Download Dual characterization of properties of risk measures on Orlicz hearts

Dual characterization of properties
of risk measures on Orlicz hearts
Patrick Cheridito
∗
Tianhui Li
Princeton University
Princeton, NJ 08544, USA
†
Cambridge University
Cambridge, 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 we
give general conditions for such risk measures to be Gâteaux-differentiable, strictly monotone with
respect to almost sure inequality, strictly convex modulo translation, strictly convex modulo comonotonicity, or monotone with respect to different stochastic orders. The theoretical results are used to
analyze various specific examples of risk measures.
Key Words: Risk measures, Gâteaux-differentiability, strict monotonicity, strict convexity, stochastic orders, Orlicz hearts
1
Introduction
The purpose of this paper is to give characterizations of properties of risk measures that can be used to
analyze particular examples. We first extend earlier representation results for risk measures on Orlicz
hearts. Then we give general conditions for monetary risk measures to be Gâteaux-differentiable,
strictly monotone with respect to almost sure inequality, strictly convex modulo translation, strictly
convex modulo comonotonicity, or monotone with respect to different stochastic orders. The theoretical
results are applied to study properties of risk measures belonging to different parametric families.
Artzner et al. (1999), Föllmer and Schied (2002a, 2002b, 2004), Frittelli and Rosazza Gianin (2002)
introduced the notions of coherent, convex and monetary risk measures. The risky objects in Artzner
et al. (1999) and Föllmer and Schied (2002a, 2002b, 2004) are uncertain financial positions modelled by
bounded random variables. Risk measures for unbounded random variables have, among others, been
studied in Frittelli and Rosazza Gianin (2002, 2004), Delbaen (2002, 2006), Cherny (2006), Rockafellar
et al. (2006), Ruszczyński and Shapiro (2006), Filipović and Kupper (2006), Cheridito and Li (2007),
Filipović and Svindland (2007).
Here, we work with risk measures for random variables belonging to an Orlicz heart. This allows
us to build on duality results of Cheridito and Li (2007). In Section 2 we introduce the setup and
extend representation results of Ruszczyński and Shapiro (2006) as well as Cheridito and Li (2007).
In Section 3, we give conditions for risk measures to be differentiable in the Gâteaux-sense. Section
4 provides characterizations of risk measures that are strictly monotone with respect to almost sure
inequality. 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 different
stochastic orders. Section 7 studies properties of risk measures that can be obtained as cash-additive
hulls of monotone convex functionals. In Section 8, we analyze five different parametric families of risk
measures. 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 random
variables on (Ω, F, P) are understood in the P-almost sure sense. L0 denotes the space of all random
variables on (Ω, F, P), where two random variables are identified if they are P-almost surely equal. For
p ∈ [1, ∞), Lp denotes the subspace of L0 consisting of all p-integrable random variables and L∞ the
subspace of L0 of essentially bounded random variables. Let Φ : [0, ∞) → [0, ∞) be a convex function
with Φ(0) = 0 and limx→∞ Φ(x) = ∞. Then Φ is automatically continuous and increasing (by which
we mean that Φ(x) ≤ Φ(y) for x ≤ y). Define the function Ψ : [0, ∞) → [0, ∞] by
Ψ(y) := sup {xy − Φ(x)} .
x≥0
The Orlicz heart
©
ª
M Φ := X ∈ L0 : EP [Φ (c|X|)] < ∞ for all c > 0
with the P-almost sure ordering and the Luxemburg norm
½
· µ
¶¸
¾
|X|
kXkΦ := inf λ > 0 : EP Φ
≤1
λ
is a Banach lattice, whose norm dual is the Orlicz space
©
ª
LΨ := ξ ∈ L0 : EP [Ψ (c|ξ|)] < ∞ for some c > 0
with the Orlicz norm
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ª
kξk∗Φ := sup EP [Xξ] : X ∈ LΦ , kXkΦ ≤ 1 ,
which is equivalent to the Luxemburg norm k·kΨ . For proofs of these facts, see, for instance, Edgar
and Sucheston (1992).
We call a mapping ρ : M Φ → (−∞, ∞] a monetary risk measure on M Φ if it has the following
properties:
(F) Finiteness at 0: ρ(0) ∈ R
(M) Monotonicity: ρ(X) ≥ ρ(Y ) for all X, Y ∈ M Φ such that X ≤ Y
(T) Translation property: ρ(X + m) = ρ(X) − m for all X ∈ M Φ and m ∈ R
We call a monetary risk measure ρ on M Φ convex if it also satisfies
(C) Convexity: ρ(λX + (1 − λ)Y ) ≤ λρ(X) + (1 − λ)ρ(Y ) for all X, Y ∈ M Φ and λ ∈ (0, 1)
A convex monetary risk measure ρ on M Φ is called coherent if it fulfills
(P) Positive homogeneity: ρ(λX) = λρ(X) for all X ∈ M Φ and λ ≥ 0.
It follows from (F), (M) and (T) that ρ(L∞ ) ⊂ R. We identify a probability measure Q on (Ω, F) that
is absolutely continuous with respect to P with its Radon–Nykodim derivative ξ = dQ/dP ∈ L1 . By
DΨ we denote the set of all Radon–Nykodim densities in LΨ :
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ª
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 identically
equal to ∞. We say a penalty function on DΨ satisfies the growth condition (G) if there exist constants
a ∈ R and b > 0 such that
γ(Q) ≥ a + b kQk∗Φ for all Q ∈ DΨ .
(2.1)
Since k.kΨ and k.k∗Φ are equivalent norms on LΨ , (2.1) holds if and only if there exist constants a0 ∈ R
and b0 > 0 such that
γ(Q) ≥ a0 + b0 kQkΨ for all Q ∈ DΨ .
For a function f : M Φ → (−∞, ∞], we denote
©
ª
dom f := X ∈ M Φ : f (X) < ∞ .
If f is convex, then so is dom f . Unless otherwise specified, we call f continuous, lower semicontinuous
or Lipschitz-continuous if it is so with respect to k.kΦ . By int(A) we denote the interior of a subset
A ⊂ M Φ with respect to the norm-topology, and by core(A) the algebraic interior, that is, the set of
all points x ∈ A with the property that for every y ∈ M Φ , there exists ε > 0 such that x + ty ∈ A for
all 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 Ruszczyński and Shapiro (2006) and Theorems 4.5 and
4.6 of Cheridito and Li (2007).
Theorem 2.1. Let γ be a penalty function on DΨ . Then
ργ (X) := sup {EQ [−X] − γ(Q)}
(2.2)
Q∈DΨ
defines a lower semicontinuous convex monetary risk measure on M Φ , 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 ∈ M Φ and every sequence (Qn )n≥1 in DΨ satisfying
lim {EQn [−X] − γ(Qn )} = ργ (X) ,
n→∞
the sequences EQn [X] and γ(Qn ), n ≥ 1, are bounded.
If (i)–(iv) hold and γ is σ(DΨ , M Φ )-lower semicontinuous, then
ργ (X) = max {EQ [−X] − γ(Q)}
Q∈DΨ
for all X ∈ M Φ .
(2.3)
Proof. That (2.2) defines a lower semicontinuous convex monetary risk measure on M Φ is clear. The
equivalence 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 we
can show (iii) ⇒ (iv). To do that, we assume (iii) and choose X ∈ M Φ and a sequence (Qn )n≥1 in DΨ
such that
EQn [−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 a
subsequence, 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 , M Φ )-lower semicontinuous. So assume
these conditions hold and let X ∈ M Φ . 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 (kQn k∗Φ )n≥1 is bounded. So it
follows from the Alaoglu–Bourbaki theorem that there exists a subsequence of (Qn )n≥1 converging to
some Q ∈ DΨ in σ(DΨ , M Φ ). Since γ is σ(DΨ , M Φ )-lower semicontinuous, one obtains
EQ [−X] − γ(Q) ≥ lim EQn [−X] − γ(Qn ) = ργ (X) ≥ EQ [−X] − γ(Q) ,
n→∞
and (2.3) is proved.
For a convex monetary risk measure ρ : M Φ → (−∞, ∞], we define the function ρ# : DΨ → (−∞, ∞]
by
ρ# (Q) := sup {EQ [−X] − ρ(X)} .
X∈M Φ
It is obviously σ(DΨ , M Φ )-lower semicontinuous. Moreover, one has:
Theorem 2.2. For a convex monetary risk measure ρ : M Φ → (−∞, ∞], 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 , M Φ )-lower semicontinuous minorant of γ.
(iii) If core(dom ρ) 6= ∅, then ρ is real-valued as well as locally Lipschitz-continuous, and
n
o
ρ(X) = max EQ [−X] − ρ# (Q)
for all X ∈ M Φ .
Q∈DΨ
(iv) If ρ is coherent and core(dom ρ) 6= ∅, then ρ is real-valued as well as Lipschitz-continuous, and
ρ(X) = max EQ [−X]
Q∈Q
for all X ∈ M Φ ,
©
ª
where Q = Q ∈ DΨ : EQ [X] + ρ(X) ≥ 0 for all X ∈ M Φ .
Proof. (i): It follows from Theorem 2.2 in Ruszczyński and Shapiro (2006) that ρ = ρρ# , which can
only hold if ρ# is a penalty function on DΨ .
(ii) follows from Theorem 2.3.4 in Zălinescu (2002) like the last part of Theorem 4.6 in Cheridito
and 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 Gâteaux-differentiability
Let f : M Φ → (−∞, ∞] be a convex function, and denote by f ∗ the convex conjugate given by
f ∗ (ξ) := sup {EP [Xξ] − f (X)} ,
X∈M Φ
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ξ ∈ LΨ .
(3.1)
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
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ª
∂f (X) := ξ ∈ LΨ : f (X + Y ) − f (X) ≥ EP [Y ξ] for all Y ∈ M Φ .
By convexity of f , the directional derivative
f 0 (X; Y ) := lim
ε↓0
f (X + εY ) − f (X)
∈ [−∞, ∞]
ε
exists in every direction Y ∈ M Φ . If there exists ξ ∈ LΨ such that
f 0 (X; Y ) = EP [Y ξ]
for all Y ∈ M Φ ,
we say f is Gâteaux-differentiable at X with Gâteaux-derivative ∇f (X) = ξ. If it exists, ∇f (X) is
unique and
EP [Y ∇f (X)] = f 0 (X; Y ) ≥ sup EP [Y ξ] for all Y ∈ M Φ ,
ξ∈∂f (X)
which implies ∂f (X) = {∇f (X)}. On the other hand, if f is continuous at X and ∂f (X) = {ξ} for
some ξ ∈ LΨ , then it follows from Theorem 2.4.10 of Zălinescu (2002) that f is Gâteaux-differentiable
at X with Gâteaux-derivative ∇f (X) = ξ.
If ρ is a convex monetary risk measure on M Φ , it follows from the properties (M) and (T) that
½ #
ρ (−ξ) for − ξ ∈ DΨ
∗
ρ (ξ) =
.
∞
for − ξ ∈ LΨ \ DΨ
The following notation will be convenient:
Definition 3.1. For a convex monetary risk measure ρ on M Φ and a penalty function γ on DΨ , we
denote
n
o
χρ (X) :=
Q ∈ DΨ : ρ(X) + ρ# (Q) = EQ [−X] , X ∈ M Φ
©
ª
χρ,γ (X) := Q ∈ DΨ : ρ(X) + γ(Q) = EQ [−X] , X ∈ M Φ
©
ª
χγ (Q) := X ∈ M Φ : ργ (X) + γ(Q) = EQ [−X] , Q ∈ DΨ
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ª
MγΦ := X ∈ M Φ : ργ (X) + γ(Q) = EQ [−X] for some Q ∈ DΨ
Note that if γ ≥ γ 0 are two penalty functions on DΨ which induce the same convex monetary risk
measure ρ on M Φ , then χρ,γ (X) ⊂ χρ,γ 0 (X) for all X ∈ M Φ and χγ (Q) ⊂ χγ 0 (Q) for all Q ∈ DΨ . In
particular, 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, Gâteaux-differentiability of convex monetary
risk measures can be characterized as follows:
Proposition 3.2. For a convex monetary risk measure ρ on M Φ and X ∈ dom ρ, the following hold:
(i) χρ (X) = −∂ρ(X)
(ii) If ρ is Gâteaux-differentiable at X, then ρ is real-valued as well as locally Lipschitz-continuous, and
χρ (X) = {−∇ρ(X)}
(iii) If X ∈ core(dom ρ) and χρ (X) = {ξ} for some ξ ∈ DΨ , then ρ is Gâteaux-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 Gâteaux-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 is
real-valued and continuous on M Φ . So Theorem 2.4.10 of Zălinescu (2002) yields that ρ is Gâteauxdifferentiable at X with ∇ρ(X) = −ξ.
Remark 3.3. Let ρ be a coherent risk measure on M Φ of the form
ρ(X) = sup EQ [−X]
Q∈Q
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 Gâteaux-differentiable at m if Q consist of just one element.
4
Strict monotonicity
M Φ if ρ(X) > ρ(Y
Definition 4.1. We call a risk measure ρ on M Φ strictly monotone on a subset A of ©
ª )
Ψ
Ψ
for all X, Y ∈ A such that X ≤ Y as well as P[X < Y ] > 0, and we denote Ds := ξ ∈ D : ξ > 0 .
Theorem 4.2. Let γ be a penalty function on DΨ . Then the following are equivalent:
(i) ργ is strictly monotone on MγΦ
(ii) ργ (X) = maxQ∈DsΨ {EQ [−X] − γ(Q)} for all X ∈ MγΦ
(iii) χργ ,γ (X) ⊂ DsΨ for all X ∈ MγΦ
(iv) χγ (Q) = ∅ for all Q ∈ DΨ \ DsΨ
Proof. The implications (iv) ⇔ (iii) ⇒ (ii) are clear. So it suffices to show (ii) ⇒ (i) ⇒ (iii).
(ii) ⇒ (i): Let X, Y ∈ MγΦ such that X ≤ Y and P[X < Y ] > 0. Then there exists Q ∈ DsΨ such
that
ργ (Y ) = EQ [−Y ] − γ(Q) < EQ [−X] − γ(Q) ≤ ργ (X) .
(i) ⇒ (iii): Assume there exist X ∈ MγΦ and Q ∈ DΨ \ DsΨ 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 ∈ MγΦ and hence contradicts (i).
Remark 4.3. If a risk measure is strictly monotone on M Φ , it is of course also relevant (see Definition
3.4 in Delbaen (2002) for the coherent case and Definition 4.32 in Föllmer and Schied (2004) for the
convex monetary case). Dual conditions for relevance of coherent risk measures are given in Theorem
3.5 of Delbaen (2002). For the convex monetary case, see Lemma 3.22 and Theorem 3.23 in Cheridito
et al. (2006).
5
Strict convexity
In order to define the notion of strict convexity for monetary risk measures, we must first dispense with
a trivial case. Call X, Y ∈ M Φ translationally equivalent (denoted X∼t Y ) if there exists m ∈ R such
6
that X = Y + m. If ρ is a monetary risk measure on M Φ and X, Y are elements of M Φ such that
X = 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 M Φ strictly convex modulo translation on a
subset A of M Φ if
ρ(λX + (1 − λ)Y ) < λρ(X) + (1 − λ)ρ(Y )
for all X, Y ∈ A and λ ∈ (0, 1) such that X6∼t Y and λX + (1 − λ)Y ∈ A.
It can be seen from (5.1) that a monetary risk measure ρ on M Φ which is strictly convex modulo
translation on a subset A of M Φ is also convex on A.
Proposition 5.2. A monetary risk measure on M Φ which is strictly convex modulo translation on a
convex subset A of M Φ is also strictly monotone on A.
Proof. Assume ρ is a monetary risk measure on M Φ that is strictly convex modulo translation but not
strictly monotone on a convex set A ⊂ M Φ . Then there exist X, Y ∈ A such that X ≤ Y , P[X < Y ] > 0
and ρ(X) = ρ(Y ). X and Y cannot be translationally equivalent. But
µ
¶
X +Y
≤ ρ(X) = ρ(Y )
ρ(Y ) ≤ ρ
2
and therefore,
µ
ρ
X +Y
2
¶
=
ρ(X) + ρ(Y )
,
2
a contradiction.
The following theorem gives dual conditions for strict convexity modulo translation. Note that for
a penalty function γ : DΨ → (−∞, ∞] and Q ∈ DΨ , a random variable X ∈ M Φ is in χγ (Q) if and only
if 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 MγΦ
(ii) χργ ,γ (X) \ χργ ,γ (Y ) 6= ∅ for all X, Y ∈ MγΦ such that X6∼t Y
(iii) χργ ,γ (X) ∩ χργ ,γ (Y ) = ∅ for all X, Y ∈ MγΦ such that X6∼t Y
(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 ∈ MγΦ , χργ ,γ (X) is not empty.
(ii) ⇒ (i): Let X, Y ∈ MγΦ and λ ∈ (0, 1) such that X6∼t Y and λX + (1 − λ)Y ∈ MγΦ . Then
λX + (1 − λ)Y 6∼t X. 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 ∈ MγΦ and Q ∈ DΨ such that X6∼t Y and Q ∈ χργ ,γ (X) ∩
χργ ,γ (Y ). Then
µ
¶
·
¸
µ
¶
X +Y
X +Y
1
X +Y
ργ
≥ EQ −
− γ(Q) = (ργ (X) + ργ (Y )) ≥ ργ
.
(5.2)
2
2
2
2
So (X + Y )/2 ∈ MγΦ , and (5.2) is in contradiction to (i). This shows that (i) implies (iii).
7
If ρ is a coherent risk measure on M Φ , it is linear on all rays {λX : λ ≥ 0}, X ∈ M Φ . So it cannot be
strictly 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 write
X∼c Y if (X(ω) − X(ω 0 ))(Y (ω) − Y (ω 0 )) ≥ 0 for P × P-almost all (ω, ω 0 ), and we define strict convexity
modulo 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 MγΦ
(ii) χργ ,γ (X) \ χργ ,γ (Y ) 6= ∅ for all X, Y ∈ MγΦ such that X6∼c Y
(iii) χργ ,γ (X) ∩ χργ ,γ (Y ) = ∅ for all X, Y ∈ MγΦ such that X6∼c Y
(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 ∈ MγΦ and λ ∈ (0, 1) such that X6∼c Y and λX + (1 − λ)Y ∈ MγΦ .
Assume that
χργ ,γ (λX + (1 − λ)Y ) ⊂ χργ ,γ (X) ∩ χργ ,γ (Y ) .
(5.3)
Then there exists
Q∈
\
χργ ,γ (µX + (1 − µ)Y ) ,
(5.4)
µ∈[0,1]
which implies
ργ (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 µ0 X + (1 − µ0 )Y 6∼c X. By (5.4), µ0 X + (1 − µ0 )Y belongs
to MγΦ . Hence, it follows from (ii) that there exists Q0 ∈ χργ ,γ (µ0 X + (1 − µ0 )Y ) \ χργ ,γ (X), and we
obtain
ργ (µ0 X + (1 − µ0 )Y ) = EQ0 [−µ0 X − (1 − µ0 )Y ] − γ(Q0 )
= µ0 (EQ0 [−X] − γ(Q0 )) + (1 − µ0 ) (EQ0 [−Y ] − γ(Q0 )) < µ0 ργ (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 ) which
does 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 F X we denote by q X the right-continuous
quantile function from (0, 1) to R given by
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ª
q X (y) := inf x ∈ R : F X (x) > y .
Viewed as a random variable on (0, 1) equipped with
Lebesgue measure,
R 1theX Borel sigma-algebra and the
X
1
q has the same distribution as X; in particular, 0 q (y)dy = EP [X] for X ∈ L .
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 Y
with 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 with
respect to S and write X ∼S Y . If X ºS Y and X 6∼S Y , we say X strictly dominates Y with respect to
S 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 Xºicv Y is
equivalent to −X¹icx − Y . Moreover, one has the following:
Xºi Y ⇔ q X (y) ≥ q Y (y) for all y ∈ (0, 1)
Rz
Rz
X ºicv Y ⇔ 0 q X (y)dy ≥ 0 q Y (y)dy for all z ∈ (0, 1) ⇔ there exists a probability
space with random variables X̃ and Ỹ such that F X̃ = F X , F Ỹ = F Y and X̃ ≥ E[Ỹ | X̃]
X ºcv Y ⇔ X ºicv Y and EP [X] = EP [Y ] ⇔ there exists a probability space with
random variables X̃ and Ỹ such that F X̃ = F X , F Ỹ = F Y and X̃ = E[Ỹ | X̃] .
(6.1)
(6.2)
(6.3)
Proofs of these facts and more on stochastic orders can, for instance, be found in Müller and Stoyan
(2002), Föllmer 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 ⇔ q X = q Y ⇔ F X = F Y ,
(6.4)
and one obtains from Jensen’s inequality for conditional expectations that EP [X | G] ºcv X for all
X ∈ L1 and every sub-sigma-algebra G ⊂ F. Furthermore, if X ∈ L1 is not G-measurable, then
EP [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 , then
by (6.2)–(6.3), there exists a probability space with random variables X̃ and Ỹ distributed as X and
Y , respectively, such that X̃ ≥ E[Ỹ | X̃]. Since F X̃ 6= F Ỹ , one has P[X̃ > E[Ỹ | X̃]] > 0 or
E[Ỹ | X̃] 6= Ỹ . This shows that EP [f (X)] > EP [f (Y )] for all strictly concave increasing functions f
such that f (X), f (Y ) ∈ L1 .
Definition 6.2. Let ρ be a monetary risk measure on M Φ and S a class of functions f : R → R.
Then we call ρ S-monotone if ρ(X) ≥ ρ(Y ) for all X, Y ∈ M Φ such that X ¹S Y . If ρ is S-monotone
and ρ(X) > ρ(Y ) for all X, Y ∈ M Φ such that X ≺S Y , we call ρ strictly S-monotone. If ρ(X) only
depends on F X , we call ρ distribution-based.
Since ºicv is weaker than ºi and ºcv , an icv-monotone monetary risk measure is also i- and cvmonotone. On the other hand, the extension of Proposition 2.1 of Dana (2005) to Orlicz hearts yields
that every cv-monotone monetary risk measure on M Φ is icv-monotone. However, an i-monotone
monetary risk measure is not necessarily cv- or icv-monotone:
Example 6.3. By (6.1), the monetary risk measure value-at-risk VaRα (X) := −q X (α) is i-monotone
and distribution-based but not strictly i-monotone. Also, for every non-constant random variable
X ∈ L1 , one has X≺cv Y := EP [X]. But there exists α ∈ (0, 1) such that VaRα (X) < VaRα (Y ). So
VaRα is not cv-monotone and therefore also not icv-monotone. It is also not convex and hence not
coherent (see Artzner et al., 1999; or Föllmer and Schied, 2004).
9
In view of (6.4), every i-, cv- or icv-monotone monetary risk measure is also distribution-based. On
the other hand, Theorem 4.1 of Dana (2005) (extended to Orlicz hearts) shows that if the probability
space is atomless, then every lower semicontinuous distribution-based convex monetary risk measure is
icv-monotone. The following example shows that this does not need to be the case if the probability
space has atoms:
Example 6.4. Consider a probability space Ω consisting of two elements, ω1 and ω2 , and two probability measures P and Q such that
P[ω1 ] = Q[ω2 ] =
1
3
and P[ω2 ] = Q[ω1 ] =
2
.
3
Then ρ(X) := EQ [−X] defines a continuous strictly monotone coherent risk measure on L1 , which
since P assigns weights unevenly, is distribution-based. Now consider the random variables X, Y, Z
given by X(ωj ) = −j, Y (ωj ) = j − 3 and Z = EP [X] = −5/3. Then X≺i Y and X≺cv Z but
ρ(X) = 4/3 < ρ(Y ) = ρ(Z) = 5/3. So ρ is not i-monotone, not cv-monotone, and therefore also not
icv-monotone.
6.1
icv-monotone monetary risk measures and sets of acceptable positions
It is well-known that a monetary risk measure ρ : M Φ → (−∞, ∞] can be reconstructed from its
acceptance set
©
ª
C := X ∈ M Φ : ρ(X) ≤ 0
through
ρC (X) := inf {m ∈ R : X + m ∈ C} ,
X ∈ MΦ .
Moreover, if B is a subset of M Φ with the following three properties:
©
ª
for all X ∈ B, the set Y ∈ M Φ : Y ≥ X is contained in B
ρB (0) ∈ R
(6.5)
(6.6)
Φ
ρB (X) ∈ (−∞, ∞] for all X ∈ M ,
(6.7)
then ρB is a monetary risk measure on M Φ . 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 M Φ has the
following property:
©
ª
for all X ∈ C, the set Y ∈ M Φ : Y ºicv X is contained in C .
(6.8)
On the other hand, for every subset B of M Φ with the properties (6.6)–(6.8), ρB is an icv-monotone
monetary risk measure on M Φ .
Proof. That the acceptance set of an icv-monotone monetary risk measure satisfies (6.8) is clear. On
the other hand, a subset B of M Φ with the properties (6.6)–(6.8) also satisfies (6.5). So ρB is a monetary
risk measure, which obviously is icv-monotone.
©
ª
For fixed X ∈©M Φ , the set Z ∈ M Φ : Z ºicv X
is convex. But this is in general not the case for
ª
sets of the form Z ∈ M Φ : Z ºicv X or Z ºicv Y for X, Y ∈ M Φ . 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 probability
space. Consider the set B = {Z ∈ L1 : Z ºicv X or Z ºicv Y }, where
½
½
6x − 1 0 < x ≤ 31
6x + 1 0 < x ≤ 13
X(x) =
Y
(x)
=
.
1
6x + 1 3 < x < 1
6x − 1 31 < x < 1
R1
R1
Rt
Rt
Set Z(x) = 12 (X + Y )(x) = 6x. Then 0 q Z (x)dx < 0 q X (x)dx as well as 0 q Z (x)dx < 0 q Y (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),
1
AVaRα (X) :=
α
Z
0
α
1
VaRy (X) dy = −
α
Z
α
q X (y) dy ,
0
is a real-valued coherent risk measure on L1 (see Föllmer and Schied, 2004). (6.2) shows that it is
icv-monotone but not strictly icv-monotone. It has been noted before that AVaR can be used as a
building block to construct other risk measures; see for instance, Kusuoka (2001), Acerbi (2002, 2004),
Föllmer and Schied (2004), Frittelli and Rosazza Gianin (2005), Leitner (2005), Dana (2005), or Jouini
et al. (2006). Here, we adapt some of the duality results of Dana (2005) to our setup and combine them
with Theorems 2.1 and 2.2 to derive representation results for icv-monotone risk measures on Orlicz
hearts. Then we provide characterizations for strict monotonicity, strict convexity modulo translation
and strict cv- and icv-monotonicity of icv-monotone risk measures. We are using the following notation:
Definition 6.7. By M Φ (0, 1) we denote the Orlicz heart corresponding to Φ over (0, 1) equipped with
the Borel sigma-algebra and Lebesgue measure. LΨ (0, 1) is the Orlicz space over (0, 1) induced by Ψ.
Furthermore, we set
©
ª
RΦ := q X : X ∈ M Φ ,
½
¾
Z 1
D̂ := l : (0, 1) → R+ : l left-continuous, decreasing and
l(y)dy = 1 ,
0
Ψ
Ψ
D̂ := D̂ ∩ L (0, 1)
and
D̂sΨ
n
o
:= l ∈ D̂Ψ : l(y) > 0 for all y ∈ (0, 1) .
If the probability space (Ω, F, P) over which M Φ is defined is atomless, it
a randomªvariable
© supports
X : X ∈ M Φ (0, 1) . But if
that is uniformly distributed on (0, 1), and RΦ© is equal to the convex
set
q
ª
(Ω, F, P) has atoms, then RΦ is smaller than q X : X ∈ M Φ (0, 1) and not necessarily convex.
For l ∈ D̂, define the right-continuous function ˜l : (0, 1] → R+ by
½
˜l(y) := l(y+) for y ∈ (0, 1) .
0
for y = 1
Then dµ(y) = −yd˜l(y) induces a probability measure µ on (0, 1] such that
Z
Z
1
1
˜
l(y) =
dµ(x) and l(y) =
dµ(x) for y ∈ (0, 1) .
[y,1] x
(y,1] x
This provides a bijection between D̂ and the set of probability measures µ on (0, 1]. For given l ∈ D̂Ψ
and X ∈ M Φ , one has
Z 1
Z
­ X ®
X
−q , l :=
−q (y)l(y)dy =
AVaRy (X)dµ(y) .
(6.9)
0
(0,1]
11
Since (6.9) defines a real-valued coherent risk measure on M Φ , it follows from Theorem 2.2 that it is
continuous in X with respect to k.kΦ . Together with (6.2), (6.9) shows that for X, Y ∈ M Φ ,
X¹icv Y
⇔
­ X ® ­ Y ®
q ,l ≤ q ,l
for all l ∈ D̂Ψ .
(6.10)
Moreover, for ξ ∈ DΨ , the function lξ given by lξ (y) := q ξ (1 − y) belongs to D̂Ψ , and by Hardy–
Littlewood’s inequality,
D
E
q X , lξ ≤ EP [Xξ] for all X ∈ M Φ ;
(6.11)
see Hardy et al. (1988) or Föllmer and Schied (2004).
Definition 6.8. We call a mapping ν : D̂Ψ → (−∞, ∞] a penalty function on D̂Ψ if it is bounded from
below and not identically equal to ∞. We say that it satisfies the growth condition (G) if there exist
constants a ∈ R and b > 0 such that
ν(l) ≥ a + b klkΨ
for all l ∈ D̂Ψ .
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 D̂Ψ . Then
©­
®
ª
ρν (X) := sup −q X , l − ν(l)
l∈D̂Ψ
defines a lower semicontinuous icv-monotone convex monetary risk measure on M Φ , and the implications
(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 ∈ M Φ and every sequence (ln )n≥1 in D̂Ψ satisfying
ª
©­ X ®
lim
−q , ln − ν(ln ) = ρν (X) ,
n→∞
®
­
the sequences q X , ln and ν(ln ), n ≥ 1, are bounded.
If (i) holds and ν is (D̂Ψ , M Φ (0, 1))-lower semicontinuous, then
©­
®
ª
ρν (X) = max −q X , l − ν(l)
for all X ∈ M Φ .
l∈D̂Ψ
(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 M Φ
®
follows from the fact that for every l ∈ D̂Ψ , −q X , l is a continuous icv-monotone coherent risk
measure on M Φ . 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 Theorem
2.1. If (i) holds, then the penalty function γ̂ : DΨ (0, 1) → (−∞, ∞] given by γ̂(ξ) := ν(lξ ) satisfies (G),
and due to (6.11), one has
©­
®
ª
ρ̂ν (X) := sup −q X , l − ν(l) = sup {EP [−Xξ] − γ̂(ξ)}
ξ∈DΨ (0,1)
l∈D̂ Ψ
12
for all X ∈ M Φ (0, 1). So we obtain from Theorem 2.1 that ρ̂ν is real-valued. But then also ρν is
real-valued. This shows (i) ⇒ (ii).
That (i) and (D̂Ψ , M Φ (0, 1))-lower semicontinuity of ν imply (6.12) follows as in the proof of Theorem 2.1.
To conclude the proof, assume that (Ω, F, P) is atomless. Then it follows from (6.11) that
ρν (X) = sup {EP [−Xξ] − γ(ξ)}
ξ∈DΨ
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 M Φ , we define
©­ X ®
ª
−q , l − ρ(X) , l ∈ D̂Ψ .
ρ† (l) := sup
X∈M Φ
Clearly, ρ† is lower semicontinuous with respect to σ(D̂Ψ , RΦ ) and therefore also with respect to
σ(D̂Ψ , M Φ (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 M Φ . Then the
following are equivalent:
(i) ρ is icv-monotone ©­
®
ª
(ii) ρ# (ξ) = supX∈M Φ −q X , lξ − ρ(X) , ξ ∈ DΨ
0
Ψ
0
(iii) ρ# (ξ) ≥ ρ# (ξ 0 ) for
that
©­ ξ, ξX ∈ξD
® such
ª ξ¹cv ξ Φ
#
(iv) ρ(X) = supξ∈DΨ −q , l − ρ (ξ) , X ∈ M
©­
®
ª
(v) ρ(X) = supl∈D̂Ψ −q X , l − ρ† (l) , X ∈ M Φ
If (i)–(v) hold, then ρ† is the smallest penalty function on D̂Ψ which induces ρ. If ρ is coherent and
(i)–(v) hold, then
D
E
­
®
ρ(X) = sup −q X , lξ = sup −q X , l
(6.13)
ξ∈Q
l∈E
for
©
ª
Q = ξ ∈ DΨ : EP [Xξ] + ρ(X) ≥ 0 for all X ∈ M Φ
and
n
o
­
®
E = l ∈ D̂Ψ : q X , l + ρ(X) ≥ 0 for all X ∈ M Φ .
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) = sup
nD
E
o
n­
o
®
−q X , lξ − ρ# (ξ) ≤ sup
−q X , l − ρ† (l) ≤ ρ(X) ,
ξ∈DΨ
l∈D̂Ψ
which implies (v).
If (i)–(v) hold, then ρ† must be a penalty function on D̂Ψ . 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 : M Φ → (−∞, ∞] be an icv-monotone convex monetary risk measure with
core(dom ρ) 6= ∅. Then
nD
E
o
n­
o
®
−q X , lξ − ρ# (ξ) = max −q X , l − ρ† (l) , X ∈ M Φ ,
(6.14)
ρ(X) = max
ξ∈DΨ
l∈D̂Ψ
and if ρ is coherent, then
D
E
­
®
ρ(X) = max −q X , lξ = max −q X , l ,
ξ∈Q
l∈E
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
n
o
n­
o
®
ρ(X) = max EP [−Xξ] − ρ# (ξ) = sup
−q X , l − ρ† (l)
for all X ∈ M Φ .
ξ∈DΨ
(6.16)
l∈D̂Ψ
­
®
Since for all ξ ∈ DΨ and X ∈ M Φ , one has EP [−Xξ] ≤ −q X , 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 0
on the sets Q and E, respectively, and ∞ otherwise.
To characterize properties of icv-monotone risk measures in terms of elements of D̂Ψ , we need the
following definitions:
Definition 6.12. Let ρ be a distribution-based convex monetary risk measure on M Φ and ν a penalty
function on D̂Ψ . Then we define the function ρ̂ : RΦ → (−∞, ∞] by ρ̂(q X ) := ρ(X), and we denote
n
o
χ̂ρ̂ (r) :=
l ∈ D̂Ψ : ρ̂(r) + ρ† (l) = h−r, li , r ∈ RΦ
n
o
χ̂ρ̂,ν (r) :=
l ∈ D̂Ψ : ρ̂(r) + ν(l) = h−r, li , r ∈ RΦ
©
ª
χ̂ν (l) := r ∈ RΦ : ρ̂ν (r) + ν(l) = h−r, li , l ∈ D̂Ψ
n
o
­
®
MνΦ :=
X ∈ M Φ : ρν (X) + ν(l) = −q X , l for some l ∈ D̂Ψ
n
o
RνΦ :=
r ∈ RΦ : ρ̂ν (r) + ν(l) = h−r, li for some l ∈ D̂Ψ .
Theorem 6.13. Let ν be a penalty function on D̂Ψ . Then the implications
(i) ⇐ (ii) ⇐ (iii) ⇐ (iv) ⇔ (v)
hold among the conditions:
(i) ρν is strictly monotone on MνΦ
(ii) ρ̂ν is strictly monotone on RνΦ
(iii) ρ̂ν (r) = maxl∈D̂Ψ {h−r, li − ν(l)} for all r ∈ RνΦ
s
(iv) χ̂ρ̂ν ,ν (r) ⊂ D̂sΨ for all r ∈ RνΦ
(v) χ̂ν (l) = ∅ for all l ∈ D̂Ψ \ D̂sΨ
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 ∈ RνΦ and l ∈ D̂Ψ \ D̂sΨ
such that ρ̂ν (r) = h−r, li − ν(l). Then there exists z ∈ (0, 1) such that l(y) = 0 for ©y ∈ (z, 1). Choose
ª
X ∈ MνΦ with q X = r. Since (Ω, F, P) has no atoms, there exists a subset A ⊂ X ≥ q X (z) with
P[A] = 1 − z. The quantile function of the random variable Y = X + 1A is equal to q X + 1[z,1) . So
­
®
­
®
ρν (Y ) ≥ −q Y , l − ν(l) = −q X , l − ν(l) = ρν (X) ≥ ρν (Y ) .
But this implies Y ∈ MνΦ and ρν (Y ) = ρν (X), a contradiction to (i).
Theorem 6.14. For a penalty function ν on D̂Ψ , the implications
(i), (ii) ⇐ (iii) ⇐ (iv) ⇔ (v)
hold among the conditions
(i) ρν is strictly convex modulo translation on MνΦ
(ii) ρ̂ν is strictly convex modulo translation on RνΦ
(iii) χ̂ρ̂ν ,ν (r) \ χ̂ρ̂ν ,ν (s) 6= ∅ for all r, s ∈ RνΦ such that r6∼t s
(iv) χ̂ρ̂ν ,ν (r) ∩ χ̂ρ̂ν ,ν (s) = ∅ for all r, s ∈ RνΦ such that r6∼t s
(v) for all l ∈ D̂Ψ , χ̂ν (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 ∈ MνΦ and λ ∈ (0, 1) such that X6∼t Y and
λX + (1 − λ)Y ∈ MνΦ . By Lemma 6.15 below, we have q λX+(1−λ)Y 6∼t q X or q λX+(1−λ)Y 6∼t q Y . Therefore,
there exists l ∈ χ̂ρ̂ν ,ν (q λX+(1−λ)Y ) which does not belong to χ̂ρ̂ν ,ν (q X ) ∩ χ̂ρ̂ν ,ν (q Y ), and we get
D
E
ρν (λX + (1 − λ)Y ) = −q λX+(1−λ)Y , l − ν(l)
­
®
­
®
≤ λ −q X , l + (1 − λ) −q Y , 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 uniformly
distributed on (0, 1). Then the mapping r 7→ r(U ) embeds RνΦ in MνΦ , 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 q X ∼t q λX+(1−λ)Y ∼t q Y . Then X∼t Y .
Proof. Denote Z = Y + EP [X − Y ]. Then
q X ∼t q λX+(1−λ)Z ∼t q Z ,
(6.17)
and EP [X] = EP [λX + (1 − λ)Z] = EP [Z], which is equivalent to
Z 1
Z 1
Z 1
X
λX+(1−λ)Z
q (y)dy =
q
(y)dy =
q Z (y)dy .
0
0
(6.18)
0
(6.17) and (6.18) imply q X = q λX+(1−λ)Z = q Z . 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∼t Y .
15
Theorem 6.16. Let ν be a penalty function on D̂Ψ . Then ρν is strictly cv-monotone on MνΦ if and
only if for all l ∈ D̂Ψ and r ∈ χ̂ν (l), r is a deterministic function of the right-continuous function
˜l(y) = l(y+).
Proof. To show the “only if”-direction, assume that ρν is strictly cv-monotone on MνΦ but there exist
˜
˜
l ∈ D̂Ψ and r ∈ χ̂ν (l) such
h that
i r is not a deterministic function of l. Then r cannot be σ(l)-measurable,
and therefore, r ≺cv E r | ˜l . Thus
D
h
i E
³ h
i´
ρ†ν (l) ≤ ν(l) = h−r, li − ρ̂ν (r) ≤ −E r | ˜l , l − ρ̂ν E r | ˜l ≤ ρ†ν (l) .
h
i
³ h
i´
But this implies E r | ˜l ∈ χ̂ν (l) and therefore, ρ̂ν (r) > ρ̂ν E r | ˜l , a contradiction.
For the “if”-part, assume ρν is not strictly cv-monotone on MνΦ . Then there exist r, s ∈ RνΦ such
that r ≺cv s and ρ̂ν (r) ≤ ρ̂ν (s). Choose l ∈ χ̂ρ̂ν ,ν (s) and observe that
ρ†ν (l) ≤ ν(l) = h−s, li − ρ̂ν (s) ≤ h−r, li − ρ̂ν (r) ≤ ρ†ν (l) .
Rz
It follows that r ∈ χ̂ν (l) and hr, li = hs, li. Since r ≺cv s, the continuous function f (z) := 0 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 a
maximizer 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 is
increasing, this implies r(z3 ) < r(z4 ). But due to hr, li = hs, li, we have
Z 1
Z 1
Z 1
˜l(y)df (y) =
˜l(y)(r(y) − s(y))dy = 0 ,
f (y)d˜l(y) = −
0
0
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 ∈ M Φ ,
©
ª ©
ª
X ∈ M Φ : Xºicv Y = X ∈ M Φ : Xºcv Y + M+Φ .
This shows that an icv-monotone monetary risk measure ρ on M Φ is strictly icv-monotone on M Φ if
and only if ρ is strictly monotone and strictly cv-monotone on M Φ .
7
Cash-additive hulls
Let V be a mapping from M Φ to (−∞, ∞] satisfying the following three properties:
(V1) V (X) ≤ V (Y ) for all X, Y ∈ M Φ such that X ≤ Y
(V2) V (λX + (1 − λ)Y ) ≤ λV (X) + (1 − λ)V (Y ) for all X, Y ∈ M Φ and λ ∈ (0, 1)
(V3) for all X ∈ M Φ , inf s∈R {V (s − X) − s} ∈ R and the infimum is attained.
Then
ρV (X) := min {V (s − X) − s}
s∈R
is the largest real-valued convex monetary risk measure on M Φ such that
ρV (X) ≤ V (−X)
for all X ∈ M Φ .
We call it the cash-additive hull of the the decreasing convex functional V (−.); see Section 5.1 of
Cheridito and Li (2007).
16
Proposition 7.1. Let X ∈ M Φ and sX ∈ R such that ρV (X) = V (sX − X) − sX . If V is Gâteauxdifferentiable at sX − X, then ρV is Gâteaux-differentiable at X with
∇ρV (X) = −∇V (sX − X) .
Proof. If V is Gâteaux-differentiable at sX − X, then
¡ V ¢0
ρV (X + εY ) − ρV (X)
ρ
(X; Y ) = lim
ε↓0
ε
V (sX − X − εY ) − V (sX − X)
≤ lim
ε↓0
ε
0
= V (sX − X; −Y ) = EP [−Y ∇V (sX − X)]
¡ ¢0
for all Y ∈ M Φ . Since ρV (X; .) is sublinear, one also has
¡ V ¢0
¡ ¢0
ρ
(X; Y ) ≥ − ρV (X; −Y ) ≥ EP [−Y ∇V (sX − X)] ,
and it follows that ρV is Gâteaux-differentiable at X with ∇ρV (X) = −∇V (sX − X).
Proposition 7.2. If V is strictly monotone on dom V , then ρV is strictly monotone on M Φ .
Proof. Let X, Y ∈ M Φ 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 dom V , then ρV is
strictly convex modulo translation (comonotonicity) on M Φ .
Proof. Let X, Y ∈ M Φ , such that X6∼t Y (X6∼c Y ) 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 − X6∼t sY − Y (sX − X6∼c sY − 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 dom V , then ρV is (strictly) icv-monotone on
M Φ . If V is distribution-based, then so is ρV .
Proof. Assume V is icx-monotone on dom V and X ¹icv Y . Then −Xºicx − Y , and there exists sX ∈ R
such that
ρV (X) = V (sX − X) − sX ≥ V (sX − Y ) − sX ≥ ρV (Y ) .
This shows that ρV is icv-monotone on M φ . 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 {H(x) − x} = ∞ .
|x|→∞
Then
V (X) = EP [H(X)]
(8.1)
is a real-valued mapping on the Orlicz heart M Φ corresponding to the function Φ(x) := H(x)−H(0). It
clearly satisfies (V1)–(V3) and is icx-monotone. So, by Proposition 7.4, ρV is a real-valued icv-monotone
convex monetary risk measure on M Φ . Its minimal penalty function is given by
·
µ
¶¸
¡ V ¢#
∗ dQ
ρ
(Q) = EP H
, Q ∈ DΨ ;
(8.2)
dP
see Section 5.4 of Cheridito and Li (2007). If H is strictly increasing, then V is strictly monotone on M Φ ,
which by Proposition 7.2 implies that also ρV is strictly monotone on M Φ . If H is strictly convex, then
V is strictly convex and strictly icx-monotone on M Φ , and so by Propositions 7.3 and 7.4, ρV is strictly
convex modulo translation and strictly icv-monotone on M Φ . If H is differentiable, then V is Gâteauxdifferentiable on M Φ with ∇V (X) = H 0 (X) and it follows from Proposition 7.1 that ρV is Gâteauxdifferentiable on M Φ with ∇ρV (X) = −H 0 (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 distance
between Q and P. Functionals of the form ρV for V equal to (8.1) have appeared in different settings
in 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:
¶
µ
H(x) + ε
(FGH1)
F
< ∞ for some x ∈ R and ε > 0
G−1 (1)
©
ª
(FGH2)
lim F ◦ H(x) − G−1 (1) x = ∞ .
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 shown
that V (X) = F (kH(X)kG ) is a well-defined mapping from M Φ to (−∞, ∞] satisfying (V1)–(V3). It
can easily be checked that it is icx-monotone. So it follows from Proposition 7.4 that ρV defines a
real-valued icv-monotone convex monetary risk measure on M Φ . Its minimal penalty function is given
in Theorem 5.3 of Cheridito and Li (2007).
Clearly, the Luxemburg norm k.kΦ is strictly monotone on M+Φ 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 7.2
that the same is true for ρV . If F and G are strictly increasing and H is strictly convex, then V is
strictly convex and strictly icx-monotone, and so by Propositions 7.3 and 7.4, ρV is strictly convex
modulo translation and strictly icv-monotone.
18
As a specific example, consider the risk measure
½
¾
°β
1°
+
°(s − X) ° − s
ρ(X) := min
p
s∈R
α
(8.3)
for (α, β, p) in (0, 1) × {1} × [1, ∞) or (0, ∞) × (1, ∞) × [1, ∞). ρ is real-valued on Lp , and if sX ∈ R
minimizes the right side of (8.3), then sX ≥ ess inf X. Moreover, for β > 1, sX is unique, sX > ess inf X,
and the minimal penalty function of ρ is given by
ρ# (Q) = c kQkdq
for q :=
β
p
, d :=
, c := αd−1 β 1−d d−1 .
p−1
β−1
For β = 1, ρ is coherent, sX is not necessarily unique, and
½
0
if kQkq ≤ α1
#
ρ (Q) =
∞ if kQkq > α1
for q =
p
p−1
(8.4)
(8.5)
(for proofs of (8.4) and (8.5), see Section 5.3 of Cheridito and Li, 2007).
β
If β, p > 1, then V (X) = α1 kX + kp is Gâteaux-differentiable on Lp with
∇V (X) =
¤ β −1 ¡ + ¢p−1
£
β
EP (X + )p p
X
.
α
Hence, it follows from Proposition 7.1 that ρ is Gâteaux-differentiable on Lp with
£
¤ β −1 ¡
¢p−1
β
∇ρ(X) = − EP ((sX − X)+ )p p
(sX − X)+
.
α
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 (.) = α1 k(.)+ kp is Gâteaux-differentiable at sX − X with
∇V (sX − X) =
¢p−1
£
¤ 1 −1 ¡
1
EP ((sX − X)+ )p p
(sX − X)+
,
α
and it follows from Proposition 7.1 that ρ is Gâteaux-differentiable at X with Gâteaux-derivative (8.6).
If β = 1, p ≥ 1 and X ∈ Lp such that P [X = ess inf X] ≥ αp , then the measure
1{X=ess inf X}
dQ
=
dP
P [X = ess inf X]
satisfies
1
α
So Q is a maximizer of the right side of
kQkq ≤
ρ(X) =
and
EQ [−X] = −ess infX .
max
Q∈Dq , kQkq ≤1/α
but not necessarily the only one.
19
EQ [−X] ,
β
For β ≥ 1, p = 1 and X ∈ L1 with P [X = sX ] = 0, V (.) = α1 EP [(.)+ ] is Gâteaux-differentiable at
sX − X with
£
¤β−1
β
∇V (sX − X) = EP (sX − X)+
1{X<sX } .
α
So it follows from Proposition 7.1 that ρ is Gâteaux-differentiable at X with
∇ρ(X) = −
1{X<sX }
.
P [X < sX ]
If β > 1, p = 1 and X ∈ L1 with P [X = sX ] > 0, the left- and right-derivative of the function
β
s 7→ α1 EP [(s − X)+ ] − s at sX are given by
£
¤β−1
β
P [X < sX ] − 1 ≤ 0 and
EP (sX − X)+
α
£
¤β−1
β
P [X ≤ sX ] − 1 ≥ 0 ,
EP (sX − X)+
α
respectively. Choose any random variable ζ such that
0≤ζ≤
£
¤β−1
β
EP (sX − X)+
1{X=sX }
α
and EP [ζ] = 1 −
£
¤β−1
β
EP (sX − X)+
P [X < sX ] .
α
Then
£
¤β−1
β
EP (sX − X)+
1{X<sX } + ζ
α
is the density of a probability measure Q such that
ξ=
EQ [−X] − αd−1 β 1−d d−1 kQkd∞
£
¤βd−d
£
¤
β
EP (sX − X)+
= EQ (sX − X)+ −
− sX
αd
£
¤β−1
£
¤β
£
¤
β
β
=
EP (sX − X)+
EP (sX − X)+ − sX
EP (sX − X)+ −
α
αd
£
¤
1
β
EP (sX − X)+ − sX = ρ(X) .
=
α
Thus, it follows from (8.4) that Q maximizes the right side of
n
o
ρ(X) = max∞ EQ [−X] − αd−1 β 1−d d−1 kQkd∞ .
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 form
1
dQ
= 1{X<sX } + ζ ,
dP
α
where ζ is a random variable satisfying
0≤ζ≤
1
1
α {X=sX }
and
EP [ζ] = 1 −
1
P [X < sX ]
α
(see, Cherny, 2006).
Since χρ (X) ⊂ Dsq does in general not hold for risk measures of the form (8.3), it follows from
Theorem 4.2, that they are not strictly monotone on Lp and hence not strictly convex modulo translation
by 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)
then
ρ(X) = sup {AVaRλ (X) − η(λ)}
(8.8)
λ∈(0,1]
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 other
hand, if the underlying probability space is atomless and (8.8) is finite for all X ∈ Lp , then (8.7) must
hold.
­
®
Proof. For each λ ∈ (0, 1], AVaRλ (X) can be written as −q X , lλ for
lλ (y) = λ−1 1(0,λ] (y) ∈ D̂ .
Set q = p/(p − 1) and define the function ν : D̂q → (−∞, ∞] by
½
η(λ) if l = lλ for some λ ∈ (0, 1]
ν(l) :=
.
∞
else
Since klλ kq = λ−1/p , the mapping ν satisfies the growth condition (G) if and only if η fulfills (8.7).
Moreover, σ(D̂q , Lp (0, 1))-lower semicontinuity of ν is equivalent to lower semicontinuity of η. Hence
the 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]. So
by Proposition 8.1,
n
o
ρ(X) = max AVaRλ (X) − αλ−1/p
(8.9)
λ∈(0,1]
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 that
´ ¯¯
d ³
−1/p ¯
AVaRλ (X) − αλ
= 0,
¯
dλ
λ=λ0
or equivalently,
AVaRλ0 (X) − VaRλ0 (X) =
α −1/p
λ
.
p 0
Since lλ ∈
/ D̂sq 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
21
for all λ ∈ (0, 1] ,
(8.10)
then
¾
½ Z λ
1
AVaRy (X)dy − η(λ)
λ∈(0,1] λ 0
ρ(X) = sup
(8.11)
defines a real-valued locally Lipschitz-continuous icv-monotone convex monetary risk measure on Lp . If
η satisfies (8.10) and is lower semicontinuous, then the supremum in (8.11) is attained. On the other
hand, 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],
1
λ
Z
0
λ
­
®
AVaRy (X)dy = −q X , lλ ,
where
(
lλ (y) =
1
λ
³ ´
log
λ
y
0
for y ≤ λ
.
for y > λ
Set q = p/(p − 1) and define the function ν : D̂q → (−∞, ∞] by
½
η(λ) if l = lλ for some λ ∈ (0, 1]
ν(l) :=
.
∞
else
With the change of variables x = log(λ/y), one obtains
Z
λ
0
So
µ ¶
Z ∞
λ
dy =
λxq e−x dx = λΓ(q + 1) .
log
y
0
q
klλ kq = Γ(q + 1)1/q λ1/q−1 = Γ(q + 1)1/q λ−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 continuous
on (0, 1]. So it follows from Proposition 8.3 that
¾
½ Z λ
1
−1/p
AVaRy (X)dy − αλ
ρ(X) = max
λ∈(0,1] λ 0
(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
µ Z
¶¯
¯
d 1 λ
−1/p ¯
= 0,
AVaRy (X)dy − αλ
¯
dλ λ 0
λ=λ0
or
1
λ0
Z
0
λ0
AVaRy (X) dy − AVaRλ0 (X) =
D̂sq
α −1/p
λ
.
p 0
Again, lλ ∈
/
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 not
identically equal to ∞. If there exist constants a ∈ R and b > 0 such that
η(λ) ≥ a + b(1 − qλ)−1/q
then
½Z
ρ(X) =
sup
λ∈[0,1/q)
0
for all λ ∈ [0, 1/q) ,
(8.13)
¾
1
AVaRy (X)(1 − λ)y
−λ
dy − η(λ)
(8.14)
is a real-valued locally Lipschitz-continuous icv-monotone convex monetary risk measure on Lp . If η
satisfies (8.13) and is lower semicontinuous, then the supremum in (8.14) is attained. On the other
hand, 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
Z
0
1
­
®
AVaRy (X)(1 − λ)y −λ dy = −q X , lλ ,
for all λ ∈ [0, 1), where
µ ¶
1
l0 (y) = log
y
and lλ (y) =
1 − λ −λ
(y − 1)
λ
for λ ∈ (0, 1) .
It can easily be checked that the mapping λ 7→ klλ kq is continuous on [0, 1/q), and for λ ↑ 1/q, one has
¶
°
°
i 1 − λ µ°
1−λh
1−λ°
1−λ
° −λ °
° −λ °
−1/q
(1 − qλ)−1/q .
(1 − qλ)
−1 =
°y ° − 1 ≤ klλ kq ≤
°y ° =
λ
λ
λ
λ
q
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
½Z 1
¾
−λ
−β
ρ(X) = max
AVaRy (X)(1 − λ)y dy − α(1 − qλ)
(8.15)
λ∈[0,1/q)
0
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
µZ 1
¶¯
¯
d
−λ
−β ¯
AVaRy (X)(1 − λ)y dy − α(1 − qλ)
= 0,
¯
dλ
0
λ=λ0
which is equivalent to
Z 1
AVaRy (X) [1 + (1 − λ0 ) log(y)] y −λ0 dy + αβq(1 − qλ0 )−β−1 = 0 .
0
Since lλ ∈ D̂sq 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 X6∼t Y in Lp for which the maximum in (8.15) is attained at the
same λ0 ∈ [0, 1/q). This means that χ̂ρ (q X ) ∩ χ̂ρ (q Y ) 6= ∅. Hence, by Theorem 6.14, ρ is in general not
strictly convex modulo translation.
23
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