X st ď Y and the distribution functions F and G of X and Y are such that F´G changes sign only finitely many times on R, then R 1 2 id pXq ď R 1 2 id pY q. A more general result was obtained in [14], which can be stated as follows: in the case when the function H is differentiable, R H pXq is nondecreasing in X with respect to the stochastic dominance of order 1 if and only if |H 1 | ď 1 2 . Cf. also [11]. The proof in [14] was rather long and involved; in addition, it used a previously obtained result of [27]. Here we are going to give (in Appendix A) a very short, direct, and simple proof of the more general In Proposition 4.2, it is not assumed that H ě 0 or that Hp0q " 0. Of course, if H is differentiable, then the 1 2 -Lipschitz condition is equivalent to the condition |H 1 | ď 1 2 in [14]. The risk measure R H pXq was called mean-risk (M-R) in [11]. It follows from [14] or Proposition 4.2 above that the risk measure R κ id pXq is coherent for any κ P r0, 1 2 s. In fact, based on Proposition 4.2, one can rather easily show more: Proposition 4.3. The risk measure R H pXq is coherent if and only if H " κ id for some κ P r0, 1 2 s.
It is possible to indicate a relation -albeit rather indirect -of the risk measure R H pXq, defined in (4.2), with the quantile bounds Q α pX; pq. Indeed, introducê assuming E X exists in R. By (2.8)-(2.9),Q α pX; pq is another majorant of Q α pX; pq, obtained by using t " E X in (2.8) as a surrogate of the minimizing value of t.
is somewhat similar to the Gini mean-difference term E Hp|X´X|q, at least when α " 1 and (the distribution of) the r.v. X is symmetric about its mean.
Moreover, if the distribution of X´E X is symmetric and stable with index γ P p1, 2s, then One may want to compare the two considered kinds of coherent measures of risk/inequality, R κ id pXq for κ P r0, 1 2 s and Q α pX; pq for α P r1, 8s and p P p0, 1q. It appears that the latter measure is more flexible, as it depends on two parameters (α and p) rather than one just one parameter (κ). Moreover, as Proposition 2.7 shows, rather generally Q α pX; pq retains a more or less close relation with the quantile Q 0 pX; pq -which, recall, is the widely used value-at-risk (VaR). On the other hand, recall here that, in contrast with the VaR, Q α pX; pq is coherent for α P r1, 8s. However, both of these kinds of coherent measures appear useful, each in its own manner, representing two different ways to express risk/inequality. [54]. Let X E be a convex cone of real-valued r.v. X P X with a finite mean E X such that X E contains all real constants.

Formulas (4.2) and (4.3) can be considered special instances of the general relation between risk measures and measures of inequality established in
Largely following [54], let us say a coherent real-valued risk measure R : X E Ñ p´8, 8s is strictly expectation-bounded if RpXq ą E X for all X P X E .`Note that here the r.v. X represents the loss, whereas in [54] it represents the gain; accordingly, X in this paper corresponds to´X in [54]; also, in [54] the cone X E was taken to be the space L 2 .˘In view of Theorem 2.4 and part (vii) of Proposition 2.1, it follows that Q α pX; pq is a coherent and strictly expectation-bounded risk measure if α P r1, 8s. Also (cf. [54, Definition 1 and Proposition 1]), let us say that a mapping D : X E Ñ r0, 8s is a deviation measure if D is subadditive, positive-homogeneous, and nonnegative with DpXq " 0 if and only if PpX " cq " 1 for some real constant c; here X is any r.v. in X E . Next provide a one-to-one correspondence between all coherent strictly expectation-bounded risk measures R : X E Ñ p´8, 8s and all upper-range dominated deviation measures D : X E Ñ r0, 8s.
In particular, it follows that the risk measureQ α p¨; pq, defined by formula (4.3), is coherent for all α P r1, 8s and all p P p0, 8q. It also follows that X Þ Ñ Q α pX´E X; pq is a deviation measure. As was noted,Q α pX; pq is a majorant of Q α pX; pq. In contrast with Q α pX; pq, in generalQ α pX; pq will not have such a close hereditary relation with the true quantile Q 0 pX; pq as e.g. the ones given in Proposition 2.7. For instance, if PpX ě xq is like x´8 then, by (2.13)-(2.14), Q α pX; pq " pÓ0 Q 0 pX; pq for each α P r0, 8s, whereasQ 8 pX; pq " 8 for all real p ą 0. On the other hand, in distinction with the definition (4.3) ofQ α pX; pq, the expression (2.8) for Q α pX; pq requires minimization in t; however, that minimization will add comparatively little to the complexity of the problem of minimizing Q α pX; pq subject to a usually large number of restrictions on the distribution of X; cf. again e.g. [52,Theorem 2].
One may also consider the following modification ofQ α pX; pq, which is still a majorant of Q α pX; pq, but is closer`thanQ α pX; pq is˘to the true quantile Q 0 pX; pq -at least when p P p0, 1q is close enough to 1`recall part (ii) of Proposition 3.7˘: The risk measureQ ă α p¨; pq is coherent and strictly expectation-bounded given the condition α P r1, 8s, which will be assumed in this paragraph. The proof of the translation invariance, positive homogeneity, and subadditivity properties ofQ ă α p¨; pq is almost the same as the proof of these properties of Q α p¨; pq, listed in Theorem 2.4. Then, to prove the monotonicity of Q α p¨; pq (with respect to the order st ď), it is enough, in view of the subadditivity ofQ ă α p¨; pq and as in the proof of [54,Theorem 2], to show that X ď 0 impliesQ ă α pX; pq ď 0, which obtains indeed, becausê Q ă α pX; pq ďQ α pX; pq andQ α pX; pq ďQ α p0; pq " 0 if X ď 0. Finally,Q ă α p¨; pq is strictly expectationbounded, because it majorizes Q α p¨; pq, which is strictly expectation-bounded, as noted. *** Recalling (1.28) and following [31,32,33], one may also consider´F p´αq X ppq as a measure of risk. Here one will need the following semigroup identity, given in [31, (8a)] (cf. e.g. [38, Remark 3.7]): whenever 0 ă ν ă α ă 8. The following proposition is well known.
Using (4.6) with ν " 2, α`1 in place of α, and´X in place of X together with Proposition 4.4, one has´F p´α´1q X ppq " 1 Γpα´1q for any α P p1, 8q. Since CVaR u pXq is a coherent risk measure, it now follows that, as noted in [32], F p´α´1q X ppq is a coherent risk measure as well, again for α P p1, 8q; by (4.7), this conclusion will hold for α " 1. However, one should remember that the expression F p´αq X ppq was defined only when the r.v. X is nonnegative (and otherwise some of the crucial considerations above will not hold). Thus, the risk measure´F p´α´1q X ppq is defined only if X ď 0 almost surely. In view of (4.9), this risk measure is a mixture of the coherent risk measures CVaR u pXq and thus a member of the general class of the so-called spectral risk measures [1], which are precisely the mixtures, over the values u P p0, 1q, of the risk measures CVaR u pXq; thus, all spectral risk measures are automatically coherent. However, in general such measures will lack such an important variational representation as the one given by formula (2.8) for the risk measure Q α pX; pq. Of course, for any "mixing" nonnegative Borel measure µ on the interval p0, 1q and the corresponding spectral risk measure CVaR µ pXq :" ż p0,1q CVaR u pXq µp duq, one can write CVaR µ pXq " ż p0,1q inf tPR`t`1 u }pX´tq`} 1˘µ p duq, (4.10) in view of (4.1) and (2.8)-(2.9). However, in contrast with (2.8), the minimization (in t P R) in (4.10) needs in general to be done for each of the infinitely many values of u P p0, 1q. If the r.v. X takes only finitely many values, then the expression of CVaR µ pXq in (4.10) can be rewritten as a finite sum, so that the minimization in t P R will be needed only for finitely many values of u; cf. e.g. the optimization problem on page 8 in [32].
On the other hand, one can of course consider arbitrary mixtures in p P p0, 1q and/or α P r1, 8q of the risk measures Q α pX; pq. Such mixtures will automatically be coherent. Also, all mixtures of the measures Q α pX; pq in p will be nondecreasing in α, and all mixtures of Q α pX; pq in α will be nonincreasing in p.
Deviation measures such as the ones studied in [54] and discussed in the paragraph containing (4.4) can be used as measures of economic inequality if the r.v. X models, say, the random income/wealth -defined as the income/wealth of an (economic) unit chosen at random from a population of such units. Then, according to the one-to-one correspondence given by (4.4), coherent risk measures R translate into deviation measures D, and vice versa.
However, the risk measures Q α p¨; pq themselves can be used to express certain aspects of economic inequality directly, without translation into deviation measures. For instance, if X stands for the random wealth then the statement Q 1 pX; 0.01q " 30 E X formalizes the common kind of expression "the wealthiest 1% own 30% of all wealth", provided that the wealthiest 1% can be adequately defined, say as follows: there is a threshold wealth value t such that the number of units with wealth greather than or equal to t is 0.01N , where N is the number of units in the entire population. Then (cf. (3.12)) 0.01 N Q 1 pX; 0.01q " 0.01 N EpX|X ě tq " N E X ItX ě tu " 0.30 N E X, whence indeed Q 1 pX; 0.01q " 30 E X. Similar in spirit expressions of economic inequality in terms of Q α pX; pq can be provided for all α P p0, 8q. For instance, suppose now that X stands for the annual income of a randomly selected household, whereas x is a particular annual household income level in question. Then, in view of (2.8)-(2.9), the inequality Q α pX; pq ě x means that for any (potential) annual household income level t less than the maximum annual household income level x˚, X in the population, the conditional α-mean E`pX´tq α |X ą t˘1 {α of the excess pX´tq`of the random income X over t is no less than`p PpXątq˘1 {α times the excess px´tq`of the income level x over t. Of course, the conditional α-mean E`pX´tq α |X ą t˘1 {α is increasing in α. Thus, using the measure Q α pX; pq of economic inequality with a greater value of α means treating high values of the economic variable X in a more progressive/sensitive manner. One may also note here that the above interpretation of the inequality Q α pX; pq ě x is a "synthetic" statement in the sense that is provides information concerning all values of potential interest of the threshold annual household income level t. Not only the upper bounds Q α pX; pq on the quantile QpX; pq, but also the upper bounds P α pX; xq on the tail probability PpX ě xq may be considered measures of risk/inequality. Indeed, if X is interpreted as the potential loss, then the tail probability PpX ě xq corresponds to the classical safety-first (SF) risk measure; see e.g. [55,22].
Using variational formulas -of which formulas (1.5), (1.7), and (2.8) are examples -to define or compute measures of risk is not peculiar to the present paper. Indeed, as mentioned previously, the special case of (2.8) with α " 1 is the well-known variational representation (3.10) of CVaR, obtained in [52,35,53]. The risk measure given by the SEC rules [3, Subsection 3.2], also mentioned before, is another example where the calculations are done, in effect, according to a certain minimization formula, which is somewhat implicit and complicated in that case.
One can now list some of the advantages of the risk/inequality measures P α pX; xq and Q α pX; pq: • P α pX; xq and Q α pX; pq are three-way monotonic and three-way stable -in α, p, and X.
• The monotonicity in X is graded continuously in α, resulting in various, controllable degrees of sensitivity of P α pX; xq and Q α pX; pq to financial risk/economic inequality. • x Þ Ñ P α pX; xq is the tail-function of a certain probability distribution. • Q α pX; pq is a p1´pq-percentile of that probability distribution.
• For small enough values of p the quantile bounds Q α pX; pq are close enough to the correspond-ing true quantiles QpX; pq provided that the right tail of the distribution of X is light enough and regular enough, depending on α. • P α pX; xq and Q α pX; pq are solutions to mutually dual optimizations problems, which can be comparatively easily incorporated into more specialized optimization problems, with additional restrictions, say on the distribution of the random variable X. • P α pX; xq and Q α pX; pq are effectively computable.
• Even when the corresponding minimizer is not identified quite perfectly -one still obtains an upper bound on the risk/inequality measures P α pX; xq or Q α pX; pq. • Optimal upper bounds on P α pX; xq and hence on Q α pX; pq over important classes of r.v.'s X represented (say) as sums of independent r.v.'s X i with restrictions on moments of the X i 's and/or sums of such moments can be given, as is done e.g. in [39,7,8,41,42,43]. • The quantile bounds Q α pX; pq with α P r1, 8s constitute a spectrum of coherent measures of financial risk and economic inequality. • The r.v.'s X of which the measures P α pX; xq and Q α pX; pq are taken are allowed to take values of both signs. In particular, if, in a context of economic inequality, X is interpreted as the net amount of assets belonging to a randomly chosen economic unit, then a negative value of X corresponds to a unit with more liabilities than paid-for assets. Similarly, if X denotes the loss on a financial investment, then a negative value of X will obtain when there actually is a net gain.
Some of these advantages, and especially their totality, appear to be unique to the bounds proposed here.
Acknowledgment. I am pleased to thank Emmanuel Rio for the mentioned communication [50], which also included a reference to [24] and in fact sparked the study presented here. Parts (ii) and (iii) follow by (1.7)-(1.9). Indeed, EpX´tq ὰ ě E X ὰ {2 pα´1q`´| t| α " 8 for all real x and t if E X ὰ " 8, and E e pX´xq{t " e´x {t E e X{t " 8 for all real x and t ą 0 if E e λX " 8 for all real λ ą 0.
Thus, part (iv) of Proposition 1.1 is proved. The proof of part (v) is rather similar to that of part (iv). Assume indeed that α " 8 and E e λ0X ă 8 for some real λ 0 ą 0. Then P 8 pX; xq ď E e λ0pX´xq " e´λ 0x E e λ0X Ñ 0 as x Ñ 8. Since P 8 pX; xq ě 0 for all real x, one indeed has P 8 pX; xq Ñ 0 as x Ñ 8.
As for the proof of the statement that P α pX; xq Ñ 1 as x Ñ´8 for α " 8, it is the same as the corresponding proof for α P p0, 8q.
Thus, part (v) of Proposition 1.1 is proved as well.
Proof of Proposition 1.2.
(i) Let us first verify part (i). For α " 0, this follows immediately from the equality in (1.6) and the definitions in (1.16).
Take then any α P p0, 8s. Take indeed any x P rx˚, 8q.
(iii) Concerning part (iii) of Proposition 1.2, consider first the case α P p0, 8q. Then, by (1.7), the function p´8, x˚s X R Q x Þ Ñ P α pX; xq´1 {α P r0, 8s is the pointwise supremum, in t P p´8, x˚q, of the family of continuous convex functions p´8, x˚s X R Q x Þ Ñ px´tq} pX´tq`}α P r0, 8s. So, the function (A.1) is convex. It is also finite on the interval p´8, x˚q, by part (ii) of Proposition 1.2. So, the function (A.1) is continuous on p´8, x˚q. Moreover, this function is lower-semicontinuous and nondecreasing, and hence continuous at the point x˚, in the case when x˚P R. Thus, the function (A.1) is continuous on the entire set p´8, x˚sXR, with respect to the natural topologies on p´8, x˚s X R and r0, 8s.
The case α " 8 is considered quite similarly. Here, instead of (A.1), one works with the function p´8, x˚s X R Q x Þ Ñ´ln P α pX; xq P p´8, 8s, which is the pointwise supremum, in real λ ą 0 such that E e λX ă 8, of the family of continuous convex (in fact, affine) functions R Q x Þ Ñ´ln E e λpX´xq " λx´ln E e λX P p´8, 8s.`Actually, the values of the latter family of functions are all real, for λ ą 0 such that E e λX ă 8, whereas all the values of the function (A.2) are in r0, 8s; however, here we take the union, p´8, 8s, of the sets R and r0, 8s as an interval which is guaranteed to contain all possible values of all the convex functions under consideration.˘Here we also use the standard conventions ln 0 :"´8 and e´8 :" 0; concerning the continuity of functions with values in the set p´8, 8s, we use the natural topology on this set. (iv) Let us now turn to part (iv) of Proposition 1.2. Consider first the case α P p0, 8q. Then, since the map r0, 8s Q r Þ Ñ r´α is continuous, it follows by part (iii) of Proposition 1.2 that P α pX; xq is indeed continuous in x P p´8, x˚q and left-continuous in x at x˚if x˚P R. The case α " 8 is quite similar; here, instead of the map r0, 8s Q r Þ Ñ r´α, one should use the continuous map p´8, 8s Q r Þ Ñ e´r.
(v) That the function R Q x Þ Ñ P α pX; xq is left-continuous follows immediately from parts (iv) and (i) if α P p0, 8s, and from the equality in (1.6) if α " 0.
On the other hand, if x P pE X, 8q then A 8 pX; xqpλq " E e λpX´xq ă E e 0pX´xq " 1 for all λ in a right neighborhood of 0 -because the right derivative of E e λpX´xq in λ at λ " 0 is EpX´xq ă 0; therefore, P 8 pX; xq ă 1, and so, again by (1.13), P α pX; xq ă 1 for all α P r1, 8s.
This completes the proof of part (vii) of Proposition 1.2.
Let us show that x α " x˚if and only if p˚" 1. By the definition of x α in (1.17) and (1.15), P α pX; xq " 1 for all x P p´8, x α q. So, by part (v) of Proposition 1.2 and the inequality x α ă 8 in part (vi) of Proposition 1.2, x α ą´8 ùñ P α pX; x α q " lim xÒxα P α pX; xq " 1.
If now x α " x˚, then x α ą´8 by the definition of x˚in (1.16); so, by part (i) of Proposition 1.2 and (A.3), p˚" P α pX; x˚q " P α pX; x α q " 1, which proves the implication x α " x˚ùñ p˚" 1. Vice versa, suppose that p˚" 1. Then necessarily x˚P R. Moreover, by part (i) of Proposition 1.1 and part (i) of Proposition 1.2, for all x P p´8, x˚s one has P α pX; xq ě P α pX; x˚q " p˚" 1, so that E α p1q Ď px˚, 8q and hence x α ě x˚. Now the conclusion x α " x˚follows by the already established inequality x α ď x˚.
(ix) By the definition of x α in (1.17) and part (i) of Proposition 1.1, the set E α p1q is an interval with endpoints x α and 8. So, by the inequality x α ă 8 in part (vi) of Proposition 1.2, E α p1q ‰ H. Thus, to verify part (ix) of Proposition 1.2, it is enough to show that x α R E α p1q. If x α "´8 then this follows immediately from the definition of E α ppq in (1.18) as a subset of R, and if x α ą´8 then the same conclusion follows by (A.3).
(x) Part (x) of Proposition 1.2 follows immediately from part (ix) of Proposition 1.2 and the inequality P α pX; xq ď 1, which latter in turn follows by (1.15).
(xi) Consider first the case α P p0, 8q. By part(i) of Proposition 1.1 and (1.15), the function (A.1) is nondecreasing, from the value 1 at´8. It is easy to see that these conditions, together with the convexity of the function (A.1), imply that this function is strictly increasing on the set tx P p´8, x˚s X R : 1 ă P α pX; xq´1 {α ă 8u. In view of part (ii) of Proposition 1.2, this implies that the function x Þ Ñ P α pX; xq is strictly decreasing on the set p´8, x˚q X tx P R : P α pX; xq ă 1u, which is the same as px α , x˚q, by part (ix) of Proposition 1.2. The conclusion in part (xi) of Proposition 1.2 for α P p0, 8q now follows by its part (iv). The case α " 8 is quite similar, where one uses, instead of (A.1), the function (A.2), whose limit at´8 is 0.
Proof of Proposition 1.4. Let α and a sequence pα n q be indeed as in Proposition 1.4. If x P rx˚,´8q then the desired conclusion P αn pX; xq Ñ P α pX; xq follows immediately from part (i) of Proposition 1.2. Therefore, assume in the rest of the proof of Proposition 1.4 that x P p´8, x˚q. Also, by (1.3), (1.2), the inequality (3.1) for α P p0, 8q, the condition X P X β , and dominated convergence, Hence, by (1.5), lim sup n P αn pX; xq ď lim sup n A αn pX; xqpλq " A α pX; xqpλq for all λ P r0, 8q, whence, again by (1.5), lim sup n P αn pX; xq ď P α pX; xq. If now α P p0, 1q then`say, by cutting off an initial segment of the sequence pα n q˘one may assume that β P p0, 1q, and then, by (A.9) with α n in place of α, the sequence`A αn pX; xqpλq˘is equicontinuous in λ P r0, 8q, uniformly in n. Therefore, by (A.5) and the Arzelà-Ascoli theorem, the convergence in (A.7) is uniform in λ P r0, λ˚s and hence the conclusion P αn pX; xq Ñ P α pX; xq follows by (A.6) -in the case when α P p0, 1q.
Quite similarly, the same conclusion holds if α " 1 " β; that is, P α pX; xq is left-continuous in α at the point α " 1 provided that E X`ă 8.
It remains to consider the case when α P r1, 8s and α n ě 1 for all n. Then, by the definition in (1.1), the functions h α and h αn are convex and hence, by (1.3), A α pX; xqpλq and A αn pX; xqpλq are convex in λ P r0, 8q. Then the conclusion P αn pX; xq Ñ P α pX; xq follows by [45,Corollary 3], the condition X P X β , (A.6), and (A.5).
Proof of Proposition 1.5. This is somewhat similar to the proof of Proposition 1.4. One difference here is the use of the uniform integrability condition, which, in view of (1.3), (3.1), and the condition X P X α , implies (see e.g. [10,Theorem 5.4]) that for all λ P r0, 8q lim nÑ8 A α pX n ; xqpλq " A α pX; xqpλq; (A. 10) here, in the case when α " 8 and λ R Λ X , one should also use the Fatou lemma for the convergence in distribution [10, Theorem 5.3], according to which one always has lim inf nÑ8 A α pX n ; xqpλq ě A α pX; xqpλq, even without the uniform integrability condition. In this entire proof, it is indeed assumed that α P p0, 8s. It follows from (A.10) and the nonnegativity of P α p¨;¨q that 0 ď lim inf nÑ8 P α pX n ; xq ď lim sup nÑ8 P α pX n ; xq ď P α pX; xq The convergence (1.21) for x P px˚, 8q follows immediately from (A.11) and part (i) of Proposition 1.2.
Using the same ingredients, it is easy to check part (ii) of Proposition 1.5 as well. Indeed, assuming that PpX n " x˚q ÝÑ nÑ8 PpX " x˚q and using also (1.6), one has PpX " x˚q " lim inf nÑ8 PpX n " x˚q ď lim inf nÑ8 PpX n ě x˚q ď lim inf nÑ8 P α pX n ; x˚q ď lim sup nÑ8 P α pX n ; x˚q ď P α pX; x˚q " PpX " x˚q, which yields (1.21) for x " x˚. Also, X n D ÝÑ nÑ8 X implies lim sup nÑ8 PpX n " x˚q ď PpX " x˚q; see e.g. [10, Theorem 2.1]. So, if PpX " x˚q " 0, then PpX n " x˚q Ñ PpX " x˚q and hence (1.21) holds for x " x˚, by the first sentence of part (ii) of Proposition 1.5.
It remains to prove part (i) of Proposition 1.5 assuming (A.4). The reasoning here is quite similar to the corresponding reasoning in the proof of Proposition 1.4, starting with (A.4). Here, instead of the continuity of λ max,α " λ max,α,X in α, one should use the convergence λ max,α,Xn Ñ λ max,α,X , which holds provided that y P px, x˚q is chosen to be such that PpX " yq " 0. Concerning the use of inequality (A.9), note that (i) the uniform integrability condition implies that EpX n´x q ὰ is bounded in n and (ii) the convergence in distribution X n D ÝÑ nÑ8 X implies that sup n P`px´X n q`ą 1 ? λ´κ˘Ý Ñ 0 as 0 ă λ´κ Ñ 0. Proposition 1.5 is now completely proved.
Proof of Theorem 1.6. The model-independence is obvious from the definition (1.5). The monotonicity in X follows immediately from (1.22), (1.10), and (1.7)-(1.9). The monotonicity in α was already given in (1.13). The monotonicity in x is part (i) of Proposition 1.1. That P α pX; xq takes on only values in the interval r0, 1s follows immediately from (1.15). The α-concavity in x and stability in x follow immediately from parts (iii) and (i) of Proposition 1.2. The stability in α and the stability in X are Propositions 1.4 and 1.5, respectively. The translation invariance, consistency, and positive homogeneity follow immediately from the definition (1.5).
Proof of Proposition 2.1.
(ii) Suppose here indeed that p P p0, p˚s X p0, 1q. Then for any x P px˚, 8q one has P α pX; xq " 0 ă p, by part (i) of Proposition 1.2, whence, by (1.18), x P E α ppq. On the other hand, for any x P p´8, x˚s one has P α pX; xq ě P α pX; x˚q " p˚ě p, by part (i) of Proposition 1.1 and part (i) of Proposition 1.2, whence x R E α ppq. So, E α ppq " px˚, 8q, and the conclusion Q α pX; pq " x˚now follows by the definition of Q α pX; pq in (2.2).
(iii) If x˚" 8 then the inequality Q α pX; pq ď x˚in part (iii) of Proposition 2.1 is trivial. If x˚ă 8 and p P pp˚, 1q, then x˚P E α ppq and hence Q α pX; pq ď x˚by (2.2). Now part (iii) of Proposition 2.1 follows from its part (ii).
(iv) Take any x P p´8, x˚q. Then P 0 pX; xq " PpX ě xq ą 0. Moreover, for all p P p0, P 0 pX; xqq one has x R E 0,X ppq. Therefore and because the set E 0,X ppq is an interval with endpoints Q 0 pX; pq and 8, it follows that x ď Q 0 pX; pq. Thus, for any given x P p´8, x˚q and for all small enough p ą 0 one has Q 0 pX; pq ě x and hence, by the already established part (iii) of Proposition 2.1, Q 0 pX; pq P rx, x˚s. This means that part (iv) of Proposition 2.1 is proved for α " 0. To complete the proof of this part, it remains to refer to the monotonicity of Q α pX; pq in α stated in (2.4) and, again, to part (iii) of Proposition 2.1.
(v) Assume indeed that α P p0, 8s. By part (viii) of Proposition 1.2, the case p˚" 1 is equivalent to x α " x˚, and in that case both mappings (2.6) and (2.7) are empty, so that part (v) of Proposition 2.1 is trivial. So, assume that p˚ă 1 and, equivalently, x α ă x˚. The function px α , x˚q Q x Þ Ñ P α pX; xq is continuous and strictly decreasing, by parts (iv) and (xi) of Proposition 1.2. At that, P α pX; x˚´q " P α pX; x˚q " p˚by parts (iv) and (i) of Proposition 1.2 if x˚ă 8, and P α pX; x˚´q " 0 " pb y (1.15) and (1.16) if x˚" 8. Also, P α pX; x α`q " P α pX; x α q " 1 by the condition x α ă xå nd parts (iv) and (x) of Proposition 1.2 if x α ą´8, and P α pX; x α`q " 1 by (1.15) if x α " 8. Therefore, the continuous and strictly decreasing function px α , x˚q Q x Þ Ñ P α pX; xq maps px α , x˚q onto pp˚, 1q, and so, formula (2.7) is correct, and there is a unique inverse function, say pp˚, 1q Q p Þ Ñ x α,p P px α , x˚q, to the function (2.7); moreover, this inverse function is continuous and strictly decreasing. It remains to show that Q α pX; pq " x α,p for all p P pp˚, 1q. Take indeed any p P pp˚, 1q. Since the function pp˚, 1q Q p Þ Ñ x α,p P px α , x˚q is inverse to (2.7) and strictly decreasing, P α pX; x α,p q " p, P α pX; xq ą p for x P px α , x α,p q, and P α pX; xq ă p for x P px α,p , x˚q. So, by part (i) of Proposition 1.1, P α pX; xq ą p for x P p´8, x α,p q and P α pX; xq ă p for x P px α,p , 8q. Now the conclusion that Q α pX; pq " x α,p for all p P pp˚, 1q follows by (2.2).
(vi) Assume indeed that α P p0, 8s and take indeed any y P`´8, Q α pX; pq˘. If P α pX; yq " 1 then the conclusion P α pX; yq ą p in part (vi) of Proposition 2.1 is trivial, in view of (2.1). So, w.l.o.g. P α pX; yq ă 1 and hence y P E α p1q " px α , 8q, by (1.18) and part (ix) of Proposition 1.2. Let now y p :" Q α pX; pq for brevity, so that y P p´8, y p q and, by the already verified part (iii) of Proposition 2.1, y p ď x˚. Therefore, x α ă y ă y p ď x˚. So, by part (v) of Proposition 2.1 and parts (iv) and (i) of Proposition 1.2, P α pX; yq ą lim xÒyp P α pX; xq " P α pX; y p q ě P α pX; x˚q " p˚, (A.12) which yields the conclusion P α pX; yq ą p in the case when p ď p˚. If now p ą p˚then p P pp˚, 1q and, by part (v) of Proposition 2.1, y p " Q α pX; pq P px α , x˚q and P α pX; y p q " p, so that the conclusion P α pX; yq ą p follows by (A.12) in this case as well.
Proof of Theorem 2.4. The model-independence, monotonicity in X, monotonicity in α, translation invariance, consistency, and positive homogeneity properties of Q α pX; pq follow immediately from (2.2) and the corresponding properties of P α pX; xq stated in Theorem 1.6.
Concerning the monotonicity of Q α pX; pq in p: that Q α pX; pq is nondecreasing in p P p0, 1q follows immediately from (2.3) for α " 0 and from (2.8) and (2.9) for α P p0, 8s. That Q α pX; pq is strictly decreasing in p P rp˚, 1q X p0, 1q if α P p0, 8s follows immediately from part (v) of Proposition 2.1 and the verified below statement on the stability in p: Q α pX; pq is continuous in p P p0, 1q if α P p0, 8s.
The monotonicity of Q α pX; pq in α follows immediately from (1.13) and (2.2). The finiteness of Q α pX; pq was already stated in part (i) of Proposition 2.1.
The concavity of Q α pX; pq in p´1 {α in the case when α P p0, 8q follows by (2.8), since B α pX; pqptq is affine (and hence concave) in p´1 {α . Similarly, the concavity of Q 8 pX; pq in ln 1 p follows by (2.8), since B 8 pX; pqptq is affine in ln 1 p . The stability of Q α pX; pq in p can be deduced from Proposition 2.1. Alternatively, the same follows from the already established finiteness and concavity of Q α pX; pq in p´1 {α or ln 1 p (cf. the proof of [53, Proposition 13]), because any finite concave function on an open interval of the real line is continuous, whereas the mappings p0, 1q Q p Þ Ñ p´1 {α P p0, 8q and p0, 1q Q p Þ Ñ ln 1 p P p0, 8q are homeomorphisms.
Concerning the stability of Q α pX; pq in X, take any real x ‰ x˚. Then the convergence P α pX n ; xq Ñ P α pX; xq holds, by Proposition 1.5. So, in view of (1.18), if x P E α,X ppq then eventually (that is, for all large enough n) x P E α,Xn ppq. Hence, by (2.2), for each real x ‰ x˚such that x ą Q α pX; pq eventually one has x ě Q α pX n ; pq. It follows that lim sup n Q α pX n ; pq ď Q α pX; pq. On the other hand, by part (vi) of Proposition 2.1, for any y P`´8, Q α pX; pq˘one has P α pX; yq ą p and hence eventually P α pX n ; yq ą p, which yields y R E α,Xn ppq and hence y ď Q α pX n ; pq. It follows that lim inf n Q α pX n ; pq ě Q α pX; pq. Recalling now the established inequality lim sup n Q α pX n ; pq ď Q α pX; pq, one completes the verification of the stability of Q α pX; pq in X.
The stability of Q α pX; pq in α is proved quite similarly, only using Proposition 1.4 in place of Proposition 1.5. Here the stipulation x ‰ x˚is not needed.
Consider now the sensitivity property. First, suppose that α P p0, 1q. Then, for all real t ă 0, the derivative of B α pX; pqptq in t is less than D :" 1´pE Y α q´1`1 {α E Y α´1 , where Y :" pX´tq`" X´t ą 0. The inequality D ď 0 can be rewritten as the true inequality τ τ`1 Lp´1q`1 τ`1 Lpτ q ě Lp0q for the convex function s Þ Ñ Lpsq :" ln E exptp1´αqs ln Y u, where τ :" α 1´α . So, the derivative is negative and hence B α pX; pqptq decreases in t ď 0 (here, to include t " 0, we also used the continuity of B α pX; pqptq in t, which follows by the condition X P X α and dominated convergence). On the other hand, if t ą 0 then B α pX; pqptq ě t ą 0. Also, B α pX; pqp0q ą 0 by (2.9) if the condition PpX ą 0q ą 0 holds. Recalling again the continuity of B α pX; pqptq in t, one completes the verification of the sensitivity property -in the case α P p0, 1q. The sensitivity property in the case α " 1 follows by (3.10). Indeed, (3.10) yields Q 1 pX; pq ě QpX; pq ą 0 if QpX; pq ą 0, and Q 1 pX; pq " 1 p E X ě 0 by the condition X ě 0 if QpX; pq " 0; moreover, one has E X ą 0 and hence Q 1 pX; pq " 1 p E X ą 0 if QpX; pq " 0 and PpX ą 0q ą 0. On the other hand, by (2.3), X ě 0 implies QpX; pq ě 0. Thus, the sensitivity property in the case α " 1 is verified is well. This and the already established monotonicity of Q α pX; pq in α implies the sensitivity property whenever α P r1, 8s. As far as this property is concerned, it remains to verify it when α " 0 -assuming that PpX ą 0q ą p. The sets E :" x P R : PpX ą xq ď p ( and E˝:" x P R : PpX ą xq ă p ( are intervals with the right endpoint 8. The condition PpX ą 0q ą p means that 0 R E. By the right continuity of PpX ą xq in x, the set E contains the closure E˝of the set E˝. So, 0 R E˝and hence 0 ă inf E˝" Q 0 pX; pq, by (2.3). Thus, the sensitivity property is fully verified.
In the presence of the positive homogeneity, the subadditivity property is easy to see to be equivalent to the convexity; cf. e.g. [51,Theorem 4.7].
Therefore, it remains to verify the convexity property. Assume indeed that α P r1, 8s. If at that α ă 8, then the function }¨} α is a norm and hence convex; moreover, this function is nondecreasing on the set of all nonnegative r.v.'s. On the other hand, the function R Q x Þ Ñ x`is nonnegative and convex. It follows by (2.9) that B α pX; pqptq is convex in the pair pX, tq. So, to complete the verification of the convexity property of Q α pX; pq in the case α P r1, 8q, it remains to refer to the well-known and easily established fact that, if f px, yq is convex in px, yq, then inf y f px, yq is convex in x; cf. e.g. [51,Theorem 5.7]. The subadditivity and hence convexity of Q α pX; pq in X in the remaining case α " 8 can now be obtained by the already established stability in α. It can also be deduced from [49,Lemma B.2] (cf. [48, Lemma 2.1]) or from by the main result in [46], in view of the inequality`L X1`¨¨¨`Xn˘˚´1 ď L X1 H¨¨¨H L Xn˘˚´1 given in the course of the discussion following [46, Corollary 2.2] therein. However, a direct proof, similar to the one above for α P r1, 8q, can be based on the observation that B 8 pX; pqptq is convex in the pair pX, tq. Since t ln 1 p is obviously linear in pX, tq, the convexity of B 8 pX; pqptq in pX, tq means precisely that for any natural number n, any r.v.'s X 1 , . . . , X n , any positive real numbers t 1 , . . . , t n , and any positive real numbers α 1 , . . . , α n with ř i α i " 1, one has the inequality t ln E e X{t ď ř i α i t i ln E e Xi{ti , where X :" ř i α i X i and t :" ř i α i t i ; but the latter inequality can be rewritten as an instance of Hölder's inequality: where Z i :" e αiXi{t and p i :" t{pα i t i q (so that ř i 1 pi " 1).`In particular, it follows that B 8 pX; pqptq is convex in t, which is useful when Q 8 pX; pq is computed by formula (2.8).T he proof of Theorem 2.4 is now complete.
Proof of Proposition 2.5. Consider first the case α P p0, 8q. Let r.v.'s X and Y be in the default domain of definition, X α , of the functional Q α p¨; pq. The condition X st ă Y and the left continuity of the function PpX ě¨q imply that for any v P R there are some u P pv, 8q and w P pv, uq such that PpX ě zq ă PpY ě zq for all z P rw, us. On the other hand, by the Fubini theorem, EpX´tq ὰ " ş R αpz´tq α´1 PpX ě zq dz for all t P R. Recalling also that X and Y are in X α , one has B α pX; pqptq ă B α pY ; pqptq for all t P R. By Proposition 3.3, Q α pY ; pq " B α pY ; pqpt opt q for some t opt P R. So, Q α pX; pq ď B α pX; pqpt opt q ă B α pY ; pqpt opt q " Q α pY ; pq.`Note that the proof of Proposition 3.3, given later in this appendix, does not use Proposition 2.5 -so that there is no vicious circle here.C oncerning the case α " 8, recall (1.16) and (1.14), and then note that the condition X st ă Y implies that x˚, Y " 8, Λ X Ě Λ Y , and B 8 pX; pqptq ă B 8 pY ; pqptq for all t P p0, 8q such that 1 t P Λ X and hence for all t P p0, 8q such that 1 t P Λ Y . Here, instead of the formula EpX´tq ὰ " ş R αpz´tq α´1 PpX ě zq dz for all t P R, one uses the formula E e pX´xq{t " ş R 1 t e pz´xq{t PpX ě zq dz for all t P p0, 8q. Using now Proposition 3.3, one sees that Q 8 pY ; pq " B 8 pY ; pqpt opt q for some t opt P p0, 8q such that 1 t P Λ Y . So, Q 8 pX; pq ď B 8 pX; pqpt opt q ă B 8 pY ; pqpt opt q " Q 8 pY ; pq. Proof of Proposition 2.6. Suppose that indeed α P r0, 1q. Let X and Y be independent r.v.'s, each with the Pareto density function given by the formula f puq " p1`uq´2 Itu ą 0u, so that PpX ě xq " PpY ě xq " p1`x`q´1 for all x P R. Then, by the condition α P r0, 1q, the condition X P X α (assumed by default in this paper and, in particular, in Proposition 2.5) holds; this is the only place in the proof of Proposition 2.6 where the condition α ă 1 is used. Also, then it is not hard to see that for all x P p0, 8q one has PpX`Y ě xq´Pp2X ě xq " 2p2`xq´2 lnp1`xq ą 0 and hence, by the definition of the relation st ă given in Proposition 2.5, Using now Proposition 2.5 together with the positive homogeneity property stated in Theorem 2.4, one concludes that Q α pX`Y ; pq ą Q α p2X; pq " 2Q α pX; pq " Q α pX; pq`Q α pY ; pq if α P p0, 1q.
Proof of Proposition 2.7.
(i) The equalities in (2.12) follow immediately from part (iv) of Proposition 2.1. The condition x˚P R in (2.12) follows from the condition x˚ă 8 -because, by the definition of x˚in (1.16), one always has x˚P p´8, 8s. Thus, part (i) of Proposition 2.7 is verified.
(ii) Take any r P pα, 8s and suppose that indeed PpX ě xq is like x´r. Then, in view of (2.11) and because the function q 0 was supposed to be positive on R, one observes that PpX ě xq ą 0 for all large enough real x. Therefore and because PpX ě xq is nondecreasing in x P R, in fact PpX ě xq ą 0 for all real x. In particular, it now follows that indeed x˚" 8. Moreover, recalling the definition (1.18) of E α,X ppq and the equality in (1.6), one sees that for any real x and all p in the (nonempty) right neighborhood`0, P 0 pX; xq˘"`0, PpX ě xq˘of 0, one has x R E 0,X ppq; therefore and because, by the definition (2.2) of Q α pX; pq, the set E 0,X ppq is an interval with endpoints Q 0 pX; pq and 8, one concludes that Q 0 pX; pq ě x for all p P`0, P 0 pX; xq˘. Thus, Q α pX; pq ÝÑ pÓ0 8 for α " 0; that the same limit relation holds for any α P r0, 8s now follows immediately by the monotonicity of Q α pX; pq in α, as stated in (2.4).
To complete the proof of Proposition 2.7, it remains to verify (2.13). First here, consider the case r ă 8, so that r P pα, 8q. For brevity, let qpxq :" PpX ě xq.
Then qpxq qpyq "`y x˘r`o p1q as x, y Ñ 8; (A.13) the latter asymptotic relation is an extension of, and proved quite similarly to, the asymptotic relation (2.11b). Introduce also xα ,p :" Q α pX; pq˘1.
Proof of Proposition 3.1. Take indeed any α P p0, 1q and p P p0, 1q. Note that there are real numbers q, r, and b such that q ą 0, r ą 0, q`r ă 1, Indeed, if 0 ă b ă 1, r " p 2 α , and q " kpbqr, where kpbq :" 2 α´p 1`bq α p1´bq α , then all of the conditions in (A.18) will be satisfied, possibly except the condition q`r ă 1, which latter will be then equivalent to the condition hpbq :" p 2 α p1`kpbqq ă 1. However, this condition can be satisfied by letting b P p0, 1q be small enough -because hp0`q " p P p0, 1q.
If now q, r, and b satisfy (A.18), then there is a r.v. X taking values´1,´b, and b with probabilities 1´q´r, q, and r, respectively. Let indeed X be such a r.v. Then for all s P p0, 8q A α pX; 0qpαsq " gpsq :" p1´q´rqp1´sq ὰ`q p1´bsq ὰ`r p1`bsq α .
Moreover, by the condition α P p0, 1q, the function g is strictly concave on each of the intervals p0, 1s, r1, 1 b s, and r 1 b , 8q. So, the minimum of gpsq in s P p0, 8q equals p and is attained precisely at two distinct positive values of s. Thus, in the case x " 0, Proposition 3.1 follows by (A.19). The case of a general x P R immediately reduces to that of x " 0 by using the shifted r.v. X`x in place of X.
Concerning part (ii) of the proposition, consider first Case 1: x˚" 8. Take then any real t 1 ą 0 such that E e X{t1 ă 8 and then any real x ą x 1 :" B 8 pX; pqpt 1 q such that q :" PpX ě xq ă p; note that q ą 0, since x˚" 8. Then for any real t ą 0 one has E e X{t ě qe x{t and hence provided that t ă t min :" x´x 1 lnpp{qq ; the latter inequality is in fact equivalent to the strict inequality in (A.20); recall here also that x ą x 1 and 0 ă q ă p, whence t min P p0, 8q. Indeed, if p˚" 0 then for each real ε ą 0 and all small enough real t ą 0, one has E e X{t ă p and hence 0 ą t ln E e X{t p ě t ln`1 p E e X{t ItX ě´εu˘ě´ε`t ln PpX ě´εq ÝÑ tÓ0´ε , which yields (A. 22) for t Ó 0, in the case when p˚" 0. As for the cases when t Ñ 8, or t Ó 0 and p˚ą 0, then (A.22) follows from (A.21) because 0 ă p ă 1.
To proceed further with the consideration of Case 2, one needs to distinguish the following three subcases. the latter equality here follows by the continuity of B 8 pX; pqptq in t P p0, 8q, which in turn takes place by the Case 2 condition x˚ă 8. This completes the consideration of Subcase 2.1 for part (ii) of the proposition. Subcase 2.2: p˚P rp, 1q. Here, note that PpX ă 0q ą 0 (since p˚ă 1) and E e X{t " p˚È e X{t ItX ă 0u. So, if t is decreasing from 8 to 0, then E e X{t is strictly decreasing and hence ln E e X{t p is strictly decreasing -to ln pp ě 0, by (A.21) and the case condition p˚P rp, 1q. Therefore, ln E e X{t p ą 0 for all t ą 0 and hence B 8 pX; pqptq " t ln E e X{t p is strictly decreasing if t is decreasing from 8 to 0. It follows that, in Subcase 2.2, inf tPTα " inf tPp0,8q in (2.8) is not attained; rather, inf tą0 B 8 pX; pqptq " lim tÓ0 B 8 pX; pqptq " 0 " x˚, in view of (A.22) and the assumption x˚" 0. It remains to consider Subcase 2.3: p˚" 1. Then PpX " 0q " 1 and hence B 8 pX; pqptq " t ln 1 p , so that, as in Subcase 2.2, inf tPTα " inf tPp0,8q in (2.8) is not attained, and inf tą0 B 8 pX; pqptq " lim tÓ0 B 8 pX; pqptq " 0 " x˚. Now Proposition 3.3 is completely proved.
Proof of Proposition 3.6.
(i) Part (i) of Proposition 3.6 follows because, as shown in the proof of Theorem 2.4, B α pX; pqptq is convex in the pair pX, tq.
(ii) Assume indeed that α P p1, 8q. It is then well known that the norm }¨} α is strictly convex, in the sense that }p1´sqY`sZ} α ă p1´sq}Y } α`s }Z} α for all s P p0, 1q and all r.v.'s Y and Z such that }Y } α`} Z} α ă 8 and PpyY`zZ ‰ 0q ą 0 for all nonzero real y and z. The strict convexity of the norm }¨} α is of course equivalent to its strict subadditivity -see e.g. [12,Corollary,page 405]. Alternatively, the strict subadditivity of the norm }¨} α can be easily discerned from a proof of Minkowski's inequality, say the classical proof based on Hölder's inequality, or the one given in [46]. Since for any t P p´8, x˚˚q the set supp`pX´tq`˘contains at least two distinct points, it follows that B α pX; pqptq is strictly convex in t P p´8, x˚˚q and hence, by continuity, in t P p´8, x˚˚s X R.
(iii) Part (iii) of Proposition 3.6 can be verified by invoking, in the proof of the subadditivity/convexity in Theorem 2.4, the well-known strictness condition for Hölder's inequality.
Proof of Proposition 3.7.
(ib) Let us now verify part (ib) of Proposition 3.7. Toward that end, assume indeed that α P p1, 8q and p P pp˚, 1q. Then, by (2.6), Q α pX; pq P p´8, x˚q. By (2.9), t ď B α pX; pqptq for all real t. So, in view of (3.15) and (3.14), α´1 QpX; pq ď Q α pX; pq ă x˚. By (3.16), one now has α´1 QpX; pq P p´8, x˚q. Since α P p1, 8q, Bptq :" B α pX; pqptq is differentiable in t P p´8, x˚q, with the derivative It follows by (3.15) that α´1 QpX; pq is a root t P p´8, x˚q of the equation B 1 ptq " 0, which can be rewritten as α´1 P pX; tq " p. Let us show that such a root t is unique and (3.20) holds. If x˚˚ă x˚, it follows by (3.19) that α´1 P pX; tq " p˚ă p for all t P rx˚˚, x˚q and hence, by (A.23), B 1 ptq ą 0 for all such t. So, all roots t P p´8, x˚q of the equation B 1 ptq " 0 are in fact in the interval p´8, x˚˚q. On the other hand, by part (ii) of Proposition 3.6, there is at most one root t P p´8, x˚˚q of the equation B 1 ptq " 0 or, equivalently, of the equation α´1 P pX; tq " p. Since α´1 QpX; pq is such a root, one obtains (3.20) -in the case when x˚˚ă x˚. Relations (3.20) hold in the remaining case when x˚˚" x˚, since, as established above, α´1 QpX; pq P p´8, x˚q. Thus, the first two sentences of part (ib) of Proposition 3.7 are verified.
Concerning the third sentence there, assume that indeed t P p´8, x˚q. Then the function g defined by the formula gpγq :" ln E`pX´tq γ |X ą t˘is convex on r0, 8q, with gp0q " 0. So, by (3.19), 24) which shows that indeed α´1 P pX; tq ď PpX ą tq. If t P p´8, x˚˚q, then the interval pt, 8q contains at least two distinct points of supp X, whence the function g is strictly convex on r0, 8q, which makes the inequality in (A.24) strict, so that the strict inequality α´1 P pX; tq ă PpX ą tq holds.
Assume now that α P p1, 8q. Note that α´1 P pX; tq is continuous in t P p´8, x˚q. So, if x˚˚ă x˚, then α´1 P pX; tq ÝÑ tÒx˚˚α´1 P pX; x˚˚q " p˚. On the other hand, by the inequality α´1 P pX; tq ď PpX ą tq for all t P p´8, x˚q, one has α´1 P pX; tq ÝÑ tÒx˚0 " p˚if p˚" 0. In the remaining case, when x˚˚" x˚and p˚ą 0, for all γ P p0, 8q, all X P X γ , all t P p´8, x˚q, and some θ γ,t P r0, 1s one can write EpX´tq γ " p˚px˚´tq γ`θ γ,t Ppt ă X ă x˚qpx˚´tq γ " tÒx˚p˚p x˚´tq γ , whence α´1 P pX; tq " p˚as t Ò x˚" x˚˚. Thus, in any case α´1 P pX; tq ÝÑ tÒx˚˚p˚.
Next, for all γ P p0, 8q, all X P X γ , and all t P p´8, 0q, by the monotone convergence EpX´tq γ " |t| γ Ep1`X{|t|q γ " tÓ´8 |t| γ , whence α´1 P pX; tq ÝÑ tÓ´8 1. So, the function t Þ ÝÑ α´1 P pX; tq maps the interval p´8, x˚˚q continuously onto the interval pp˚, 1q. Moreover, by part (ib) of Proposition 3.7, this function is one-to-one. Furthermore, by (A.23), this function is nonincreasing, because B is convex and hence B 1 is nondecreasing. It then follows that this function is strictly decreasing on the interval p´8, x˚˚q. One concludes that the function pp˚, 1q Q p Þ ÝÑ α´1 QpX; pq " t α,p P p´8, x˚˚q is a bijection, which is the strictly decreasing continuous inverse to the strictly decreasing continuous bijection p´8, x˚˚q Q t Þ ÝÑ α´1 P pX; tq P pp˚, 1q. This completes the proof of part (ii) of Proposition 3.7.
(iii) Consider now part (iii) of Proposition 3.7. The first equality in (3.21) is (3.17), and the second equality there follows by the definition of QpX; pq in (2.3).
(iv) If X " c for some c P R then x˚" c and p˚" 1, so that part (iv) of Proposition 3.7 follows immediately from its part (ia).
(v) Concerning the positive homogeneity of α´1 QpX; pq stated in part (v) of Proposition 3.7, the case κ " 0 follows immediately by the consistency of α´1 QpX; pq and (3.16). The case of any real κ ą 0 follows by (3.15) and the identity B α pκX; pqpκtq " κB α pX; pqptq for all t P R.
(vi) The translation invariance of α´1 QpX; pq stated in part (vi) of Proposition 3.7, follows immediately by (3.15) and the identity B α pX`c; pqpt`cq " B α pX; pqptq`c for all t P R.
(ix) The case α " 1 of part (ix) of Proposition 3.7 follows, in view of (3.17), because Q 0 pX; pq is well known not to be subadditive or convex. Take now any α P p1, 8q and any p P p0, 1q. To verify part (ix) of Proposition 3.7 for such α and p, let us use an idea from [54], which allows one to show that the non-subadditivity follows from the non-monotonicity and partial monotonicity. Thus, let Y and X be as in the above proof of part (viii) of Proposition 3.7. Let V :" Y´X, so that V ď 0 and hence, by part (vii) of Proposition 3.7, α´1 QpV ; pq ď 0. It follows that α´1 QpV ; pq`α´1QpX; pq ď α´1 QpX; pq ă α´1 QpY ; pq " α´1 QpV`X; pq. So, α´1 Qp¨; pq is not subadditive. Since α´1 Qp¨; pq is positive homogeneous, it is not convex either.
The proof of Proposition 3.7 is now quite complete.
Proof of Proposition 4.2. To prove the "if" part of the proposition, suppose that H is 1 2 -Lipschitz and take any r.v.'s X and Y such that X st ď Y . We have to show that then R H pXq ď R H pY q. By (1.25) and because R H pXq depends only on the distribution of X, w.l.o.g. X ď Y . Let pX,Ỹ q be an independent copy of the pair pX, Y q. Then, by (4.2), the 1 2 -Lipschitz condition, the triangle inequality, and the condition X ď Y , R H pXq´R H pY q " EpX´Y q`E Hp|X´X|q´E Hp|Y´Ỹ |q ď EpX´Y q`1 2 Ep|X´X|´|Y´Ỹ |q ď EpX´Y q`1 2 E |X´X´Y`Ỹ | ď EpX´Y q`1 2 Ep|X´Y |`|X´Ỹ |q " EpX´Y q`E |X´Y | " EpX´Y q`EpY´Xq " 0, so that the "if" part of Proposition 4.2 is verified.
To prove the "only if" part of the proposition, suppose that R H pXq is nondecreasing in X with respect to the stochastic dominance of order 1 and take any x and y in r0, 8q such that x ă y. It is enough to show that then |Hpxq´Hpyq| ď 1 2 py´xq. Take also an arbitrary p P p0, 1q. Let X and Y be such r.v.'s that PpX " 0q " 1 if x " 0, PpX " xq " p " 1´PpX " 0q if x P p0, 8q, and PpY " yq " p " 1´PpY " 0q. Then X st ď Y , whence, by (4.2), 0 ě 1 p rR H pXq´R H pY qs " x´y`2p1´pqrHpxq´Hpyqs, which yields Hpxq´Hpyq ď 1 2p1´pq py´xq for an arbitrary p P p0, 1q and hence Hpxq´Hpyq ď 1 2 py´xq.
So, the convexity property of R H pXq is verified, which, as noted earlier, is equivalent to the subadditivity given the positive homogeneity. Now, to finish the proof of "if" part of Proposition 4.3, it remains to notice that the monotonicity property of R κ id pXq for κ P r0, 1 2 s follows immediately from Proposition 4.2.
To prove the "only if" part of the proposition, suppose that the function H is such that R H pXq is coherent and thus positive homogeneous, monotonic, and subadditive (as noted before, R H pXq is translation invariant for any H). Take any p P p0, 1q and let X here be a r.v. such that PpX " 1q " p " 1´PpX " 0q. Then, by the positive homogeneity, for any real u ą 0 one has 0 " R H puXq´uR H pXq " aA`B, where B :" p1´uqHp0q, A :" Hpuq´uHp1q´B, and a :" 2pp1´pq, so that the range of values of a is the entire interval p0, 1 2 q as p varies in the interval p0, 1q. Thus, aA`B " 0 for all a P p0, 1 2 q.
On the other hand, aA`B is a polynomial in a, with coefficients A and B not depending on a. It follows that A " B " 0, which yields Hpuq " uHp1q for all u P p0, 8q and Hp0q " 0. Hence, Hpuq " uHp1q for all real u ě 0. In other words, H " κ id, with κ :" Hp1q. Then the monotonicity property and Proposition 4.2 imply that |κ| ď 1 2 . It remains to show that necessarily κ ě 0. Take here X and Y to be independent standard normal r.v.'s. Then, by the subadditivity, 2κ E |X| " R κ id pX`Y q ď R κ id pXq`R κ id pY q " 2 ? 2 κ E |X|, whence indeed κ ě 0.