Dual characterization of properties

of risk measures on Orlicz hearts

Patrick Cheridito ∗

Princeton UniversityPrinceton, NJ 08544, USA

Tianhui Li †

Cambridge UniversityCambridge, CB3 0DS, UK

First version: February 26, 2008. This version: July 18, 2008

Abstract

We extend earlier representation results for monetary risk measures on Orlicz hearts. Then wegive general conditions for such risk measures to be Gateaux-differentiable, strictly monotone withrespect to almost sure inequality, strictly convex modulo translation, strictly convex modulo comono-tonicity, or monotone with respect to different stochastic orders. The theoretical results are used toanalyze various specific examples of risk measures.

Key Words: Risk measures, Gateaux-differentiability, strict monotonicity, strict convexity, stochas-tic orders, Orlicz hearts

1 Introduction

The purpose of this paper is to give characterizations of properties of risk measures that can be used toanalyze particular examples. We first extend earlier representation results for risk measures on Orliczhearts. Then we give general conditions for monetary risk measures to be Gateaux-differentiable,strictly monotone with respect to almost sure inequality, strictly convex modulo translation, strictlyconvex modulo comonotonicity, or monotone with respect to different stochastic orders. The theoreticalresults are applied to study properties of risk measures belonging to different parametric families.

Artzner et al. (1999), Follmer and Schied (2002a, 2002b, 2004), Frittelli and Rosazza Gianin (2002)introduced the notions of coherent, convex and monetary risk measures. The risky objects in Artzneret al. (1999) and Follmer and Schied (2002a, 2002b, 2004) are uncertain financial positions modelled bybounded random variables. Risk measures for unbounded random variables have, among others, beenstudied in Frittelli and Rosazza Gianin (2002, 2004), Delbaen (2002, 2006), Cherny (2006), Rockafellaret al. (2006), Ruszczynski and Shapiro (2006), Filipovic and Kupper (2006), Cheridito and Li (2007),Filipovic and Svindland (2007).

Here, we work with risk measures for random variables belonging to an Orlicz heart. This allowsus to build on duality results of Cheridito and Li (2007). In Section 2 we introduce the setup andextend representation results of Ruszczynski and Shapiro (2006) as well as Cheridito and Li (2007).In Section 3, we give conditions for risk measures to be differentiable in the Gateaux-sense. Section4 provides characterizations of risk measures that are strictly monotone with respect to almost sureinequality. In Section 5 we give conditions for risk measures to be strictly convex modulo translation

We thank Andreas Hamel and Michael Kupper for fruitful discussions and helpful comments.∗Supported by NSF Grant DMS-0642361, a Rheinstein Award and a Peek Fellowship.†Supported by a Marshall Scholarship and a Merage Fellowship.

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or comonotonicity. In Section 6 we discuss monotonicity of risk measures with respect to differentstochastic orders. Section 7 studies properties of risk measures that can be obtained as cash-additivehulls of monotone convex functionals. In Section 8, we analyze five different parametric families of riskmeasures. Two of them were introduced in Cheridito and Li (2007), the others are new.

2 Definitions and preliminaries

We fix a probability space (Ω,F ,P). Equalities X = Y and inequalities X ≥ Y between randomvariables on (Ω,F ,P) are understood in the P-almost sure sense. L0 denotes the space of all randomvariables on (Ω,F ,P), where two random variables are identified if they are P-almost surely equal. Forp ∈ [1,∞), Lp denotes the subspace of L0 consisting of all p-integrable random variables and L∞ thesubspace of L0 of essentially bounded random variables. Let Φ : [0,∞) → [0,∞) be a convex functionwith Φ(0) = 0 and limx→∞Φ(x) = ∞. Then Φ is automatically continuous and increasing (by whichwe mean that Φ(x) ≤ Φ(y) for x ≤ y). Define the function Ψ : [0,∞) → [0,∞] by

Ψ(y) := supx≥0

xy − Φ(x) .

The Orlicz heartMΦ :=

X ∈ L0 : EP [Φ (c|X|)] < ∞ for all c > 0

with the P-almost sure ordering and the Luxemburg norm

‖X‖Φ := inf

λ > 0 : EP

[Φ

( |X|λ

)]≤ 1

is a Banach lattice, whose norm dual is the Orlicz space

LΨ :=ξ ∈ L0 : EP [Ψ (c|ξ|)] < ∞ for some c > 0

with the Orlicz norm‖ξ‖∗Φ := sup

EP [Xξ] : X ∈ LΦ , ‖X‖Φ ≤ 1

,

which is equivalent to the Luxemburg norm ‖·‖Ψ. For proofs of these facts, see, for instance, Edgarand Sucheston (1992).

We call a mapping ρ : MΦ → (−∞,∞] a monetary risk measure on MΦ if it has the followingproperties:

(F) Finiteness at 0: ρ(0) ∈ R(M) Monotonicity: ρ(X) ≥ ρ(Y ) for all X, Y ∈ MΦ such that X ≤ Y(T) Translation property: ρ(X + m) = ρ(X)−m for all X ∈ MΦ and m ∈ RWe call a monetary risk measure ρ on MΦ convex if it also satisfies(C) Convexity: ρ(λX + (1− λ)Y ) ≤ λρ(X) + (1− λ)ρ(Y ) for all X,Y ∈ MΦ and λ ∈ (0, 1)

A convex monetary risk measure ρ on MΦ is called coherent if it fulfills(P) Positive homogeneity: ρ(λX) = λρ(X) for all X ∈ MΦ and λ ≥ 0.

It follows from (F), (M) and (T) that ρ(L∞) ⊂ R. We identify a probability measure Q on (Ω,F) thatis absolutely continuous with respect to P with its Radon–Nykodim derivative ξ = dQ/dP ∈ L1. ByDΨ we denote the set of all Radon–Nykodim densities in LΨ:

DΨ :=ξ ∈ LΨ : ξ ≥ 0 , EP [ξ] = 1

,

2

and of course,Dq := ξ ∈ Lq : ξ ≥ 0 , EP [ξ] = 1 , for q ∈ [1,∞] .

We call a mapping γ : DΨ → (−∞,∞] a penalty function if it is bounded from below and not identicallyequal to ∞. We say a penalty function on DΨ satisfies the growth condition (G) if there exist constantsa ∈ R and b > 0 such that

γ(Q) ≥ a + b ‖Q‖∗Φ for all Q ∈ DΨ . (2.1)

Since ‖.‖Ψ and ‖.‖∗Φ are equivalent norms on LΨ, (2.1) holds if and only if there exist constants a′ ∈ Rand b′ > 0 such that

γ(Q) ≥ a′ + b′ ‖Q‖Ψ for all Q ∈ DΨ .

For a function f : MΦ → (−∞,∞], we denote

dom f :=X ∈ MΦ : f(X) < ∞

.

If f is convex, then so is dom f . Unless otherwise specified, we call f continuous, lower semicontinuousor Lipschitz-continuous if it is so with respect to ‖.‖Φ. By int(A) we denote the interior of a subsetA ⊂ MΦ with respect to the norm-topology, and by core(A) the algebraic interior, that is, the set ofall points x ∈ A with the property that for every y ∈ MΦ, there exists ε > 0 such that x + ty ∈ A forall t ∈ [0, ε]. int(A) is always contained in core(A), but not necessarily the other way around.

We need the following two results on the dual representation of risk measures on Orlicz hearts.They summarize and extend Theorem 2.2 of Ruszczynski and Shapiro (2006) and Theorems 4.5 and4.6 of Cheridito and Li (2007).

Theorem 2.1. Let γ be a penalty function on DΨ. Then

ργ(X) := supQ∈DΨ

EQ [−X]− γ(Q) (2.2)

defines a lower semicontinuous convex monetary risk measure on MΦ, and the following are equivalent:(i) γ satisfies the growth condition (G)(ii) core(dom ργ) 6= ∅(iii) ργ is real-valued and locally Lipschitz-continuous(iv) For each X ∈ MΦ and every sequence (Qn)n≥1 in DΨ satisfying

limn→∞ EQn [−X]− γ(Qn) = ργ(X) ,

the sequences EQn [X] and γ(Qn), n ≥ 1, are bounded.If (i)–(iv) hold and γ is σ(DΨ,MΦ)-lower semicontinuous, then

ργ(X) = maxQ∈DΨ

EQ [−X]− γ(Q) for all X ∈ MΦ . (2.3)

Proof. That (2.2) defines a lower semicontinuous convex monetary risk measure on MΦ is clear. Theequivalence of (i)–(iii) is shown in Theorem 4.5 of Cheridito and Li (2007). Clearly, it follows from (iv)that ργ is real-valued. In particular, one has (iv) ⇒ (ii). So the equivalence of (i)–(iv) is proved if wecan show (iii) ⇒ (iv). To do that, we assume (iii) and choose X ∈ MΦ and a sequence (Qn)n≥1 in DΨ

such thatEQn [−X]− γ(Qn) → ργ(X) ∈ R . (2.4)

Suppose that (γ(Qn))n≥1 is unbounded. Since γ is bounded from below, we can, by possibly passing to asubsequence, assume that γ(Qn) ≥ n, for all n ≥ 1. Then it follows from (2.4) that EQn [−X] /γ(Qn) →1. In particular, EQn [−X] →∞, and therefore,

ργ(2X) ≥ EQn [−2X]− γ(Qn) = EQn [−X] + EQn [−X]− γ(Qn) →∞ ,

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a contradiction to (iii). Hence, (γ(Qn))n≥1 must be bounded, which, by (2.4), implies that also(EQn [X])n≥1 is bounded.

It remains to show (2.3) if γ satisfies (i)–(iv) and is σ(DΨ,MΦ)-lower semicontinuous. So assumethese conditions hold and let X ∈ MΦ. Choose a sequence (Qn)n≥1 in DΨ such that

EQn [−X]− γ(Qn) → ργ(X) .

By (iv), (γ(Qn))n≥1 is bounded, which together with (i) implies that (‖Qn‖∗Φ)n≥1 is bounded. So itfollows from the Alaoglu–Bourbaki theorem that there exists a subsequence of (Qn)n≥1 converging tosome Q ∈ DΨ in σ(DΨ,MΦ). Since γ is σ(DΨ,MΦ)-lower semicontinuous, one obtains

EQ [−X]− γ(Q) ≥ limn→∞EQn [−X]− γ(Qn) = ργ(X) ≥ EQ [−X]− γ(Q) ,

and (2.3) is proved.

For a convex monetary risk measure ρ : MΦ → (−∞,∞], we define the function ρ# : DΨ → (−∞,∞]by

ρ#(Q) := supX∈MΦ

EQ [−X]− ρ(X) .

It is obviously σ(DΨ,MΦ)-lower semicontinuous. Moreover, one has:

Theorem 2.2. For a convex monetary risk measure ρ : MΦ → (−∞,∞], the following hold:(i) If ρ is lower semicontinuous, then ρ# is a penalty function on DΨ with ρ = ρρ#.(ii) If ρ = ργ for a penalty function γ on DΨ, then ρ# is the largest convex, σ(DΨ,MΦ)-lower semi-continuous minorant of γ.(iii) If core(dom ρ) 6= ∅, then ρ is real-valued as well as locally Lipschitz-continuous, and

ρ(X) = maxQ∈DΨ

EQ [−X]− ρ#(Q)

for all X ∈ MΦ .

(iv) If ρ is coherent and core(dom ρ) 6= ∅, then ρ is real-valued as well as Lipschitz-continuous, and

ρ(X) = maxQ∈Q

EQ [−X] for all X ∈ MΦ ,

where Q =Q ∈ DΨ : EQ [X] + ρ(X) ≥ 0 for all X ∈ MΦ

.

Proof. (i): It follows from Theorem 2.2 in Ruszczynski and Shapiro (2006) that ρ = ρρ# , which canonly hold if ρ# is a penalty function on DΨ.

(ii) follows from Theorem 2.3.4 in Zalinescu (2002) like the last part of Theorem 4.6 in Cheriditoand Li (2007).

(iii) follows from Theorems 4.5 and 4.6 of Cheridito and Li (2007).(iv) follows from Corollaries 4.7 and 4.8 of Cheridito and Li (2007).

3 Subdifferentiability and Gateaux-differentiability

Let f : MΦ → (−∞,∞] be a convex function, and denote by f∗ the convex conjugate given by

f∗(ξ) := supX∈MΦ

EP [Xξ]− f(X) , ξ ∈ LΨ . (3.1)

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It is immediate from (3.1) that for fixed X ∈ dom f ,

f(X) + f∗(ξ) ≥ EP [Xξ] for all ξ ∈ LΨ ,

with equality if and only ξ is in the subdifferential

∂f(X) :=ξ ∈ LΨ : f(X + Y )− f(X) ≥ EP [Y ξ] for all Y ∈ MΦ

.

By convexity of f , the directional derivative

f ′(X; Y ) := limε↓0

f(X + εY )− f(X)ε

∈ [−∞,∞]

exists in every direction Y ∈ MΦ. If there exists ξ ∈ LΨ such that

f ′(X; Y ) = EP [Y ξ] for all Y ∈ MΦ ,

we say f is Gateaux-differentiable at X with Gateaux-derivative ∇f(X) = ξ. If it exists, ∇f(X) isunique and

EP [Y∇f(X)] = f ′(X; Y ) ≥ supξ∈∂f(X)

EP [Y ξ] for all Y ∈ MΦ ,

which implies ∂f(X) = ∇f(X). On the other hand, if f is continuous at X and ∂f(X) = ξ forsome ξ ∈ LΨ, then it follows from Theorem 2.4.10 of Zalinescu (2002) that f is Gateaux-differentiableat X with Gateaux-derivative ∇f(X) = ξ.

If ρ is a convex monetary risk measure on MΦ, it follows from the properties (M) and (T) that

ρ∗(ξ) =

ρ# (−ξ) for − ξ ∈ DΨ

∞ for − ξ ∈ LΨ \ DΨ .

The following notation will be convenient:

Definition 3.1. For a convex monetary risk measure ρ on MΦ and a penalty function γ on DΨ, wedenote

χρ(X) :=Q ∈ DΨ : ρ(X) + ρ#(Q) = EQ [−X]

, X ∈ MΦ

χρ,γ(X) :=Q ∈ DΨ : ρ(X) + γ(Q) = EQ [−X]

, X ∈ MΦ

χγ(Q) :=X ∈ MΦ : ργ(X) + γ(Q) = EQ [−X]

, Q ∈ DΨ

MΦγ :=

X ∈ MΦ : ργ(X) + γ(Q) = EQ [−X] for some Q ∈ DΨ

Note that if γ ≥ γ′ are two penalty functions on DΨ which induce the same convex monetary riskmeasure ρ on MΦ, then χρ,γ(X) ⊂ χρ,γ′(X) for all X ∈ MΦ and χγ(Q) ⊂ χγ′(Q) for all Q ∈ DΨ. Inparticular, χρ,γ(X) ⊂ χρ(X) and χγ(Q) ⊂ χρ#(Q) since ρ# is the minimal penalty function of ρ.

In terms of the notions introduced in Definition 3.1, Gateaux-differentiability of convex monetaryrisk measures can be characterized as follows:

Proposition 3.2. For a convex monetary risk measure ρ on MΦ and X ∈ dom ρ, the following hold:(i) χρ(X) = −∂ρ(X)(ii) If ρ is Gateaux-differentiable at X, then ρ is real-valued as well as locally Lipschitz-continuous, andχρ(X) = −∇ρ(X)(iii) If X ∈ core(dom ρ) and χρ(X) = ξ for some ξ ∈ DΨ, then ρ is Gateaux-differentiable at X with∇ρ(X) = −ξ.

5

Proof. (i) is immediate from the definition of χρ and the fact that ∂ρ(X) ⊂ −DΨ.To prove (ii), note that X has to be in core(dom ρ) if ρ is Gateaux-differentiable at X. Then it

follows from Theorem 2.2 that ρ is real-valued and locally Lipschitz-continuous. χρ(X) = −∇ρ(X)follows from ∂ρ(X) = ∇ρ(X) and (i).

If the assumptions of (iii) hold, it follows from (i) that ∂ρ(X) = −ξ. By Theorem 2.2, ρ isreal-valued and continuous on MΦ. So Theorem 2.4.10 of Zalinescu (2002) yields that ρ is Gateaux-differentiable at X with ∇ρ(X) = −ξ.

Remark 3.3. Let ρ be a coherent risk measure on MΦ of the form

ρ(X) = supQ∈Q

EQ [−X]

for a set Q ⊂ DΨ. Then ρ#(Q) = 0 for all Q ∈ Q, and therefore, Q ⊂ χρ(m) for all m ∈ R. So by (ii)of Proposition 3.2, ρ can only be Gateaux-differentiable at m if Q consist of just one element.

4 Strict monotonicity

Definition 4.1. We call a risk measure ρ on MΦ strictly monotone on a subset A of MΦ if ρ(X) > ρ(Y )for all X,Y ∈ A such that X ≤ Y as well as P[X < Y ] > 0, and we denote DΨ

s :=ξ ∈ DΨ : ξ > 0

.

Theorem 4.2. Let γ be a penalty function on DΨ. Then the following are equivalent:(i) ργ is strictly monotone on MΦ

γ

(ii) ργ(X) = maxQ∈DΨsEQ [−X]− γ(Q) for all X ∈ MΦ

γ

(iii) χργ ,γ(X) ⊂ DΨs for all X ∈ MΦ

γ

(iv) χγ(Q) = ∅ for all Q ∈ DΨ \ DΨs

Proof. The implications (iv) ⇔ (iii) ⇒ (ii) are clear. So it suffices to show (ii) ⇒ (i) ⇒ (iii).(ii) ⇒ (i): Let X, Y ∈ MΦ

γ such that X ≤ Y and P[X < Y ] > 0. Then there exists Q ∈ DΨs such

thatργ(Y ) = EQ [−Y ]− γ(Q) < EQ [−X]− γ(Q) ≤ ργ(X) .

(i) ⇒ (iii): Assume there exist X ∈ MΦγ and Q ∈ DΨ \ DΨ

s such that ργ(X) = EQ [−X] − γ(Q).Then there exists A ∈ F with P[A] > 0 and Q[A] = 0. So

ργ(X + 1A) ≥ EQ [−X − 1A]− γ(Q) = EQ [−X]− γ(Q) = ργ(X) ≥ ργ(X + 1A) .

This implies X + 1A ∈ MΦγ and hence contradicts (i).

Remark 4.3. If a risk measure is strictly monotone on MΦ, it is of course also relevant (see Definition3.4 in Delbaen (2002) for the coherent case and Definition 4.32 in Follmer and Schied (2004) for theconvex monetary case). Dual conditions for relevance of coherent risk measures are given in Theorem3.5 of Delbaen (2002). For the convex monetary case, see Lemma 3.22 and Theorem 3.23 in Cheriditoet al. (2006).

5 Strict convexity

In order to define the notion of strict convexity for monetary risk measures, we must first dispense witha trivial case. Call X, Y ∈ MΦ translationally equivalent (denoted X∼tY ) if there exists m ∈ R such

6

that X = Y + m. If ρ is a monetary risk measure on MΦ and X,Y are elements of MΦ such thatX = Y + m for m ∈ R, then

ρ(λX + (1− λ)Y ) = ρ(λ(Y + m) + (1− λ)Y ) = ρ(Y )− λm = λρ(X) + (1− λ)ρ(Y ) (5.1)

for all 0 ≤ λ ≤ 1. So ρ cannot be strictly convex between X and Y .

Definition 5.1. We call a monetary risk measure ρ on MΦ strictly convex modulo translation on asubset A of MΦ if

ρ(λX + (1− λ)Y ) < λρ(X) + (1− λ)ρ(Y )

for all X,Y ∈ A and λ ∈ (0, 1) such that X 6∼tY and λX + (1− λ)Y ∈ A.

It can be seen from (5.1) that a monetary risk measure ρ on MΦ which is strictly convex modulotranslation on a subset A of MΦ is also convex on A.

Proposition 5.2. A monetary risk measure on MΦ which is strictly convex modulo translation on aconvex subset A of MΦ is also strictly monotone on A.

Proof. Assume ρ is a monetary risk measure on MΦ that is strictly convex modulo translation but notstrictly monotone on a convex set A ⊂ MΦ. Then there exist X, Y ∈ A such that X ≤ Y , P[X < Y ] > 0and ρ(X) = ρ(Y ). X and Y cannot be translationally equivalent. But

ρ(Y ) ≤ ρ

(X + Y

2

)≤ ρ(X) = ρ(Y )

and therefore,

ρ

(X + Y

2

)=

ρ(X) + ρ(Y )2

,

a contradiction.

The following theorem gives dual conditions for strict convexity modulo translation. Note that fora penalty function γ : DΨ → (−∞,∞] and Q ∈ DΨ, a random variable X ∈ MΦ is in χγ(Q) if and onlyif X + m is in χγ(Q) for all m ∈ R.

Theorem 5.3. For a penalty function γ on DΨ, the following are equivalent:(i) ργ is strictly convex modulo translation on MΦ

γ

(ii) χργ ,γ(X) \ χργ ,γ(Y ) 6= ∅ for all X, Y ∈ MΦγ such that X 6∼tY

(iii) χργ ,γ(X) ∩ χργ ,γ(Y ) = ∅ for all X,Y ∈ MΦγ such that X 6∼tY

(iv) for all Q ∈ DΨ, χγ(Q) contains at most one element modulo translation.

Proof. The equivalence of (iii) and (iv) is obvious. So it is enough to show (iii) ⇒ (ii) ⇒ (i) ⇒ (iii).(iii) ⇒ (ii) holds since for X ∈ MΦ

γ , χργ ,γ(X) is not empty.(ii) ⇒ (i): Let X, Y ∈ MΦ

γ and λ ∈ (0, 1) such that X 6∼tY and λX + (1 − λ)Y ∈ MΦγ . Then

λX + (1− λ)Y 6∼tX. Thus there exists Q ∈ χργ ,γ(λX + (1− λ)Y ) \ χργ ,γ(X), and we have

ργ(λX + (1− λ)Y ) = EQ [−λX − (1− λ)Y ]− γ(Q)= λ (EQ [−X]− γ(Q)) + (1− λ) (EQ [−Y ]− γ(Q)) < λργ(X) + (1− λ)ργ(Y ) .

(i) ⇒ (iii): Assume there exist X,Y ∈ MΦγ and Q ∈ DΨ such that X 6∼tY and Q ∈ χργ ,γ(X) ∩

χργ ,γ(Y ). Then

ργ

(X + Y

2

)≥ EQ

[−X + Y

2

]− γ(Q) =

12

(ργ(X) + ργ(Y )) ≥ ργ

(X + Y

2

). (5.2)

So (X + Y )/2 ∈ MΦγ , and (5.2) is in contradiction to (i). This shows that (i) implies (iii).

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If ρ is a coherent risk measure on MΦ, it is linear on all rays λX : λ ≥ 0, X ∈ MΦ. So it cannot bestrictly convex modulo translation. But it can be strictly convex modulo weaker equivalence relations,such as, for instance, comonotonicity. We call two random variables X and Y comonotone and writeX∼cY if (X(ω)−X(ω′))(Y (ω)−Y (ω′)) ≥ 0 for P×P-almost all (ω, ω′), and we define strict convexitymodulo comonotonicity analogously to strict convexity modulo translation (see Definition 5.1 above).

Theorem 5.4. For a penalty function γ on DΨ, the following are equivalent:(i) ργ is strictly convex modulo comonotonicity on MΦ

γ

(ii) χργ ,γ(X) \ χργ ,γ(Y ) 6= ∅ for all X, Y ∈ MΦγ such that X 6∼cY

(iii) χργ ,γ(X) ∩ χργ ,γ(Y ) = ∅ for all X,Y ∈ MΦγ such that X 6∼cY

(iv) for all Q ∈ DΨ, χγ(Q) contains at most one element modulo comonotonicity.

Proof. The implications (iii)⇔ (iv), (iii)⇒ (ii) and (i)⇒ (iii) follow exactly as in Theorem 5.3. To prove(ii) ⇒ (i), suppose there exist X, Y ∈ MΦ

γ and λ ∈ (0, 1) such that X 6∼cY and λX + (1− λ)Y ∈ MΦγ .

Assume thatχργ ,γ(λX + (1− λ)Y ) ⊂ χργ ,γ(X) ∩ χργ ,γ(Y ) . (5.3)

Then there existsQ ∈

⋂

µ∈[0,1]

χργ ,γ(µX + (1− µ)Y ) , (5.4)

which impliesργ(µX + (1− µ)Y ) = µργ(X) + (1− µ)ργ(Y ) for all µ ∈ [0, 1] . (5.5)

But for µ0 ∈ (0, 1) small enough, one has µ0X + (1 − µ0)Y 6∼cX. By (5.4), µ0X + (1 − µ0)Y belongsto MΦ

γ . Hence, it follows from (ii) that there exists Q0 ∈ χργ ,γ(µ0X + (1 − µ0)Y ) \ χργ ,γ(X), and weobtain

ργ(µ0X + (1− µ0)Y ) = EQ0 [−µ0X − (1− µ0)Y ]− γ(Q0)= µ0 (EQ0 [−X]− γ(Q0)) + (1− µ0) (EQ0 [−Y ]− γ(Q0)) < µ0ργ(X) + (1− µ0)ργ(Y ) ,

a contradiction to (5.5). So (5.3) cannot hold, that is, there exists Q ∈ χργ ,γ(λX + (1 − λ)Y ) whichdoes not belong to χργ ,γ(X) ∩ χργ ,γ(Y ), and we obtain

ργ(λX + (1− λ)Y ) = EQ [−λX − (1− λ)Y ]− γ(Q)= λ (EQ [−X]− γ(Q)) + (1− λ) (EQ [−Y ]− γ(Q)) < λργ(X) + (1− λ)ργ(Y ) .

6 Risk measures and stochastic orders

For a random variable X ∈ L0 with distribution function FX we denote by qX the right-continuousquantile function from (0, 1) to R given by

qX(y) := infx ∈ R : FX(x) > y

.

Viewed as a random variable on (0, 1) equipped with the Borel sigma-algebra and the Lebesgue measure,qX has the same distribution as X; in particular,

∫ 10 qX(y)dy = EP [X] for X ∈ L1.

We call a function f : R→ R increasing (decreasing) if f(x) ≤ f(y) (f(x) ≥ f(y)) for x ≤ y.

8

Definition 6.1. Let X,Y ∈ L1 and S a class of functions f : R → R. Then we say X dominates Ywith respect to S and write X ºS Y if

EP [f(X)] ≥ EP [f(Y )]

for all f ∈ S such that f(X), f(Y ) ∈ L1. If X ºS Y and X ¹S Y , we call X equivalent to Y withrespect to S and write X ∼S Y . If X ºS Y and X 6∼S Y , we say X strictly dominates Y with respect toS and write X ÂS Y . By i we denote the class of all increasing functions, by cv all concave functions,by icv all increasing concave functions, and by icx all increasing convex functions.

It is immediate from Definition 6.1 that ºi and ºcv are stronger than ºicv and that XºicvY isequivalent to −X¹icx − Y . Moreover, one has the following:

XºiY ⇔ qX(y) ≥ qY (y) for all y ∈ (0, 1) (6.1)

X ºicv Y ⇔ ∫ z0 qX(y)dy ≥ ∫ z

0 qY (y)dy for all z ∈ (0, 1) ⇔ there exists a probabilityspace with random variables X and Y such that F X = FX , F Y = F Y and X ≥ E[Y | X]

(6.2)

X ºcv Y ⇔ X ºicv Y and EP [X] = EP [Y ] ⇔ there exists a probability space withrandom variables X and Y such that F X = FX , F Y = F Y and X = E[Y | X] .

(6.3)

Proofs of these facts and more on stochastic orders can, for instance, be found in Muller and Stoyan(2002), Follmer and Schied (2004) or Shaked and Shanthikumar (2007). It is clear from (6.1)–(6.3)that

X ∼i Y ⇔ X ∼cv Y ⇔ X ∼icv Y ⇔ qX = qY ⇔ FX = F Y , (6.4)

and one obtains from Jensen’s inequality for conditional expectations that EP [X | G] ºcv X for allX ∈ L1 and every sub-sigma-algebra G ⊂ F . Furthermore, if X ∈ L1 is not G-measurable, thenEP [f(EP [X | G])] > EP [f(X)] for all strictly concave functions f such that f(X) ∈ L1. In particular,EP [X | G] Âcv X as well as EP [X | G] Âicv X. On the other hand, if X Âcv Y or X Âicv Y , thenby (6.2)–(6.3), there exists a probability space with random variables X and Y distributed as X andY , respectively, such that X ≥ E[Y | X]. Since F X 6= F Y , one has P[X > E[Y | X]] > 0 orE[Y | X] 6= Y . This shows that EP [f(X)] > EP [f(Y )] for all strictly concave increasing functions fsuch that f(X), f(Y ) ∈ L1.

Definition 6.2. Let ρ be a monetary risk measure on MΦ and S a class of functions f : R → R.Then we call ρ S-monotone if ρ(X) ≥ ρ(Y ) for all X,Y ∈ MΦ such that X ¹S Y . If ρ is S-monotoneand ρ(X) > ρ(Y ) for all X, Y ∈ MΦ such that X ≺S Y , we call ρ strictly S-monotone. If ρ(X) onlydepends on FX , we call ρ distribution-based.

Since ºicv is weaker than ºi and ºcv, an icv-monotone monetary risk measure is also i- and cv-monotone. On the other hand, the extension of Proposition 2.1 of Dana (2005) to Orlicz hearts yieldsthat every cv-monotone monetary risk measure on MΦ is icv-monotone. However, an i-monotonemonetary risk measure is not necessarily cv- or icv-monotone:

Example 6.3. By (6.1), the monetary risk measure value-at-risk VaRα(X) := −qX(α) is i-monotoneand distribution-based but not strictly i-monotone. Also, for every non-constant random variableX ∈ L1, one has X≺cvY := EP [X]. But there exists α ∈ (0, 1) such that VaRα(X) < VaRα(Y ). SoVaRα is not cv-monotone and therefore also not icv-monotone. It is also not convex and hence notcoherent (see Artzner et al., 1999; or Follmer and Schied, 2004).

9

In view of (6.4), every i-, cv- or icv-monotone monetary risk measure is also distribution-based. Onthe other hand, Theorem 4.1 of Dana (2005) (extended to Orlicz hearts) shows that if the probabilityspace is atomless, then every lower semicontinuous distribution-based convex monetary risk measure isicv-monotone. The following example shows that this does not need to be the case if the probabilityspace has atoms:

Example 6.4. Consider a probability space Ω consisting of two elements, ω1 and ω2, and two proba-bility measures P and Q such that

P[ω1] = Q[ω2] =13

and P[ω2] = Q[ω1] =23

.

Then ρ(X) := EQ [−X] defines a continuous strictly monotone coherent risk measure on L1, whichsince P assigns weights unevenly, is distribution-based. Now consider the random variables X, Y, Zgiven by X(ωj) = −j, Y (ωj) = j − 3 and Z = EP [X] = −5/3. Then X≺iY and X≺cvZ butρ(X) = 4/3 < ρ(Y ) = ρ(Z) = 5/3. So ρ is not i-monotone, not cv-monotone, and therefore also noticv-monotone.

6.1 icv-monotone monetary risk measures and sets of acceptable positions

It is well-known that a monetary risk measure ρ : MΦ → (−∞,∞] can be reconstructed from itsacceptance set

C :=X ∈ MΦ : ρ(X) ≤ 0

throughρC(X) := inf m ∈ R : X + m ∈ C , X ∈ MΦ .

Moreover, if B is a subset of MΦ with the following three properties:

for all X ∈ B, the setY ∈ MΦ : Y ≥ X

is contained in B (6.5)

ρB(0) ∈ R (6.6)ρB(X) ∈ (−∞,∞] for all X ∈ MΦ , (6.7)

then ρB is a monetary risk measure on MΦ. In regard to the icv-order, one has the following

Proposition 6.5. The acceptance set C of an icv-monotone monetary risk measure ρ on MΦ has thefollowing property:

for all X ∈ C, the setY ∈ MΦ : YºicvX

is contained in C . (6.8)

On the other hand, for every subset B of MΦ with the properties (6.6)–(6.8), ρB is an icv-monotonemonetary risk measure on MΦ.

Proof. That the acceptance set of an icv-monotone monetary risk measure satisfies (6.8) is clear. Onthe other hand, a subset B of MΦ with the properties (6.6)–(6.8) also satisfies (6.5). So ρB is a monetaryrisk measure, which obviously is icv-monotone.

For fixed X ∈ MΦ, the setZ ∈ MΦ : Z ºicv X

is convex. But this is in general not the case for

sets of the formZ ∈ MΦ : Z ºicv X or Z ºicv Y

for X, Y ∈ MΦ. This allows us to construct the

following example of a non-convex icv-monotone monetary risk measure.

10

Example 6.6. Let (0, 1) with the Borel sigma-algebra and the Lebesgue measure be our probabilityspace. Consider the set B = Z ∈ L1 : Z ºicv X or Z ºicv Y , where

X(x) =

6x− 1 0 < x ≤ 13

6x + 1 13 < x < 1

Y (x) =

6x + 1 0 < x ≤ 13

6x− 1 13 < x < 1

.

Set Z(x) = 12(X + Y )(x) = 6x. Then

∫ 10 qZ(x)dx <

∫ 10 qX(x)dx as well as

∫ t0 qZ(x)dx <

∫ t0 qY (x)dx for

0 < t < 23 . Hence, Z /∈ B, and ρB is a non-convex icv-monotone monetary risk measure on L1.

6.2 Dual representations of icv-monotone convex monetary risk measures

Average value-at-risk at level α ∈ (0, 1),

AVaRα(X) :=1α

∫ α

0VaRy(X) dy = − 1

α

∫ α

0qX(y) dy ,

is a real-valued coherent risk measure on L1 (see Follmer and Schied, 2004). (6.2) shows that it isicv-monotone but not strictly icv-monotone. It has been noted before that AVaR can be used as abuilding block to construct other risk measures; see for instance, Kusuoka (2001), Acerbi (2002, 2004),Follmer and Schied (2004), Frittelli and Rosazza Gianin (2005), Leitner (2005), Dana (2005), or Jouiniet al. (2006). Here, we adapt some of the duality results of Dana (2005) to our setup and combine themwith Theorems 2.1 and 2.2 to derive representation results for icv-monotone risk measures on Orliczhearts. Then we provide characterizations for strict monotonicity, strict convexity modulo translationand strict cv- and icv-monotonicity of icv-monotone risk measures. We are using the following notation:

Definition 6.7. By MΦ(0, 1) we denote the Orlicz heart corresponding to Φ over (0, 1) equipped withthe Borel sigma-algebra and Lebesgue measure. LΨ(0, 1) is the Orlicz space over (0, 1) induced by Ψ.Furthermore, we set

RΦ :=qX : X ∈ MΦ

,

D :=

l : (0, 1) → R+ : l left-continuous, decreasing and∫ 1

0l(y)dy = 1

,

DΨ := D ∩ LΨ(0, 1) and DΨs :=

l ∈ DΨ : l(y) > 0 for all y ∈ (0, 1)

.

If the probability space (Ω,F ,P) over which MΦ is defined is atomless, it supports a random variablethat is uniformly distributed on (0, 1), and RΦ is equal to the convex set

qX : X ∈ MΦ(0, 1)

. But if

(Ω,F ,P) has atoms, then RΦ is smaller thanqX : X ∈ MΦ(0, 1)

and not necessarily convex.

For l ∈ D, define the right-continuous function l : (0, 1] → R+ by

l(y) :=

l(y+) for y ∈ (0, 1)0 for y = 1

.

Then dµ(y) = −ydl(y) induces a probability measure µ on (0, 1] such that

l(y) =∫

[y,1]

1x

dµ(x) and l(y) =∫

(y,1]

1x

dµ(x) for y ∈ (0, 1) .

This provides a bijection between D and the set of probability measures µ on (0, 1]. For given l ∈ DΨ

and X ∈ MΦ, one has

⟨−qX , l⟩

:=∫ 1

0−qX(y)l(y)dy =

∫

(0,1]AVaRy(X)dµ(y) . (6.9)

11

Since (6.9) defines a real-valued coherent risk measure on MΦ, it follows from Theorem 2.2 that it iscontinuous in X with respect to ‖.‖Φ. Together with (6.2), (6.9) shows that for X, Y ∈ MΦ,

X¹icvY ⇔ ⟨qX , l

⟩ ≤ ⟨qY , l

⟩for all l ∈ DΨ . (6.10)

Moreover, for ξ ∈ DΨ, the function lξ given by lξ(y) := qξ(1 − y) belongs to DΨ, and by Hardy–Littlewood’s inequality, ⟨

qX , lξ⟩≤ EP [Xξ] for all X ∈ MΦ ; (6.11)

see Hardy et al. (1988) or Follmer and Schied (2004).

Definition 6.8. We call a mapping ν : DΨ → (−∞,∞] a penalty function on DΨ if it is bounded frombelow and not identically equal to ∞. We say that it satisfies the growth condition (G) if there existconstants a ∈ R and b > 0 such that

ν(l) ≥ a + b ‖l‖Ψ for all l ∈ DΨ .

The following is a variant of Theorem 2.1 that will be useful to construct examples in Section 8.

Theorem 6.9. Let ν be a penalty function on DΨ. Then

ρν(X) := supl∈DΨ

⟨−qX , l⟩− ν(l)

defines a lower semicontinuous icv-monotone convex monetary risk measure on MΦ, and the implica-tions

(i) ⇒ (ii) ⇔ (iii) ⇔ (iv)

hold among the conditions:(i) ν satisfies the growth condition (G)(ii) core(dom ρν) 6= ∅(iii) ρν is real-valued and locally Lipschitz-continuous(iv) For each X ∈ MΦ and every sequence (ln)n≥1 in DΨ satisfying

limn→∞

⟨−qX , ln⟩− ν(ln)

= ρν(X) ,

the sequences⟨qX , ln

⟩and ν(ln), n ≥ 1, are bounded.

If (i) holds and ν is (DΨ, MΦ(0, 1))-lower semicontinuous, then

ρν(X) = maxl∈DΨ

⟨−qX , l⟩− ν(l)

for all X ∈ MΦ . (6.12)

If the underlying probability space (Ω,F ,P) is atomless, then the conditions (i)–(iv) are equivalent.

Proof. That ρν defines a lower semicontinuous icv-monotone convex monetary risk measure on MΦ

follows from the fact that for every l ∈ DΨ,⟨−qX , l

⟩is a continuous icv-monotone coherent risk

measure on MΦ. By Theorem 2.2, ρ# is a penalty function on DΨ with ρν = ρρ# . So, (ii) ⇔ (iii)follows from Theorem 2.1. (iv) ⇒ (ii) is clear, and (iii) ⇒ (iv) can be shown as in the proof of Theorem2.1. If (i) holds, then the penalty function γ : DΨ(0, 1) → (−∞,∞] given by γ(ξ) := ν(lξ) satisfies (G),and due to (6.11), one has

ρν(X) := supl∈DΨ

⟨−qX , l⟩− ν(l)

= sup

ξ∈DΨ(0,1)

EP [−Xξ]− γ(ξ)

12

for all X ∈ MΦ(0, 1). So we obtain from Theorem 2.1 that ρν is real-valued. But then also ρν isreal-valued. This shows (i) ⇒ (ii).

That (i) and (DΨ,MΦ(0, 1))-lower semicontinuity of ν imply (6.12) follows as in the proof of The-orem 2.1.

To conclude the proof, assume that (Ω,F ,P) is atomless. Then it follows from (6.11) that

ρν(X) = supξ∈DΨ

EP [−Xξ]− γ(ξ)

for the penalty function γ : DΨ → (−∞,∞] given by γ(ξ) := ν(lξ). By Theorem 2.1, condition (ii)holds if and only if γ satisfies (G), which is equivalent to saying that ν satisfies (G).

For every convex monetary risk measure ρ on MΦ, we define

ρ†(l) := supX∈MΦ

⟨−qX , l⟩− ρ(X)

, l ∈ DΨ .

Clearly, ρ† is lower semicontinuous with respect to σ(DΨ, RΦ) and therefore also with respect toσ(DΨ,MΦ(0, 1)).

The following is an adaption of Theorem 3.1 in Dana (2005) to our setup:

Theorem 6.10. Let ρ be a lower semicontinuous convex monetary risk measure on MΦ. Then thefollowing are equivalent:

(i) ρ is icv-monotone(ii) ρ#(ξ) = supX∈MΦ

⟨−qX , lξ⟩− ρ(X)

, ξ ∈ DΨ

(iii) ρ#(ξ) ≥ ρ#(ξ′) for ξ, ξ′ ∈ DΨ such that ξ¹cvξ′

(iv) ρ(X) = supξ∈DΨ

⟨−qX , lξ⟩− ρ#(ξ)

, X ∈ MΦ

(v) ρ(X) = supl∈DΨ

⟨−qX , l⟩− ρ†(l)

, X ∈ MΦ

If (i)–(v) hold, then ρ† is the smallest penalty function on DΨ which induces ρ. If ρ is coherent and(i)–(v) hold, then

ρ(X) = supξ∈Q

⟨−qX , lξ

⟩= sup

l∈E

⟨−qX , l⟩

(6.13)

forQ =

ξ ∈ DΨ : EP [Xξ] + ρ(X) ≥ 0 for all X ∈ MΦ

andE =

l ∈ DΨ :

⟨qX , l

⟩+ ρ(X) ≥ 0 for all X ∈ MΦ

.

Proof. The equivalence of (i)–(iv) follows as in the proof of Theorem 3.1 in Dana (2005). The implication(v) ⇒ (i) is a consequence of Theorem 6.9. On the other hand, if (i)–(iv) hold, one has

ρ#(ξ) = ρ†(lξ) for all ξ ∈ DΨ ,

and it follows that

ρ(X) = supξ∈DΨ

⟨−qX , lξ

⟩− ρ#(ξ)

≤ sup

l∈DΨ

⟨−qX , l⟩− ρ†(l)

≤ ρ(X) ,

which implies (v).If (i)–(v) hold, then ρ† must be a penalty function on DΨ. That it is the smallest one which induces

ρ is clear. If ρ is coherent and (i)–(v) hold, then ρ# and ρ† are equal to 0 on the sets Q and E ,respectively, and ∞ otherwise. This shows (6.13).

13

Corollary 6.11. Let ρ : MΦ → (−∞,∞] be an icv-monotone convex monetary risk measure withcore(dom ρ) 6= ∅. Then

ρ(X) = maxξ∈DΨ

⟨−qX , lξ

⟩− ρ#(ξ)

= max

l∈DΨ

⟨−qX , l⟩− ρ†(l)

, X ∈ MΦ , (6.14)

and if ρ is coherent, then

ρ(X) = maxξ∈Q

⟨−qX , lξ

⟩= max

l∈E⟨−qX , l

⟩, X ∈ MΦ , (6.15)

for Q and E as in Theorem 6.10.

Proof. By Theorems 2.2 and 6.10, ρ# and ρ† are penalty functions, and

ρ(X) = maxξ∈DΨ

EP [−Xξ]− ρ#(ξ)

= sup

l∈DΨ

⟨−qX , l⟩− ρ†(l)

for all X ∈ MΦ . (6.16)

Since for all ξ ∈ DΨ and X ∈ MΦ, one has EP [−Xξ] ≤ ⟨−qX , lξ⟩

by Hardy–Littlewood’s inequality(6.11) and ρ#(ξ) = ρ†(lξ) by (ii) of Theorem 6.10, the supremum in (6.16) is attained. This shows(6.14). (6.15) follows from (6.14) since for coherent ρ, the penalty functions ρ# and ρ† are equal to 0on the sets Q and E , respectively, and ∞ otherwise.

To characterize properties of icv-monotone risk measures in terms of elements of DΨ, we need thefollowing definitions:

Definition 6.12. Let ρ be a distribution-based convex monetary risk measure on MΦ and ν a penaltyfunction on DΨ. Then we define the function ρ : RΦ → (−∞,∞] by ρ(qX) := ρ(X), and we denote

χρ(r) :=

l ∈ DΨ : ρ(r) + ρ†(l) = 〈−r, l〉

, r ∈ RΦ

χρ,ν(r) :=

l ∈ DΨ : ρ(r) + ν(l) = 〈−r, l〉

, r ∈ RΦ

χν(l) :=r ∈ RΦ : ρν(r) + ν(l) = 〈−r, l〉 , l ∈ DΨ

MΦν :=

X ∈ MΦ : ρν(X) + ν(l) =

⟨−qX , l⟩

for some l ∈ DΨ

RΦν :=

r ∈ RΦ : ρν(r) + ν(l) = 〈−r, l〉 for some l ∈ DΨ

.

Theorem 6.13. Let ν be a penalty function on DΨ. Then the implications

(i) ⇐ (ii) ⇐ (iii) ⇐ (iv) ⇔ (v)

hold among the conditions:(i) ρν is strictly monotone on MΦ

ν

(ii) ρν is strictly monotone on RΦν

(iii) ρν(r) = maxl∈DΨs〈−r, l〉 − ν(l) for all r ∈ RΦ

ν

(iv) χρν ,ν(r) ⊂ DΨs for all r ∈ RΦ

ν

(v) χν(l) = ∅ for all l ∈ DΨ \ DΨs

If the underlying probability space (Ω,F ,P) has no atoms, then all conditions (i)–(v) are equivalent.

14

Proof. (v) ⇔ (iv) ⇒ (iii) ⇒ (ii) follow as the corresponding implications of Theorem 4.2. (ii) ⇒ (i)is clear. To complete the proof it suffices to show (i) ⇒ (iv) when the underlying probability space(Ω,F ,P) has no atoms. So assume this is the case and (i) holds but there exist r ∈ RΦ

ν and l ∈ DΨ \DΨs

such that ρν(r) = 〈−r, l〉 − ν(l). Then there exists z ∈ (0, 1) such that l(y) = 0 for y ∈ (z, 1). ChooseX ∈ MΦ

ν with qX = r. Since (Ω,F ,P) has no atoms, there exists a subset A ⊂ X ≥ qX(z)

with

P[A] = 1− z. The quantile function of the random variable Y = X + 1A is equal to qX + 1[z,1). So

ρν(Y ) ≥ ⟨−qY , l⟩− ν(l) =

⟨−qX , l⟩− ν(l) = ρν(X) ≥ ρν(Y ) .

But this implies Y ∈ MΦν and ρν(Y ) = ρν(X), a contradiction to (i).

Theorem 6.14. For a penalty function ν on DΨ, the implications

(i), (ii) ⇐ (iii) ⇐ (iv) ⇔ (v)

hold among the conditions(i) ρν is strictly convex modulo translation on MΦ

ν

(ii) ρν is strictly convex modulo translation on RΦν

(iii) χρν ,ν(r) \ χρν ,ν(s) 6= ∅ for all r, s ∈ RΦν such that r 6∼ts

(iv) χρν ,ν(r) ∩ χρν ,ν(s) = ∅ for all r, s ∈ RΦν such that r 6∼ts

(v) for all l ∈ DΨ, χν(l) contains at most one element modulo translation in RΦ.If the underlying probability space (Ω,F ,P) has no atoms, then the conditions (i)–(v) are equivalent.

Proof. The implications (v) ⇔ (iv) ⇒ (iii) are obvious. (iii) ⇒ (ii) follows as the implication (ii) ⇒ (i)of Theorem 5.3. To prove (iii) ⇒ (i), assume there exist X, Y ∈ MΦ

ν and λ ∈ (0, 1) such that X 6∼tY andλX +(1−λ)Y ∈ MΦ

ν . By Lemma 6.15 below, we have qλX+(1−λ)Y 6∼tqX or qλX+(1−λ)Y 6∼tq

Y . Therefore,there exists l ∈ χρν ,ν(qλX+(1−λ)Y ) which does not belong to χρν ,ν(qX) ∩ χρν ,ν(qY ), and we get

ρν(λX + (1− λ)Y ) =⟨−qλX+(1−λ)Y , l

⟩− ν(l)

≤ λ⟨−qX , l

⟩+ (1− λ)

⟨−qY , l⟩− ν(l) < λρν(X) + (1− λ)ρν(Y ) .

If the probability space (Ω,F ,P) has no atoms, it supports a random variable U that is uniformlydistributed on (0, 1). Then the mapping r 7→ r(U) embeds RΦ

ν in MΦν , and (i) implies (ii). Moreover,

RΦ is convex, and (ii) ⇒ (iv) follows as the implication (i) ⇒ (iii) of Theorem 5.3.

Lemma 6.15. Let X,Y ∈ L1 and λ ∈ (0, 1) such that qX∼tqλX+(1−λ)Y∼tq

Y . Then X∼tY .

Proof. Denote Z = Y + EP [X − Y ]. Then

qX∼tqλX+(1−λ)Z∼tq

Z , (6.17)

and EP [X] = EP [λX + (1− λ)Z] = EP [Z], which is equivalent to∫ 1

0qX(y)dy =

∫ 1

0qλX+(1−λ)Z(y)dy =

∫ 1

0qZ(y)dy . (6.18)

(6.17) and (6.18) imply qX = qλX+(1−λ)Z = qZ . So one has

EP [f(X)] = EP [f(λX + (1− λ)Z)] ≥ λEP [f(X)] + (1− λ)EP [f(Z)] = EP [f(X)]

for all concave functions f : R → R such that f(X) ∈ L1. This shows that X = Z and hence,X∼tY .

15

Theorem 6.16. Let ν be a penalty function on DΨ. Then ρν is strictly cv-monotone on MΦν if and

only if for all l ∈ DΨ and r ∈ χν(l), r is a deterministic function of the right-continuous functionl(y) = l(y+).

Proof. To show the “only if”-direction, assume that ρν is strictly cv-monotone on MΦν but there exist

l ∈ DΨ and r ∈ χν(l) such that r is not a deterministic function of l. Then r cannot be σ(l)-measurable,and therefore, r ≺cv E

[r | l

]. Thus

ρ†ν(l) ≤ ν(l) = 〈−r, l〉 − ρν(r) ≤⟨−E

[r | l

], l

⟩− ρν

(E

[r | l

])≤ ρ†ν(l) .

But this implies E[r | l

]∈ χν(l) and therefore, ρν(r) > ρν

(E

[r | l

]), a contradiction.

For the “if”-part, assume ρν is not strictly cv-monotone on MΦν . Then there exist r, s ∈ RΦ

ν suchthat r ≺cv s and ρν(r) ≤ ρν(s). Choose l ∈ χρν ,ν(s) and observe that

ρ†ν(l) ≤ ν(l) = 〈−s, l〉 − ρν(s) ≤ 〈−r, l〉 − ρν(r) ≤ ρ†ν(l) .

It follows that r ∈ χν(l) and 〈r, l〉 = 〈s, l〉. Since r ≺cv s, the continuous function f(z) :=∫ z0 s(y) −

r(y)dy is non-negative and satisfies f(0) = f(1) = 0 as well as max0≤z≤1 f(z) > 0. Let z0 ∈ (0, 1) be amaximizer of f and denote

z1 := max z ≤ z0 : f(z) = f(z0)/2 and z2 := min z ≥ z0 : f(z) = f(z0)/2 .

Then there exist z3 ∈ [z1, z0] and z4 ∈ [z0, z2] such that s(z3) > r(z3) and s(z4) < r(z4). Since s isincreasing, this implies r(z3) < r(z4). But due to 〈r, l〉 = 〈s, l〉, we have

∫ 1

0f(y)dl(y) = −

∫ 1

0l(y)df(y) =

∫ 1

0l(y)(r(y)− s(y))dy = 0 ,

and it follows that l(z1) = l(z3) = l(z4) = l(z2). So r cannot be a deterministic function of l.

Remark 6.17. Lemma 2.3 of Dana (2005) extended to Orlicz hearts yields that for fixed Y ∈ MΦ,X ∈ MΦ : XºicvY

=

X ∈ MΦ : XºcvY

+ MΦ

+ .

This shows that an icv-monotone monetary risk measure ρ on MΦ is strictly icv-monotone on MΦ ifand only if ρ is strictly monotone and strictly cv-monotone on MΦ.

7 Cash-additive hulls

Let V be a mapping from MΦ to (−∞,∞] satisfying the following three properties:

(V1) V (X) ≤ V (Y ) for all X, Y ∈ MΦ such that X ≤ Y(V2) V (λX + (1− λ)Y ) ≤ λV (X) + (1− λ)V (Y ) for all X,Y ∈ MΦ and λ ∈ (0, 1)(V3) for all X ∈ MΦ, infs∈R V (s−X)− s ∈ R and the infimum is attained.

ThenρV (X) := min

s∈RV (s−X)− s

is the largest real-valued convex monetary risk measure on MΦ such that

ρV (X) ≤ V (−X) for all X ∈ MΦ .

We call it the cash-additive hull of the the decreasing convex functional V (−.); see Section 5.1 ofCheridito and Li (2007).

16

Proposition 7.1. Let X ∈ MΦ and sX ∈ R such that ρV (X) = V (sX − X) − sX . If V is Gateaux-differentiable at sX −X, then ρV is Gateaux-differentiable at X with

∇ρV (X) = −∇V (sX −X) .

Proof. If V is Gateaux-differentiable at sX −X, then

(ρV

)′(X; Y ) = lim

ε↓0ρV (X + εY )− ρV (X)

ε

≤ limε↓0

V (sX −X − εY )− V (sX −X)ε

= V ′(sX −X;−Y ) = EP [−Y∇V (sX −X)]

for all Y ∈ MΦ. Since(ρV

)′ (X; .) is sublinear, one also has

(ρV

)′(X; Y ) ≥ − (

ρV)′

(X;−Y ) ≥ EP [−Y∇V (sX −X)] ,

and it follows that ρV is Gateaux-differentiable at X with ∇ρV (X) = −∇V (sX −X).

Proposition 7.2. If V is strictly monotone on domV , then ρV is strictly monotone on MΦ.

Proof. Let X, Y ∈ MΦ with X ≤ Y and P[X < Y ] > 0. Then there exists sX ∈ R such that

ρV (X) = V (sX −X)− sX > V (sX − Y )− sX ≥ ρV (Y ) .

Proposition 7.3. If V is strictly convex modulo translation (comonotonicity) on domV , then ρV isstrictly convex modulo translation (comonotonicity) on MΦ.

Proof. Let X, Y ∈ MΦ, such that X 6∼tY (X 6∼cY ) and λ ∈ (0, 1). There exist sX , sY ∈ R such that

ρV (X) = V (sX −X)− sX and ρV (Y ) = V (sY − Y )− sY .

Then sX −X 6∼tsY − Y (sX −X 6∼csY − Y ) , and therefore

λρV (X) + (1− λ)ρV (Y )= λ V (sX −X)− sX+ (1− λ) V (sY − Y )− sY > V (λsX + (1− λ)sY − [λX + (1− λ)Y ])− [λsX + (1− λ)sY ]≥ ρV (λX + (1− λ)Y ) .

Proposition 7.4. If V is (strictly) icx-monotone on domV , then ρV is (strictly) icv-monotone onMΦ. If V is distribution-based, then so is ρV .

Proof. Assume V is icx-monotone on domV and X ¹icv Y . Then −Xºicx−Y , and there exists sX ∈ Rsuch that

ρV (X) = V (sX −X)− sX ≥ V (sX − Y )− sX ≥ ρV (Y ) .

This shows that ρV is icv-monotone on Mφ. The other claims follow analogously.

17

8 Examples

8.1 Transformed loss risk measures

Let H : R→ R be an increasing convex function with the property

lim|x|→∞

H(x)− x = ∞ .

ThenV (X) = EP [H(X)] (8.1)

is a real-valued mapping on the Orlicz heart MΦ corresponding to the function Φ(x) := H(x)−H(0). Itclearly satisfies (V1)–(V3) and is icx-monotone. So, by Proposition 7.4, ρV is a real-valued icv-monotoneconvex monetary risk measure on MΦ. Its minimal penalty function is given by

(ρV

)#(Q) = EP

[H∗

(dQdP

)], Q ∈ DΨ ; (8.2)

see Section 5.4 of Cheridito and Li (2007). If H is strictly increasing, then V is strictly monotone on MΦ,which by Proposition 7.2 implies that also ρV is strictly monotone on MΦ. If H is strictly convex, thenV is strictly convex and strictly icx-monotone on MΦ, and so by Propositions 7.3 and 7.4, ρV is strictlyconvex modulo translation and strictly icv-monotone on MΦ. If H is differentiable, then V is Gateaux-differentiable on MΦ with ∇V (X) = H ′(X) and it follows from Proposition 7.1 that ρV is Gateaux-differentiable on MΦ with ∇ρV (X) = −H ′(sX −X) for sX ∈ R such that ρV (X) = V (sX −X)− sX .

For H∗(1) = 0, (8.2) is an f-divergence after Csiszar (1967) and can be interpreted as a distancebetween Q and P. Functionals of the form ρV for V equal to (8.1) have appeared in different settingsin Ben-Tal and Teboulle (1987), Schied (2007), Cheridito and Li (2007), Cherny and Kupper (2007).

8.2 Transformed norm risk measures

Let F be a left-continuous increasing convex function from [0,∞) to (−∞,∞] such that limx→∞ F (x) =∞, G : [0,∞) → [0,∞) a convex function with G(0) = 0 and limx→∞G(x) = ∞, and H : R → [0,∞)an increasing convex function with limx→∞H(x) = ∞. Assume the following two conditions hold:

(FGH1) F

(H(x) + ε

G−1(1)

)< ∞ for some x ∈ R and ε > 0

(FGH2) limx→∞

F H(x)−G−1(1) x

= ∞ .

Define H0(x) := H(x) − H(0) for x ≥ 0. Then Φ := G H0 is a convex function from [0,∞) to[0,∞) with Φ(0) = 0 and limx→∞Φ(x) = ∞. In Section 5.2 of Cheridito and Li (2007) it is shownthat V (X) = F (‖H(X)‖G) is a well-defined mapping from MΦ to (−∞,∞] satisfying (V1)–(V3). Itcan easily be checked that it is icx-monotone. So it follows from Proposition 7.4 that ρV defines areal-valued icv-monotone convex monetary risk measure on MΦ. Its minimal penalty function is givenin Theorem 5.3 of Cheridito and Li (2007).

Clearly, the Luxemburg norm ‖.‖Φ is strictly monotone on MΦ+ if and only if Φ is strictly increasing.

So if F, G,H are strictly increasing, then V is strictly monotone, and it follows from Proposition 7.2that the same is true for ρV . If F and G are strictly increasing and H is strictly convex, then V isstrictly convex and strictly icx-monotone, and so by Propositions 7.3 and 7.4, ρV is strictly convexmodulo translation and strictly icv-monotone.

18

As a specific example, consider the risk measure

ρ(X) := mins∈R

1α

∥∥(s−X)+∥∥β

p− s

(8.3)

for (α, β, p) in (0, 1) × 1 × [1,∞) or (0,∞) × (1,∞) × [1,∞). ρ is real-valued on Lp, and if sX ∈ Rminimizes the right side of (8.3), then sX ≥ ess infX. Moreover, for β > 1, sX is unique, sX > ess inf X,and the minimal penalty function of ρ is given by

ρ#(Q) = c ‖Q‖dq for q :=

p

p− 1, d :=

β

β − 1, c := αd−1β1−dd−1 . (8.4)

For β = 1, ρ is coherent, sX is not necessarily unique, and

ρ#(Q) =

0 if ‖Q‖q ≤ 1α

∞ if ‖Q‖q > 1α

for q =p

p− 1(8.5)

(for proofs of (8.4) and (8.5), see Section 5.3 of Cheridito and Li, 2007).If β, p > 1, then V (X) = 1

α ‖X+‖βp is Gateaux-differentiable on Lp with

∇V (X) =β

αEP

[(X+)p

]βp−1 (

X+)p−1

.

Hence, it follows from Proposition 7.1 that ρ is Gateaux-differentiable on Lp with

∇ρ(X) = −β

αEP

[((sX −X)+)p

]βp−1 (

(sX −X)+)p−1

.

By Proposition 3.2, ∇ρ(X) is in −DΨ. So it can be written as

∇ρ(X) = − ((sX −X)+)p−1

E [((sX −X)+)p−1]. (8.6)

For β = 1, p > 1 and X ∈ Lp with P [X = ess inf X] < αp, one easily checks that P [X < sX ] > 0.Hence, V (.) = 1

α ‖(.)+‖p is Gateaux-differentiable at sX −X with

∇V (sX −X) =1α

EP[((sX −X)+)p

] 1p−1 (

(sX −X)+)p−1

,

and it follows from Proposition 7.1 that ρ is Gateaux-differentiable at X with Gateaux-derivative (8.6).If β = 1, p ≥ 1 and X ∈ Lp such that P [X = ess inf X] ≥ αp, then the measure

dQdP

=1X=ess inf X

P [X = ess inf X]

satisfies‖Q‖q ≤

1α

and EQ [−X] = −ess infX .

So Q is a maximizer of the right side of

ρ(X) = maxQ∈Dq , ‖Q‖q≤1/α

EQ [−X] ,

but not necessarily the only one.

19

For β ≥ 1, p = 1 and X ∈ L1 with P [X = sX ] = 0, V (.) = 1αEP [(.)+]β is Gateaux-differentiable at

sX −X with

∇V (sX −X) =β

αEP

[(sX −X)+

]β−1 1X<sX .

So it follows from Proposition 7.1 that ρ is Gateaux-differentiable at X with

∇ρ(X) = − 1X<sXP [X < sX ]

.

If β > 1, p = 1 and X ∈ L1 with P [X = sX ] > 0, the left- and right-derivative of the functions 7→ 1

αEP [(s−X)+]β − s at sX are given by

β

αEP

[(sX −X)+

]β−1 P [X < sX ]− 1 ≤ 0 andβ

αEP

[(sX −X)+

]β−1 P [X ≤ sX ]− 1 ≥ 0 ,

respectively. Choose any random variable ζ such that

0 ≤ ζ ≤ β

αEP

[(sX −X)+

]β−1 1X=sX and EP [ζ] = 1− β

αEP

[(sX −X)+

]β−1 P [X < sX ] .

Thenξ =

β

αEP

[(sX −X)+

]β−1 1X<sX + ζ

is the density of a probability measure Q such that

EQ [−X]− αd−1β1−dd−1 ‖Q‖d∞

= EQ[(sX −X)+

]− β

αdEP

[(sX −X)+

]βd−d − sX

=β

αEP

[(sX −X)+

]β−1 EP[(sX −X)+

]− β

αdEP

[(sX −X)+

]β − sX

=1α

EP[(sX −X)+

]β − sX = ρ(X) .

Thus, it follows from (8.4) that Q maximizes the right side of

ρ(X) = maxQ∈D∞

EQ [−X]− αd−1β1−dd−1 ‖Q‖d

∞

.

But it is not necessarily the only measure in D∞ with this property.For β = p = 1 and X ∈ L1 such that P [X = sX ] > 0, it is well known that the maximizing measures

for ρ at X are of the formdQdP

=1α

1X<sX + ζ ,

where ζ is a random variable satisfying

0 ≤ ζ ≤ 1α

1X=sX and EP [ζ] = 1− 1αP [X < sX ]

(see, Cherny, 2006).Since χρ(X) ⊂ Dq

s does in general not hold for risk measures of the form (8.3), it follows fromTheorem 4.2, that they are not strictly monotone on Lp and hence not strictly convex modulo translationby Proposition 5.2.

20

8.3 Delta spectral measures

Proposition 8.1. Let p ∈ [1,∞) and η : (0, 1] → (−∞,∞] a function that is not identically equal to∞. If there exist constants a ∈ R and b > 0 such that

η(λ) ≥ a + bλ−1/p for all λ ∈ (0, 1] , (8.7)

thenρ(X) = sup

λ∈(0,1]AVaRλ(X)− η(λ) (8.8)

defines a real-valued locally Lipschitz-continuous icv-monotone convex monetary risk measure on Lp.If η satisfies (8.7) and is lower semicontinuous, then the supremum in (8.8) is attained. On the otherhand, if the underlying probability space is atomless and (8.8) is finite for all X ∈ Lp, then (8.7) musthold.

Proof. For each λ ∈ (0, 1], AVaRλ(X) can be written as⟨−qX , lλ

⟩for

lλ(y) = λ−11(0,λ](y) ∈ D .

Set q = p/(p− 1) and define the function ν : Dq → (−∞,∞] by

ν(l) :=

η(λ) if l = lλ for some λ ∈ (0, 1]∞ else

.

Since ‖lλ‖q = λ−1/p, the mapping ν satisfies the growth condition (G) if and only if η fulfills (8.7).Moreover, σ(Dq, Lp(0, 1))-lower semicontinuity of ν is equivalent to lower semicontinuity of η. Hencethe proposition follows from Theorem 6.9.

Example 8.2. For α > 0 and p ∈ [1,∞), η(λ) = αλ−1/p satisfies (8.7) and is continuous on (0, 1]. Soby Proposition 8.1,

ρ(X) = maxλ∈(0,1]

AVaRλ(X)− αλ−1/p

(8.9)

defines a real-valued locally Lipschitz-continuous icv-monotone convex monetary risk measure on Lp.If VaRλ(X) is continuous in λ, then the maximum in (8.9) is either attained at λ = 1 or at λ = λ0

such thatd

dλ

(AVaRλ(X)− αλ−1/p

) ∣∣∣∣λ=λ0

= 0 ,

or equivalently,AVaRλ0(X)−VaRλ0(X) =

α

pλ−1/p0 .

Since lλ /∈ Dqs for λ ∈ (0, 1), it follows from Theorem 6.13 that ρ is in general not strictly monotone.

By Proposition 5.2, it is not strictly convex modulo translation either.

8.4 Uniform spectral measures

Proposition 8.3. Let p ∈ (1,∞) and η : (0, 1] → (−∞,∞] a function which is not identically equal to∞. If there exist constants a ∈ R and b > 0 such that

η(λ) ≥ a + bλ−1/p for all λ ∈ (0, 1] , (8.10)

21

then

ρ(X) = supλ∈(0,1]

1λ

∫ λ

0AVaRy(X)dy − η(λ)

(8.11)

defines a real-valued locally Lipschitz-continuous icv-monotone convex monetary risk measure on Lp. Ifη satisfies (8.10) and is lower semicontinuous, then the supremum in (8.11) is attained. On the otherhand, if the underlying probability space is atomless and (8.11) is finite for all X ∈ Lp, then (8.10)must hold.

Proof. By (6.9), one has for all λ ∈ (0, 1],

1λ

∫ λ

0AVaRy(X)dy =

⟨−qX , lλ⟩

,

where

lλ(y) =

1λ log

(λy

)for y ≤ λ

0 for y > λ.

Set q = p/(p− 1) and define the function ν : Dq → (−∞,∞] by

ν(l) :=

η(λ) if l = lλ for some λ ∈ (0, 1]∞ else

.

With the change of variables x = log(λ/y), one obtains

∫ λ

0logq

(λ

y

)dy =

∫ ∞

0λxqe−xdx = λΓ(q + 1) .

So‖lλ‖q = Γ(q + 1)1/qλ1/q−1 = Γ(q + 1)1/qλ−1/p ,

and the proposition follows from Theorem 6.9 like Proposition 8.1.

Example 8.4. For α > 0 and p ∈ (1,∞), the function η(λ) = αλ−1/p satisfies (8.10) and is continuouson (0, 1]. So it follows from Proposition 8.3 that

ρ(X) = maxλ∈(0,1]

1λ

∫ λ

0AVaRy(X)dy − αλ−1/p

(8.12)

is a real-valued locally Lipschitz-continuous icv-monotone convex monetary risk measure on Lp.Since for each X ∈ Lp, AVaRλ(X) is continuous in λ, the maximum in (8.12) is either attained at

λ = 1 or λ = λ0 satisfying

d

dλ

(1λ

∫ λ

0AVaRy(X)dy − αλ−1/p

) ∣∣∣∣λ=λ0

= 0 ,

or1λ0

∫ λ0

0AVaRy(X) dy −AVaRλ0(X) =

α

pλ−1/p0 .

Again, lλ /∈ Dqs for λ ∈ (0, 1). Hence, by Theorem 6.13, ρ is in general not strictly monotone, and by

Proposition 5.2, not strictly convex modulo translation either.

22

8.5 Power spectral measures

Proposition 8.5. Let p ∈ (1,∞), q = p/(p− 1) and η a mapping from [0, 1/q) to (−∞,∞] that is notidentically equal to ∞. If there exist constants a ∈ R and b > 0 such that

η(λ) ≥ a + b(1− qλ)−1/q for all λ ∈ [0, 1/q) , (8.13)

then

ρ(X) = supλ∈[0,1/q)

∫ 1

0AVaRy(X)(1− λ)y−λdy − η(λ)

(8.14)

is a real-valued locally Lipschitz-continuous icv-monotone convex monetary risk measure on Lp. If ηsatisfies (8.13) and is lower semicontinuous, then the supremum in (8.14) is attained. On the otherhand, if the underlying probability space is atomless and (8.14) is finite for all X ∈ Lp, then (8.13)must hold.

Proof. By (6.9), one has ∫ 1

0AVaRy(X)(1− λ)y−λdy =

⟨−qX , lλ⟩

,

for all λ ∈ [0, 1), where

l0(y) = log(

1y

)and lλ(y) =

1− λ

λ(y−λ − 1) for λ ∈ (0, 1) .

It can easily be checked that the mapping λ 7→ ‖lλ‖q is continuous on [0, 1/q), and for λ ↑ 1/q, one has

1− λ

λ

[(1− qλ)−1/q − 1

]=

1− λ

λ

(∥∥∥y−λ∥∥∥

q− 1

)≤ ‖lλ‖q ≤

1− λ

λ

∥∥∥y−λ∥∥∥

q=

1− λ

λ(1− qλ)−1/q .

So the proposition follows from Proposition 6.9 like Propositions 8.1 and 8.3.

Example 8.6. Let p ∈ (1,∞), q = p/(p − 1), α > 0 and β ≥ 1/q. Then η(λ) = α(1 − qλ)−β satisfies(8.13) and is continuous on [0, 1/q). Hence, it follows from Proposition 8.5 that

ρ(X) = maxλ∈[0,1/q)

∫ 1

0AVaRy(X)(1− λ)y−λdy − α(1− qλ)−β

(8.15)

is a real-valued locally Lipschitz-continuous icv-monotone convex monetary risk measure on Lp.The maximum in (8.15) is either attained at λ = 0 or λ = λ0 satisfying

d

dλ

(∫ 1

0AVaRy(X)(1− λ)y−λdy − α(1− qλ)−β

) ∣∣∣∣λ=λ0

= 0 ,

which is equivalent to∫ 1

0AVaRy(X) [1 + (1− λ0) log(y)] y−λ0dy + αβq(1− qλ0)−β−1 = 0 .

Since lλ ∈ Dqs for all λ ∈ [0, 1/q), it follows from Theorem 6.13 that ρ is strictly monotone on Lp.

However, it is possible that there exist X 6∼tY in Lp for which the maximum in (8.15) is attained at thesame λ0 ∈ [0, 1/q). This means that χρ(qX)∩ χρ(qY ) 6= ∅. Hence, by Theorem 6.14, ρ is in general notstrictly convex modulo translation.

23

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