4.6Concave distortions

Let us now have a closer look at the coherent risk measures

which appear in the Representation Theorem 4.62 for law-invariant convex risk measures. We are going to characterize these risk measures ρμ in two ways, first as Choquet integrals with respect to some concave distortion of the underlying probability measure P, and then, in the next section, by a property of comonotonicity.

Again, we will assume throughout this section that the underlying probability space (Ω,F, P) is atomless. Since AV@Rλ is coherent, continuous from below, and law-invariant, any mixture ρμ for some probability measure μ on (0, 1] has the same properties. According to Remark 4.50, we may set AV@R0(X) = − ess inf X so that we can extend the definition (4.54) to probability measures μ on the closed interval [0, 1]. However, ρμ will only be continuous from above and not from below if μ({0}) > 0, because AV@R0 is not continuous from below.

Our first goal is to show that ρμ(X) can be identified with the Choquet integral of the loss −X with respect to the set function cψ(A) := ψ(P[ A ]), where ψ is the concave function defined in the following lemma. Choquet integrals were introduced in Example 4.14, and the risk measure MINVAR of Exercise 4.1.8 provides a first example for a risk measure arising as the Choquet integral of a set function cψ. Recall that every concave function ψ admits a right-continuous right-hand derivative see Proposition A.7.

Lemma 4.69. The identity

defines a one-to-one correspondence between probability measures μ on [0, 1] and increasing concave functions ψ : [0,1] → [0, 1] with ψ(0) = 0 and ψ(1) = 1. Moreover, we have ψ(0+) = μ({0}).

Proof. Suppose first that μ is given and ψ is defined by ψ(1) = 1 and (4.55). Then ψ is concave and increasing on (0, 1]. Moreover,

Hence, we may set ψ(0) := 0 and obtain an increasing concave function on [0, 1].

Conversely, if ψ is given, then is a decreasing right-continuous function on (0, 1) and can be written as ψʹ+(t) = ν((t, 1]) for some positive Radon measure ν on (0, 1]. We first define μ on (0, 1] by μ(dt) = t ν(dt). Then (4.55) holds and, by Fubini’s theorem,

Hence, setting μ({0}) := ψ(0+) defines a probability measure μ on [0, 1].

Theorem 4.70. For a probability measure μ on [0, 1], let ψ be the concave function defined in Lemma 4.69. Then, for X ∈ X ,

Proof. Using the fact that we get as in (4.52) that

Hence, we obtain the first identity. For the second one, we will first assume X ≥ 0. Then

where F X is the distribution function of X. Using Fubini’s theorem, we obtain

since This proves the second identity for X ≥ 0, since ψ(0+) = μ({0}) and ess sup X = AV@R0(−X). If X ∈ L∞ is arbitrary, we consider X + C, where C := − ess inf X. The cash invariance of ρμ yields

Example 4.71. Clearly, the risk measure AV@Rλ is itself of the form ρμ where μ = δλ. For λ > 0, the corresponding concave distortion function is given by

Thus, we obtain yet another representation of AV@Rλ:

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Exercise 4.6.1. Find the probability measure μ on [0, 1] such that the coherent risk measure MINVAR introduced in Exercise 4.1.8 is of the form (4.54).

◊

Exercise 4.6.2. Suppose that there exists a set A ∈ F with 0 < P[ A ] < 1 such that ρμ(−A) = −ρμ(A ). Use the representation (4.56) to deduce that μ = δ1, i.e., ρμ(X) = E[ −X ] for all X ∈ L∞. More generally, show that μ = δ1 if there exists a nonconstant X ∈ L∞ such that ρμ(−X) = −ρμ(X).

◊

Corollary 4.72. If μ({0}) = 0 in Theorem 4.70, then

where φ is an inverse function of ψ, taken in the sense of Definition A.18.

Proof. Due to Lemma A.19, the distribution of φ under the Lebesgue measure has the distribution function ψ and hence the density ψʹ. Therefore

where we have used Lemma A.27 in the last step. An application of Theorem 4.70 concludes the proof.

Let us continue with a brief discussion of the set function cψ(A) = ψ(P[ A ]).

Definition 4.73. Let ψ : [0,1] → [0, 1] be an increasing function such that ψ(0) = 0 and ψ(1) = 1. The set function

is called the distortion of the probability measure P with respect to the distortion function ψ.

◊

Definition 4.74. A set function c : F → [0, 1] is called monotone if

and normalized if

c(∅) = 0 and c(Ω) = 1.

A monotone set function is called submodular or strongly subadditive if

Clearly, any distortion cψ is normalized and monotone.

Proposition 4.75. Let cψ be the distortion of P with respect to the distortion function ψ. If ψ is concave, then cψ is submodular. Moreover, if the underlying probability space is atomless, then also the converse implication holds.

Proof. Suppose first that ψ is concave. Take A, B ∈ F with P[ A ]≤ P[ B ]. We must show that c := cψ satisfies

This is trivial if r = 0, where

For r > 0 the concavity of ψ yields via (A.5) that

Multiplying both sides with r gives the result.

Now suppose that c = cψ is submodular and assume that (Ω,F, P) is atomless. By Exercise A.1.2 below it is sufficient to show that ψ(y) ≥ (ψ(x) + ψ(z))/2 whenever 0 ≤ x ≤ z ≤ 1 and y = (x + z)/2. To this end, we will construct two sets A, B ⊂ F such that P[ A ] = P[ B ] = y, P[ A ∩ B ] = x, and P[ A ∪ B ] = z. Submodularity then gives ψ(x) + ψ(z) ≤ 2ψ(y) and in turn the concavity of ψ.

In order to construct the two sets A and B, take a random variable U with a uniform distribution on [0, 1], which exists by Proposition A.31. Then

are as desired.

Let us now recall the notion of a Choquet integral, which was introduced in Example 4.14.

Definition 4.76. Let c : F → [0, 1] be any set function which is normalized and monotone. The Choquet integral of a bounded measurable function X on (Ω,F) with respect to c is defined as

Note that the Choquet integral coincides with the usual integral as soon as c is a σ-additive probability measure; see also Lemma 4.97 below.

With this definition, Theorem 4.70 allows us to identify the risk measure ρμ as the Choquet integral of the loss with respect to a concave distortion cψ of the underlying probability measure P:

Corollary 4.77. For a probability measure μ on [0, 1], let ψ be the concave distortion function defined in Lemma 4.69, and let cψ denote the distortion of P with respect to ψ. Then, for X ∈ L∞,

Combining Corollary 4.77 with Theorem 4.62, we obtain the following characterization of law-invariant convex risk measures in terms of concave distortions:

Corollary 4.78. A convex risk measure ρ is law-invariant and continuous from above if and only if

where the supremum is taken over the class of all concave distortion functions ψ and

Exercise 4.6.3. Let c : F → [0, 1] be a normalized and monotone set function and define the dual set function by (A) := 1 − c(Ω A). Show that the Choquet integrals with respect to c and satisfy

for bounded and measurable X.

◊

Exercise 4.6.4. For λ ∈ ℝ consider the functional

where is an independent copy of X. Recall from Exercise 4.1.7 that ρλ(X) is a coherent risk measure if and only if 0 ≤ λ ≤ 1/2. Show that ρλ(X) can be represented in the form

for a certain function fλ and use this representation to give another proof of the fact that ρλ is a law-invariant coherent risk measure if and only if 0 ≤ λ ≤ 1/2. Then identify the corresponding distortion function in the representation of ρλ as a Choquet integral for 0 ≤ λ ≤ 1/2.

◊

The following series of exercises should be compared with Exercises 4.1.8 and 4.6.1.

Exercise 4.6.5. For β ≥ 1 consider the concave distortion function Show that for β ∈ ℕ the corresponding risk measure

has the property that MAXVARβ(X) = −E[ Y1 ], if Y1, . . . , Yβ are independent and identically distributed (i.i.d.) random variables for which max(Y1, . . . , Yβ) ∼ X.

◊

Exercise 4.6.6. For β ≥ 1 consider the distortion functionShow that for β ∈ ℕ the corresponding risk measure

has the property that MAXMINVARβ(X) = −E[ Y1 ], if Y1, . . . , Yβ are i.i.d. random variables for which max(Y1, . . . , Yβ) ∼ min(X1, . . . , Xβ), where X1, . . . , Xβ are independent copies of X.

◊

Exercise 4.6.7. For β ≥ 1 consider the distortion functionShow that for β ∈ ℕ the corresponding risk measure

has the property that MINMAXVARβ(X) = −E[ min(Y1, . . . , Yβ) ], if Y1, . . . , Yβ are i.i.d. random variables for which max(Y1, . . . , Yβ) ∼ X.

◊

As another consequence of Theorem 4.70, we obtain an explicit description of the maximal representing set Qμ ⊂ M1(P) for the coherent risk measure ρμ.

Theorem 4.79. Let μ be a probability measure on [0, 1], and let ψ be the corresponding concave function defined in Lemma 4.69. Then ρμ can be represented as

where the set Qμ is given by

Moreover, Qμ is the maximal subset of M1(P) that represents ρμ.

Proof. The risk measure ρμ is coherent and continuous from above. By Corollary 4.37, it can be represented by taking the supremum of expectations over the set Qmax = {Q ∈ M1(P) | αmin(Q) = 0}. Using (4.50) and Theorem 4.70, we see that a measure Q ∈ M1(P) with density Z = dQ/dP belongs to Qmax if and only if

for all X ∈ L∞. Let X be a Bernoulli-distributed random variables X with parameter 1 −t, i.e., X takes the two values 0 and 1 and P[ X = 1] = 1 −t. Then we have qX = [t,1] a.e., and so we obtain

for all t ∈ (0, 1). Hence Qmax ⊂ Qμ. For the proof of the converse inclusion, we show that the density Z of a fixed measure Q ∈ Qμ satisfies (4.57) for any given X ∈ L∞. We may assume without loss of generality that X ≥ 0. Let ν be the positive finite measure on [0, 1] such that = ν([0, s]). Using Fubini’s theorem and the definition of Qμ, we get

which coincides with the right-hand side of (4.57).

Corollary 4.80. In the context of Theorem 4.79, the following conditions are equivalent.

(a) ρμ is continuous from below.

(b) μ({0}) = 0.

If these equivalent conditions are satisfied, then, for any X ∈ L∞, the supremum in the representation

is attained by the measure QX ∈ Qμ with density dQX / dP = f (X), where f is a decreasing function such that

Moreover, with λ denoting the Lebesgue measure on (0, 1),

Proof. If (b) holds, then (a) follows from Theorem 4.52 and monotone convergence. For the proof of the converse implication, we suppose by way of contradiction that δ := μ({0}) > 0. In this case, we can write

where μʹ := μ( · |(0, 1]). Then ρμis continuous from below since μʹ({0}) = 0, but on our atomless probability space, AV@R0 is not, and so ρμ does not satisfy (a); see Remark 4.50.

Let us now prove the remaining assertions. Since ψ(0+) = μ({0}) = 0, a measure Q with density Z = dQ/dP belongs to Qμ if and only if ds for all t. Since ψʹ(1−t) is a quantile function for the law of ψʹ under λ, part (e) of Theorem 2.57 implies (4.58). The problem of identifying the maximizing measure Q X is hence equivalent to minimizing E[ ZX ] under the constraint that Z is a density function such that P ◦ Z −1 ≽uni λ ◦(ψʹ)−1. Let us first assume that X ≥ 0. Then it follows from Theorem 3.44 that f (X) minimizes E[ YX ] over all Y ∈ L1+ such that P ◦ Y−1 ≽uni λ ◦ (ψʹ)−1. Moreover, Remark 3.46 shows that and so ZX := f (X) ≥ 0 is the density of an optimal probability measure Q X ∈ Qμ. If X is not positive, then we may take a constant c such that X + c ≥ 0 and apply the preceding argument. The formula for f then follows from the fact that F X+c(x + c) = F X(x).

Exercise 4.6.8. For a concave distortion function ψ on [0, 1] define its convex conjugate function as

Show that the set Qμ in Theorem 4.79 can be represented as

Remark 4.81. As long as we are interested in a law-invariant risk assessment, we can represent a financial position X ∈ L∞ by its distribution function F X or, equivalently, by the function

If we only consider positions X with values in [0, 1] then their proxies GX vary in the class of right-continuous decreasing functions G on [0, 1] such that G(1) = 0 and G(0) ≤ 1. Due to Theorem 4.70, a law-invariant coherent risk measure ρμ induces a functional U on the class of proxies via

Since ψ is increasing and concave, the functional U has the form of a von Neumann-Morgenstern utility functional on the probability space given by Lebesgue measure on the unit interval [0, 1]. As such, it can be characterized by the axioms in Section 2.3, and this is the approach taken in Yaari’s “dual theory of choice” [276]. More generally, we can introduce a utility function u on [0, 1] with u(0) = 0 and consider the functional

introduced by Quiggin [227]. For u(x) = x this reduces to the “dual theory”, for ψ(x) = x we recover the classical utility functionals

discussed in Section 2.3.

◊

4.7Comonotonic risk measures

In many situations, the risk of a combined position X +Y will be strictly lower than the sum of the individual risks, because one position serves as a hedge against adverse changes in the other position. If, on the other hand, there is no way for X to work as a hedge for Y then we may want the risk simply to add up. In order to make this idea precise, we introduce the notion of comonotonicity. Our main goal in this section is to characterize the class of all convex risk measures that share this property of comonotonicity.

As in the first two sections of this chapter, we will denote by X the linear space of all bounded measurable functions on the measurable space (Ω,F).

Definition 4.82. Two measurable functions X and Y on (Ω,F) are called comono-tone if

A monetary risk measure ρ on X is called comonotonic if

whenever X, Y ∈ X are comonotone.

◊

Lemma 4.83. If ρ is a comonotonic monetary risk measure on X , then ρ is positively homogeneous.

Proof. Note that (X, X) is a comonotone pair. Hence ρ(2X) = 2ρ(X). An iteration of this argument yields ρ(rX) = rρ(X) for all rational numbers r ≥ 0. Positive homogeneity now follows from the Lipschitz continuity of ρ; see Lemma 4.3.

We will see below that every comonotonic monetary risk measure on X arises as the Choquet integral with respect to a certain set function on (Ω,F). In the sequel, c : F → [0, 1] will always denote a set function that is normalized and monotone; see Definition 4.74. Unless otherwise mentioned, we will not assume that c enjoys any additivity properties. Recall from Definition 4.76 that the Choquet integral of X ∈ X with respect to c is defined as

The proof of the following proposition was already given in Example 4.14.

Proposition 4.84. The Choquet integral of the loss,

is a monetary risk measure on X which is positively homogeneous.

Definition 4.85. Let X be a measurable function on (Ω,F). An inverse function rX : (0, 1) → ℝ of the increasing function GX(x) := 1 − c(X > x), taken in the sense of Definition A.18, is called a quantile function for X with respect to c.

◊

If c is a probability measure, then GX(x) = c(X ≤ x). Hence, the preceding definition extends the notion of a quantile function given in Definition A.24. The following proposition yields an alternative representation of the Choquet integral in terms of quantile functions with respect to c.

Proposition 4.86. Let rX be a quantile function with respect to c for X ∈ X . Then

Proof. We have and one easily checks that rX+m = rX + m a.e. for all m ∈ ℝ and each quantile function rX+m of X + m. Thus, we may assume without loss of generality that X ≥ 0. In this case, Remark A.20 and Lemma A.19 imply that the largest quantile function is given by

Since a.e. on (0, 1), Fubini’s theorem implies

The preceding proposition yields the following generalization of Corollary 4.72 when applied to a continuous distortion of a probability measure as defined in Definition 4.73.

Corollary 4.87. Let cψ(A) = ψ(P[ A ]) be the distortion of the probability measure P with respect to the continuous distortion function ψ. If φ is an inverse function for the increasing function ψ in the sense of Definition A.18, then the Choquet integral with respect to cψ satisfies

where qX is a quantile function for X ∈ X , taken with respect to P.

Proof. Due to the continuity of ψ, we have ψ(a) ≤ t if and only if a ≤ φ+(t) = inf{x | ψ(x) > t}. Thus, we can compute the lower quantile function of X with respect to cψ:

Next note that φ+(t) = φ(t) for a.e. t. Moreover, φ has the continuous distribution function ψ under the Lebesgue measure, and so we can replace by the arbitrary quantile function q X.

Theorem 4.88. A monetary risk measure ρ on X is comonotonic if and only if there exists a normalized monotone set function c on (Ω,F) such that

In this case, c is given by c(A) = ρ(−A ).

The preceding theorem implies in view of Corollary 4.77 that all mixtures

are comonotonic. We will see in Theorem 4.93 below that these are in fact all convex risk measures that are law-invariant and comonotonic. The proof of Theorem 4.88 requires a further analysis of comonotone random variables.

Lemma 4.89. For two measurable functions X and Y on (Ω,F), the following conditions are equivalent.

(a) X and Y are comonotone.

(b) There exists a measurable function Z on (Ω,F) and increasing functions f and g on ℝ such that X = f (Z) and Y = g(Z).

(c) There exist increasing functions f and g on ℝ such that X = f (X+Y) and Y = g(X+Y).

Proof. The implications “(c) ⇒ (b)” and “(b) ⇒ (a)” are obvious. To prove “(a) ⇒ (c)”, suppose that X and Y are comonotone and define Z by Z := X +Y.We show that z := Z(ω) has a unique decomposition as z = x + y, where (x, y) = (X(ωʹ), Y(ωʹ)) for some ωʹ ∈ Ω. Having established this, we can put f (z) := x and g(z) := y, and we will show later on that these functions are indeed increasing. The existence of the decomposition as z = x + y follows by taking x := X (ω) and y := Y(ω), so it remains to show that these are the only possible values x and y. To this end, let us suppose that X(ω) + Y(ω) = z = X(ωʹ) + Y(ωʹ) for some ωʹ ∈ Ω. Then

and comonotonicity implies that this expression vanishes. Hence x = X(ωʹ) and y = Y(ωʹ).

Next, we check that both f and g are increasing functions on Z(Ω). So let us suppose that

This implies

Comonotonicity thus yields that X(ω1)− X(ω2)≤ 0 and Y(ω1)− Y(ω2)≤ 0, whence f (z1)≤ f (z2) and g(z1)≤ g(z2). Thus, f and g are increasing on Z (Ω), and it is straightforward to extend them to increasing functions defined on ℝ.

Lemma 4.90. If X, Y ∈ X is a pair of comonotone functions, and rX, rY, rX+Y are quantile functions with respect to c, then

Proof. Write X = f (Z) and Y = g(Z) as in Lemma 4.89. The same argument as in the proof of Lemma A.27 shows that f (rZ) and g(rZ) are quantile functions for X and Y under c if rZ is a quantile function for Z. An identical argument applied to the increasing function h := f + g shows that h(rZ) = f (rZ)+ g(rZ) is a quantile function for X + Y. The assertion now follows from the fact that all quantile functions of a random variable coincide almost everywhere, due to Lemma A.19.

Remark 4.91. Applied to the special case of quantile function with respect to a probability measure, the preceding lemma yields that V@Rλ and AV@Rλ are comonotonic.

◊

Proof of Theorem 4.88. We already know from Proposition 4.84 that the Choquet integral of the loss is a monetary risk measure. Comonotonicity follows by combining Proposition 4.86 with Lemma 4.90.

Conversely, suppose now that ρ is comonotonic. Then ρ is positively homogeneous according to Lemma 4.83. In particular we have ρ(−m) = m for m ≥ 0. Thus, we obtain a normalized monotone set function by letting c(A) := ρ(−A ). Moreover, is a comonotonic monetary risk measure on X that coincides with ρ on indicator functions: ρ(−A) = c(A) = ρc(−A ). Let us now show that ρ and ρc coincide on simple random variables of the form

Since these random variables are dense in L∞, Lemma 4.3 will then imply that ρ = ρc. In order to show that ρc(X) = ρ(X) for X as above, we may assume without loss of generality that x1 ≥ x2 ≥· ··≥ xn and that the sets Ai are disjoint. By cash invariance, we may also assume X ≥ 0, i.e., xn ≥ 0. Thus, we can write where bi := xi − xi+1 ≥ 0, xn+1 := 0, and Note that biBi and bkBk is a pair of comonotone functions. Hence, alsoand b kBk are comonotone, and we get inductively

Remark 4.92. The argument at the end of the preceding proof shows that the Choquet integral of a simple random variable

and disjoint sets A1, . . . , An can be computed as

where B0 := ∅ and for i = 1, . . . , n.

◊

Exercise 4.7.1. For X, Y ∈ X and a uniform random variable U, define the comono-tone copies := q X(U) and := qY (U) and show that the law of X + Y dominates the law of + with respect to second-order stochastic dominance, ≽uni. Hint: Use Remark 4.49 and the comonotonicity of AV@Rλ.

◊

So far, we have shown that comonotonic monetary risk measures can be identified with Choquet integrals of normalized monotone set functions. Our next goal is to characterize those set functions that induce risk measures with the additional property of convexity. To this end, we will first consider law-invariant risk measures. The following result shows that the risk measures AV@Rλ may be viewed as the extreme points in the convex class of all law-invariant convex risk measures on L∞ that are comonotonic.

Theorem 4.93. On an atomless probability space, the class of risk measures

is precisely the class of all law-invariant convex risk measures on L∞ that are comono-tonic. In particular, any convex risk measure that is law-invariant and comonotonic is also coherent and continuous from above.

Proof. Comonotonicity of ρμ follows from Corollary 4.77 and Theorem 4.88. Conversely, let us assume that ρ is a law-invariant convex risk measure that is also comonotonic. By Theorem 4.88, ). The law-invariance of ρ implies that c(A) is a function of the probability P[ A ], i.e., there exists an increasing function ψ on [0, 1] such that ψ(0) = 0, ψ(1) = 1, and c(A) = ψ(P[ A ]). Note that A∪B and A∩B is a pair of comonotone functions for all A, B ∈ F. Hence, comonotonicity and subadditivity of ρ imply

Proposition 4.75 thus implies that ψ is concave. Corollary 4.77 finally shows that the Choquet integral with respect to c can be identified with a risk measure ρμ, where μ is obtained from ψ via Lemma 4.69.

Now we turn to the characterization of all comonotonic convex risk measures on X . Recall that, for a positively homogeneous monetary risk measure, convexity is equivalent to subadditivity. Also recall that M1,f := M1,f (Ω,F) denotes the set of all finitely additive normalized set functions Q : F → [0, 1], and that EQ[ X ] denotes the integral of X ∈ X with respect to Q ∈ M1,f , as constructed in Theorem A.54.

Theorem 4.94. For the Choquet integral with respect to a normalized monotone set function c, the following conditions are equivalent.

(d) The set function c is submodular.

In this case, Qc is equal to the maximal representing set Qmax for ρ.

Before giving the proof of this theorem, let us state the following corollary, which gives a complete characterization of all comonotonic convex risk measures, and a remark concerning the set Qc in part (c), which is usually called the core of c.

Corollary 4.95. A convex risk measure on X is comonotonic if and only if it arises as the Choquet integral of the loss with respect to a submodular, normalized, and monotone set function c. In this case, c is given by c(A) = ρ(−A ), and ρ has the representation

where Qc = {Q ∈ M1,f | Q[ A ]≤ c(A) for all A ∈ F} is equal to the maximal representing set Qmax.

Proof. Theorems 4.88 and 4.94 state that dc is a comonotonic coherent risk measure, which can be represented as in the assertion, as soon as c is a submo-dular, normalized, and monotone set function. Conversely, any comonotonic convex risk measure ρ is coherent and arises as the Choquet integral of c(A) := ρ(−A ), due to Theorem 4.88. Theorem 4.94 then gives the submodularity of c.

Remark 4.96. Let c be a normalized, monotone, submodular set function. Theorem 4.94 implies in particular that the core Qc of c is nonempty. Moreover, c can be recovered from Qc:

If c has the additional continuity property that c(An) → 0 for any decreasing sequence (An) of events such that then this property is shared by any Q ∈ Qc, and it follows that Q is σ-additive. Thus, the corresponding coherent risk measure admits a representation in terms of σ-additive probability measures. It follows by Lemma 4.21 that ρ is continuous from above.

◊

The proof of Theorem 4.94 requires some preparations. The assertion of the following lemma is not entirely obvious, since Fubini’s theorem may fail if Q ∈ M1,f is not σ-additive.

Lemma 4.97. For X ∈ X and Q ∈ , M1,f the integral EQ[ X ] is equal to the Choquet integral

Proof. It is enough to prove the result for X ≥ 0. Suppose first that is as in Remark 4.92. Then

The result for general X ∈ X follows by approximating X uniformly with Xn which take only finitely many values, and by using the Lipschitz continuity of both EQ[ · ] and with respect to the supremum norm.

Lemma 4.98. Let A1, . . . , An be a partition of Ω into disjoint measurable sets, and suppose that the normalized monotone set function c is submodular. Let Q be the probability measure on F0 := σ(A1, . . . , An) with weights

Then for all F0-measurable and equality holds if the values of X are arranged in decreasing order: x1 ≥· ··≥ xn.

Proof. Clearly, it suffices to consider only the case X ≥ 0. Then Remark 4.92 implies as soon as the values of X are arranged in decreasing order.

Now we prove for arbitrary F0-measurable X ≥ 0. To this end, note that any permutation σ of {1, . . . , n} induces a probability measure Qσ on F0 by applying the definition of Q to the re-labeled partition Aσ(1) . . . , Aσ(n). If σ is a permutation such that xσ(1) ≥· ··≥ xσ(n), then we have and so the assertion will follow if we can prove that EQσ [ X ] ≥ EQ[ X ]. To this end, it is enough to show that EQτ [ X ] ≥ EQ[ X ] if τ is the transposition of two indices i and i + 1 which are such that xi < xi+1, because σ can be represented as a finite product of such transpositions.

Note next that

To compute the probabilities Qτ[ Ak ], let us introduce

Then for k k≠ τi. Hence,

Moreover, and hence due to the submodularity of c. Thus,

Using (4.62), (4.63), and our assumption xi < xi+1 thus yields EQτ [ X ]≥ EQ[ X ].

Proof of Theorem 4.94. (a)⇔(b): According to Proposition 4.84, the property of positive homogeneity is shared by all Choquet integrals, and the implication (b)⇒(a) is obvious.

(b)⇒(c): By Corollary 4.19, ρ(−X) = maxQ∈Qmax EQ[ X ], where Q ∈ M1,f belongs to Qmax if and only if

We will now show that this set Qmax coincides with the set Qc. If Q ∈ Qmax then, in ∫ particular, Q[ A ]≤ A dc = c(A) for all A ∈ F. Hence Q ∈ Qc. Conversely, suppose Q ∈ Qc. If X ≥ 0 then

where we have used Lemma 4.97. Cash invariance yields (4.64).

(c)⇒(b) is obvious.

(b)⇒(d): This follows precisely as in (4.60).

(d)⇒(b): We have to show that the Choquet integral is subadditive. By Lemma 4.3, it is again enough to prove this for random variables which only take finitely many values. Thus, let A1, . . . , An be a partition of Ω into finitely many disjoint measurable sets. Let us write X = i xi, Y = i yi, and let us assume that the indices i = 1, . . . , n are arranged such that x1 + y1 ≥· ··≥ xn + yn. Then the probability measure Q constructed in Lemma 4.98 is such that

But this is the required subadditivity of the Choquet integral.