4.2Robust representation of convex risk measures

In this section, we consider a situation of Knightian uncertainty, where no probability measure P is fixed on the measurable space (Ω,F). Let X denote the space of all bounded measurable functions on (Ω,F). Recall that X is a Banach space if endowed with the supremum norm · . As in Section 2.5, we denote by M1 := M1(Ω,F) the set of all probability measures on (Ω,F) and by M1,f := M1,f (Ω,F) the set of all finitely additive set functions Q : F → [0, 1] which are normalized to Q[ Ω ] = 1. By EQ[ X ] we denote the integral of X with respect to Q ∈ M1,f ; see Appendix A.6.

If ρ is a coherent risk measure on X , thenwe are in the context of Proposition 2.84, i.e., the functional ϕ defined by ϕ(X) := −ρ(X) satisfies the four properties listed in Proposition 2.83. Hence, we have the following result:

Proposition 4.15. A functional ρ : X → ℝ is a coherent risk measure if and only if there exists a subset Q of M1,f such that

Moreover, Q can be chosen as a convex set for which the supremum in (4.16) is attained.

Our first goal in this section is to obtain an analogue of this result for convex risk measures. Applied to a coherent risk measure, it will yield an alternative proof of Proposition 4.15, which does not depend on the discussion in Chapter 2. Moreover, it will provide a description of the maximal set Q in (4.16). Our second goal will be to obtain criteria which guarantee that a risk measure can be represented in terms of σ-additive probability measures.

Let α : M1,f → ℝ ∪ {+∞} be any functional such that

For each Q ∈ M1,f the functional is convex, monotone, and cash invariant on X , and these three properties are preserved when taking the supremum over Q ∈ M1,f . Hence,

defines a convex risk measure on X such that

The functional α will be called a penalty function for ρ on M1,f , and we will say that ρ is represented by α on M1,f .

Theorem 4.16. Any convex risk measure ρ on X is of the form

where the penalty function αmin is given by

Moreover, αmin is the minimal penalty function which represents ρ, i.e., any penalty function α for which (4.17) holds satisfies α(Q) ≥ αmin(Q) for all Q ∈ M1,f .

Proof. In a first step, we show that

To this end, recall that Xʹ := ρ(X) + X ∈ Aρ by (4.1). Thus, for all Q ∈ M1,f

From here, our claim follows.

For X given, we will now construct some Q X ∈ M1,f such that

which, in view of the previous step, will prove our representation (4.18). By cash invariance it suffices to prove this claim for X ∈ X with ρ(X) = 0. Moreover, we may assume without loss of generality that ρ(0) = 0. Then X is not contained in the nonempty convex set

Since B is open due to Lemma 4.3, we may apply the separation argument in the form of Theorem A.58. It yields a nonzero continuous linear functional on X such that

We claim that (Y) ≥ 0 if Y ≥ 0. Monotonicity and cash invariance of ρ imply that 1 + λY ∈ B for any λ > 0. Hence,

which could not be true if (Y) < 0.

Our next claim is that (1) > 0. Since does not vanish identically, there must be some Y such that 0 < (Y) = (Y+)− (Y−). We may assume without loss of generality that Y< 1. Positivity of implies (Y+) > 0 and (1−Y+)≥ 0. Hence (1) = (1−Y+)+ (Y+) > 0.

By the two preceding steps and Theorem A.54, we conclude that there exists some QX ∈ M1,f such that

Note that B ⊂ Aρ, and so

On the other hand, Y +ε ∈ B for any Y ∈ Aρ and each ε > 0. This shows that αmin(QX) is in fact equal to −b/(1). It follows that

Thus, QX is as desired, and the proof of the representation (4.18) is complete.

Finally, let α be any penalty function for ρ. Then, for all Q ∈ M1,f and X ∈ X

and hence

Thus, α dominates αmin.

Remark 4.17. (a) If we take α = αmin in (4.19), then all inequalities in (4.19) must be identities. Thus, we obtain an alternative formula for the minimal penalty function αmin:

(b) Note that αmin is convex and lower semicontinuous for the total variation distance on M1,f as defined in Definition A.53, since it is the supremum of affine continuous functions on M1,f .

(c) Suppose ρ is defined via ρ := ρA for a given acceptance set A ⊂ X . Then A determines αmin:

This follows from the fact that X ∈ Aρ implies ε + X ∈ A for all ε > 0.

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Remark 4.18. Equation (4.20) shows that the penalty function αmin corresponds to the Fenchel–Legendre transform, or conjugate function, of the convex function ρ on the Banach space X . More precisely,

where ρ∗ : Xʹ → ℝ ∪ {+∞} is defined on the dual Xʹ of X by

and where Q ∈ Xʹ is given by Q(X) = EQ[ −X ] for Q ∈ M1,f . This suggests an alternative proof of Theorem 4.16. First note that, by Theorem A.54, Xʹ can be identified with the space ba := ba(Ω,F) of finitely additive set functions with finite total variation. Moreover, ρ is lower semicontinuous with respect to the weak topology σ(X , Xʹ), since any set {ρ ≤ c} is convex, strongly closed due to Lemma 4.3, and hence weakly closed by Theorem A.63. Thus, the general duality theorem for conjugate functions as stated in Theorem A.65 yields

where ρ∗∗ denotes the conjugate function of ρ∗, i.e.,

In a second step, using the arguments in the second part of the proof of Theorem 4.16, we can now check that monotonicity and cash invariance of ρ imply that ≤ 0 and (1) = −1 for any ∈ Xʹ = ba such that ρ∗() < ∞. Identifying − with Q ∈ M1,f and using equation (4.21), we see that (4.22) reduces to the representation

Moreover, the supremum is actually attained: M1,f is weak∗ compact in Xʹ = ba due to the Banach-Alaoglu theorem stated in Theorem A.66, and so the upper semicontinuous functional attains its maximum on M1,f.

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The representation

of a coherent risk measure ρ via some set Q ⊂ M1,f , as formulated in Proposition 4.15, is a particular case of the representation theorem for convex risk measures, since it corresponds to the penalty function

The following corollary shows that the minimal penalty function of a coherent risk measure is always of this type.

Corollary 4.19. The minimal penalty function αmin of a coherent risk measure ρ takes only the values 0 and +∞. In particular,

for the convex set

and Qmax is the largest set for which a representation of the form (4.23) holds.

Proof. Recall from Proposition 4.6 that the acceptance set Aρ of a coherent risk measure is a cone. Thus, the minimal penalty function satisfies

for all Q ∈ M1,f and λ > 0. Hence, αmin can take only the values 0 and +∞.

Exercise 4.2.1. Let ρ be a coherent risk measure on X and assume that ρ admits a representation

with some class Q ⊂ M1,f .

(a) Show that ρ(X) + ρ(−X) ≥ 0 for all X ∈ X .

(b) Show that the following conditions are equivalent.

(i) ρ is additive, i.e., ρ(X + Y) = ρ(X) + ρ(Y) for all X, Y ∈ X .

(ii) ρ(X) + ρ(−X) = 0 for all X ∈ X .

(iii) The class Q reduces to a single element Q, i.e., ρ is simply the expected loss with respect to Q.

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The preceding exercise can be generalized as follows.

Exercise 4.2.2. Consider a normalized convex risk measure on X with representation

for some penalty function α. Suppose that ρ is additive on some linear subspace Y ⊂ X , that is, ρ(X + Y) = ρ(X) + ρ(Y) for X, Y ∈ Y .

(a) Show that

(b) Conclude that ρ is Y -additive on X , that is,

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The penalty function α arising in the representation (4.17) is not unique, and it is often convenient to represent a convex risk measure by a penalty function that is not the minimal one. For instance, the minimal penalty function may be finite for certain finitely additive set functions while another α is concentrated only on probability measures as in the case of Example 4.8. Another situation of this type occurs for risk measures which are constructed as the supremum of a family of convex risk measures:

Proposition 4.20. Suppose that for every i in some index set I we are given a convex risk measure ρi on X with associated penalty function αi. If supi∈I ρi(0) < ∞ then

is a convex risk measure that can be represented with the penalty function

Proof. The condition ρ(0) = supi∈I ρi(0) < ∞ implies that ρ takes only finite values. Moreover,

and the assertion follows.

In the sequel, we are particularly interested in those convex measures of risk which admit a representation in terms of σ-additive probability measures. Such a risk measure ρ can be represented by a penalty function α which is infinite outside the set M1 := M1(Ω,F):

In this case, one can no longer expect that the supremum above is attained. This is illustrated by Example 4.8 if X does not take on its infimum.

A representation (4.24) in terms of probability measures is closely related to certain continuity properties of ρ. We first examine a necessary condition of “continuity from above”.

Lemma 4.21. A convex risk measure ρ which admits a representation (4.24) on M1 is continuous from above in the sense that

Moreover, continuity from above is equivalent to the “Fatou property” of lower semicontinuity with respect to bounded pointwise convergence: If (Xn) is a bounded sequence in X which converges pointwise to X ∈ X , then

Proof. First we show (4.26) under the assumption that ρ has a representation in terms of probability measures. Dominated convergence implies that EQ[ Xn ] → EQ[ X ] for each Q ∈ M1. Hence,

In order to show the equivalence of (4.26) and (4.25), let us first assume (4.26). By monotonicity, ρ(Xn)≤ ρ(X) for each n if Xn ↘ X, and so ρ(Xn) ↗ ρ(X) follows.

Now we assume continuity from above. Let (Xn) be a bounded sequence in X which converges pointwise to X. Define Ym := supn≥m Xn ∈ X . Then Ym decreases to X. Since ρ(Xn)≥ ρ(Yn) by monotonicity, condition (4.25) yields that

The following theorem gives a strong sufficient condition which guarantees that any penalty function for ρ is concentrated on the set M1 of probability measures. This condition is “continuity from below” rather than from above; we will see a class of examples in Section 4.9.

Theorem 4.22. For a convex risk measure ρ on X , the following two conditions are equivalent.

(a) ρ is continuous from below in the sense that

(b) The minimal penalty function αmin (and hence every other penalty function representing ρ) is concentrated on the class M1 of probability measures, i.e.,

In particular we have

whenever one of these two equivalent conditions is satisfied.

For the proof of this theorem we need the following lemma.

Lemma 4.23. Let ρ be a convex risk measure on X which is represented by the penalty function α on M1,f , and consider the level sets

For any sequence (Xn) in X such that 0 ≤ Xn ≤ 1, the following two conditions are

Proof. (a)⇒(b): In a first step, we show that for all Y ∈ X

Indeed, since α represents ρ, we have for Q ∈ Λc

and dividing by −λ yields (4.27).

Now consider a sequence (Xn) which satisfies (a). Then (4.27) shows that for all λ ≥ 1

Taking λ ↑ ∞and assuming X n ≤ 1 proves (b).

(b)⇒(a): Clearly, for all n

Since EQ[ −λXn ]≤ 0 for all Q, only those Q can contribute to the supremum on the right-hand side for which

Hence, for all n

But condition (b) implies that EQ[ −λXn ] converges to −λ uniformly in Q ∈ Λc, and so (a) follows.

The proof of our theorem will also rely on Dini’s lemma, which we recall here for the convenience of the reader.

Lemma 4.24. On a compact set, a sequence of continuous functions fn increasing to a continuous function f converges even uniformly.

Proof. For ε > 0, the compact sets satisfy Hence there must be some n0 ∈ ℕ such that Kn = ∅ for all n ≥ n0.

Proof of Theorem 4.22. To prove the implication (a)⇒(b), recall that Q is σ-additive if and only if Q[ An ] ↗ 1 for any increasing sequence of events An ∈ F such that Thus, our claim is implied by the implication (a)⇒(b) of Lemma 4.23 if we take Xn := . An

We now prove the implication (b)⇒(a) of our theorem. Suppose X n ↗ X pointwise on Ω. We need to show that ρ(Xn) ↘ ρ(X). By cash invariance, wemay assume without loss of generality that X n ≥ 0 for all n. As in the proof of the implication (a)⇒(b) of Lemma 4.23, we see that

where c := 1 − ρ(X) and Λc = {Q ∈ M1,f | αmin(Q) ≤ c}. We will show below that Λc ⊂ M1 implies that

Together with the representation (4.28), this will imply the desired convergence ρ(Xn) → ρ(X).

To prove (4.29), recall first from Appendix A.6, and in particular from Definition A.53, that M1,f belongs to the larger vector space ba := ba(Ω,F) of all finitely additive set functions μ : F → ℝ with finite total variation μvar. In fact, ba can be identified with the topological dual of the Banach space X with respect to · ; see Theorem A.54. Since we can write

it is clear that M1,f is a bounded and weak∗ closed set in ba. Hence M1,f is weak∗ compact by the Banach-Alaoglu theorem (see Theorem A.66). Since αmin is weak∗ lower semicontinuous on M1,f as supremum of the weak∗ continuous maps with Y ∈ Aρ, the level set Λc is also weak∗ compact.

After these preparations, we can now prove (4.29). Clearly, the functions n(Q) := EQ[ −Xn ] form a decreasing sequence of weak∗ continuous functions on Λc. Moreover,if Λc ⊂ M1, we even have n(Q) ↘ (Q) := EQ[ −X ] for each Q ∈ Λc, due to the monotone convergence theorem. By the established compactness of Λc, (4.29) thus follows from Dini’ s lemma.

Remark 4.25. Let ρ be a convex risk measure which is continuous from below. Then ρ is also continuous from above, as can be seen by combining Theorem 4.22 and Lemma 4.21.

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Exercise 4.2.3. Show that for a convex risk measure ρ on X the following two conditions are equivalent.

(a) ρ is continuous from below.

(b) ρ satisfies the following “Lebesgue property”: whenever (Xn) is a bounded sequence in X which converges pointwise to X.

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Exercise 4.2.4. Show that for a convex risk measure ρ on X the following two conditions are equivalent.

(a) ρ is continuous from below.

(b) For every c > −ρ(0), the coherent risk measures

are continuous from below, where Λc = {Q ∈ M1,f | αmin(Q) ≤ c}.

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Example 4.26. Let us consider a utility function u on ℝ, a probability measure Q ∈ M1(Ω,F), and fix some threshold c ∈ ℝ. As in Example 4.10, we suppose that a position X is acceptable if its expected utility E Q[ u(X) ] is bounded from below by u(c). Alternatively, we can introduce the convex increasing loss function (x) = −u(−x) and define the convex set of acceptable positions

where x0 := −u(c). Let ρ := ρA denote the convex risk measure induced by A . In Section 4.9, we will show that ρ is continuous from below, and we will derive a formula for its minimal penalty function.

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Let us now continue the discussion in a topological setting. More precisely, we will assume for the rest of this section that Ω is a separable metric space and that F is the σ-field of Borel sets. As before, X is the linear space of all bounded measurable functions on (Ω,F). We denote by Cb(Ω) the subspace of bounded continuous functions on Ω, and we focus on the representation of convex risk measures viewed as functionals on Cb(Ω).

Proposition 4.27. Let ρ be a convex risk measure on X such that

Then there exists a penalty function α on M1 such that

In fact, one can take

Proof. Let αmin be the minimal penalty function of ρ on M1,f . We show that for any with αmin() < ∞ there exists Q ∈ M1 such that E[ X ] = EQ[ X ] for all X ∈ Cb(Ω). Take a sequence (Yn) in C b(Ω) which increases to some Y ∈ C b(Ω), and choose δ > 0 such that Xn := 1 + δ(Yn − Y) ≥ 0 for all n. Clearly, (Xn) satisfies condition (a) of Lemma 4.23, and so E[ Xn ] → 1, i.e.,

This continuity property of the linear functional EQ[ · ] on Cb(Ω) implies, via the Daniell–Stone representation theorem as stated in Appendix A.6, that it coincides on C b(Ω) with the integral with respect to a σ-additive measure Q. Taking α as in (4.32) gives the result.

Remark 4.28. If Ω is compact then any convex risk measure admits a representation (4.31) on the space C b(Ω) = C(Ω). Indeed, if (Xn) is a sequence in C b(Ω) that increases to a constant λ, then this convergence is even uniform by Lemma 4.24. Since ρ is Lipschitz continuous on C(Ω) by Lemma 4.3, it satisfies condition (4.30).

Alternatively, we could argue as in Remark 4.18 and apply the general duality theorem for the Fenchel–Legendre transform to the convex functional ρ on the Banach space C(Ω). Just note that any continuous functional on C(Ω) which is positive and normalized is of the form (X) = EQ[ X ] for some probability measure Q ∈ M1; see Theorem A.51.

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Definition 4.29. A convex risk measure ρ on X is called tight if there exists an increasing sequence K1 ⊂ K2 ⊂ · · · of compact subsets of Ω such that

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Note that every convex risk measure is tight if Ω is compact.

Proposition 4.30. Suppose that the convex risk measure ρ on X is tight. Then (4.30) holds and the conclusion of Proposition 4.27 is valid. Moreover, if Ω is a Polish space and α is a penalty function on M1 such that

then the level sets Λc = {Q ∈ M1 | α(Q) ≤ c} are relatively compact for the weak topology on M1.

Proof. First we show (4.30). Suppose Xn ∈ Cb(Ω) are such that Xn ↗ λ > 0. We may assume without loss of generality that ρ is normalized. Convexity and normalization guarantee that condition (4.30) holds for all λ > 0 as soon as it holds for all λ ≥ c where c is an arbitrary constant larger than 1. Hence, the cash invariance of ρ implies that there is no loss of generality in assuming Xn ≥ 0 for all n. We must show that ρ(Xn)≤ ρ(λ) + 2ε eventually, where we take ε ∈ (0, λ − 1).

By assumption, there exists a compact set K N such that

By Dini’s lemma, as recalled in Lemma 4.24, there exists some n0 ∈ ℕ such that λ − ε ≤ X n on K N for all n ≥ n0. Finally, monotonicity implies

To prove the relative compactness of Λc, we will show that for any ε > 0 there exists a compact set Kε ⊂ Ω such that for all c > 0

The relative compactness of Λc will then be an immediate consequence of Prohorov’s characterization of weakly compact sets in M1, as stated in Theorem A.45. We fix a countable dense set {ω1, ω2, . . . } ⊂ Ω and a complete metric δ which generates the topology of Ω. For r > 0 we idefine rcontinuous functions Δon Ω by

The function is dominated by the indicator function of the closed metric ball

Let

Clearly,

is continuous and satisfies as well as for n ↑ ∞.

According to (4.27), we have for all λ > 0

Now we take λk := 2k/ε and rk := 1 /k. The first part of this proof and (4.30) yield the existence of n k ∈ ℕ such that

and thus

We let

Then, for each Q ∈ Λc

The reader may notice that Kε is closed, totally bounded and, hence, compact. A short proof of this fact goes as follows: Let (xj) be a sequence in Kε. We must show that (xj) has a convergent subsequence. Since Kε is covered by Brk (ω1), . . . , Brk (ωnk ) for each k, there exists some ik ≤ nk such that infinitely many xj are contained in Brk (ωik ). A diagonalization argument yields a single subsequence (xj) which for each k is contained in some Brk (ωik ). Thus, (xj) is a Cauchy sequence with respect to the complete metric δ and, hence, converging to some element ω ∈ Ω.

Remark 4.31. Note that the representation (4.31) does not necessarily extend from C b(Ω) to the space X of all bounded measurable functions. Suppose in fact that Ω is compact but not finite. Then condition (4.30) holds as explained in Remark 4.28. Moreover, there is a finitely additive Q0 ∈ M1,f which does not belong to M1; see Example A.56. The proof of Proposition 4.27 shows that there is some ∈ M1 such that the coherent risk measure ρ defined by ρ(X) := EQ0[ −X ] coincides with EQ[ −X ] for X ∈ C b(Ω). But ρ does not admit a representation of the form

In fact, this would imply

for Q ∈ M1 and any X ∈ X , hence α(Q) = ∞for any Q ∈ M1.

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