arX

iv:2

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RM

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Feb

202

1

Inf-convolution and optimal risk sharing with countable sets of

risk measures

Marcelo Brutti Righia,∗

marcelo.righi@ufrgs.br

Marlon Ruoso Morescoa

marlon.moresco@ufrgs.br

aBusiness School, Federal University of Rio Grande do Sul, Washington Luiz, 855, Porto Alegre,

Brazil, zip 90010-460

Abstract

The inf-convolution of risk measures is directly related to risk sharing and general equi-

librium, and it has attracted considerable attention in mathematical finance and insurance

problems. However, the theory is restricted to finite sets of risk measures. In this study,

we extend the inf-convolution of risk measures in its convex-combination form to a count-

able (not necessarily finite) set of alternatives. The intuitive principle of this approach a

generalization of convex weights in the finite case. Subsequently, we extensively generalize

known properties and results to this framework. Specifically, we investigate the preservation

of properties, dual representations, optimal allocations, and self-convolution.

Keywords: Risk measures, Inf-convolution, Risk sharing, Representations, Optimal alloca-

tions.

1 Introduction

The theory of risk measures has attracted considerable attention in mathematical finance and

insurance since the seminal paper by Artzner et al. (1999). The books by Pflug and Romisch

(2007), Delbaen (2012), Ruschendorf (2013), and Follmer and Schied (2016) are comprehensive

expositions of this subject. In these studies, a key topic is the inf-convolution of risk measures,

which is directly related to risk sharing and general equilibrium. These problems may be

connected with regulatory capital reduction, risk transfer in insurance–reinsurance contracts,

and several other applications in classic studies such as Borch (1962), Arrow (1963), Gerber

(1978), and Buhlmann (1982), as well as more recent research as in Landsberger and Meilijson

(1994), Dana and Meilijson (2003), and Heath and Ku (2004).

Formally, the inf-convolution of risk measures is defined as

ni=1ρ

i(X) = inf

n∑

i=1

ρi(Xi) :

n∑

i=1

Xi = X

,

∗Corresponding author. We would like to thank the editor, anonymous associate editor, and reviewer forconstructive comments and suggestions, which have been very useful to improve the technical quality of themanuscript. We are grateful for the financial support of FAPERGS (Rio Grande do Sul State Research Council)project number 17/2551-0000862-6 and CNPq (Brazilian Research Council) projects number 302369/2018-0 and407556/2018-4.

1

whereX andXi, i = 1, · · · , n, belong to some linear space of random variables over a probability

space, and ρi, i = 1, · · · , n, are risk measures, which are functionals on this linear space. By

using a slightly modified version, convex combinations, which represent weighting schemes,

may be considered as follows: µ = µ1, · · · , µn ∈ [0, 1]n,∑n

i=1 µi = 1; this modified version is

defined as

ρµ,nconv(X) = inf

n∑

i=1

µiρi(Xi) :

n∑

i=1

µiXi = X

.

Letting ρi = µiρi, i = 1, · · · , n immediately implies that ρ

µ,nconv(X) shares some properties as

the standard ni=1ρ

i. See Starr (2011) for details of the use of such formulation in general

equilibrium theory to obtain all Pareto-optimal allocations.

Convex risk measures, as initially proposed by Follmer and Schied (2002) and Frittelli and Rosazza Gianin

(2002), have recently attracted considerable attention in the context of inf-convolutions, as in

several other areas of risk management. This subject is explored in Barrieu and El Karoui

(2005), Burgert and Ruschendorf (2006), Burgert and Ruschendorf (2008), Jouini et al. (2008),

Filipovic and Svindland (2008), Ludkovski and Ruschendorf (2008), Ludkovski and Young (2009),

Acciaio and Svindland (2009), Acciaio (2009), Tsanakas (2009), Dana and Le Van (2010), Delbaen

(2012), and Kazi-Tani (2017). These studies present a detailed investigation of the properties

of inf-convolution as a risk measure per se, as well as optimality conditions for the resulting

allocations.

Beyond the usual approach of convex risk measures, some studies have been concerned with

inf-convolution in relation to specific properties, as in Acciaio (2007), Grechuk et al. (2009),

Grechuk and Zabarankin (2012), Carlier et al. (2012), Mastrogiacomo and Rosazza Gianin (2015),

and Liu et al. (2020), particular risk measures, as the recent quantile risk sharing in Embrechts et al.

(2018), Embrechts et al. (2020), Weber (2018), Wang and Ziegel (2018), and Liu et al. (2019),

or even specific topics, as in Liebrich and Svindland (2019). However, these studies are re-

stricted to finite sets of risk measures.

In this study, we extend the convex combination-based inf-convolution of risk measures to

a infinite countable set of alternatives. Specifically, we consider a collection of risk measures

ρI = ρi, i ∈ I, where I is a nonempty countable set. Then, we obtain the generalized version

of the convex inf-convolution as follows:

ρµconv(X) = inf

∑

i∈I

ρi(Xi)µi :∑

i∈I

Xiµi = X

.

The intuitive principle of this approach is to regard µ = µii∈I ⊂ [0, 1] such that∑

i∈I µi = 1 as

a generalization of convex weights in the finite case. We extensively generalize known properties

and results to this framework. More specifically, we investigate the preservation of properties

of ρI , dual representations, optimal allocations, and self-convolution. Of course, owing to the

extent of the related literature, we do not intend to be exhaustive. To the best of our knowledge,

there is no study in this direction.

Regarding to interpretation, it is the minimum amount of risk, which may represent capital

requirement for instance, among all possible ways of dividing a risk X into countable fragments

and distributing the capital into countable units under weighting scheme. Such units could be

2

business lines, agents, lotteries, liquidation times etc. Thus, cuch formulation arises naturally

when the agent is allowed to pulverize its position in as many fragments as desired. Weights

could be the importance of each ρi in the global decision. The key point to be noted, is that in

this case the split of the position X can be taken at any number of fragments as desired instead

of a fixed finite one as in the usual approach. Thus, it is expected that the value generated for

the resulting risk measure be smaller than the one resulting from any fixed finite inf-convolution.

The work of Wang (2016) considers a countable allocation, but with ρi = ρ for any i ∈ I,

i.e., a fixed risk measure. In his paper, the regulatory arbitrage, which occurs when dividing a

position into several fragments results in a reduced capital requirement in relation to the joint

position. In our framework, we generalize such reasoning by allowing distinct risk measures.

In this sense, our approach identifies the limit case regarding all possibilities for the agent

concerning to the division of a position. We also allow for weights in order to allow for distinct

degrees of importance for each risk measure.

The study by Righi (2019b) considers an arbitrary set of risk measures and investigates the

properties of combinations of the form ρ = f(ρI), where f is a combination function over a

linear space generated by the outcomes of ρI(X) = ρi(X), i ∈ I. Under the lack of a universal

choice of a best risk measure from a set of alternatives, one can think into considering the joint

use of many candidates in the goal of benefit from distinct qualities. Clearly, the inf-convolution

is not suitable for such a framework, as X is fixed. Thus, under our approach for a countable

set of candidates it is possible to split the position in order to obtain the best allocation.

From a mathematical point of view, the countable I is a limiting case. The consideration

of an arbitrary (not necessarily finite or countable) set of risk measures done by considering a

measure (probability) µ over a suitable sigma algebra G of I. The problem then becomes

ρµconv(X) = inf

∫

Iρi(Xi)dµ :

∫

IXidµ = X

.

However, it would be necessary to impose assumptions on G in order to avoid measurability

issues. Such assumptions, that the maps i → Xi(ω) are measurable for any ω ∈ Ω and every

family Xi ∈ L∞, i ∈ I, would imply that G is the power set, which would leave us without

meaningful choices for probability measures. In fact, as a consequence of Ulam’s theorem every

probability measure on the power set of set with cardinality as the powerset of N is a discrete

probability measure.

The remainder of this paper is organized as follows. In Section 2, we present preliminaries

regarding notation, and briefly provide background material on the theory of risk measures. In

Section 3, we present the proposed approach and results regarding the preservation of financial

and continuity properties of the set of risk measures. In Section 4, we prove results regarding

dual representations for the convex, coherent, and law-invariant cases. In Section 5, we ex-

plore optimal allocations by considering general results regarding the existence, comonotonic

improvement, and law invariance of solutions, as well as the comonotonicity and flatness of

distributions. In Section 6, we explore the special topic of self-convolution and its relation to

regulatory arbitrage.

3

2 Preliminaries

We consider a probability space (Ω,F ,P). All equalities and inequalities are in the P−a.s. sense.

Let L0 = L0(Ω,F ,P) and L∞ = L∞(Ω,F ,P) be the spaces of (equivalence classes under P−a.s.

equality of) finite and essentially bounded random variables, respectively. When not explicit,

we consider in L∞ its strong topology. We define 1A as the indicator function for an event

A ∈ F . We identify constant random variables with real numbers. A pair X,Y ∈ L0 is called

comonotone if (X(w) −X(w′)) (Y (w) − Y (w′)) ≥ 0, w,w′

∈ Ω holds P ⊗ P − a.s. We denote

by Xn → X convergence in the L∞ essential supremum norm ‖·‖∞, whereas limn→∞

Xn = X

indicates P − a.s. convergence. The notation X Y , for X,Y ∈ L∞, indicates second-order

stochastic dominance, that is, E[f(X)] ≤ E[f(Y )] for any increasing convex function f : R → R.

In particular, E[X|F ′] X for any σ-algebra F ′ ⊆ F .

Let P be the set of all probability measures on (Ω,F). We denote, by EQ[X] =∫

ΩXdQ,

FX,Q(x) = Q(X ≤ x), and F−1X,Q(α) = inf x : FX,Q(x) ≥ α, the expected value, the (increasing

and right-continuous) probability function, and its left quantile for X ∈ L∞ with respect to

Q ∈ P. We write XQ∼ Y when FX,Q = FY,Q. We drop subscripts indicating probability

measures when Q = P. Furthermore, let Q ⊂ P be the set of probability measures Q that are

absolutely continuous with respect to P, with Radon–Nikodym derivative dQdP

. We denote the

topological dual (L∞)∗ of L∞ by ba, which is defined as the space of finitely additive signed

measures (with finite total variation norm ‖·‖TV ) that are absolutely continuous with respect

to P; moreover, we let ba1,+ = m ∈ ba : m ≥ 0,m(Ω) = 1 and by abuse of notation, we define

Em[X] =∫

ΩXdm as the bilinear-form integral of X ∈ L∞ with respect to m ∈ ba1,+.

Definition 2.1. A functional ρ : L∞ → R is called a risk measure. It may have the following

properties:

(i) Monotonicity: If X ≤ Y , then ρ(X) ≥ ρ(Y ), ∀X,Y ∈ L∞.

(ii) Translation invariance: ρ(X + C) = ρ(X) − C, ∀X,Y ∈ L∞, ∀ C ∈ R.

(iii) Convexity: ρ(λX + (1− λ)Y ) ≤ λρ(X) + (1− λ)ρ(Y ), ∀X,Y ∈ L∞, ∀ λ ∈ [0, 1].

(iv) Positive homogeneity: ρ(λX) = λρ(X), ∀X,Y ∈ L∞, ∀ λ ≥ 0.

(v) Law invariance: If FX = FY , then ρ(X) = ρ(Y ), ∀X,Y ∈ L∞.

(vi) Comonotonic additivity: ρ(X + Y ) = ρ(X) + ρ(Y ), ∀X,Y ∈ L∞ with X,Y comonotone.

(vii) Loadedness: ρ(X) ≥ −E[X], ∀X ∈ L∞.

(viii) Limitedness: ρ(X) ≤ − ess infX, ∀X ∈ L∞.

A risk measure ρ is called monetary if it satisfies (i) and (ii), convex if it is monetary

and satisfies (iii), coherent if it is convex and satisfies (iv), law invariant if it satisfies (v),

comonotone if it satisfies (vi), loaded if it satisfies (vii), and limited if it satisfies (viii). Unless

otherwise stated, we assume that risk measures are normalized in the sense that ρ(0) = 0. The

acceptance set of ρ is defined as Aρ = X ∈ L∞ : ρ(X) ≤ 0.

4

In addition to the usual norm-based continuity notions, P−a.s. pointwise continuity notions

are relevant in the context of risk measures.

Definition 2.2. A risk measure ρ : L∞ → R is called

(i) Fatou continuous: If limn→∞

Xn = X implies that ρ(X) ≤ lim infn→∞

ρ(Xn), ∀ Xn∞n=1 bounded

in L∞ norm and for any X ∈ L∞.

(ii) Continuous from above: If limn→∞

Xn = X, with Xn being decreasing, implies that ρ(X) =

limn→∞

ρ(Xn), ∀ Xn∞n=1,X ∈ L∞.

(iii) Continuous from below: If limn→∞

Xn = X, with Xn being increasing, implies that ρ(X) =

limn→∞

ρ(Xn), ∀ Xn∞n=1,X ∈ L∞.

(iv) Lebesgue continuous: If limn→∞

Xn = X implies that ρ(X) = limn→∞

ρ(Xn), ∀Xn∞n=1 bounded

in L∞ norm and X ∈ L∞.

For more details regarding these properties, we refer to the classic books mentioned in the

introduction. We also have the following dual representations.

Theorem 2.3 (Theorem 2.3 in Delbaen (2002b), Theorem 4.33 in Follmer and Schied (2016)).

Let ρ : L∞ → R be a risk measure. Then,

(i) ρ is a convex risk measure if and only if it can be represented as

ρ(X) = maxm∈ba1,+

Em[−X]− αminρ (m)

, ∀X ∈ L∞, (2.1)

where αminρ : ba1,+ → R+ ∪ ∞, defined as αmin

ρ (m) = supX∈L∞

Em[−X]− ρ(X) =

supX∈Aρ

Em[−X], is a lower semi-continuous (in the total-variation norm) convex function

that is called penalty term.

(ii) ρ is a Fatou-continuous coherent risk measure if and only if it can be represented as

ρ(X) = maxm∈Qρ

Em[−X], ∀X ∈ L∞, (2.2)

where Qρ ⊆ ba1,+ is a nonempty, closed, and convex set that is called the dual set of ρ.

Remark 2.4. With the assumption of Fatou continuity, the representations in the previous

theorem could be considered over Q instead of ba1,+, but with the supremum not necessarily

being attained. Moreover, for convex risk measures, we can define certain subgradients using

Legendre–Fenchel duality (i.e., convex conjugates), as follows:

∂ρ(X) = m ∈ ba1,+ : ρ(Y )− ρ(X) ≥ Em[−(Y −X)] ∀ Y ∈ L∞

=

m ∈ ba1,+ : Em[−X]− αminρ (m) ≥ ρ(X)

,

∂αminρ (m) =

X ∈ L∞ : αminρ (n)− αmin

ρ (m) ≥ E(n−m)[−X] ∀ n ∈ ba

=

X ∈ L∞ : Em[−X]− ρ(X) ≥ αminρ (m)

,

5

The negative sign in the expectation above is used to maintain the (anti) monotonicity pattern

of risk measures. We note that these subgradient sets could be empty if we consider only Q

instead of ba1,+. Moreover, by Theorem 2.3, we could replace the inequalities in definition

of sub-gradients by equalities. Further, it is immediate that X ∈ ∂αminρ (m) if and only if

m ∈ ∂ρ(X).

Example 2.5. Examples of risk measures:

(i) Expected loss (EL): This is a Fatou-continuous, law-invariant, comonotone, coherent

risk measure defined as EL(X) = −E[X] = −∫ 10 F−1

X (s)ds. We have that AEL =

X ∈ L∞ : E[X] ≥ 0 and QEL = P.

(ii) Value at risk (VaR): This is a Fatou-continuous, law-invariant, comonotone, monetary

risk measure defined as V aRα(X) = −F−1X (α), α ∈ [0, 1]. We have the acceptance set

AV aRα = X ∈ L∞ : P(X < 0) ≤ α.

(iii) Expected shortfall (ES): This is a Fatou-continuous, law-invariant, comonotone, coher-

ent risk measure defined as ESα(X) = 1α

∫ α

0 V aRs(X)ds, α ∈ (0, 1] and ES0(X) =

V aR0(X) = − ess infX. We have AESα =

X ∈ L∞ :∫ α

0 V aRs(X)ds ≤ 0

and QESα =

Q ∈ Q : dQdP

≤ 1α

.

(iv) Entropic risk measure (Ent): This is a Fatou-continuous, law-invariant, convex risk mea-

sure defined as Entγ(X) = 1γlog(

E[

e−γX])

, γ ≥ 0. Its acceptance set is AEntγ =

X ∈ L∞ : E[e−γX ] ≤ 1

, and the penalty term is αminEntγ (Q) = 1

γE[

dQdP

log(

dQdP

)]

.

(v) Maximum loss (ML): This is a Fatou-continuous, law-invariant, coherent risk measure

defined as ML(X) = −ess infX = −F−1X (0). We have AML = X ∈ L∞ : X ≥ 0 and

QML = Q.

If law invariance is satisfied, as is the case in most practical applications, interesting features

are present. In this paper, when dealing with law invariance we always assume that our base

probability space (Ω,F ,P) is atomless.

Theorem 2.6 (Theorem 2.1 in Jouini et al. (2006) and Proposition 1.1 in Svindland (2010)).

Let ρ : L∞ → R be a law-invariant, convex risk measure. Then, ρ is Fatou continuous.

Theorem 2.7 (Theorem 4.3 in Bauerle and Muller (2006), Corollary 4.65 in Follmer and Schied

(2016)). Let ρ : L∞ → R be a law-invariant, convex risk measure. Then, X Y implies that

ρ(X) ≤ ρ(Y ).

Theorem 2.8 (Theorems 4 and 7 in Kusuoka (2001), Theorem 4.1 in Acerbi (2002), Theorem

7 in Fritelli and Rosazza Gianin (2005)). Let ρ : L∞ → R be a risk measure. Then

(i) ρ is a law-invariant, convex risk measure if and only if it can be represented as

ρ(X) = supm∈M

∫

(0,1]ESα(X)dm − βmin

ρ (m)

, ∀X ∈ L∞, (2.3)

6

where M is the set of probability measures on (0, 1], and βminρ : M → R+∪∞ is defined

as βminρ (m) = sup

X∈Aρ

∫

(0,1]ESα(X)dm.

(ii) ρ is a law-invariant, coherent risk measure if and only if it can be represented as

ρ(X) = supm∈Mρ

∫

(0,1]ESα(X)dm, ∀X ∈ L∞, (2.4)

where Mρ =

m ∈ M :∫

(u,1]1vdm = F−1

dQdP

(1− u), Q ∈ Qρ

.

(iii) ρ is a law-invariant, comonotone, coherent risk measure if and only if it can be represented

as

ρ(X) =

∫

(0,1]ESα(X)dm (2.5)

=

∫ 1

0V aRα(X)φ(α)dα (2.6)

=

∫ 0

−∞(g(P(−X ≥ x)− 1)dx+

∫ ∞

0g(P(−X ≥ x))dx, ∀X ∈ L∞, (2.7)

where m ∈ Mρ, φ : [0, 1] → R+ is decreasing and right-continuous, with φ(1) = 0 and∫ 10 φ(u)du = 1, and g : [0, 1] → [0, 1], called distortion, is increasing and concave, with

g(0) = 0 and g(1) = 1. We have that∫

(u,1]1vdm = φ(u) = g′+(u) ∀ u ∈ [0, 1].

Remark 2.9. (i) Functionals with representation as in (iii) of the last theorem are called

spectral or distortion risk measures. This concept is related to capacity set functions and

Choquet integrals. Note that Comonotonic Additivity implies coherence for convex risk

measures, see Lemma 4.83 of Follmer and Schied (2016) for instance. In this case, ρ can

be represented by Qρ = Q ∈ Q : Q(A) ≤ g(P(A)), ∀ A ∈ F, which is the core of g, if

and only if g is its distortion function. If φ is not decreasing (and thus g is not concave),

then the risk measure is not convex and cannot be represented as combinations of ES.

(ii) Without law invariance, we can (see, for instance, Theorem 4.94 and Corollary 4.95 in

Follmer and Schied (2016)) represent a convex, comonotone risk measure ρ by a Choquet

integral as follows:

ρµconv(X) =

∫

(−X)dc

=

∫ 0

−∞(c(−X ≥ x)− 1)dx +

∫ ∞

0c(−X ≥ x)dx

= maxm∈bac1,+

Em[−X], ∀X ∈ L∞,

where c : F → [0, 1] is a normalized (c(∅) = 0 and c(Ω) = 1), monotone (if A ⊆ B then

c(A) ≤ c(B)), submodular (c(A∪B)+ c(A∩B) ≤ c(A)+ c(B)) set function that is called

capacity and is defined as c(A) = ρ(−1A) ∀ A ∈ F , and bac1,+ = m ∈ ba1,+ : m(A) ≤

c(A) ∀A ∈ F.

7

3 Proposed approach

Let ρI = ρi : L∞ → R, i ∈ I be some (a priori specified) collection of normalized monetary

risk measures, where I is a nonempty infinite countable set. We define the set of weighting

schemes V =

µii∈I ⊂ [0, 1] :∑

i∈I µi = 1

. Otherwise stated we fix µ ∈ V and denote

Iµ = i ∈ I : µi > 0. We use the notations Xi ∈ L∞, i ∈ I = Xi, i ∈ I = Xi i∈I = Xi

for families indexed over I; these families should be understood as generalizations of n-tuples.

For any X ∈ L∞, we define its allocations as

A(X) =

Xi i∈I :∑

i∈I

Xiµi = X, Xi is bounded

.

Evidently, ω →∑

i∈I Xi(ω)µi defines a random variable in L∞ for any Xii∈I ∈ A(X), X ∈

L∞. We note that the identity∑

i∈I Xiµi = X should then be understood in the P − a.s.

sense. The restriction to bounded allocations Xi is to circumvent technical issues, such as

convergence for instance. Note that this is always the case for finite I.

The countable case we study can be regarded as I = N where the set of allocations A(X)

consists of all sequences Xii∈N ⊂ L∞ such that the associated sequence∑n

i=1 µiXi converges

to X in the P− a.s. sense. Note that if∑n

i=1Xiµi = X for some n ∈ N, then

∑n+ki=1 Xiµi = X

for any k ∈ N by taking Xi = 0 for i > n. In particular X1, . . . ,Xn, 0, . . . ∈ A(X). We

have that A(X) 6= ∅ for any X because we can select Xi = X, ∀ i ∈ I. We also note that

Xii∈I ∈ A(X + Y ) is equivalent to Xi − Y i∈I ∈ A(X) for any X,Y ∈ L∞. Furthermore, if

Xii∈I ∈ A(X) and Y ii∈I ∈ A(Y ), then aXi + bY i ∈ A(aX + bY ) for any a, b ∈ R and

X,Y ∈ L∞. We now define the core functional in our study.

Definition 3.1. Let ρI = ρi : L∞ → R, i ∈ I be a collection of monetary risk measures and

µ ∈ V. The µ-weighted inf-convolution risk measure is a functional ρµconv : L∞ → R ∪ −∞

defined as

ρµconv(X) = inf

∑

i∈I

ρi(Xi)µi : Xii∈I ∈ A(X)

. (3.1)

Remark 3.2. We defined risk measures as functionals that only assume finite values. By abuse

of notation, we will also consider ρµconv to be a risk measure, and we will provide conditions

whereby it is finite. Since ρI consists of monetary risk measures, then ρµconv(X) < ∞ because

for any X ∈ L∞ we have that ρµconv(X) ≤∑

i∈I ρi(X)µi ≤ ‖X‖∞ < ∞. Moreover, we note that

normalization is not directly inherited from ρI ; indeed, ρµconv(0) ≤ 0. When ρ

µconv is convex it

is finite if and only if ρµconv(0) > −∞, which is a well-known fact from convex analysis that a

convex function that does assume ∞ is either finite or −∞ point-wise (see Lemma 16 of Delbaen

(2012) for instance).

The following proposition provides useful results regarding well definiteness, representations

and properties of ρµconv.

Proposition 3.3. We have that

(i) ρµconv is well defined.

8

(ii) For any X ∈ L∞ holds

ρµconv(X) = inf

∑

i∈I

ρi(X −Xi)µi : Xii∈I ∈ A(0)

= limn→∞

inf

n∑

i=1

ρi(Xi)µi :n∑

i=1

Xiµi = X

= inf

n∑

i=1

ρi(Xi)µi : n ∈ N,

n∑

i=1

Xiµi = X

.

(iii) If ρI consists of risk measures satisfying positive homogeneity, then ρµconv(X) ≤ ρi(X)∀i ∈

Iµ, ∀X ∈ L∞. In particular, ρµconv < ∞.

Proof. (i) We must to show that∑

i∈I ρi(Xi)µi converges for any Xi ∈ A(X). Due to

translation invariance, is enough to show that∑∞

i ρ(Xi)µi converges for any Xi such

that ρ(Xi) ≥ 0. As Xi is bounded, there is Y ∈ L∞ such that y := ess inf Y ≤ Y ≤

Xi,∀ i ∈ I. By monotonicity it follows that 0 ≤ ρ(Xi) ≤ ρ(Y ) ≤ ρ(y) = −y. Thus,

0 ≤∑

i∈I

ρ(Xi)µi ≤ −∑

i∈I

yµi = −y∑

i∈I

µi = −y.

Hence, the claim follows by monotone convergence.

(ii) For the first relation, we note that∑

i∈I Xiµi = X if and only if

∑

i∈I(X − Xi)µi =∑

i∈I(Xi −X)µi = 0. Thus, by letting Y i = X −Xi, ∀ i ∈ I, we have that

ρµconv(X) = inf

∑

i∈I

ρi(X − Y i)µi : Yii∈I ∈ A(0)

.

Regarding the second relation, first notice that

limn→∞

inf

n∑

i=1

ρ(Xi)µi :n∑

i=1

Xiµi = X

= inf⋃

n∈N

n∑

i=1

ρ(Xi)µi :n∑

i=1

Xiµi = X

.

Let Bn :=∑n

i=1 ρ(Xi)µi :

∑ni=1X

iµi = X

and B :=∑∞

i=1 ρ(Xi)µi : X

i ∈ A(X)

.

Such sets depend on X and Bn ⊆ Bn+1 ⊆ B ⊆ R,∀ n ∈ N. And as any convergent infinite

sum is the limit of finite sums, we obtain B ⊆ cl(∪n∈NBn). Thus, cl(B) = cl(∪n∈NBn).

If both B and ∪n∈NBn are unbounded from below then their infimum coincide to −∞.

If they both are bounded from below, we have that ρµconv(X) = inf B = inf cl(B) =

inf(cl(∪n∈NBn)) = inf ∪n∈NBn = limn inf∑n

i ρ(Xi)µi :

∑ni X

iµi = X

. Therefore, we

only need to show that B is unbounded if and only if ∪n∈NBn is unbounded from below.

Since ∪n∈NBn ⊆ B we clearly have that if ∪n∈NBn is unbounded from below so is B. For

the converse, let B be unbounded from below. Then there is a sequence bj ⊆ B such

that bj ↓ −∞. As any bj is a limit point of a sequence (in n) ajn ⊆ ∪n∈NBn, we can

find another sequence (in j) ajn(j)

⊆ ∪n∈NBn, where n(j) is sufficiently large natural

number, such that aj

n(j) → −∞. This fact implies that ∪n∈NBn is also unbounded from

9

below.

For the third relation, we show that n → inf∑n

i=1 ρi(Xi)µi :

∑ni=1 X

iµi = X

is de-

creasing. We have that

inf

n+1∑

i=1

ρi(Xi)µi :

n+1∑

i=1

Xiµi = X

≤ inf

n+1∑

i=1

ρi(Xi)µi :

n∑

i=1

Xiµi = X,Xn+1 = 0

= inf

n∑

i=1

ρi(Xi)µi :n∑

i=1

Xiµi = X

.

Hence, the infimum with respect to n ∈ N can be replaced by a limit.

(iii) We assume, toward a contradiction, that there is X ∈ L∞ such that ρµconv(X) > ρj(X)

for some j ∈ Iµ. Let Yii∈I be such that Y i = (µj)

−1X, recalling that µj > 0, for i = j,

and Y i = 0 otherwise. Then, Y ii∈I ∈ A(X). Thus, by positive homogeneity and the

definition of ρµconv we have that

ρj(X) < ρµconv(X) ≤∑

i∈I

ρi(Y i)µi = ρj(X),

which is a contradiction. In this case, for any X ∈ L∞, we have that ρµconv(X) ≤

inf i∈Iµ ρi(X) < ∞.

We now present a result regarding the preservation by ρµconv of financial properties of ρI .

Proposition 3.4. ρµconv is monetary. Moreover, If ρI consists of risk measures with convexity,

positive homogeneity, law invariance, loadedness, or limitedness, then each property is inherited

by ρµconv.

Proof. (i) Monotonicity: Let X ≥ Y . Then, there is Z ≥ 0 such that X = Y + Z. Since

A(Y ) + A(Z) ⊆ A(Y + Z), we have

ρµconv(X) = inf

∑

i∈I

ρi(Xi)µi : Xii∈I ∈ A(Y + Z)

≤ inf

∑

i∈I

ρi(Y i + Zi)µi : Yii∈I ∈ A(Y ), Zii∈I ∈ A(Z), Zi ≥ 0 ∀ i ∈ I

≤ inf

∑

i∈I

ρi(Y i + Z)µi : Yii∈I ∈ A(Y )

≤ inf

∑

i∈I

ρi(Y i)µi : Yii∈I ∈ A(Y )

= ρµconv(Y ).

10

(ii) Translation invariance: For any C ∈ R, we have that

ρµconv(X + C) = inf

∑

i∈I

ρi(Xi)µi : Xi − Ci∈I ∈ A(X)

= inf

∑

i∈I

ρi(Y i + C)µi : Yii∈I ∈ A(X)

= ρµconv(X) −C.

(iii) Convexity: For any λ ∈ [0, 1], we have that

λρµconv(X) + (1− λ)ρµconv(Y ) = infXi∈A(X),Y i∈A(Y )

∑

i∈I

[λρi(Xi) + (1− λ)ρi(Y i)]µi

≥ infXi∈A(X),Y i∈A(Y )

∑

i∈I

ρi(λXi + (1− λ)Y i)µi

≥ inf

∑

i∈I

ρi(Zi)µi : Zii∈I ∈ A(λX + (1− λ)Y )

= ρµconv(λX + (1− λ)Y ).

(iv) Positive homogeneity: For any λ ≥ 0, we have that

λρµconv(X) = inf

∑

i∈I

ρi(λY i)µi : Yii∈I ∈ A(X)

= inf

∑

i∈I

ρi(λY i)µi : λYii∈I ∈ A(λX)

= ρµconv(λX).

(v) Law invariance: We begin by showing that ρµconv inherits law invariance on the sub-domain

L∞⊥ := X ∈ L∞ : ∃uniform on [0, 1] r.v. independent ofX. To that, let X,Y ∈ L∞

⊥ with

X ∼ Y and take some arbitrary Xii∈I ∈ A(X). Note that we can have a countable

set U i, i ∈ I of i.i.d. uniform on [0, 1] random variables independent of Y because

our probability space is atomless, see Theorem 1 of Delbaen (2002b) for instance. Now

take X0 = X, Y 0 = Y and let Y i = F−1Xi|Xi−1,··· ,X0(U

i|Y i−1, · · · , Y 0) ∀ i ∈ I, which is the

conditional quantile function. We thus get that (Y, Y 1, · · · , Y n) ∼ (X,X1, · · · ,Xn)∀n ∈ I

and Y ii∈I ∈ A(Y ). In this sense we obtain ρµconv(Y ) ≤

∑

i∈I ρi(Y i)µi =

∑

i∈I ρi(Xi)µi.

Taking the infimum over A(X) we get ρµconv(Y ) ≤ ρµconv(X). By reversing roles ofX and Y ,

we obtain Law Invariance on L∞⊥ . Now, let X,Y ∈ L∞ withX ∼ Y and define Xn ⊂ L∞

as Xn = 1n⌊nX⌋ and Yn ⊂ L∞ as Yn = 1

n⌊nY ⌋, where ⌊·⌋ is the floor function. Moreover,

it is easy to show that Xn ∼ Yn, ∀ n ∈ N. By Lemma 3 in Liu et al. (2020) we have that

if X ∈ L∞ takes values in a countable set, then X ∈ L∞⊥ . Thus Xn ⊂ L∞

⊥ and

ρµconv(Xn) = ρ

µconv(Yn), ∀ n ∈ N. From item (i) in Proposition 3.6 we have that ρ

µconv is

Lipschitz continuous, in particular posses continuity in ‖·‖∞ norm. Note that X2n → X.

Thus |ρµconv(X)−ρµconv(Y )| ≤ lim

n→∞|ρµconv(X)−ρ

µconv(X2n)|+ lim

n→∞|ρµconv(Y )−ρ

µconv(Y2n)| =

0. Hence ρµconv(X) = ρ

µconv(Y ).

11

(vi) Loadedness: We fix X ∈ L∞ and note that for any Xii∈I ∈ A(X) we have that

∑

i∈I

ρi(Xi)µi ≥∑

i∈I

E[−Xi]µi = E[−X].

By taking the infimum over A(X), we obtain that ρµconv(X) ≥ −E[X].

(vii) Limitedness: We fix X ∈ L∞. By the monotonicity of the countable sum, we have that

ρµconv(X) ≤∑

i∈I

ρi(X)µi ≤ − ess infX.

Remark 3.5. (i) Concerning the preservation of subadditivity, that is, ρ(X + Y ) ≤ ρ(X) +

ρ(Y ), the result follows by an argument analogous to that for convexity, but with X + Y

instead of λX + (1 − λ)Y . We note that in this case, we have normalization because

ρµconv(0) ≤ 0, whereas ρµconv(X) ≤ ρ

µconv(X) + ρ

µconv(0), which implies ρµconv(0) ≥ 0. If the

risk measures of ρI are loaded, we also have normalization because 0 = ρ(0) ≥ ρµconv(0) ≥

E[−0] = 0. Of course, in the case of positive homogeneity, we also obtain normalization.

(ii) Regarding the preservation of comonotonic additivity, let X,Y ∈ L∞ be a comonotone

pair. Then, λX,(1 − λ)Y is also comonotone for any λ ∈ [0, 1]. We note that for any

monetary risk measure ρ, comonotonic additivity implies positive homogeneity. Then, by

arguing as in the proof of (iii) and (iv) of the last proposition with λ = 12 , we have that

ρµconv(X + Y ) = 2ρµconv

(

X

2+

Y

2

)

≤ 2

(

1

2ρµconv(X) +

1

2ρµconv(Y )

)

= ρµconv(X) + ρµconv(Y ).

Thus, we obtain subadditivity for comonotone pairs. If, additionally, convexity (and hence

coherence) for ρI comonotonic additivity is preserved, as shown in Theorem 4.7.

In the following, we focus on the preservation by ρµconv of continuity properties of ρI .

Proposition 3.6. We have that

(i) ρµconv is Lipschitz continuous.

(ii) If ρI consists of continuous from below risk measures, then ρµconv is continuous from below.

Proof. (i) For each i ∈ I, we have that |ρi(X)− ρi(Y )| ≤ ‖X − Y ‖∞. Thus,

|ρµconv(X)− ρµconv(Y )|

=

∣

∣

∣

∣

∣

inf

∑

i∈I

ρi(X −Xi)µi : Xii∈I ∈ A(0)

− inf

∑

i∈I

ρi(Y −Xi)µi : Xii∈I ∈ A(0)

∣

∣

∣

∣

∣

≤ sup

∣

∣

∣

∣

∣

∑

i∈I

[

ρi(X −Xi)− ρi(Y −Xi)]

µi

∣

∣

∣

∣

∣

: Xii∈I ∈ A(0)

≤ ‖X − Y ‖∞.

(ii) Let Xn∞n=1 ⊂ L∞ be increasing such that lim

n→∞Xn = X ∈ L∞. By the monotonicity

of ρI we have that each ρi(Xn − Xi) is decreasing in n. Moreover, i → ρi(Xn − Xi) is

12

bounded above by supi∈I‖X1−Xi‖∞ < ∞. Thus, by the monotone convergence Theorem

we have that

limn→∞

ρµconv(Xn) = infn

infXi∈A(0)

∑

i∈I

ρi(Xn −Xi)µi

= infXi∈A(0)

infn

∑

i∈I

ρi(Xn −Xi)µi

= infXi∈A(0)

∑

i∈I

[

infn

ρi(Xn −Xi)]

µi

= ρµconv(X).

Remark 3.7. (i) It is important to note that Fatou continuity is not preserved even when I

is finite, as limn→∞

Xn = X does not imply the existence of Xini∈I ∈ A(Xn) ∀ n ∈ N and

Xii∈I ∈ A(X) such that limn→∞

Xin = Xi, ∀ i ∈ I. See Example 9 in Delbaen (2002a),

for instance. Accordingly, one should be careful when dual representations that depends

of such continuity property are considered.

(ii) If ρI consists of convex risk measures that are continuous from below, then ρµconv is convex

Lebesgue continuous. This is true because continuity from below is equivalent to Lebesgue

continuity for convex risk measures, see Theorem 4.22 in Follmer and Schied (2016) for

instance.

Robustness is a key concept in the presence of model uncertainty. It implies small variation

in the output functional when there is bad specification. See, for instance, Cont et al. (2010),

Kratschmer et al. (2014), and Kiesel et al. (2016). We now present a formal definition.

Definition 3.8. Let d be a pseudo-metric on L∞. Then, a risk measure ρ : L∞ → R is called

d-robust if it is continuous with respect to d.

Remark 3.9. Convergence in distribution in the set of bounded random variables (i.e., con-

vergence with respect to the Levy metric) is pivotal in the presence of uncertainty regarding

distributions, as in the model risk framework. It is well known that convex risk measures are

not upper semicontinuous with respect to the Levy metric. By Proposition 3.4, if each member

of ρI is a convex risk measure, then so is ρµconv. Thus, robustness with respect to this metric is

ruled out.

In light of Proposition 3.6, we have that the continuity of the risk measures in ρI with re-

spect to d is not generally preserved by ρµconv. Consequently, the same is true for d-robustness.

Nonetheless, under stronger assumptions, we have the following corollary regarding the preser-

vation of robustness.

Corollary 3.10. If ρI consists of risk measures that are Lipschitz continuous (with constant

C) with respect to d, then ρµconv is d-robust.

Proof. The proof is analogous to that of (i) in Proposition 3.6.

13

4 Dual representations

We now present the main results regarding the representation of ρµconv for convex cases.

Theorem 4.1. Let ρI be a collection of convex risk measures. We have that

(i) The acceptance set of ρµconv is

Aρµconv

= cl(Aµ), (4.1)

where Aµ =

X ∈ L∞ : ∃ Xii∈I ∈ A(X) s.t. Xi ∈ Aρi ∀ i ∈ Iµ

=∑

i∈I Aρiµi. More-

over, Aµ is not dense in L∞ if and only if Aµ 6= L∞.

(ii) The minimal penalty term of ρµconv is

αminρµconv

(m) =∑

i∈I

αminρi (m)µi, ∀m ∈ ba1,+. (4.2)

Proof. We have that ρµconv is a convex risk measure that is either finite or identically −∞.

Moreover, it is well known for any n ∈ N that

n∑

i=1

αminρi µi = sup

X∈L∞

Em[−X]− inf

n∑

i=1

ρi(Xi)µi :n∑

i=1

Xiµi = X

,

which generates the acceptance set cl(∑n

i=1Aρiµi

)

with µ1, . . . , µn ⊂ [0, 1]n. We now demon-

strate the claims.

(i) Let X ∈ Aµ. Then, there is Xii∈I ∈ A(X) such that Xi ∈ Aρi ∀ i ∈ Iµ. Thus,

ρµconv(X) ≤

∑

i∈I ρi(Xi)µi ≤ 0. This implies X ∈ Aρ

µconv

. By taking closures we get

cl(Aµ) ⊆ Aρµconv

. We note that Aµ is not necessarily closed (for reasons similar to those

for which ρµconv does not inherit Fatou continuity). For the converse relation, let X ∈

int(Aρµconv

). Then there is Xi ∈ A(X) such that k =∑

i∈I ρi(Xi)µi < 0. Since

ρi(Xi + ρi(Xi)− k) < 0 for any i ∈ I, we have that Y i = Xi + ρi(Xi)− k ∈ int(Aρi) for

any i ∈ I. Moreover,∑

i∈I Yiµi = X + k − k = X. Then Y i ∈ A(X), which implies

X ∈ Aµ ⊆ cl(Aµ). Thus, int(Aρµconv

) ⊆ Aµ. By taking closures we get cl(int(Aρµconv

)) =

Aρµconv

⊆ cl(Aµ), which gives the required equality. Moreover, let Aµ be norm-dense in

L∞. Thus, for any X ∈ L∞ and k > 0, there is Y ∈ Aµ such that ‖X − Y ‖∞ ≤ k.

Thus, X + k ≥ Y and, as Aµ is monotone, we obtain X + k ∈ Aµ. As both X and k are

arbitrary, Aµ = L∞. The converse relation is trivial.

(ii) We have for any m ∈ ba1,+ that

αminρµconv

(m) = supX∈L∞

Em[−X]− limn→∞

inf

n∑

i=1

ρi(Xi)µi :

n∑

i=1

Xiµi = X

≤ limn→∞

supX∈L∞

Em[−X]− inf

n∑

i=1

ρi(Xi)µi :n∑

i=1

Xiµi = X

= limn→∞

n∑

i=1

αminρi (m)µi =

∑

i∈I

αminρi (m)µi.

14

For the converse, by Theorem 2.3 and Remark 2.4, we have that

αminρµconv

(m) = supX∈A

ρµconv

Em[−X], Aρµconv

=

X ∈ L∞ : αminρµconv

(X) ≥ Em[−X] ∀m ∈ ba1,+

.

Moreover, note that if X ∈∑n

i=1 Aρiµi, then X =∑n

i=1 Xiµi with Xi ∈ Aρi with

i = 1, . . . , n. Thus, X ∈ Aµ since X1, . . . ,Xn, 0, 0, . . . ∈ A(X) and 0 is acceptable for

any ρi. Then for any m ∈ ba1,+ we have

∑

i∈I

αminρi (m)µi = lim

n→∞

n∑

i=1

sup

Em[−X]µi : X ∈ Aρi

= limn→∞

sup

n∑

i=1

Em[−Xi]µi : Xi ∈ Aρii=1,...,n

= limn→∞

sup

Em[−X] : X ∈n∑

i=1

Aρiµi

≤ limn→∞

sup Em[−X] : X ∈ Aµ

= sup

Em[−X] : X ∈ Aρµconv

= αminρµconv

(m).

Hence, αminρµconv

=∑

i∈I αminρi

µi. Regarding the properties of m →∑

i∈I αminρi

(m)µi, non-

negativity is straightforward, whereas convexity follows from the monotonicity of the

integral and the convexity of each αminρi

because for any λ ∈ [0, 1] and m1,m2 ∈ ba1,+, we

have that

∑

i∈I

αminρi (λm1 + (1− λ)m2)µi ≤

∑

i∈I

[

λαminρi (m1) + (1− λ)αmin

ρi (m2)]

µi

= λ∑

i∈I

αminρi (m1)µi + (1− λ)

∑

i∈I

αminρi (m2)µi.

Furthermore, by Fatou’s lemma (which can be used because each αminρi

is bounded from

below by 0) and by the lower semi-continuity of each αminρi

with respect to the total

variation norm on ba, for any mn such that mn → m, we have that

∑

i∈I

αminρi (m)µi ≤

∑

i∈I

lim infn→∞

αminρi (mn)µi ≤ lim inf

n→∞

∑

i∈I

αminρi (mn)µi.

Remark 4.2. (i) Under the assumption of Fatou continuity for both the risk measures in ρI

and ρµconv, the claims in Theorem 4.1 could be adapted by replacing the finitely additive

measures m ∈ ba+,1 by probabilities Q ∈ Q. Moreover, weak∗ topological concepts could

replace the corresponding strong (norm) topological concepts.

(ii) The weighted risk measure is a functional ρµ : L∞ → R defined as ρµ(X) =∑

i∈I ρi(X)µi.

Theorem 4.6 in Righi (2019b) states that, assuming Fatou continuity, ρµ can be repre-

15

sented using a convex (not necessarily minimal) penalty defined as

αρµ(Q) = inf

∑

i∈I

αminρi

(

Qi)

µi :∑

i∈I

Qiµi = Q, Qi ∈ Q ∀ i ∈ I

.

By the duality of convex conjugates, αρµ = αminρµ if and only if αρµ is lower semi-continuous.

Nonetheless, this represents a connection between weighted and inf-convolution functions

for countable I, as in the traditional finite case.

The following corollary provides interesting properties regarding the normalization, finite-

ness, preservation, dominance and sub-gradients of ρµconv.

Corollary 4.3. Let ρI be a collection of convex risk measures. Then

(i)

m ∈ ba1,+ :∑

i∈I αminρi

(m)µi < ∞

6= ∅ if and only if then ρµconv is finite. In this case

Aµ is not dense in L∞.

(ii) ρµconv is normalized if and only if

m ∈ ba1,+ : αminρi

(m) = 0 ∀ i ∈ Iµ

6= ∅.

(iii) If ρ : L∞ → R is a convex risk measure with ρ(X) ≤ (≥ or =)ρi(X) ∀ i ∈ Iµ, ∀X ∈ L∞,

then ρ(X) ≤ (≥ or =)ρµconv(X), ∀X ∈ L∞.

(iv)

m ∈ ba1,+ : m ∈ ∂ρi(Xi) ∀ i ∈ Iµ

=⋂

i∈Iµ∂ρi(Xi) ⊆ ∂ρ

µconv(X) for any Xii∈I ∈

A(X) and X ∈ L∞.

(v)

X ∈ L∞ : ∃ Xii∈I ∈ A(X) s.t. Xi ∈ ∂αminρi

(m) ∀ i ∈ Iµ

=∑

i∈I ∂αminρi

(m)µi ⊆ ∂αminρµconv

(m)

for any m ∈ ba1,+.

Proof. (i) By Proposition 3.4, we have ρµconv < ∞. If

m ∈ ba1,+ :∑

i∈I αminρi

(m)µi < ∞

6=

∅, then there exists m ∈ ba1,+ such that

−∞ < Em[−X]−∑

i∈I

αminρi (m)µi ≤ Em[−X]− αmin

ρµconv

(m) ≤ ρµconv(X).

If

m ∈ ba1,+ :∑

i∈I αminρi

(m)µi < ∞

= ∅, then αminρµconv

(m) =∑

i∈I αminρi

(m)µi = ∞,∀m ∈

ba1,+. Hence, ρµconv(X) = −∞, ∀X ∈ L∞. Furthermore, if Aµ is dense in L∞, then by

(i) in Theorem 4.1, we have Aµ = L∞. Thus, we have ρµconv(X) = infm ∈ R : X +m ∈

L∞ = −∞, ∀X ∈ L∞.

(ii) Let

m ∈ ba1,+ : αminρi

(m) = 0 ∀ i ∈ Iµ

6= ∅ and m′ in this set. Then, αminρµconv

(m′) =∑

i∈I αminρi

(m′)µi = 0. Hence,

ρµconv(0) = − minm∈ba1,+

αminρµconv

(m) = 0.

For the converse relation, let

m ∈ ba1,+ : αminρi

(m) = 0 ∀ i ∈ Iµ

= ∅. Thus, for any

m ∈ ba1,+, we have that

i ∈ Iµ : αminρi

(m) > 0

6= ∅. Then,

ρµconv(0) = − minm∈ba1,+

αminρµconv

(m) < 0,

16

which implies that ρµconv is not normalized.

(iii) We prove for the leq relation since the others are quite similar. By Theorem 2.3 and and

Remark 2.4, we have that αminρ (m) ≥ αmin

ρi(m) ∀ i ∈ Iµ for any m ∈ ba1,+. Thus, by

Theorem 4.1, we obtain that

αminρ (m) ≥

∑

i∈I

αminρi (m)µi = αmin

ρµconv

(m).

Hence, ρ(X) ≤ ρµconv(X), ∀X ∈ L∞.

(iv) Let m′ ∈

m ∈ ba1,+ : m ∈ ∂ρi(Xi) ∀ i ∈ Iµ

. Then,

Em′ [−X]− αminρµconv

(m′) ≥∑

i∈I

(

Em′ [−Xi]− αminρi (m′)

)

µi ≥∑

i∈I

ρi(Xi)µi ≥ ρµconv(X),

which implies m′ ∈ ∂ρµconv(X).

(v) We fix m ∈ ba1,+. If X ∈

X ∈ L∞ : ∃ Xii∈I ∈ A(X) s.t. Xi ∈ ∂αminρi

(m) ∀ i ∈ Iµ

, let

Xii∈I ∈ A(X) such that Xi ∈ ∂αminρi

(m) ∀ i ∈ Iµ Then,

Em[−X]− ρµconv(X) ≥∑

i∈I

(

Em[−Xi]− ρi(Xi))

µi =∑

i∈I

αminρi (m)µi ≥ αmin

ρµconv

(m).

Thus, X ∈ ∂αminρµconv

(m).

Remark 4.4. In the context of item (iii) and under coherence of ρI , by Proposition 3.3 we have

that ρ(X) ≤ ρµconv(X) ≤ ρi(X) ∀ i ∈ Iµ, ∀ X ∈ L∞ for any convex risk measure ρ such that

ρ(X) ≤ ρi(X) ∀ i ∈ Iµ, ∀ X ∈ L∞. In this sense, we can understand ρµconv as the “lower-

convexification” of the non-convex risk measure inf i∈Iµ ρi in the sense that the former is the

largest convex risk measure that is dominated by the latter.

We now present the main results regarding the representation of ρµconv for coherent cases.

Theorem 4.5. Let ρI be a collection of coherent risk measures. Then,

(i) ρµconv is finite, and its dual set is

Qρµconv

=

m ∈ ba1,+ : m ∈ Qρi ∀ i ∈ Iµ

=⋂

i∈Iµ

Qρi . (4.3)

In particular, Qρµconv

is non empty.

(ii) The acceptance set of ρµconv is

Aρµconv

= clconv (A∪) = cl(Aµ), A∪ =⋃

i∈Iµ

Aρi , (4.4)

where clconv denotes the closed convex hull.

17

Proof. (i) By Proposition 3.4, we have that ρµconv is a coherent risk measure. Since ρµconv(0) =

0, it is finite. In this case, its dual set is composed by the measures m ∈ ba1,+ such that

αminρµconv

(m) = 0. By Theorems 2.3 and 4.1, we obtain that

Qρµconv

=

m ∈ ba1,+ :∑

i∈I

αminρi (m)µi = 0

=

m ∈ ba1,+ : αminρi (m) = 0 ∀ i ∈ Iµ

=

m ∈ ba1,+ : m ∈ Qρi ∀ i ∈ Iµ

=⋂

i∈Iµ

Qρi .

The convexity and closedness of Qρµconv

follow from the convexity and lower semicontinuity

of αminρµconv

. Furthermore, if Qρµconv

=

m ∈ ba1,+ : αminρi

(m) = 0 ∀ i ∈ Iµ

= ∅, then by

Corollary 4.3, ρµconv(0) < 0, which contradicts coherence.

(ii) We recall that, by Theorem 2.3 and Remark 2.4, for any coherent risk measure ρ : L∞ → R,

we have X ∈ Aρ if and only if Em[−X] ≤ 0 ∀m ∈ Qρ. Thus,

Qρµconv

=⋂

i∈Iµ

m ∈ ba1,+ : Em[−X] ≤ 0 ∀X ∈ Aρi

=

m ∈ Q : Em[−X] ≤ 0, ∀X ∈ ∪i∈IµAρi

= m ∈ Q : Em[−X] ≤ 0, ∀X ∈ clconv(A∪) = Qρclconv(A∪).

By considering the closed convex hull does not affect is because the map X → Em[X]

is linear and continuous for any m ∈ Q. Thus, ρµconv = ρclconv(A∪). Hence, Aρµconv

=

Aρclconv(A∪)= clconv (A∪). We note that A∪ is nonempty, monotone (in the sense that

X ∈ A∪ and Y ≥ X implies Y ∈ A∪), and a cone, as this is true for any Aρi . Moreover,

it is evident that A∪ ⊆ Aµ for normalized risk measures in ρI . Thus, by Theorem 4.1, we

have that Aρµconv

= clconv (A∪) ⊆ cl(Aµ) ⊆ Aρµconv

.

Remark 4.6. (i) Similarly to Remark 4.2, under the assumption of Fatou continuity for both

the risk measures in ρI and ρµconv, the claims in Theorem 4.1 could be adapted by replac-

ing the finitely additive measures m ∈ ba+,1 by probabilities Q ∈ Q. Moreover, weak∗

topological concepts could replace the corresponding strong (norm) topological concepts.

(ii) In light of Corollary 4.3, we have, under the hypotheses of Theorem 4.5, that ρµconv is

finite and normalized if and only if the condition

m ∈ ba1,+ : αminρi

(m) = 0 ∀ i ∈ Iµ

6= ∅

is satisfied. Moreover, both assertions are equivalent to Aµ not being dense in L∞. The

intuition for A∪ is that some position is acceptable if it is acceptable for any relevant (in

the µ sense) member of ρI .

We now focus on dual representations under the assumption of law invariance and comono-

tonic additivity.

Theorem 4.7. Let ρI be a collection of convex, law-invariant risk measures. Then,

18

(i) ρµconv is finite, normalized and has penalty term

βminρµconv

(m) =∑

i∈I

βminρi (m)µi, ∀m ∈ M. (4.5)

(ii) If, in addition, ρI consists of comonotone risk measures, then ρµconv is comonotone, and

its distortion function is

g = infi∈Iµ

gi, (4.6)

where gi is the distortion of ρi for each i ∈ I.

Proof. (i) By Theorem 2.6 and Proposition 3.4, we have that ρµconv is a Fatou continuous

convex risk measure with ρµconv < ∞. Moreover, ρµconv is either finite or identically −∞.

Regarding finiteness and normalization, we note that ρ(X) ≥ −E[X], ∀ X ∈ L∞ for

normalized, convex, law-invariant risk measures by second-order stochastic dominance.

Thus,

αminρi (P) = sup

X∈L∞

E[−X] − ρ(X) ≤ supX∈L∞

ρ(X)− ρ(X) = 0, ∀ i ∈ I.

By the non-negativity of the penalty terms, we have αminρi

(P) = 0, ∀ i ∈ I. Hence, by

item (ii) of Corollary 4.3, we conclude that ρµconv is normalized and, consequently, finite.

Moreover, the penalty term can be obtained by an argument similar to that in (ii) of

Theorem 4.1 by considering the m →∫

(0,1]ESα(X)dm linear and playing the role of

m → Em[−X] and recalling that acceptance sets of law invariant risk measures are law

invariant in the sense that X ∈ Aρ and X ∼ Y implies Y ∈ Aρ.

(ii) By Theorems 2.8, 4.1, and 4.5, as well as Remark 2.9, we have, recalling that ρµconv is

finite and Fatou continuous, that

Qρµconv

= Q ∈ Q : Q(A) ≤ gi(P(A)) ∀ i ∈ Iµ ∀A ∈ F

=

Q ∈ Q : Q(A) ≤ infi∈Iµ

gi(P(A)) ∀ A ∈ F

.

By the properties of the infimum and gii∈I , we obtain that g : [0, 1] → [0, 1] is increasing

and concave, and it satisfies g(0) = 0 and g(1) = 1. Thus, ρµconv can be represented as a

Choquet integral using (2.7), which implies that it is comonotone.

Remark 4.8. As a direct consequence of (ii) in the last theorem, if ρi = ESαi, αi ∈ [0, 1]∀ i ∈ I,

with α = supi∈Iµ αi, then ρ

µconv(X) = ESα(X), ∀X ∈ L∞. The financial intuition is that the

inf convolution of countable many ES at distinct significance levels provides the same risk as

the less conservative option.

Regarding comonotonic additivity, (ii) in Theorem 4.7 remains true if we drop the law

invariance of ρI , as shown in the following corollary.

19

Corollary 4.9. Let ρI = ρi : L∞ → R, i ∈ I be a collection of convex, comonotone risk

measures. Then, ρµconv is finite, normalized, and comonotone, and its capacity function is

c(A) = infi∈Iµ

ci(A), ∀ A ∈ F , (4.7)

where ci is the capacity of ρi for each i ∈ I.

Proof. We note that, by Proposition 3.4, we have ρµconv < ∞ and ρ

µconv(0) = 0 > −∞. Let the

set function c : F → [0, 1] be defined as c(A) = inf i∈Iµ ci(A), where ci is the capacity related

to ρi for each i ∈ I. Thus, by an argument similar to that in Theorem 4.7, if we consider

bac1,+ = m ∈ ba1,+ : m(A) ≤ c(A) ∀A ∈ F, then the reasoning in Remark 2.9 implies that the

claim is true.

5 Optimal allocations

An interesting feature of traditional finite inf-convolution is capital allocation. Highly relevant

concept is Pareto optimality, which is defined as follows.

Definition 5.1. We call Xii∈I ∈ A(X)

(i) Optimal for X ∈ L∞ if∑

i∈I ρi(Xi)µi = ρ

µconv(X).

(ii) Pareto optimal for X ∈ L∞ if for any Y ii∈I ∈ A(X) such that ρi(Y i) ≤ ρi(Xi)∀ i ∈ Iµ,

we have ρi(Y i) = ρi(Xi) ∀ i ∈ Iµ.

Remark 5.2. (i) If ρµconv is normalized, Xi = 0i∈I is optimal for 0. This implies the con-

dition that if Xii∈I ∈ A(0) and ρi(Xi) ≤ 0 ∀ i ∈ Iµ, then ρi(Xi) = 0 ∀ i ∈ Iµ, and

therefore Xi = 0i∈I is also Pareto optimal for 0. This can be understood as a non-

arbitrage condition. We note that any optimal allocation must be Pareto optimal, and

that a risk sharing rule is also a Pareto-optimal allocation.

If ρI consists of monetary risk measures and I is finite, Theorem 3.1 in Jouini et al. (2008)

shows that optimal and Pareto-optimal allocations coincide. In the following proposition, we

extend this result to the context of countable I.

Proposition 5.3. We have that

(i) Xii∈I ∈ A(X) is optimal for X ∈ L∞ if and only if it is Pareto optimal for X ∈ L∞.

(ii) if Xii∈I is optimal for X ∈ L∞, then so is Xi + Cii∈I , where Ci ∈ R ∀ i ∈ Iµ, and∑

i∈I Ciµi = 0.

(iii) Under sub-additivity of ρI and normalization of ρµconv we have that if Xii∈I is optimal

for X ∈ L∞, then so is Xi + Y i for any Y i that is optimal for 0.

Proof. (i) The “only if” part is straightforward, as in Remark 5.2. For the “if” part, let

Xii∈I ∈ A(X) be not optimal for X ∈ L∞. Then, there is Y ii∈I ∈ A(X) such that

20

∑

i∈I ρi(Y i)µi <

∑

i∈I ρi(Xi)µi. Let k

i = ρi(Xi)−ρi(Y i)∀ i ∈ Iµ and k =∑

i∈I kiµi > 0.

Moreover, Y i − ki + ki∈I ∈ A(X) and

∑

i∈I

ρi(Y i − ki + k)µi <∑

i∈I

ρi(Y i − ρi(Xi) + ρi(Y i))µi =∑

i∈I

ρi(Xi)µi.

Hence, Xii∈I is not Pareto optimal for X.

(ii) We note that

∑

i∈I

ρi(Xi + Ci)µi =∑

i∈I

ρi(Xi)µi −∑

i∈I

Ciµi = ρµconv(X).

Thus, Xi + Cii∈I is also Pareto optimal.

(iii) This claim follows by

∑

i∈I

ρ(Xi + Y i) ≤∑

i∈I

ρ(Xi) +∑

i∈I

ρ(Y i) = ρµconv(X) + ρµconv(0) = ρµconv(X).

Hence, Xi + Y i ∈ A(X) is Pareto optimal for X.

We now determine a necessary and sufficient condition for optimality in the case of convex

risk measures.

Theorem 5.4. Let ρI be a family of convex risk measures. Then Xii∈I ∈ A(X) is optimal

for X ∈ L∞ if and only if⋂

i∈Iµ∂ρi(Xi) = ∂ρ

µconv(X) 6= ∅.

Proof. By Corollary 4.3 we have⋂

i∈Iµ∂ρi(Xi) ⊆ ∂ρ

µconv(X). We assume

⋂

i∈Iµ∂ρi(Xi) 6= ∅.

Then, let m′ ∈

m ∈ ba1,+ : m ∈ ∂ρi(Xi) ∀ i ∈ Iµ

⊆ ∂ρµconv(X). We have that

∑

i∈I

ρi(Xi)µi =∑

i∈I

(

Em′ [−Xi]− αminρi (m′)

)

µi ≤ Em′ [−X]− αminρµconv

(m′) = ρµconv(X).

Hence, Xii∈I is optimal for X ∈ L∞. Regarding the converse, for any m′ ∈ ∂ρµconv(X), we

obtain by an argument similar to that in Theorem 4.1 the following:

ρµconv(X) = Em′ [−X]−∑

i∈I

αminρi (m′)µi

= limn→∞

(

Em′ [−X] +n∑

i=1

infY ∈L∞

Em′ [Y ] + ρi (Y )

µi

)

≤ limn→∞

(

Em′ [−X] + infY1,...,Yn⊂L∞

n∑

i=1

Em′ [Y i] + ρi(

Y i)

µi

)

≤ limn→∞

inf

n∑

i=1

Em′ [Y i −X] + ρi(

Y i)

µi :n∑

i=1

Y iµi = X

= limn→∞

inf

n∑

i=1

ρi(

Y i)

µi :

n∑

i=1

Y iµi = X

= ρµconv(X).

21

Thus, ρµconv(X) =∑

i∈Iµρi(Xi)µi if and only if ∃ m′ ∈ ba1,+ such that ρi(Xi) = Em′ [−X] +

αminρi

(m′) ∀ i ∈ Iµ. Hence, m′ ∈

m ∈ ba1,+ : m ∈ ∂ρi(Xi) ∀ i ∈ Iµ

.

We have the following corollary regarding subdifferential and optimality conditions.

Corollary 5.5. Let ρI be a collection convex risk measures. If for any X ∈ L∞ there is an

optimal allocation, then ∂αminρµconv

(m) =∑

i∈Iµ∂αmin

ρi(m)µi for any m ∈ ba1,+.

Proof. From Corollary 4.3 we have∑

i∈Iµ∂αmin

ρi(m)µi ⊆ ∂αmin

ρµconv

(m) for any m ∈ ba1,+. For

the converse relation, if ∂αminρµconv

(m) = ∅, then the claim is immediately obtained. Let then

X ∈ ∂αminρµconv

(m). By the definition of Legendre–Fenchel convex-conjugate duality, the optimality

condition is equivalent to the existence of m ∈ ba1,+ such that Xi ∈ ∂αminρi

(m) ∀ i ∈ Iµ. By

Theorem 5.4, we have that Xi ∈ ∂αminρi

(m) ∀ i ∈ Iµ. Then, X ∈∑

i∈Iµ∂αmin

ρi(m)µi.

Under the assumption of law invariance, it is well known that, for finite I, the mini-

mization problem has a solution under co-monotonic allocations (see, for instance, Theorem

3.2 in Jouini et al. (2008), Proposition 5 in Dana and Meilijson (2003), or Theorem 10.46 in

Ruschendorf (2013)). For the extension to general I, we should extend some definitions and

results regarding comonotonicity. We note that if I is finite, these are equivalent to their

traditional counterparts.

Definition 5.6. Xii∈I is called I-comonotone if (Xi,Xj) is comonotone ∀ (i, j) ∈ Iµ × Iµ.

Lemma 5.7. Xii∈I ∈ A(X), X ∈ L∞, is I-comonotone if and only if there exists a class

of functions hi : R → R, i ∈ I that are (∀ i ∈ Iµ) Lipschitz continuous and increasing, and

they satisfy Xi = hi (X) and∑

i∈I hi(x)µi = x, ∀ x ∈ R. In particular, if Xii∈I ∈ A(X) is

I-comonotone, then F−1X (α) =

∑

i∈I F−1Xi (α)µi, ∀ α ∈ [0, 1].

Proof. For the “if” part, let Xi,Xj ∈ Iµ such that Xi = hi(X) and Xj = hj(X) for hii ∈ I

satisfying the assumptions. Then Xi,Xj are comonotone. For the “only if” part, let Xii∈I be

I-comonotone and X(Ω) = x ∈ R : ∃ ω ∈ Ω s.t. X(w) = x. Then, for any fixed ω ∈ Ω, there

is a family xi = Xi(ω) ∈ R : i ∈ I such that X(ω) = x =∑

i∈I xiµi ∈ X(Ω). Moreover, we

define hi(x) = xi ∀ i ∈ Iµ. If there are ω, ω′ ∈ Ω such that∑

i∈I Xi(ω)µi = x =

∑

i∈I Xi(ω′)µi,

we then obtain∑

i∈I

(

Xi(ω)−Xi(ω′))

µi = 0. Assuming I-comonotonicity, we have that

Xi(ω) = Xi(ω′) ∀ i ∈ Iµ. Consequently, the map x →∑

i∈I hi(x)µi = Id(x) is well defined.

Regarding the increasing behavior of hi, let x, y ∈ X(Ω) with x ≤ y. Then, there are ω, ω′ such

that∑

i∈I Xi(ω)µi = x ≤ y =

∑

i∈I Xi(ω′)µi, which implies

∑

i∈I

(

Xi(ω)−Xi(ω′))

µi ≤ 0.

Comonotonicity implies that this relation is equivalent to hi(x) = Xi(ω) ≤ Xi(ω′) = hi(y)∀ i ∈

Iµ. Concerning Lipschitz continuity, for any x, x + δ ∈ X(Ω) with δ > 0 we obtain 0 ≤

hi(x+ δ) − hi(x) ≤ (µi)−1δ, ∀ i ∈ Iµ. The first inequality is due to the increasing behavior of

hi. The second is because for any i ∈ I we have x+ δ =∑

j∈I\i hj(x+ δ)µj + hi(x+ δ)µi ≥

∑

j∈I\i hj(x)µj + hi(x)µi = hi(x + δ)µi + x− hi(x)µi. It remains to extend hi from X(Ω)

to R. We first extend it to cl(X(Ω)). If x ∈ bd(X(Ω)) is only a one-sided boundary point, then

the continuous extension poses no problem, as increasing functions are involved. If x can be

approximated from both sides, then Lipschitz continuity implies that the left- and right-sided

continuous extensions coincide. The extension to R is performed linearly in each connected

22

component of R\cl(X(Ω)) so that the condition∑

i∈I hi(x) = x is satisfied. Then, the main

claim is proved. Moreover, let Xii∈I ∈ A(X) be I-comonotone. Then, x →∑

i∈I hi(x)µi

is also Lipschitz continuous and increasing. We recall that F−1g(X) = g(F−1

X ) for any increasing

function g : R → R. Then, for any α ∈ [0, 1], we obtain

F−1X (α) = F−1∑

i∈Ihi(X)µi

(α) =∑

i∈I

hi(F−1X (α))µi =

∑

i∈I

F−1hi(X)

(α)µi =∑

i∈I

F−1Xi (α)µi.

We now prove the following comonotonic-improvement theorem for arbitrary I.

Theorem 5.8. Let X ∈ L∞. Then, for any Xii∈I ∈ A(X), there is an I-comonotone

Y ii∈I ∈ A(X) such that Y i Xi ∀ i ∈ Iµ

Proof. Let Fn be the σ-algebra generated by ω : k2−n ≤ X(ω) ≤ k2n ⊂ Ω for k > 0,

Xn = E[X|Fn], and Xin = E[Xi|Fn] ∀ i ∈ I. Then, lim

n→∞Xn = X, lim

n→∞Xi

n = Xi for any i ∈ I,

andXin Xi for each n each i. By the arguments in Proposition 1 in Landsberger and Meilijson

(1994) or Proposition 10.46 in Ruschendorf (2013), we can conclude that every allocation of X

taking a countable number of values is dominated by a comonotone allocation. Thus, by Lemma

5.7, for any n ∈ N, there are Lipschitz continuous, increasing functions hin : R → R, i ∈ I with∑

i∈I hinµi = Id such that Y i

n = hin(Xn) Xin ∀ i ∈ Iµ We note that these functions constitute

a bounded, closed, equicontinuous family. Then, by Ascoli’s theorem, there is a subsequence of

hin that converges uniformly on [ess infX, ess supX] to the Lipschitz-continuous and increasing

hi in the ∀ i ∈ Iµ sense. Thus,∑

i∈I hiµi = Id on [ess infX, ess supX]. We then have Y i =

hi(X) Xi ∀ i ∈ Iµ by considering uniform limits. Finally, by Lemma 5.7, we obtain that

Y i i∈I is I-comonotone. It remains to show that Y i i∈I belongs to A(X). We have that∑

i∈I Yiµi =

∑

i∈I hi(X)µi = X P− a.s. Hence, Y i i∈I ∈ A(X).

We are now in a position to extend the existence of optimal allocations to our framework of

law-invariant, convex risk measures.

Theorem 5.9. Let ρI be a collection of law-invariant, convex risk measures. Then,

(i) For any X ∈ L∞, there is an I-comonotone optimal allocation.

(ii) In addition to initial hypotheses, if ρI consists of risk measures that are strictly monotone

with respect to , then every optimal allocation for any X ∈ L∞ is I-comonotone.

(iii) In addition to initial hypotheses, if ρI consists of strictly convex functionals, we have

uniqueness of the optimal allocation up to scaling (if Xii∈I is optimal for X ∈ L∞,

then so is Xi + Cii∈I , where Ci ∈ R ∀ i ∈ Iµ and∑

i∈I Ciµi = 0).

Proof. (i) By Theorem 5.8, we can restrict the minimization problem to I-comonotonic al-

locations, as, by Theorem 2.7, law-invariant risk measures preserve second-order stochas-

tic dominance. Let Y in = hin(X) ∈ L∞, i ∈ In be an optimal sequence for X,

i.e. limn→∞

∑

i∈I ρi(Y i

n)µi = ρµconv(X), where hin : [ess infX, ess supX] → R are increas-

ing, bounded, and Lipschitz-continuous functions. Such sequence always exist because

23

ρI is monetary and we can take hin = −xn for any i ∈ I, where xn → ρµconv(X). Thus,

each hin is increasing, bounded and Lipschitz continuous while limn→∞

∑

i∈I ρi(hin(X))µi =

limn→∞

∑

i∈I ρi(−xn)µi = lim

n→∞xn = x. By an argument similar to that in Theorem 5.8, we

have that hi is the uniform limit (after passing to a subsequence if necessary) of hin.

Thus, Y in = hin(X) → hi(X) = Y i. By continuity in the essential supremum norm, we

have that limn→∞

∣

∣ρi(Y in)− ρi(Y i)

∣

∣ = 0. As hi is the uniform limit of hin, we have by

dominated convergence, since |ρi(Y in)| ≤ ‖X‖∞ < ∞ for any n ∈ N, that

ρµconv(X) = limn→∞

∑

i∈I

ρi(Y in)µi =

∑

i∈I

limn→∞

ρi(Y in)µi =

∑

i∈I

ρi(Y i)µi.

Hence, Y ii∈I is the desired optimal allocation.

(ii) We recall that strict monotonicity implies that if X Y and X 6∼ Y , then ρi(X) <

ρi(Y ) ∀ i ∈ Iµ for any X,Y ∈ L∞. Let Xii∈I be an optimal allocation for X ∈ L∞.

Then, by Theorem 5.8, there is an I-comonotone allocation Y ii∈I ∈ A(X) such that

ρµconv(X) =∑

i∈I

ρi(Xi)µi ≥∑

i∈I

ρi(Y i)µi.

Thus, Y ii∈I is also optimal. If Xi = Y i ∀ i ∈ Iµ, then we have the claim. If there

is i ∈ Iµ such that Xi 6= Y i, then we have by strictly monotonicity regarding that

ρi(

Y i)

< ρi(Xi), contradicting the optimality of Xii∈I . Hence, every optimal allocation

for X is I-comonotone.

(iii) We assume, toward a contradiction, that both Xii∈I and Y ii∈I are optimal allocations

for X ∈ L∞ such that there is i ∈ Iµ with Xi 6= Y i and there is no Ci ∈ R ∀ i ∈ Iµ such

that∑

i∈I Ciµi = 0 and Xi + Cii∈I (otherwise, item (ii) in Proposition 5.3 assures

optimality). We note that for any λ ∈ [0, 1], the family Zi = λXi + (1 − λ)Y ii∈I is in

A(X). However, we would have

∑

i∈I

ρi(Zi)µi < λ∑

i∈I

ρi(Xi)µi + (1− λ)∑

i∈I

ρi(Y i)µi = ρµconv(X),

which contradicts the optimality of both Xii∈I and Y ii∈I for X.

Remark 5.10. The examples in Jouini et al. (2008) and Delbaen (2006) show that law invari-

ance is essential to ensure the existence of an I-comonotone solution as above. However, the

uniqueness of this optimal allocation is not ensured outside the scope of strict convexity as in

item (iii) of the last Theorem.

If ρI consists of comonotone, law-invariant, convex risk measures, then we can prove an

additional result regarding the connection between optimal allocations and the notion of flatness

for quantile functions. To this end, the following definitions and lemma are required. We recall

that dF−1X is the differential of F−1

X .

24

Definition 5.11. Let g1, g2 be two distortions with g1 ≤ g2. A quantile function F−1X , X ∈ L∞,

is called flat on x ∈ [0, 1] : g1(x) < g2(x) if dF−1X = 0 almost everywhere on g1 < g2 and

(F−1X (0+)− F−1

X (0))(g2(0+)− g1(0+)) = 0.

Lemma 5.12 (Lemmas 4.1 and 4.2 in Jouini et al. (2008)). Let ρ : L∞ → R be a law-invariant,

comonotone, convex risk measure with distortion g; moreover, let m ∈ ba1,+ has a Lebesgue

decomposition m = ZmP + ms into a regular part with density Zm and a singular part ms.

Then,

(i) gm : [0, 1] → R defined as gm(0) = 0 and gm(t) = ‖ms‖TV +∫ t

0 F−1Zm

(1 − s)ds, 0 < t ≤ 1,

is a concave distortion.

(ii) For any m ∈ ∂ρ(X), we have that X and −Zm are comonotone. Moreover, the measure

m′ such that Zm′ = E[Zm|X] belongs to ∂ρ(X).

(iii) ∂ρ(X) =

m ∈ ba1,+ : gm ≤ g, F−1X is flat on gm < g

.

Theorem 5.13. Let ρI consist of law-invariant, comonotone, convex risk measures with dis-

tortions gii∈I , g = infi∈Iµ gi, and Xii∈I ∈ A(X) be I-comonotone. If F−1

Xi is flat on

g < gi ∩ dF−1X > 0 ∀ i ∈ Iµ, then Xii∈I is an optimal allocation for X ∈ L∞. The

converse is true if (i, α) → V aRα(Xi)g′m(α) is bounded.

Proof. By Theorem 4.7, g is the distortion of ρµconv. Let U be a [0, 1]-uniform random variable

(the existence of which is ensured because the space is atomless) such that X = F−1X (U). We

define m ∈ ba1,+ by m = g(0+)δ0(U) + g′(U)1(0,1](U), where δ0 is the Dirac measure at 0. It is

easily verified using (i) in Lemma 5.12 that gm = g. Moreover, let Xii∈I be an I-comonotone

optimal allocation for X (the existence of which is ensured by Theorem 5.9). Thus, in the

∀ i ∈ Iµ sense, gm ≤ gi, −Zm is comonotone with Xi, and, by hypothesis, F−1Xi is flat on

gm < gi ∩ dF−1X > 0. By Lemma 5.7, we have dF−1

X = 0 = α ∈ [0, 1] : dF−1Xi (α) =

0 ∀ i ∈ Iµ. Thus, gm < gi ∩ dF−1X = 0 = ∅ ∀ i ∈ Iµ By (iii) of Lemma 5.12, we have that

m ∈ ∂ρi(Xi) ∀ i ∈ Iµ Then, by Theorem 5.4, we obtain that Xii∈I is an optimal allocation.

For the converse, by Theorem 5.4 and Corollary 5.5, we have that Xi ∈ ∂αminρi

(m′) ∀ i ∈ Iµ

for some m′ ∈ ba1,+. By Corollary 5.5, we have that X ∈ ∂αminρµconv

(m′), and by convex-conjugate

duality, m′ ∈ ∂ρµconv(X). Thus, by (ii) in Lemma 5.12, we obtain that m ∈ ba1,+ such that

Zm = E[Zm′ |X] belongs to ∂ρµconv(X) =

m ∈ ba1,+ : m ∈ ∂ρi(Xi) ∀ i ∈ Iµ

. By Theorem 2.8

and (iii) of Lemma 5.12, we have that ρi(Xi) =∫ 10 V aRα(Xi)g′m(α)dα ∀ i ∈ Iµ. As Xii∈I is

an optimal allocation, Theorem 4.7 and Lemma 5.7 imply that

∫ 1

0V aRα(X)g′(α)dα = ρµconv(X)

=∑

i∈I

∫ 1

0V aRα(Xi)g′m(α)dαµi

=

∫ 1

0

∑

i∈I

V aRα(Xi)µig′m(α)dα

=

∫ 1

0V aRα(X)g′m(α)dα.

25

We can make the interchange of sum and integral for dominated convergence because of the

boudedness assumption. By continuity, V aRα(X)dα = −dF−1X (α). Then, we have

∫ 10 (gm(α) −

g(α))dF−1X (α) = 0. As m ∈ ∂ρ

µconv(X), (iii) in Lemma 5.12 implies that gm ≤ g, and therefore

gm = g in dF−1X > 0. Hence, F−1

Xi is flat on g < gi ∩ dF−1X > 0 ∀ i ∈ Iµ.

Remark 5.14. We also have in this context that for any optimal allocation Xii∈I , F−1Xi is flat

on gi 6= gj∩dF−1X = 0 for any j 6= i in Iµ sense. To see this, let t ∈ gj < gi∩dF−1

X = 0.

We note that, by Lemma 5.7, dF−1X = 0 = α ∈ [0, 1] : dF−1

Xi (α) = 0 ∀ i ∈ Iµ. Then, by

Lemma 5.12, gm(t) < gi(t), and thus dF−1Xi is flat at gm < gi. By comonotonicity and Lemma

5.7, the same is true for dF−1Xj . By repeating the argument for t ∈ gj > gi ∩ dF−1

X = 0, we

prove the claim.

A relevant concept in the present context is the dilated risk measure, which is stable under

inf-convolution and has a dilatation property with respect to the size of a position. In this

particular situation, we can provide explicit solutions for optimal allocations even without law

invariance. We now define this concept and extend some interesting related results to our

framework.

Definition 5.15. Let ρ : L∞ → R be a risk measure, and γ > 0 be a real parameter. The dilated

risk measure with respect to ρ and γ is a functional ργ : L∞ → R defined as

ργ(X) = γρ

(

1

γX

)

. (5.1)

Remark 5.16. A typical example of dilated measure is the entropic Entγ with Ent1 as basis.

It is evident that, for convex risk measures, αminργ = γαmin

ρ . Moreover, a convex risk measure

is coherent if and only if ρ = ργ pointwise for any γ > 0. Under normalization, limγ→∞

ργ defines

the smallest coherent risk measure that dominates ρ.

Proposition 5.17. We have that

(i) (ρµconv)γ = inf∑

i∈I(ρi)γ(X

i)µi : Xii∈I ∈ A(X)

for any γ > 0.

(ii) Let ρ be a convex risk measure and γi > 0i∈I , such that∑

i∈I γiµi = γ. If ρi = ργi ∀ i ∈

Iµ, then

γi

γX

i∈Iis optimal for X ∈ L∞ and ρ

µconv = ργ .

Proof. (i) For any γ > 0 and X ∈ L∞, we have that

inf

∑

i∈I

(ρi)γ(Xi)µi : X

ii∈I ∈ A(X)

= γ inf

∑

i∈I

ρi(

1

γXi

)

µi : Xii∈I ∈ A(X)

= γ inf

∑

i∈I

ρi(

Y i)

µi : Yii∈I ∈ A

(

1

γX

)

= γρµconv

(

1

γX

)

= (ρµconv)γ(X).

(ii) We obtain from Theorem 4.1 that

αminρµconv

(m) =∑

i∈I

αminργi

(m)µi =∑

i∈I

γiαminρ (m)µi = γαmin

ρ (m) = αminργ (m), ∀m ∈ ba.

26

Thus, ρµconv = ργ . It is straightforward to note that

γi

γX

i∈I∈ A(X). We then obtain

that

ρµconv(X) ≤∑

i∈I

ργi

(

γi

γX

)

µi =∑

i∈I

γiρ

(

1

γiγi

γX

)

µi = γρ

(

1

γX

)

= ργ(X) ≤ ρµconv(X).

Hence,

γi

γX

i∈Iis optimal for X ∈ L∞.

Remark 5.18. We note that for any X ∈ L∞,

γiγ−1X

i∈Iis I-comonotone, which is in

consonance with Theorem 5.9 when the risk measures in ρI are law invariant. For instance, we

have that if ρi = Entγi , γi > 0, ∀ i ∈ Iµ, then ρµconv = Entγ .

We now follow the approach of Embrechts et al. (2018) by focusing in robustness of optimal

allocations instead of the one for ρµconv. Intuitively, if an optimal allocation is robust then under

a small model misspecification, the true aggregate risk value would be close from the obtained

one. We now formally define such concept and prove a result that relate robustness and upper

semi continuity. To that, we need the concept of allocation principle, which is defined in the

following.

Definition 5.19. We define the following:

(i) hi : R → R, i ∈ I is an allocation principle if ∀ i ∈ Iµ: hi(X) ∈ L∞ ∀X ∈ L∞, hi has

at most finitely points of discontinuity, and∑

i∈I hiµi is the identity function. We denote

by H the set of allocation principles.

(ii) let d be a pseudo-metric on L∞ and hii∈I ∈ H. Then hi(X)i∈I is d-robust if the map

on L∞ defined as Y →∑

i∈I ρi(hi(Y ))µi is continuous at X in respect to d.

Remark 5.20. Note that from Lemma 5.7, I-comonotonicity implies the existence of allocation

principles.

Proposition 5.21. Let d be a pseudo-metric on L∞. If hi(X)i∈I is a d-robust optimal

allocation of X ∈ L∞, then ρµconv is upper semi continuous at X with respect to d.

Proof. Let hi(X)i∈I be a d-robust optimal allocation for X ∈ L∞, and Xn ⊂ L∞ be such

that Xn → X in d. Since ρµconv(Xn) ≤

∑

i∈I ρi(hi(Xn))µi, by convergence regarding d we get

that

lim supn→∞

ρµconv(Xn) ≤ lim supn→∞

∑

i∈I

ρi(hi(Xn))µi = ρµconv(X).

Remark 5.22. Regarding the converse statement for proposition 5.21, Theorem 5 of Embrechts et al.

(2018), which is focused in quantiles, asserts that even continuity in respect to d is not sufficient

for the existence of a robust optimal allocation. Furthermore, from Remark 3.9, when each

member of ρI is a convex risk measure there is not robust optimal allocations regarding the

Levy metric. In fact, this is a relatively new concept and even in the finite I case there is still

not a general sufficient condition. We left for future research a more complete study of general

sufficient conditions for robust allocations.

27

6 Self-convolution and regulatory arbitrage

In this section, we consider the special case ρi = ρ, ∀ i ∈ I. In this situation, we have that ρµconv

is a self-convolution. This concept is highly important in the context of regulatory arbitrage (as

in Wang (2016)), where the goal is to reduce the regulatory capital of a position by splitting it.

The difference between ρ(X) and ρµconv(X) is then obtained by a simple rearrangement (sharing)

of risk. We now adjust this concept to our framework.

Definition 6.1. The regulatory arbitrage of a risk measure ρ is a functional τρ : L∞ → R+∪∞

defined as

τρ(X) = ρ(X) − ρµconv(X), ∀X ∈ L∞. (6.1)

Moreover, ρ is called

(i) free of regulatory arbitrage if τρ(X) = 0, ∀X ∈ L∞;

(ii) of finite regulatory arbitrage if τρ(X) < ∞, ∀X ∈ L∞;

(iii) of partially infinite regulatory arbitrage if τρ(X) = ∞ for some X ∈ L∞;

(iv) of infinite regulatory arbitrage if τρ(X) = ∞, ∀X ∈ L∞.

Remark 6.2. As ρµconv ≤ ρ < ∞, we have that τρ is well defined. Our approach is different from

that in Wang (2016) because we consider “convex” inf-convolutions instead of direct sums. More

specifically, the approach by Wang (2016) is

R(X) = inf

n∑

i=1

ρ(Xi), n ∈ N,Xi ∈ L∞, i = 1, · · · , n,n∑

i=1

Xi = X

= limn→∞

inf

n∑

i=1

ρ(Xi),Xi ∈ L∞, i = 1, · · · , n,n∑

i=1

Xi = X

,

while in our case we have the role for µi. This distinction leads to differences specially, as is

explored in Theorem 6.6 below, because in our approach convexity rules out regulatory aritrage,

while in Wang (2016) sub-additivity plays this role. For instance, in that approach, Entγ is of

limited regulatory arbitrage, whereas in ours (see Proposition 6.6 below), we have τEntγ = 0,

that is, Entγ is free of regulatory arbitrage. Moreover, it is evident that τEL = 0 because∑

i∈I EL(Xi)µi = EL(X), ∀Xii∈I ∈ A(X).

It was proved in Wang (2016) that V aRα is of infinite regulatory arbitrage. The following

proposition adapts this to our framework.

Proposition 6.3. Let α ∈ (0, 1]. If cardinality of Iµ is at least k + 1 such that 1k< α, then

V aRα is of infinite regulatory arbitrage.

Proof. Let ijk+1j=1 be members of Iµ such that 1

k< α. Then k > 1. Moreover, let Bj , j =

1, · · · , k be a partition of Ω such that P(Bj) = 1kfor any j = 1, · · · , k. We note that as

(Ω,F ,P) is atomless, such a partition always exists. For fixed X ∈ L∞ and some arbitrary real

28

number m > 0, let Xii∈I be defined as

Xi(ω) =

m(1− k1Bj(ω))

(k − 1)µi, for i = ij, j = 1, · · · , k,

X(ω)

µik+1

, for i = ik+1,

0, otherwise.

for any ω ∈ Ω. Thus, Xii∈I ∈ A(X) because for any ω ∈ Ω, the following is true:

∑

i∈I

Xi(ω)µi =k∑

j=1

[

m(1− k1Bj(ω))

(k − 1)µij

]

µij +

[

X(ω)

µik+1

]

µik+1

=m

k − 1

k∑

j=1

(1− k1Bj(ω)) +X(ω)

=m

k − 1(k − k) +X(ω) = X(ω).

Furthermore, we note that for i = ij , j = 1, · · · , k, we have that

P(Xi < 0) = P

(

1Bj>

1

k

)

= P(

1Bj= 1)

= P (Bj) =1

k< α.

Thus, V aRα(Xi) < 0. In fact, we have that V aRα(−1Bj) = 0 and thus

V aRα(Xi) =m

(k − 1)µij

(

kV aRα(−1Bj)− 1

)

= −m

(k − 1)µij

< 0.

As∑

i∈I

V aRα(Xi)µi =

k∑

j=1

V aRα(Xi)µij + V aRα(Xik+1)µik+1= V aRα(X) −

mk

k − 1,

V aRα(X) < ∞, and m > 0 is arbitrary, we obtain that

ρµconv(X) ≤ V aRα(X)− limm→∞

mk

k − 1= −∞.

Hence, we conclude that τρ(X) = ∞ for any X ∈ L∞, which implies that V aRα is of infinite

regulatory arbitrage.

Remark 6.4. (i) The idea is that if any position X could be split into k+1 random variables

with 1k< α, then we would obtain an arbitrarily smaller weighted V aRα. In the framework

in Wang (2016), it is always possible to obtain a countable division of any position owing

to the nature of the functional R in Remark 6.2.

(ii) We note that smaller values of α require a richer structure on I to allow the regulatory

arbitrage strategy. This is in fact desired, as such values represent riskier scenarios. For

example, for α = 0.01, which is the demanded level for regulatory capital in Basel accords,

a cardinality of at least k + 1 = 102 would be necessary for the required partition.

29

(iii) This result can be extended to the more general framework of ρI = V aRαi, i ∈ I when

α∗ = inf i∈Iµ αi > 0. As V aRαi

≤ V aRα∗

∀ i ∈ Iµ, if cardinality of Iµ is at least k + 1

with 1k< α∗, then

ρµconv(X) = infXi∈A(X)

∑

i∈I

V aRαi

(Xi)µi

≤ infXi∈A(X)

∑

i∈I

V aRα∗

(Xi)µi

= −∞.

In fact, analogous reasoning is valid for any choice of risk measures ρI dominated by

V aRα∗

.

We now state more general results regarding τρ in our framework. To this end, the following

property of risk measures is required.

Definition 6.5. A risk measure ρ : L∞ → R is called I-convex if ρ(∑

i∈I Xiµi

)

≤∑

i∈I ρ(

Xi)

µi

for any Xii∈I ∈ ∪X∈L∞A(X).

Theorem 6.6. We have the following for a risk measure ρ:

(i) ρ is I-convex if and only if it is free of regulatory arbitrage.

(ii) If ρ is a convex risk measure, then it is free of regulatory arbitrage.

(iii) If ρ is subadditive, then it is at most of finite regulatory arbitrage.

(iv) Let ρ1, ρ2 : L∞ → R be risk measures such that ρ1 ≤ ρ2. If ρ1 is of finite regulatory

arbitrage, then ρ2 is not of infinite regulatory arbitrage. Moreover, if ρ2 is of infinite (or

partially infinite) regulatory arbitrage, then so is ρ1.

(v) If ρ satisfies the positive homogeneity condition, then τρ(0) > 0 if and only if τρ(0) = ∞.

(vi) If ρ is loaded, then it is not of infinite regulatory arbitrage. If, in addition, it has the

limitedness property, then it is of finite regulatory arbitrage.

Proof. We note that as we consider finite risk measures, it holds that τρ(X) = ∞ if and only if

ρµconv(X) = −∞. Then,

(i) We assume that ρ is I-convex, and let X ∈ L∞. Then, ρ(X) ≤∑

i∈I ρ(Xi)µi for any

Xii∈I ∈ A(X). By taking the infimum over A(X), we obtain ρµconv(X) ≤ ρ(X) ≤

ρµconv(X). For the converse, we obtain ρ(X) = ρ

µconv(X) ≤

∑

i∈I ρ(Xi)µi for any X

ii∈I ∈

A(X), which is I-convexity.

(ii) By Corollary 4.3, we have that ρµconv(X) = ρ(X), ∀X ∈ L∞. As a direct consequence, we

obtain that ρ is free of regulatory arbitrage.

(iii) We begin with the claim that if ρ is subadditive, then it is of partially infinite regulatory

arbitrage if and only if it is of infinite regulatory arbitrage. By Proposition 3.4 and Remark

3.5, we have that ρµconv is also subadditive and normalized. We need only show that

partially infinite regulatory arbitrage implies infinite regulatory arbitrage. Let X ∈ L∞

be such that τρ(X) = ∞. As ρ is finite, it holds that ρµconv(X) = −∞. Let now Y ∈ L∞.

30

We have that ρµconv(Y ) ≤ ρµconv(X)+ρ

µconv(Y −X) = −∞. Thus, ρ is of infinite regulatory

arbitrage. However, we have that τρ(0) = ρ(0) − ρµconv(0) = 0 < ∞. Then, ρ is not

of infinite regulatory arbitrage and, by the previous claim, it is not of partially infinite

regulatory arbitrage either. Thus, ρ is at most of finite regulatory arbitrage.

(iv) It is evident that, in this case, we have, by abuse of notation, (ρ1)µconv ≤ (ρ2)

µconv. If ρ1 is

of finite regulatory arbitrage, then −∞ < (ρ1)µconv(X) ≤ (ρ2)

µconv(X), ∀X ∈ L∞. Thus,

ρ2 is also of infinite regulatory arbitrage. If now ρ2 is of infinite regulatory arbitrage,

then (ρ1)µconv(X) ≤ (ρ2)

µconv(X) = −∞, ∀X ∈ L∞. Thus, ρ1 is also of finite regulatory

arbitrage. For partially infinite regulatory arbitrage, the reasoning is analogous.

(v) We need only prove the “only if” part because the converse is automatically obtained.

As ρ(0) = 0, τρ(0) > 0 implies ρµconv(0) < 0. Then, there is Xii∈I ∈ A(0) such that

ρµconv(0) ≤

∑

i∈I ρ(Xi)µi < 0. As λXii∈I ∈ A(0)∀λ ∈ R+, by the positive homogeneity

of ρ, we obtain that

ρµconv(0) ≤ limλ→∞

∑

i∈I

ρ(λXi)µi = limλ→∞

λ∑

i∈I

ρ(Xi)µi = −∞.

Hence, τρ(0) = ρ(0)− ρµconv(0) = ∞.

(vi) By Proposition 3.4, we have that ρµconv inherits loadedness and limitedness from ρ. The

loadedness of ρ implies normalization of ρµconv, and therefore τρ(0) = 0. Thus, ρ is not of

infinite regulatory arbitrage. If, in addition, ρ is limited, then for any X ∈ L∞, we have

that τρ(X) ≤ E[X]− ess infX < ∞. Hence, we obtain finite regulatory arbitrage for ρ.

Remark 6.7. (i) In the approach in Wang (2016), τρ is always subadditive, whereas in our

case, this is not ensured. This fact alters most results and arguments, as it is crucial in his

study. Moreover, (iii), (iv), (v), and (vi) would remain true if we considered the general

framework of arbitrary ρI and made the adaption τρI =∑

i∈I ρiµi−ρ

µconv = ρµ−ρ

µconv ≥ 0,

where ρµ is as Remark 4.2.

(ii) A remarkable feature is that is possible to identify τρ as a deviation measure in the sense

of Rockafellar et al. (2006), Rockafellar and Uryasev (2013), Righi and Ceretta (2016),

Righi (2019a), and Righi et al. (2019). For instance, the bound for τρ in the proof of (vi)

is known as lower-range dominance for deviation measures.

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