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However, such choice can lead to multidimensional or even infinite dimensional problems that bring a level for complexity that may make the treatment of risk measurement impossible to handle. For example, in a portfolio optimization problem in order to take into account this variety of features from distinct risk measures the agent may end up with a very complex multi-objective function or even an exorbitant volume of constraints. Such situation would lead to elevated computational cost or even the impossibility for a feasible solution. Hence, the alternative is to consider a combination of all the candidates instead of all of them individually. In the context for portfolio optimization, we would have a single constraint or objective with a larger space of feasibility. The drawback for considering such a combination is that we may end up without the main characteristics that define risk measures. More precisely, the axiomatic theory of risk measures strongly relies on a set of financial and mathematical properties that are related to dual repre- sentations and acceptance sets. Thus, the understanding on how to preserve such properties in a general fashion is crucial to guarantee the usefulness of combinations. In this sense, the devel- opment of a theoretical body for combinations of risk measures is pivotal. A general approach must deal with arbitrary sets of candidates, in particular uncountable ones. Such situation may arise when the parameter that defines the candidates relies on subset of the real line, such as the significance level for Value at Risk or some probability of default in credit risk. The main challenge relies in the generality that is needed to perform this kind of task since we are not able to rely in methods for finite dimensional spaces that appear on the literature. For instance, when dealing with some uncountable set of candidates we are not able to even consider usual summation, crucial for averaging, having to replace it by integration. However, in this case we have measurability issues to take into account that may become complex. Also, set operations may not be preserving topological properties, such as uncountable union of closed sets does not have to be closed. Even the choice of suitable domains for combination functions in infinite dimensional spaces can be per se a source of complexity since there is no canonical functional space. Another source of difficult and complexity is that combinations may assume any functional form, such as averaging or supremum based worst cases. In fact, any function applied over the set of candidate risk measures can be considered. Thus, to study the impact of such combinations in properties, acceptance sets, dual representations among other features is not straightforward. Since specific combination functions can be more suitable to distinct contexts, to have a general treatment is very beneficial. In this sense, the existence of a theory to provide a solid and practical manner to preserve desired properties, obtain acceptance sets or dual represent a general combination of risk measures can help to improve other fields in mathematical finance as well.

Under this background, in this paper we study risk measures of the form ρ = f(ρI), where i ρI = {ρ , i ∈ I} is a set of alternative risk measures and f is some combination function. We propose a framework whereby no assumption is made on the index set I, apart from non- emptiness. Typically, this kind of procedure uses a finite set of candidates, leading the domain of f to be some Euclidean space. In our case, the domain of f is taken by a subset of the random variables over a suitable measurable space created on I. From that, our main goal is to

2 establish general results on properties, develop dual representations and study acceptance sets for such composed risk measures based on the properties of both ρI and f in a general sense. For this purpose, we expose results for some featured special cases, which are also of particular interest, such as a worst case and mixtures of risk measures.

There are studies regarding particular cases for f, I and ρI , such as the worst case in F¨ollmer and Schied (2002), the sum of monetary and deviation measures in Righi (2019), fi- nite convex combinations in Ang et al. (2018), scenario-based aggregation in Wang and Ziegel (2018), model risk-based weighting over a non-additive measure in Jokhadze and Schmidt (2020). Our main contribution is do not impose a restriction on the set of alternative risk measures and the generality of combination functions we consider. Furthermore, none of such papers does consider all the features we do take into account. It is worthy to mention that the well known concept of inf-convolution of risk measures as in Barrieu and El Karoui (2005) or Jouini et al. (2008) is not suitable to the approach in this paper since even in the most sim- ple case of two risk measures we cannot write it as a direct combination as f(ρ1(X), ρ2(X)) = infY {ρ1(X −Y )+ρ2(X)} since it is not only a function of X. More precisely, the inf-convolution depends of every allocation X1 +X2 = X. The work of Righi (2020) is focused to inf-convolution and optimal risk sharing for arbitrary sets of risk measures. The classic theory for risk measures is developed under the assumption of a given probability measure that represents beliefs of the world. However, we frequently do not know if there is a correct probability measure. We then apply our framework to the special situation where I is a subset of probability measures since we frequently do not know if there is a correct one, but we have instead a set of candidates. We consider the concept of probability-based risk measurement, which is a collection of risk measures from a functional on the set of distributions generated by probabilities in I. Because the choice of probabilities may affect the risk value for a large amount, we would like to have risk measures that are robust in the sense that functionals do not suffer relevant variations in their value from changes on the chosen probability measure. This is the case for combinations in our approach since they depend on the whole set of alternative probabilities. The probability based risk measurement is directly related to the notion of model un- certainty, or model risk, relative to uncertainty concerning the choice of a model, which is studied in detail for risk measures by Kerkhof et al. (2010), Bernard and Vanduffel (2015), Barrieu and Scandolo (2015), Danielsson et al. (2016), Kellner and Rosch (2016), Frittelli and Maggis (2018), Maggis et al. (2018) among others. From a practical point of view, the Turner Review raises the issue of model ambiguity, often called Knightian uncertainty. Despite there is a consensus that better methods for dealing with model uncertainty are pivotal to improving risk management, measures to quantify it are not consolidated at the same level as those for market risk. We then explores how our probability based approach can deal with such model uncertainty measurement and proposes a flexible class of measures. Works such as those of Bartl et al. (2019), Bellini et al. (2018), and Guo and Xu (2019) focus on particular risk measures, instead of a general framework. Laeven and Stadje (2013), Frittelli and Maggis (2018), Jokhadze and Schmidt (2020), Wang and Ziegel (2018) and Qian et al. (2019) explore frameworks for risk measures considering multiple probabilities. However, they

3 do not develop exactly the same features we do. In these studies, restrictive assumptions are made on the set I, such as it being finite and possessing a reference measure. We in our turn solely assume non-emptiness. The stream of robust risk measures approached in papers like as Cont et al. (2010), Kratschmer et al. (2014) and Kiesel et al. (2016) explores such uncertainty in a more statistical fashion, while our approach is probabilistic. We have structured the rest of this paper as follows: in section 2 we expose preliminaries regarding notation, a brief background on the theory of risk measures in order to support our framework and our proposed approach with some examples; in section 3 we present results regarding properties of combination functions and how they affect the resulting risk measures in both financial and continuity properties; in section 4 we develop and prove our results on representations of resulting risk measures in terms of properties from both the set of candidates and the combination for the general convex and law invariant cases, as well we address a repre- sentation for the worst-case risk measure; in section 5 we present results on how properties of combination functions affect the resulting risk measures acceptance set and provide a general characterization for the convex and coherent cases; in section 6 we explore the special frame- work of probability-based risk measurement, exposing results specific to this context such as representations, stochastic orders and elicitability; in section 7 we address the issue of model uncertainty by exposing how our probabilistic framework can be considered for this task and we propose a new class of model uncertainty measures.

2 Preliminaries

2.1 Notation

Consider the atom-less probability space (Ω, F, P). All equalities and inequalities are in the P-a.s. sense. We have that L0 = L0(Ω, F, P) and L∞ = L∞(Ω, F, P) are, respectively, the spaces of (equivalent classes under P-a.s. equality of) finite and essentially bounded random variables. We define 1A as the indicator function for an event A ∈ F. We identify con- stant random variables with real numbers. We say that a pair X,Y ∈ L0 is comonotone if ′ ′ (X(w) − X(w )) (Y (w) − Y (w )) ≥ 0 holds P × P-a.s. We denote by Xn → X convergence in ∞ the L essential supremum norm k·k∞, while lim Xn = X means P-a.s. convergence. Let P be n→∞ the set of all probability measures on (Ω, F). We denote EQ[X]= Ω XdQ, FX,Q(x)= Q(X ≤ x) −1 and FX,Q(α) = inf {x : FX,Q(x) ≥ α}, respectively, the expected value,R the (non-decreasing and Q right-continuous) probability function and its inverse for X under Q ∈ P. We write X ∼ Y when FX,Q = FY,Q. We drop subscripts regarding probability measures when Q = P. Further- more, let Q ⊆ P be the set of probability measures that are absolutely continuous in relation dQ to P with Radon-Nikodym derivatives dP .

2.2 Background

Definition 2.1. A functional ρ : L∞ → R is a risk measure. Its acceptance set is defined as ∞ Aρ = {X ∈ L : ρ(X) ≤ 0}. ρ may possess the following properties:

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

4 (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) Fatou continuity: if lim Xn = X ∈ L and {Xn} ⊆ L bounded, then ρ(X) ≤ n→∞ n=1 lim inf ρ(Xn). n→∞

We have that ρ is called monetary if it fulfills Monotonicity and Translation Invariance, convex if it is monetary and respects Convexity, coherent if it is convex and fulfills Positive Homogeneity, law invariant if it has Law Invariance, comonotone if it attends Comonotonic Additivity, and Fatou continuous if it possesses Fatou continuity. In this paper, we are working with normalized risk measures in the sense of ρ(0) = 0.

Beyond usual norm and Fatou-based continuities, (a.s.) point-wise are relevant for risk measures.

Definition 2.2. A risk measure ρ : L∞ → R is said to be:

(i) Continuous from above: lim Xn = X and {Xn} non-increasing implies in ρ(X) = n→∞ ∞ ∞ lim ρ(Xn), ∀ {Xn}n , X ∈ L . n→∞ =1

(ii) Continuous from below: lim Xn = X and {Xn} non-decreasing implies in ρ(X) = n→∞ ∞ ∞ lim ρ(Xn), ∀ {Xn} , X ∈ L . n→∞ n=1 ∞ ∞ (iii) Lebesgue continuous: lim Xn = X implies in ρ(X) = lim ρ(Xn) ∀ {Xn} ⊆ L n→∞ n→∞ n=1 bounded and ∀ X ∈ L∞.

For a review on details regarding the interpretation of such properties, we recommend the mentioned books on the classic theory. Regarding acceptance sets, we have the following direct implications.

Theorem 2.3 (Proposition 4.6 in F¨ollmer and Schied (2016)). Let Aρ be the acceptance set defined by ρ: L∞ → R. Then:

∞ (i) If ρ fulfills Monotonicity, then X ∈ Aρ, Y ∈ L and Y ≥ X implies in Y ∈ Aρ. In ∞ particular, L+ ⊆Aρ.

(ii) If ρ fulfills Translation Invariance, then ρ(X) = inf {m ∈ R: X + m ∈Aρ}.

(iii) If ρ is a monetary risk measure, then Aρ is non-empty, closed with respect to the supremum ∞ norm, Aρ ∩ {X ∈ L : X < 0} = ∅, and inf{m ∈ R: m ∈Aρ} > −∞.

(iv) If ρ attends Convexity, then Aρ is a convex set.

(v) If ρ fulfills Positive Homogeneity, then Aρ is a cone.

5 Under convexity and Fatou continuity, the well known convex duality plays an inportant role for risk measures. In fact, we have the following.

Theorem 2.4 (Theorem 2.3 of Delbaen (2002), Theorem 4.33 of F¨ollmer and Schied (2016)). Let ρ : L∞ → R be a risk measure. Then:

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

min ∞ ρ(X)= sup EQ[−X] − αρ (Q) , ∀ X ∈ L , (2.1) Q∈Q  min min where αρ : Q → R+ ∪ {∞}, defined as αρ (Q) = sup EQ[−X], is a lower semi- X∈Aρ continuous convex function called penalty term. This is equivalent to Aρ be weak* closed, i.e. σ(L∞,L1) closed.

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

∞ ρ(X)= sup EQ[−X], ∀ X ∈ L , (2.2) Q∈Qρ

where Qρ ⊆ Q is non-empty, closed and convex called the dual set of ρ.

Example 2.5. Examples of risk measures:

(i) Expected Loss (EL): This is a Fatou continuous law invariant comonotone coherent 1 −1 risk measure defined as EL(X) = −E[X] = − 0 FX (s)ds. We have that AEL = ∞ {X ∈ L : E[X] ≥ 0} and QEL = {P}. R (ii) Value at Risk (VaR): This is a Fatou continuous law invariant comonotone monetary α −1 risk measure defined as V aR (X) = −FX (α), α ∈ [0, 1]. We have that AV aRα = {X ∈ L∞ : P(X < 0) ≤ α}.

(iii) Expected Shortfall (ES): This is a Fatou continuous law invariant comonotone coher- α 1 α s 0 ent risk measure defined as ES (X) = α 0 V aR (X)ds, α ∈ (0, 1] and ES (X) = α 0 α ∞ s α V aR (X) = − ess inf X. We have AES = RX ∈ L : 0 V aR (X)ds ≤ 0 and QES = Q dQ 1 ∈ Q : dP ≤ α .  R n o (iv) Maximum loss (ML): This is a Fatou continuous law invariant coherent risk measure −1 ∞ defined as ML(X) = −ess inf X = FX (0). We have AML = {X ∈ L : X ≥ 0} and QML = Q.

When there is Law Invariance, which is the case in most practical applications, interesting features are present.

Theorem 2.6 (Theorem 2.1 of Jouini et al. (2006) and Proposition 1.1 of Svindland (2010)). Let ρ : L∞ → R be a law invariant convex risk measure. Then ρ is Fatou continuous.

Theorem 2.7 (Theorems 4 and 7 of Kusuoka (2001), Theorem 4.1 of Acerbi (2002), Theorem 7 of Fritelli and Rosazza Gianin (2005)). Let ρ : L∞ → R be a risk measure. Then:

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

α min ∞ ρ(X)= sup ES (X)dm − βρ (m) , ∀ X ∈ L , (2.3) m∈M (Z(0,1] )

min where M is the set of probability measures on (0, 1] and βρ : M→ R+ ∪ {∞}, defined min α as βρ (m)= sup (0,1] ES (X)dm. X∈Aρ R (ii) ρ is a law invariant coherent risk measure if and only if it can be represented as:

ρ(X)= sup ESα(X)dm, ∀ X ∈ L∞, (2.4) m∈Mρ Z(0,1]

1 −1 where Mρ = m ∈ M: (u,1] v dm = F dQ (1 − u), Q ∈ Qρ .  dP  R (iii) ρ is a law invariant comonotone coherent risk measure if and only if it can be represented as: ρ(X)= ESα(X)dm, ∀ X ∈ L∞, (2.5) Z(0,1] where m ∈ Mρ.

2.3 Proposed approach

i ∞ Let ρI = {ρ : L → R, i ∈ I} be some (a priori specified) collection of risk measures, where ∞ i I is a non-empty set. We write, for fixed X ∈ L , ρI (X) = {ρ (X), i ∈ I}. We would like to define risk measures as ρ(X) = f(ρI(X)), where f is some combination (aggregation) function. When I is finite with dimension n, we have that f : Rn → R. This situation, which is common in practical matters, brings simplification to the framework. However, as exposed in the introduction when I is an arbitrary set we need a more complex setup. Examples of such complexity are the fact that in this case we may have to deal with integration and measurability, some set operations may note preserve topological properties and the absence of a canonical functional space to be the domain of f. Consider the measurable space (I, G), where G is a sigma-algebra of sub-sets from I. We define K0 = K0(I, G) and K∞ = K∞(I, G) as the spaces of point-wise finite and bounded random variables on (I, G), respectively. In these spaces we understand equalities, inequalities and limits in the point-wise sense. We define V as the set of probability measures in (I, G). In order to avoid measurability issues we make the following assumption.

i Assumption 2.8. The maps RX : I → R, defined as RX (i)= ρ (X), are G-measurable for any X ∈ L∞.

Remark 2.9. A possible, but not unique, choice is when G = σ({i → ρi(X): X ∈ L∞}) = −1 ∞ σ({RX (B): B ∈ B(R), X ∈ L }), where B(R) is the Borel set of R. Other possibility is, of course, the power set 2I . A situation of interest is when G is a Borel sigma-algebra and i → ρi(X) continuous for any X. This is the case of I = R and ρi composed by V aRi or ESi, for instance.

7 0 ∞ Thus we can associate the domain of f with X = X ρI = span({R ∈ K : ∃X ∈ L s.t.R(i)= i 0 ∞ ρ (X), ∀i ∈ I}∪{1})= span({R ∈ K : R = RX , X ∈ L }∪{1}). The linear span is in order to ∞ preserve vector space operations. We can identify ρI (X) with RX by ρI : L →X . From nor- malization, we have R0 = 0. When ρI(X) is bounded, which is the case for any monetary risk measure since ρ(X) ≤ ρ(ess inf X)= − ess inf X < ∞, similarly for ρ(X) ≥− ess sup X > −∞, we have that X ⊆ K∞. Under this framework, the composition is a functional f : X → R. We use, when necessary, the canonical extension convention that f(R) = ∞ for R ∈ K0\X . We consider normalized combination functions as f(R0)= f(0) = 0.

Example 2.10. The worst-case risk measure is a functional ρW C : L∞ → R defined as

ρW C (X) = sup ρi(X). (2.6) i∈I

This risk measure is typically considered when the agent (investor, regulator etc.) seeks pro- tection. When I is finite the supremum is, of course, a maximum. This combination is the point-wise supremum f W C(R) = sup{R(i): i ∈ I}. If X ⊆ K∞, then ρW C < ∞. When I = Q Q and ρ (X) = EQ[−X] − α(Q), with α: Q → R+ ∪ {∞} such that inf{α(Q): Q ∈ Q} = 0, we have that ρW C becomes a Fatou continuous convex risk measure as (2.1) in Theorem 2.4. Q W C Analogously, for a non-empty closed convex I ⊆ Q and ρ (X) = EQ[−X], we have that ρ becomes a Fatou continuous coherent risk measure as (2.2) in this same Theorem 2.4. Analo- gous analysis can be made to obtain law invariant convex and coherent risk measures as (2.4) and (2.3), respectively, as in Theorem 2.7.

Example 2.11. The weighted risk measure is a functional ρµ : L∞ → R defined as

ρµ(X)= ρi(X)dµ, (2.7) ZI where µ is a probability on (I, G). This risk measure represents an expectation of RX regarding µ. Since i → ρi(X) is G-measurable, the integral is well-defined. In addition, it is finite when ∞ µ X ∈ K . When I is finite, ρ is a convex mixture of the functionals which compose ρI . The µ µ1 µ2 W C combination function is f (R) = I Rdµ. We have that |ρ (X) − ρ (X)| ≤ |ρ (X)|kµ1 − µ2kT V , where k·kT V is the totalR variation norm. Hence it is somehow continuous (robust) regarding choice of the probability measure µ ∈V. If I = (0, 1] and ρi(X)= ESi(X), we have that ρµ(X) defines a law invariant comonotone convex risk measure as (2.5), which is Fatou continuous due to Theorem 2.6.

Example 2.12. A spectral (distortion) risk measure is a functional ρφ : L∞ → R defined as

1 ρφ(X)= V aRα(X)φ(α)dα, (2.8) Z0 1 where φ : [0, 1] → R+ is a non-increasing functional such that 0 φ(u)du = 1. Any law invariant comonotone convex risk measure can be expressed in this fashR ion. The relationship between 1 this representation and the one in (2.5) is given by (u,1] v dm = φ(u), where m ∈ Mρ. When φ is not non-increasing, we have that the risk measureR is not convex and the representation

8 as combinations of ES does not hold. Let I = [0, 1], λ the Lebesgue measure, and µ ≪ λ −1 φ i i with φ(i) = F dµ (1 − i). Thus I φdλ = 1 and ρ (X) = I ρ (X)φ(i)dλ. By choosing ρ (X) = dλ V aRi(X) we have that any spectralR risk measure is a specialR case of ρµ.

Example 2.13. Consider the risk measure ρu(X): L∞ → R defined as

u ρ (X)= u(ρI (X)), (2.9) where u : X → R is a monetary utility in the sense that if R ≥ S, then u(R) ≥ u(S) and u(R + C)= u(R)+ C, C ∈ R. In this case the combination is f u = u. Note that u(R) can be identified with π(−R), where π is a risk measure on X . For instance, one can pick π as EL, VaR, ES or ML. In these cases we would obtain for some base probability µ, respectively the µ −1 1 α −1 following combinations: f , FR,µ(1 − α), α 0 FR,µ(1 − s)ds, and esssupµ R.

Example 2.14. For this example considerRL∞ the space of point-wise bounded random vari- ∞ ables on (Ω, F). We denote F = {FX,Q : X ∈ L , Q ∈ P}. In this case, uncertainty is linked to probabilities in the sense that I ⊆ P and we can define risk measurement under the intuitive idea that we obtain the same functional from distinct probabilities that repre- sent scenarios. In this case a probability-based risk measurement is a family of risk measures Q ∞ ρI = {ρ : L → R, Q ∈ I} such that

Q ρ ∞ ρ (X)= R (FX,Q), ∀ X ∈ L , ∀ Q ∈I, (2.10) where Rρ : F → R is called risk functional. This is the framework of sections 6 and 7. It can be utilized to deal with probabilistic issues and uncertainty. In fact, under a suitable choice for f connected to dispersion and deviations it is possible to use our approach to quantify model uncertainty.

3 Properties

3.1 Properties of combinations

In this section, we expose results regarding the preservation of properties for composed risk measures ρ = f(ρI) based on the properties of both ρI and f. We begin by defining properties of the composition f.

Definition 3.1. A combination f : X → R may have the following properties:

(i) Monotonicity: if R ≥ S, then f(R) ≥ f(S), ∀ R,S ∈X .

(ii) Translation Invariance: f(R + C)= f(R)+ C, ∀ R,S ∈X , ∀ C ∈ R.

(iii) Positive Homogeneity: f(λS)= λf(S), ∀ R ∈X , ∀ λ ≥ 0.

(iv) Convexity: f(λR + (1 − λ)S) ≤ λf(R) + (1 − λ)f(S), ∀ λ ∈ [0, 1], ∀ R,S ∈X .

(v) Additivity: f(R + S)= f(R)+ f(S), ∀ R,S ∈X .

9 ∞ ∞ (vi) Fatou Continuity: If lim Rn = R ∈ K , with {Rn} ⊆ X bounded, then f(R) ≤ n→∞ n=1 lim inf f(Rn). n→∞ Remark 3.2. Such properties for the combination function f are in parallel to those of risk measures, exposed in Definition 2.1. Note the adjustment in signs from there. We use the same terms indiscriminately for both f and ρ with reasoning to fit the context. We could have imposed a determined set of properties for the combination. However, we choose to keep a more general framework where it may or may not possess such properties.

Proposition 3.3. Let X ⊆ K∞. We have that:

(i) f W C defined as in Example 2.10 fulfills Monotonicity, Translation Invariance, Positive Homogeneity, Convexity and Fatou continuity.

(ii) f µ defined as in Example 2.11 fulfills Monotonicity, Translation Invariance, Positive Ho- mogeneity, Convexity, Additivity and Fatou continuity.

Proof. (i) Monotonicity, Translation Invariance, Positive Homogeneity and Convexity are obtained directly from the definition of supremum. Regarding Fatou continuity, let ∞ ∞ {Rn} ⊆X bounded such that lim Rn = R ∈ K . Then we have that n=1 n→∞

W C W C f (R) =sup lim Rn ≤ lim inf sup Rn = lim inf f (Rn). n→∞ n→∞ n→∞

(ii) From the properties of integration, we have that f µ respects Monotonicity, Translation Invariance, Positive Homogeneity, Convexity and Additivity. For Fatou continuity, let ∞ ∞ {Rn}n ⊆ X bounded such that lim Rn = R ∈ K . Then we have from Dominated =1 n→∞ Convergence that

µ µ f (R)= lim Rndµ ≤ lim inf Rndµ = lim inf f (Rn). n→∞ n→∞ n→∞ ZI ZI

Remark 3.4. Note that for any combination f with the property of Boundedness, i.e |f(R)|≤ W C W C f (R), ∀ R ∈ X , we have ρ(X) ≤ ρ (X). Consequently AρWC ⊆ Aρ. From Theorem 2.4 ∞ µ applied to functionals over K , we have that {f }µ∈V are the only combination functions that fulfill all properties in Definition 3.1.

3.2 Financial properties

We now focus on the preservation of financial properties.

i ∞ Proposition 3.5. Let ρI = {ρ : L → R, i ∈ I} be a collection of risk measures, f : X → R, ∞ and ρ: L → R a risk measure defined as ρ(X)= f(ρI(X)). Then:

(i) If ρI is composed of risk measures with Monotonicity and f possesses this same property, then also does ρ.

10 (ii) If ρI is composed of risk measures with Translation Invariance and f possesses this same property, then also does ρ.

(iii) If ρI is composed of risk measures with Convexity and f possesses this same property in pair with Monotonicity, then ρ fulfills Convexity.

(iv) If ρI is composed of risk measures with Positive Homogeneity and f possesses this same property, then also does ρ.

(v) If ρI is composed of law invariant risk measures, then ρ fulfills Law Invariance.

(vi) If ρI is composed of comonotone risk measures and f fulfills Additivity, then ρ has Comonotonic Additivity.

(vii) If ρI is composed of Fatou continuous point-wise bounded risk measures and f has Fatou continuity in pair with Monotonicity, then also does ρ.

∞ i i Proof. (i) Let X,Y ∈ L with X ≥ Y . Then ρ (X) ≤ ρ (Y ), ∀ i ∈I. Thus, RX ≤ RY and

ρ(X)= f(RX ) ≤ f(RY )= ρ(Y ).

(ii) Let X ∈ L∞ and C ∈ R. Then ρi(X + C) = ρi(X) − C, ∀ i ∈ I. Thus, ρ(X + C) =

f(RX+C )= f(RX − C)= f(RX ) − C = ρ(X) − C.

(iii) Let X,Y ∈ L∞ and λ ∈ [0, 1]. Then ρi(λX + (1 − λ)Y ) ≤ λρi(X) + (1 − λ)ρi(Y ), ∀ i ∈I.

Thus, ρ(λX + (1 − λY )) = f(RλX+(1−λY )) ≤ f(λRX + (1 − λ)RY ) ≤ λρ(X)+ (1 − λ)ρ(Y ).

∞ i i (iv) Let X ∈ L and λ ≥ 0. Then ρ (λX = λρ (X), ∀ i ∈ I. Thus ρ(λX) = f(RλX ) =

f(λRX )= λf(RX )= λρ(X).

∞ i i (v) Let X,Y ∈ L such that FX = FY . Then ρ (X) = ρ (Y ), ∀ i ∈ I. Thus RX = RY

point-wisely. Hence, RX and RY belong to the same equivalence class on X and ρ(X)=

f(RX )= f(RY )= ρ(Y ).

(vi) Let X,Y ∈ L∞ be a comonotone pair. Then ρi(X + Y )= ρi(X)+ ρi(Y ), ∀ i ∈I. Thus

ρ(X + Y )= f(RX+Y )= f(RX + RY )= f(RX )+ f(RY )= ρ(X)+ ρ(Y ).

∞ ∞ ∞ i i (vii) Let{Xn}n ⊆ L bounded with lim Xn = X ∈ L . Then ρ (X) ≤ lim inf ρ (Xn), ∀ i ∈ =1 n→∞ n→∞ I. Since supn∈N ρI (Xn) ≤ supn∈NkXnk∞ < ∞, we get that {RXn } is bounded. Thus ρ(X)= f(RX ) ≤ f(lim inf RX ) ≤ lim inf f(RX ) = lim inf ρ(Xn). n→∞ n n→∞ n n→∞

Remark 3.6. Converse relations are not always guaranteed. For instance, spectral risk measures in Example 2.12 are convex despite the collection {V aRα, α ∈ [0, 1]} is not in general. Moreover, The preservation of Subadditivity, i.e. ρ(X + Y ) ≤ ρ(X)+ ρ(Y ), ∀ X,Y ∈ L∞, is quite similar to the one for Convexity, obtained by replacing the properties. The same for Comonotone Convexity and Comonotone Subadditivity, see Kou et al. (2013), which are relaxed counterparts for Comonotonic pairs. For completeness sake, we now investigate the preservation of other properties in the liter- ature of risk measures.

11 i ∞ Proposition 3.7. Let ρI = {ρ : L → R, i ∈ I} be a collection of risk measures, f : X → R, ∞ and ρ: L → R a risk measure defined as ρ(X)= f(ρI(X)). Then:

(i) If ρI is composed of risk measures with Convexity and f possesses Monotonicity and Quasi-convexity, i.e. f(λR + (1 − λ)S) ≤ max{f(R),f(S)}, ∀ λ ∈ [0, 1], ∀ R,S ∈X , then ρ fulfills Quasi-convexity.

i i (ii) If ρI is composed of risk measures with Cash-subadditivity, i.e. ρ (X + C) ≥ ρ (X) − ∞ C, ∀ C ∈ R+, ∀ X ∈ L , and f possesses Monotonicity and Translation Invariance, then ρ has Cash-subadditivity.

(iii) If ρI is composed of risk measures with Relevance, i.e. X ≤ 0 and P(X < 0) > 0 imply ρi(X) > 0, ∀ X ∈ L∞, and f has strict Monotonicity, i.e. R>S imply f(R) > f(S), ∀ R,S ∈X , then ρ has Relevance.

i − − (iv) If ρI is composed of risk measures with Surplus Invariance, i.e. ρ (X) ≤ 0 and Y ≤ X imply ρi(Y ) ≤ 0, ∀ X,Y ∈ L∞, and f has Monotonicity together to f ≥ f W C, then ρ possesses Surplus Invariance.

∞ Proof. (i) Let X,Y ∈ L and λ ∈ [0, 1]. Then ρ(λX + (1 − λ)Y ) = f(RλX+(1−λY )) ≤ f(λRX + (1 − λ)RY ) ≤ max{ρ(X), ρ(Y )}.

∞ (ii) Take X ∈ L and C ∈ R+. Then ρ(X + C) ≥ f(RX − C)= f(RX ) − C = ρ(X) − C.

∞ (iii) Let X ∈ L such that X ≤ 0 and P(X < 0) > 0. Then ρI(X) > 0 point-wisely in X .

Thus, from strict Monotonocity of f we get ρ(X)= f(RX ) > 0.

(iv) Let X,Y ∈ L∞ such that ρ(X) ≤ 0 and Y − ≤ X−. Thus, from hypotheses we get i W C ρ (X) ≤ f (RX ) ≤ f(RX ) ≤ 0 for any i ∈ I. This leads to ρI(Y ) ≤ 0 point-wisely in

X . Hence, ρ(Y )= f(RY ) ≤ 0.

Remark 3.8. See Cerreia-Vioglio et al. (2011), El Karoui and Ravanelli (2009), Delbaen (2012) and Koch-Medina et al. (2017) for details on Quasi-convexity, Cash-subadditivty, Relevance and Surplus Invariance, respectively. Unfortunately, Quasi-convexity of ρI is not preserved if f is monotone and quasi-convex (or even convex). The proof of such claim relies on the fact that X does not have a total order, in contrast to the case of the real line where f ◦ ρ would be quasi-convex. Moreover, even in the case exposed in item (i) of the last Proposition, in order to guarantee preservation for Cash subadditivity, which is the typical companion for Quasi- convexity, we must to assume that f possesses Translation Invariance. Since it would imply Convexity, we would be back at the original framework of the paper.

3.3 Continuity properties

In this subsection, the goal is the preservation of continuity properties beyound Fatou continuity.

i ∞ Proposition 3.9. Let ρI = {ρ : L → R, i ∈ I} be a collection of point-wise bounded risk ∞ measures, f : X → R, and ρ: L → R a risk measure defined as ρ(X)= f(ρI(X)). Then:

12 (i) If ρI is composed of Lipschitz continuous risk measures and f fulfills Monotonicity, Sub- additivity, i.e. f(R + S) ≤ f(R)+ f(S), ∀ R,S ∈X , and Boundedness, then ρ is Lipschitz continuous.

(ii) If ρI is composed of risk measures that posses continuity from above, below or Lebesgue and f is Fatou continuous, then ρ is Fatou continuous for non-increasing sequences, non- decreasing sequences, or any sequences, respectively.

(iii) If ρI is composed of risk measures that posses any property among continuity from above,

below or Lebesgue, and f is Lebesgue continuous, i.e. lim Rn = R implies f(R) = n→∞ ∞ ∞ lim f(Rn), ∀ {Rn} ⊆X bounded and any R ∈X∩ K , then ρ also does. n→∞ n=1

Proof. (i) From Monotonicity and Subadditivity of f and RX ≤ RY + |RX − RY | we have i i that |f(RX ) − f(RY )|≤ f(|RX − RY |). Moreover, |ρ (X) − ρ (Y )| ≤ kX − Y k∞, ∀ X,Y ∈ L∞, ∀ i ∈I. Then from boundedness of f we get

W C ∞ |ρ(X) − ρ(Y )|≤ f(|RX − RY |) ≤ f (|RX − RY |) ≤ kX − Y k∞, ∀ X,Y ∈ L .

∞ ∞ ∞ i (ii) Let {Xn} ⊆ L bounded such that lim Xn = X ∈ L and ρ Lebesgue continuous n=1 n→∞ for any i ∈I. Then we have {RX } bounded and lim RX = RX point-wise. Hence n n→∞ n

ρ(X)= f lim RX ≤ lim inf f(RX ) = lim inf ρ(Xn). n→∞ n n→∞ n n→∞   When each ρi is continuous from above or below, the same reasoning which is restricted to non-decreasing or non-increasing sequences, respectively, is valid.

(iii) Similar to (ii), but in this case f lim RX = lim f(RX ). n→∞ n n→∞ n  

Remark 3.10. Item (i) can be generalized to ρI uniformly equicontinuous, i.e. the δ − ǫ criteria does not depend on i ∈ I, from which Lipschitz continuity is a special case. Moreover, let Q W C I = Q, ρ (X)= EQ[−X] and f = f . In this case we have ρ = ML, which is not continuous from below even ρQ possessing such property. However, ML is Fatou continuous. This example illustrates the item (ii) in the last Proposition. Moreover, f µ satisfies Lebesgue continuity as in (iii) when X ⊆ K∞, which is the case for monetary risk measures for instance.

4 Representations

4.1 General result

In this section, we expose results regarding the representation of composed risk measures ρ = f(ρI) based on the properties of both ρI and f. The goal is to highlight the role of such terms. We begin the preparation with a Lemma for representation of f, without dependence on the properties of ρI .

13 Lemma 4.1. Let X ⊆ K∞. A functional f : X → R, posses Monotonicity, Translation Invari- ance, Convexity and Fatou continuity if and only if it can be represented as

f(R) = sup Rdµ − γf (µ) , ∀ R ∈X , (4.1) µ∈V ZI  where γf : V → R+ ∪ {∞} is defined as

γf (µ)= sup Rdµ − f(R) . (4.2) R∈X ZI  Proof. The fact that (4.1) possesses Monotonicity, Translation Invariance, Convexity and Fatou continuity is straightforward. For the only if direction, one can understand f(R) as π(−R), where π is a Fatou continuous convex risk measure on K∞. Note that it is finite. Thus, from Theorem 4.22 of F¨ollmer and Schied (2016), which is similar to Theorem 2.4 but without a base probability, and f(R)= ∞ for any R ∈ K∞\X we have that

f(R)= π(−R) = sup Rdµ − sup Rdµ − f(R) = sup Rdµ − γf (µ) . µ∈V ZI R∈X ZI  µ∈V ZI 

∞ Remark 4.2. When R = RX point-wisely for some X ∈ L , we have that the representation becomes µ f(RX ) = sup {ρ (X) − γf (µ)} . µ∈V

If f possesses Positive Homogeneity, then γf assumes value 0 in Vf = {µ ∈ V : f(R) ≥

µ I Rdµ, ∀ R ∈ X } and ∞ otherwise. For instance, Vf = {µ} and Vf WC = V. Note that inf γf (µ) = 0 from the assumption of normalization for f. Rµ∈V We need the following auxiliary result, which may be of individual interest, for interchanging the supremum and integral in a specific case that it is useful in posterior results. The following Lemma is a generalization of the (countable) additive property of supremum over the sum of sets.

Lemma 4.3. Let (I, G,µ) be a probability space, hi : Y → R, i ∈ I, a collection of bounded i functionals over a non-empty space Y such that i → h (yi) is G-measurable for any {yi ∈Y, i ∈ I}. Then i i sup h (y)dµ = sup h (yi)dµ ZI y∈Yi {yi∈Yi,i∈I} ZI i where, for any i ∈I: Yi ⊆Y, Yi 6= ∅, and sup h (y) is G-measurable. y∈Yi

Proof. Note that the axiom of choice guarantees the existence of {yi ∈ Yi, i ∈ I}. For every

ǫ> 0 we have that there exists {yi ∈Yi, i ∈ I} such that

i i i sup h (y) − ǫ ≤ h (yi) ≤ sup h (y). y∈Yi y∈Yi

14 Then, by integrating over I in relation to µ we obtain

i i i sup h (y)dµ − ǫ ≤ h (yi)dµ ≤ sup h (y)dµ. ZI y∈Yi ZI ZI y∈Yi Thus, we get that

i i i sup h (y)dµ − ǫ ≤ sup h (yi)dµ ≤ sup h (y)dµ. ZI y∈Yi {yi∈Yi,i∈I} ZI ZI y∈Yi Since one can take ǫ arbitrarily, the result follows.

We also need an assumption to circumvent some measurability issues in order to avoid indefiniteness of posterior measure related concepts, such as integration.

i ∞ Assumption 4.4. When ρI = {ρ : L → R, i ∈ I} is a collection of Fatou continuous convex risk measures we assume that the following maps are G-measurable for any collection {Qi ∈ P, i ∈ I}:

∞ (i) i → EQi [X], ∀ X ∈ L .

min i (ii) i → αρi (Q ).

Remark 4.5. Similarly to Assumption 2.8, as a single, but not unique, example for G one could consider the sigma-algebra generated by all such maps. Note that the maps i → Qi(A), ∀A ∈ F, are also G-measurable since by choosing X = 1A, A ∈ F, we get EQ[1A]= Q(A) for any Q ∈ P. In fact, since the simple functions are dense in L∞ the converse implication is also true and the assumption of measurability could be made for the maps i → Qi(A), ∀ A ∈ F. Moreover, also min i are G-measurable the maps i → αρi (Q), ∀ Q ∈ P, by picking Q = Q ∀ i ∈ I. Note that we min can consider the extension αρ (Q)= ∞, ∀ Q ∈ P\Q, for any minimal penalty term.

i ∞ Lemma 4.6. Let ρI = {ρ : L → R, i ∈ I} be a collection of Fatou continuous convex risk i i measures. Then A → I Q (A)dµ defines a probability measure for any {Q ∈ P, i ∈ I}. i ∞ Moreover, E Qidµ[X]= I EQi [X]dµ for any {Q ∈ P, i ∈ I} and any X ∈ L . RI R i R µ i Proof. For some {Q ∈ P, i ∈ I} let Q (A) = I Q (A)dµ, ∀ A ∈ F. It is direct that both µ µ Q (∅) = 0 and Q (Ω) = 1. For countably additivity,R let {An}n∈N be a collection of mutually disjoint sets. Then, since i → Qi(A) is bounded ∀ A ∈ F we have

∞ ∞ ∞ µ ∞ i i µ Q (∪n=1An)= Q (An)dµ = Q (An)dµ = Q (An). I n n I n Z X=1 X=1 Z X=1 Hence Qµ is a probability measure. Regarding expectation interchange we have for any X ∈ L∞ i that x → Q (X ≤ x) = FX,Qi (x) is monotone and right-continuous for any i ∈ I and, from Assumption 4.4, i → Qi(X ≤ x) is G-measurable. Then (x, i) → Qi(X ≤ x) is B(R) ⊗ G-

15 measurable, indeed integrable. Hence we have

∞ 0 i i E Qidµ[−X]= (1 − Q (X ≤ x))dµdx + Q (X ≤ x)dµdx RI Z0 ZI Z−∞ ZI ∞ 0 = (1 − Qi(X ≤ x))dx + Qi(X ≤ x)dx dµ ZI Z0 Z−∞ 

= EQi [−X]dµ. ZI By changing signs we get the claim.

The role played by ρµ becomes clear since it can be understood as the expectation under µ of elements RX ∈ X . Thus, it is important to know how the properties of ρI affect the repre- sentation of ρµ. Proposition 2.1 of Ang et al. (2018) explores a case with finite number coherent risk measures, while we address a situation with an arbitrary set of convex risk measures.

i ∞ Theorem 4.7. Let ρI = {ρ : L → R, i ∈ I} be a collection of Fatou continuous convex risk measures and ρµ : L∞ → R defined as in (2.7). Then:

(i) ρµ can be represented as:

µ ∞ ρ (X)= sup {EQ[−X] − αρµ (Q)} , ∀ X ∈ L , (4.3) Q∈Q

with convex αρµ : Q→ R+ ∪ {∞} defined as

min i i i αρµ (Q) = inf αρi Q dµ: Q dµ = Q, Q ∈Q∀ i ∈I . (4.4) ZI ZI 
(ii) If in addition ρi fulfills, for every i ∈I, Positive Homogeneity, then the representation is

µ ∞ ρ (X)= sup EQ[−X], ∀ X ∈ L , (4.5) Q∈cl(Qρµ )

µ i i with Qρ = Q ∈ Q: Q = I Q dµ, Q ∈ Qρi ∀ i ∈I convex and non-empty. Proof. (i) From Propositions R3.3 and 3.5 we have that ρµ is a Fatou continuous convex risk measure. Thus we have

µ min ρ (X)= sup EQ [−X] − αρi (Q) dµ I Q∈Q ! Z n o min i = sup EQi [−X] − αρi Q dµ i {Q ∈Q,i∈I} ZI  
 min i i i = sup EQ[−X] − inf αρi Q dµ: Q dµ = Q, Q ∈Q∀ i ∈I Q∈Q  ZI ZI 
= sup {EQ[−X] − αρµ (Q)} . Q∈Q

We have used Lemma 4.3 for the interchange of supremum and integral with Lemma 4.6 for expectations. Assumption 4.4 is used to guarantee that integrals make sense. Note

16 that αρµ is well-defined since the infimum is not altered for the distinct choices of possible

i µ combinations that lead to I Q dµ = Q. Non-negativity for αρ is straightforward. We also have αρµ (Q) = ∞ forR any Q ∈ P\Q. For convexity, let λ ∈ [0, 1], Q1, Q2 ∈ Q, i i Q3 = λQ1 + (1 − λ)Q2, Qj = {Q ∈ Q, i ∈ I}: I Q dµ = Qj , j ∈ {1, 2, 3}, and the Qi , Qi , i λQi λ Qi dµ Q auxiliary set Qλ = {( 1 2) ∈Q×Q ∈ I}: IR( 1 + (1 − ) 2) = 3 . We thus obtain  R

min i i λαρµ (Q1) + (1 − λ)αρµ (Q2) ≥ inf αρi (λQ1 + (1 − λ)Q2)dµ Qi ,i 1, Qi ,i 2 { 1∈Q ∈I}∈Q { 2∈Q ∈I}∈Q ZI min i i ≥ inf αρi (λQ1 + (1 − λ)Q2)dµ Qi ,Qi ,i {( 1 2)∈Q×Q ∈I}∈Qλ ZI min i ≥ inf αρi (Q )dµ {Qi∈Q,i∈I}∈Q3 ZI = αρµ (Q3).

(ii) In this framework we obtain

µ ρ (X)= sup EQ[−X]dµ Q∈Q ZI ρi

= sup EQi [−X]dµ Qi I n ∈Qρi ,i∈Io Z

= sup EQ[−X] Q∈Qρµ

= sup EQ[−X]. Q∈cl(Qρµ )

∞ µ To demonstrate that taking closure does not affect the supremum, let {Qn}n=1 ∈ Qρ such that Qn → Q in the total variation norm. Then we have

EQ[−X] = lim EQ [−X] ≤ sup EQ [−X] ≤ sup EQ[−X]. n→∞ n n n Q∈Qρµ

Since every Qρi is non-empty, we have that Qρµ posses at least one element Q ∈ Q such

i i µ that Q = I Q dµ, Q ∈ Qρi ∀ i ∈ I. Let Q1, Q2 ∈ Qρ . Then, we have for any λ ∈ [0, 1] i i λQ λ Q λQ λ Q dµ i i that 1 +R (1 − ) 2 = I 1 + (1 − ) 2 . Since Qρ is convex for any ∈ I min we have that λQ1 + (1 − Rλ)Q 2 ∈ Qρµ as desired.
Still remains to show that αρµ is µ min i i i µ an indicator function on cl (Qρ ). Note that αρi (Q ) = 0, ∀ Q ∈ Qρ. Thus, αρ is well min defined on Qρµ and we have that 0 ≤ αρµ (Q) ≤ αρµ (Q) = 0, ∀Q ∈ Qρµ . Due to the lower min semi-continuity property, we have that αρµ (Q) = 0 for any limit point Q of sequences in min Qρµ . If Q ∈ Q\cl (Qρµ ), then αρµ (Q)= ∞, otherwise the dual representation would be violated.

Remark 4.8. We have that αρµ in this case can be understood as some extension of the concept of inf-convolution for arbitrary terms represented by theoretical measure and integral concepts.

17 In fact, the sum of finite risk measures leads to the inf-convolutions of their penalty functions. By extrapolating the argument, such a result is also useful regarding general conjugate for an arbitrary mixture of convex functionals.

i Remark 4.9. Note that we could consider the families {Q ∈ P, i ∈ I} that define both αρµ and Qρµ by belonging to determined sets in therms of µ-a.s. instead of point-wise in I. This is because the criteria is Lebesgue integral with respect to each specified µ ∈V. We choose for the point-wise option in order to keep pattern since we have not assumed fixed probability on

(I, G) alongside the text. Moreover, the integrals that define both αρµ and Qρµ may also be understood in the sense of Bochner integral, see Aliprantis and Border (2006) chapter 11 for details.

min Remark 4.10. We have that αρµ is lower semi-continuous if and only if it coincides with αρµ . This because when αρµ is lower semi-continuous, by bi-duality regarding Legendre-Fenchel con- µ ∗ ∗∗ µ ∗ min min jugates and the fact that ρ = (αρµ ) , we obtain αρµ = (αρµ ) = (ρ ) = αρµ . Thus, αρµ is the lower semi-continuous hull of αρµ in the sense that we can obtain the first by closing the epigraph of αρµ in Q×R+ ∪{∞}. In the case of finite cardinality for I, Qρµ is closed as exposed in Proposition 2.1 of Ang et al. (2018), which makes possible to drop the closure on (4.5) in such situation. We now have the necessary conditions to enunciate the main result in this section, which is a representation for composed risk measures in the usual framework of Theorem 2.4.

i ∞ Theorem 4.11. Let ρI = {ρ : L → R, i ∈ I} be a collection of Fatou continuous convex risk measures, f : X → R possessing Monotonicity, Translation Invariance, Convexity and Fatou ∞ continuity, and ρ: L → R defined as ρ(X)= f(ρI(X)). Then:

(i) ρ can be represented as

∞ ρ(X) = sup {EQ[−X] − αρ(Q)} , ∀ X ∈ L , (4.6) Q∈Q

where αρ(Q) = inf {αρµ (Q)+ γf (µ)}, with γf and αρµ defined as in (4.2) and (4.4), µ∈V respectively.

(ii) If in addition to the initial hypotheses f possesses Positive Homogeneity, then the penalty

term becomes αρ(Q)= inf αρµ (Q), where Vf is as in Remark 4.2. µ∈Vf

(iii) If in addition to the initial hypotheses ρi possess, for any i ∈ I, Positive Homogeneity,

then αρ(Q)= ∞∀ Q ∈ Q\∪µ∈V cl (Qρµ ).

(iv) If in addition to the initial hypotheses we have the situations in (ii) and (iii), then the representation of ρ becomes

∞ ρ(X)= sup EQ[−X], ∀ X ∈ L , (4.7) Vf Q∈Qρ

Vf µ where Qρ is the closed convex hull of ∪µ∈Vf cl(Qρ ).

18 Proof. From the hypotheses and Proposition 3.5 we have that ρ is a Fatou continuous convex risk measure.

(i) From Lemma 4.1 and Theorem 4.7 we have that

ρ(X) = sup sup [EQ[−X] − αρµ (Q)] − γf (µ) µ∈V (Q∈Q )

= sup EQ[−X] − inf [αρµ (Q)+ γf (µ)] µ∈V Q∈Q   = sup {EQ[−X] − αρ(Q)} . Q∈Q

(ii) If f possesses Positive Homogeneity, then γf assumes value 0 in Vf and ∞ otherwise. Thus, we get

αρ(Q)= inf {αρµ (Q)+ γf (µ)} = inf αρµ (Q). µ∈V µ∈Vf

(iii) When each element of ρI fulfills Positive Homogeneity, we have that αρµ (Q)= ∞∀ µ ∈V

for any Q ∈ Q\∪µ∈V cl(Qρµ ). By adding the non-negative term γf (µ) and taking the infimum over V, one gets the claim.

(iv) In this context, the generated ρ is coherent from Theorem 3.5. Moreover, in this case from Lemma 4.1 and Proposition 4.7 together to items (ii) and (iii) we have that

ρ(X)= sup sup EQ[−X] µ∈Vf Q∈cl(Qρµ )

= sup EQ[−X] Q cl µ ∈∪µ∈Vf (Qρ )

= sup EQ[−X]. Vf Q∈Qρ

In order to verify that the supremum is not altered by considering the closed convex hull,

µ let Q1, Q2 ∈∪µ∈Vf cl(Qρ ) and Q = λQ1 + (1 − λ)Q2, λ ∈ [0, 1]. Then

EQ[−X] ≤ max(EQ1 [−X], EQ2 [−X]) ≤ sup EQ[−X], Q cl µ ∈∪µ∈Vf (Qρ )

thus convex combinations do not alter the supremum. For closure, the deduction is quite min similar to that used in the proof of Theorem 4.7. We also have that αρ is an indicator Vf on Qρ . This fact is true because in this case αρµ (Q) = 0 for at least one µ ∈Vf for any

µ Q ∈∪µ∈Vf cl(Qρ ). The same is valid for convex combinations and limit points is due to min convexity and lower semi-continuity of αρ .

µ Vf Remark 4.12. Note that when f = f we recover the result in Theorem 4.7. Moreover, Qρ ⊆ W C QρWC since f ≤ f for any bounded combination f. Furthermore, when αρµ is lower semi-

19 continuous we have that αρ coincides with the minimal penalty term because

min αρ (Q)= sup {EQ[−X] − f(RX )} X∈L∞

µ = sup EQ[−X] − sup {ρ (X) − γf (µ)} X∈L∞ ( µ∈V )

µ = inf γf (µ)+ sup {EQ[−X] − ρ (X)} µ∈V ∞  X∈L  = αρ(Q).

Hence, the reasoning in Remark 4.10 is also valid in here. Remark 4.13. Under Cash-subadditivity or Relevance, the supremum over Q can be replaced, respectively, by sub-probabilities (measures on (Ω, F) with Q(Ω) ≤ 1) or probabilities equivalent to P. Under Quasi-convexity for f and dropping its Convexity and Translation Invariance µ we get the representation ρ(X) = supµ∈V Rf (ρ (X),µ), where Rf : R ×V → R is defined as

Rf (x,µ) = inf f(R): I Rdµ = x . See the respective papers in Remark 3.8 for details. Regarding the specificR case of ρ W C, Proposition 9 in F¨ollmer and Schied (2002) states that min it an be represented by, the non necessarily convex, αρWC (Q) = inf α i (Q), ∀ Q ∈ Q. Under i∈I ρ n coherence, Theorem 2.1 of Ang et al. (2018) claims that QρWC = conv(∪i=iQρi ) when I is finite with cardinality n. We now expose a result that states the equivalence between these facts and our approach, and generalize them.

Proposition 4.14. Let {ρi : L∞ → R, i ∈ I} be a collection of Fatou continuous convex risk measures, and ρW C : L∞ → R defined as in (2.10). Then:

min (i) αρWC (Q)= inf αρµ (Q) = inf α i (Q), ∀ Q ∈ Q. µ∈V i∈I ρ

(ii) If in addition to the initial hypotheses ρi possess, for any i ∈ I, Positive Homogeneity, V then QρWC = Qρ , which is the closed convex hull of ∪i∈IQρi .

Proof. From Propositions 3.3 and 3.5 we have that ρW C is a Fatou continuous convex risk i W C measure when all ρ also are. Moreover, from the fact that |ρ (X)| ≤ kXk∞ < ∞ we have that ρW C takes only finite values.

(i) For fixed Q ∈ P, we have that for any ǫ> 0, there is j ∈I such that

min min min inf αρi (Q) ≤ αρj (Q) ≤ inf αρi (Q)+ ǫ. i∈I i∈I

min min Recall that inf α i (Q) ≤ αρµ (Q) ≤ α i (Q)dµ for any µ ∈V. Then it is true that for i∈I ρ I ρ any ǫ> 0, there is µ ∈V such that R

min min inf αρi (Q) ≤ αρµ (Q) ≤ inf αρi (Q)+ ǫ. i∈I i∈I

By taking the infimum over V and since ǫ was taken arbitrarily, we get that αρWC (Q) = min inf α i (Q). i∈I ρ

20 (ii) From Propositions 3.3 and 3.5 we have that ρW C is a Fatou continuous coherent risk measure when all ρi also are. Thus, in light of Theorem 2.4, it has a dual representation. We then have W C ρ (X) = sup sup EQ [−X]= sup EQ[−X]. i∈I Q∈Qρi Q∈∪i∈I Qρi The fact that supremum is not altered by considering the closed convex hull follows similar

steps as those in the proof of Theorem 4.11. We have that ∪i∈I Qρi is non-empty because min every Qρi contains at least one element. Moreover, αρWC is an indicator function since min min for Q ∈∪i∈IQρi , then αρi (Q) = 0 for at least one i ∈I. Consequently 0 ≤ αρWC (Q) ≤ αρWC (Q) = 0. We have to consider the cases where Q is a convex combination or a min limit point from the union. In both cases αρWC (Q) = 0 from the convexity and lower min semi-continuity of αρWC , respectively. Hence, QρWC coincides to the closed convex hull V of ∪i∈IQρi . Regarding the equivalence with Qρ , note that for any i ∈ I we have that

Q ∈ Qρi if and only if Q ∈ clQρδi , where δi ∈ V is defined as δi(A) = 1A(i), ∀ A ∈ G. Thus, we get that ∪i∈IQρi ⊆∪µ∈V cl(Qρµ ). By considering closed convex hulls we obtain

V µ min QρWC ⊆ Qρ . For the converse relation note that if Q ∈ ∪µ∈V cl(Qρ ), then αρWC (Q) ≤ min inf αρµ (Q) = inf α i (Q) = 0. Hence, Q ∈ QρWC as desired. µ∈V i∈I ρ

4.2 Law invariant case

Under Law Invariance of the components in ρI , the generated ρ is representable in light of those formulations in Theorem 2.7. We begin addressing the measurability issues.

i ∞ Lemma 4.15. Let ρI = {ρ : L → R, i ∈ I} be a collection of law invariant convex risk measures. Then the following maps are G-measurable for any {mi ∈ M, i ∈ I}:

α i ∞ (i) i → (0,1] ES (X)dm , ∀ X ∈ L .

(ii) i → mR i(B), ∀ B ∈B(0, 1], where B(0, 1] is the Borel sigma-algebra on (0, 1].

min i (iii) i → βρi (m ).

Proof. We have for any fixed {mi ∈ M, i ∈ I} the following:

mi Qi 1 dmi F −1 u (i) For any ∈ M there is ∈ Q such that (u,1] v = dQi (1 − ). Then, from the dP definition of ES and the Hardy-Littlewood inequality,R the following relation holds:

1 α i −1 −1 i ES (X)dm = − FX (α)F dQi (1 − α)dα = sup EQ [−Y ]. Z(0,1] Z0 dP Y ∼X

From Assumption 4.4 we have that i → EQi [−Y ] is G-measurable for any Y ∼ X. More- over, for any n ∈ N, there is Yn ∼ X such that

1 gn(i)= EQi [−Yn]+ ≥ sup EQi [−Y ] ≥ EQi [−Yn]= hn(i). n Y ∼X

21 Note that both gn and hn are G-measurable for all n ∈ N and µ-integrable for any µ ∈V

since EQi [−Yn] is bounded. The same is true for g = lim inf gn and h = limsup hn. In n→∞ n→∞ min this case we have that g(i) ≥ αρi (Q) ≥ h(i), ∀ i ∈I. Moreover, we have

1 (g − h )dµ ≤ , ∀ n ∈ N, ∀ µ ∈V. n n n ZI From Fatou lemma it is true that

0 ≤ (g − h)dµ ≤ lim inf (gn − hn)dµ = 0. n→∞ ZI ZI

Thus, g = h µ-a.s. for any µ ∈V. Since δi ∈V, ∀ i ∈I, with δi(A) = 1A(i), ∀ A ∈ G, we α i then have that g = h point-wisely. Then i → (0,1] ES (X)dm = sup EQi [−Y ]= g(i)= Y ∼X h(i) is G-measurable for any {mi ∈ M, i ∈ I}R.

(ii) Let B ∈ B(0, 1]. Then we can write B = ∪n(an, bn], i.e. a countable union of intervals ∞ (an, bn] ⊆ (0, 1]. Now, for each n, we define a pair Xn,Yn ∈ L such that P(Xn = −1) = i bn, P(Xn = 0) = 1 − bn, P(Yn = −1) = an, P(Yn = 0) = 1 − an. Thus, for any m ∈ M we have that

i i m (B)= 1(an,bn]dm n (0,1] X Z i i = 1(0,bn]dm − 1(0,an]dm n (0,1] (0,1] ! X Z Z α i α i = ES (Xn)dm − ES (Yn)dm . n (0,1] (0,1] ! X Z Z

i α i α i From item (i), we have that i → m (B) = n (0,1] ES (Xn)dm − (0,1] ES (Yn)dm i is G-measurable for any {m ∈ M, i ∈ I}. P R R  (iii) From item (i) jointly to Theorems 2.4 and 2.7, we have that

i min i min dQ dQ 1 i −1 i min i β i (m ) = sup α i (Q): ∼ , dm = F (1 − u), Q ∈ Q = α i (Q ). ρ ρ dP dP v dQi ρ ( Z(u,1] dP )

min i min i Hence, from Assumption 4.4 the maps i → βρi (m ) = αρi (Q ) are G-measurable for any {mi ∈ M, i ∈ I}.

We now need an auxiliary result for the representation when ρµ is law invariant. The next Proposition follows in this direction.

i ∞ Proposition 4.16. Let ρI = {ρ : L → R, i ∈ I} be a collection of law invariant convex risk measures and ρµ : L∞ → R defined as in (2.7). Then:

22 (i) ρµ can be represented as:

µ α ∞ ρ (X)= sup ES (X)dm − βρµ (m) , ∀ X ∈ L , (4.8) m∈M (Z(0,1] )

with convex βρµ : M→ R+ ∪ {∞}, defined as

min i i i βρµ (m) = inf βρi (m )dµ: m dµ = m, m ∈M∀ i ∈I . (4.9) ZI ZI 

(ii) If in addition ρi fulfills, for every i ∈I, Positive Homogeneity, then the representation is

ρµ(X)= sup ESα(X)dm, ∀ X ∈ L∞, (4.10) m∈cl(Mρµ ) Z(0,1]

µ i i with Mρ = m ∈ M: m = I m dµ,m ∈ Mρi ∀ i ∈I non-empty and convex.

(iii) If ρi also is, for every i ∈I,R comonotone, then the representation is

ρµ(X)= ESα(X)dm, ∀ X ∈ L∞, (4.11) Z(0,1]

where m ∈ cl(Mρµ ).

Proof. From the hypotheses and Proposition 3.5 together to Theorem 2.6 we have that ρµ is a law invariant convex risk measure. Theorem 2.6 assures its Fatou continuity. The proof follows i α i similar steps to those of Theorem 4.7 with m → (0,1] ES (X)dm linear and playing the role i α i Q E i X i ES X dm of → Q [− ]. Note that from Lemma 4.15, R → (0,1] ( ) is G-measurable. For the comonotonic case in (iii), the result is due to the supremR um in (2.5) be attained for each ρi.

Remark 4.17. The representation in item (iii) on (4.11) is equivalent to the spectral one as

1 ρµ(X)= V aRα(X)φµ(α)dα, (4.12) Z0 µ i i where φ (α)= I φ (α)dµ, and φ : [0, 1] → [0, 1] is as in Example 2.12 for any i ∈I. The map i i → φ (α) is G-measurableR to any α ∈ [0, 1]. To verify this claim, note that for each i ∈ I it i 1 i i dνi 1 is true that φ (α) = (α,1] s dm (s), m ∈ M. Then dµi = s defines a finite measure on (0, 1]. i i i i Thus, there is η ∈ MR such that ν = βη , β ∈ R+. Since, from Lemma 4.15, i → η (α, 1] is G-measurable for any α ∈ [0, 1], then also is i → νi(α, 1] = φi(α). We are now able to propose a result for the dual representation under Law Invariance. The next Corollary exposes such content.

i ∞ Corollary 4.18. Let ρI = {ρ : L → R, i ∈ I} be a collection of law invariant convex risk measures, f : X → R possessing Monotonicity, Translation Invariance, Convexity and Fatou ∞ continuity, and ρ: L → R defined as ρ(X)= f(ρI(X)). Then:

23 (i) ρ can be represented as

α ∞ ρ(X)= sup ES (X)dm − βρ(m) , ∀ X ∈ L , (4.13) m∈M (Z(0,1] )

where βρ(m) = inf {βρµ (m)+ γf (µ)}, with γf and βρµ defined as in (4.2) and (4.9), µ∈V respectively.

(ii) If in addition to the initial hypotheses f possesses Positive Homogeneity, then the penalty

term becomes βρ(m)= inf βρµ (m), where Vf is as in Remark 4.2. µ∈Vf

(iii) If in addition to the initial hypotheses ρi possess, for any i ∈ I, Positive Homogeneity,

then βρ(m)= ∞, ∀ m ∈ M\∪µ∈V cl(Mρµ ).

(iv) If in addition to the initial hypotheses we have the situations in (ii) and (iii), then the representation of ρ becomes

ρ(X)= sup ESα(X)dm, ∀ X ∈ L∞, (4.14) Vf (0,1] m∈Mρ Z

Vf µ where Mρ is the closed convex hull of ∪µ∈Vf cl(Mρ ).

(v) If in addition to the initial hypotheses ρi possess, for any i ∈I, Comonotonic Additivity, µ then βρ(m)= ∞, ∀ m ∈ M\∪µ∈V {mc }, where

µ α mc = argmax ES (X)dm, ∀ µ ∈V. m∈cl(Mρµ ) Z(0,1]

(vi) If in addition to the initial hypotheses we have (ii) and (v), then the representation of ρ becomes ρ(X)= sup ESα(X)dm, ∀ X ∈ L∞, (4.15) Vf (0,1] m∈Mρ,c Z

Vf µ where Mρ,c is the closed convex hull of ∪µ∈Vf {mc }.

Proof. Direct from Theorem 4.11 and Proposition 4.16.

Remark 4.19. Since the comonotonicity of a pair X,Y does not imply the same property for the pair RX ,RY , the only situation where ρ is surely comonotone occurs, from Proposition 3.5 µ and Lemma 4.1, when {f }µ∈V . In this case we have

µ α µ ∞ ρ(X)= ρ (X)= ES (X)dmc , ∀ X ∈ L . (4.16) Z(0,1]

µ 1 µ µ From (4.12), we have that φ (α)= (α,1] s dmc (s), where φ is as in Remark 4.17. Remark 4.20. It would be possible alsoR to investigate a situation where f possesses representa- tion in terms of ES over X as in Example 2.13, but we do not consider any base probability on (I, G). However, note that those would be special cases of our framework.

24 5 Acceptance sets

5.1 Properties

In this section, we expose results regarding the acceptance sets of composed risk measures

ρ = f(ρI) based on the properties of both ρI and f. For that, we make use of results from the previous sections. Of course, when f possesses Monotonicity we have

∞ −1 AρI ⊇ X ∈ L : ∃ R ∈ f (0) s.t. ρI (X) ≤ R ,  where the inequality is in the point-wise order of X . If in addition f is injective function, then from normalization, f(0) = 0, we have

∞ −1 ∞ X ∈ L : ∃ R ∈ f (0) s.t. ρI(X) ≤ R = {X ∈ L : ρI(X) ≤ 0} = AρWC .  However, since the point-wise order in X is not total, the set AρI can be much larger than the positions that leads to elements in the non-positive cone of X . Thus, to provide a general

characterization for AρI is not trivial. We begin by translating the role of financial properties preservation from section 3 for acceptance sets.

i ∞ Corollary 5.1. Let ρI = {ρ : L → R, i ∈ I} be a collection of risk measures, f : X → R, ∞ and ρ: L → R a risk measure defined as ρ(X)= f(ρI(X)). Then:

(i) If ρI is composed of risk measures with Monotonicity and f possesses this same property, ∞ then Aρ is monotone, i.e. X ∈Aρ, Y ∈ L and Y ≥ X implies in Y ∈Aρ. In particular, ∞ L+ ⊆Aρ.

(ii) If ρI is composed of risk measures with Translation Invariance and f possesses this same

property, then ρ(X) = inf {m ∈ R: X + m ∈Aρ}.

(iii) If the conditions in items (i) and (ii) are fulfilled, then Aρ is non-empty, closed with respect ∞ to the supremum norm, Aρ ∩ {X ∈ L : X < 0} = ∅, and inf{m ∈ R: m ∈Aρ} > −∞.

(iv) If ρI is composed of risk measures with Convexity and f possesses this same property in

pair with Monotonicity, then Aρ is a convex set.

(v) If ρI is composed of risk measures with Positive Homogeneity and f possesses this same

property, then Aρ is a cone.

(vi) If ρI is composed of law invariant risk measures, then Aρ is law invariant in the sense of

X ∈Aρ and X ∼ Y imply Y ∈Aρ.

(vii) If ρI is composed of comonotone risk measures and f fulfills Additivity, then Aρ is stable for sums of comonotonic pairs of random variables.

(viii) If the conditions in items (i), (ii) and (iv) are fulfilled, ρI is composed of Fatou continuous

risk measures and f has Fatou continuity and Monotonicity, then Aρ is weak* closed.

25 Proof. The claims are directly obtained by noticing that they are implications from Theorems 2.3 and 2.4 together to Proposition 3.5.

i ∞ Corollary 5.2. Let ρI = {ρ : L → R, i ∈ I} be a collection of risk measures, f : X → R, ∞ and ρ: L → R a risk measure defined as ρ(X)= f(ρI(X)). Then:

(i) If ρI is composed of risk measures with Convexity and f possesses Monotonicity and

Quasi-convexity, then Aρ is a convex set.

(ii) If ρI is composed of risk measures with Monotonicity and Cash-subadditivity and f pos- sesses Monotonicity and Translation Invariance, then

ρ(X) ≥ inf {m ∈ R: X + m ∈Aρ} , if ρ(X) ≤ 0,

(ρ(X) ≤ inf {m ∈ R: X + m ∈Aρ} , if ρ(X) ≥ 0.

(iii) If ρI is composed of risk measures with Relevance and f has strict Monotonicity, then ∞ Aρ ∩ {X ∈ L− : P(X < 0) > 0} = ∅.

(iv) If ρI is composed of risk measures with Surplus Invariance and f has Monotonicity together W C − − ∞ to f ≥ f , then X ∈Aρ and Y ≤ X imply Y ∈Aρ, ∀ X,Y ∈ L .

Proof. Items (i), (iii) and (iv) are direct consequences from Theorem 2.3 and Proposition 3.7. For item (ii), Proposition 3.7 implies ρ has Cash-subadditivity. Note that it may be restated as ∞ ρ(X − C) ≤ ρ(X)+ C, ∀ C ∈ R+, ∀ X ∈ L or m → ρ(X + m)+ m be non-decreasing in R+ for any X ∈ L∞. With Monotonicity, Proposition 2.1 in Cerreia-Vioglio et al. (2011) assures ρ is Lipschitz continuous. Fix X ∈ L∞. If ρ(X) ≤ 0, then ρ(X + ρ(X)) = ρ(X − (−ρ(X))) ≤

ρ(X)−ρ(X) = 0. Thus, X +ρ(X) ∈Aρ and ρ(X) ≥ inf {m ∈ R: X + m ∈Aρ}. If ρ(X) ≥ 0, let k = inf {m ∈ R: X + m ∈Aρ}. Thus, k ≥ 0. For any m ∈ R with X +m ∈Aρ, we obtain ρ(X + k)+k ≤ ρ(X +m)+m ≤ m. Then, it is true that ρ(X +k)+k ≤ inf {m ∈ R: X + m ∈Aρ} = k. Thus, ρ(X + k) ≤ 0. Hence, k ≥ ρ(X + k)+ k ≥ ρ(X).

For particular cases, a characterization of the acceptance sets can be made explicit.

Example 5.3. We get the following examples for Af(ρI ):

W C (i) For f(R) = supi∈I R(i) we obtain f(ρI)= ρ . In this case we get

∞ i AρWC = {X ∈ L : ρ (X) ≤ 0 ∀ i ∈ I} = Aρi . i \∈I

µ (ii) For f(R)= I Rdµ we obtain f(ρI)= ρ . We then have that R ∞ i i Aρµ = X ∈ L : ρ (X)dµ ≤− ρ (X)dµ . i i ( Z{ρ (X)≤0} Z{ρ (X)>0} )

Note that from Assumption 2.8 both {ρi(X) ≤ 0} and {ρi(X) > 0} are in G for any X ∈ L∞.

26 φ 1 α (iii) We have that any spectral (distortion) risk measure ρ (X) = 0 V aR (X)φ(α)dα is a µ i i −1 special case of ρ by choosing ρ (X)= V aR (X) and µ ≪ λ withRφ(i)= F dµ (1−i). Since dλ α both α → V aR and α → φ(α) are non-increasing, we can pick αX ∈ [0, 1] dependent of ∞ α α X ∈ L such that V aR (X)φ(α) ≥ 0 for any α < αX and V aR (X)φ(α) ≤ 0 for any

α > αX . In this case we get

1 αX ∞ α α Aρφ = X ∈ L : V aR φ(α)dα ≤− V aR φ(α)dα .  ZαX Z0  From the properties of VaR in Example 2.5, Proposition 3.5 and Corollary 5.1 this set is norm closed, monotone, law invariant, a cone and stable for addition of comonotone pairs. If we also have that φ is non-increasing, then the acceptance set is convex and weak* closed.

Nonetheless, a direct general characterization of AρI is not so easy from the complexity that arises from the combination. On the next subsection we provide a general characterization for the case of convex risk measures.

5.2 General result

We now explore a more informative characterization for the acceptance sets of f(ρI) for the case of convex risk measures from section 4. In this sense, the next Theorem explores the role of Aρµ in such framework.

i ∞ Theorem 5.4. Let ρI = {ρ : L → R, i ∈ I} be a collection of Fatou continuous convex risk measures, f : X → R possessing Monotonicity, Translation Invariance, Convexity and Fatou ∞ continuity, and ρ: L → R defined as ρ(X)= f(ρI(X)). Then: (i) The acceptance set of ρ is given by

Aρ = {Aρµ − γf (µ)}. (5.1) µ \∈V

(ii) if in addition to initial hypotheses f fulfills Positive Homogeneity, then the acceptance set of ρ is given by

Aρ = Aρµ . (5.2) µ \∈Vf Proof. (i) We recall that the acceptance set of any Fatou continuous convex risk measure ρ can be obtained through its penalty term as

∞ min Aρ = X ∈ L : sup EQ[−X] − α (Q) ≤ 0 ( Q∈Q ) ∞  min = X ∈ L : EQ[−X] ≤ α (Q) ∀ Q ∈ Q .  Note that this is equivalent to

∞ Aρ = {X ∈ L : EQ[−X] ≤ α(Q) ∀ Q ∈ Q}

27 for any, not necessarily minimal, penalty term αρ that represents ρ. Thus, from Theorem 4.11 we obtain

∞ Aρ = X ∈ L : EQ[−X] ≤ inf {αρµ (Q)+ γf (µ)} ∀ Q ∈ Q µ∈V   ∞ = {X ∈ L : EQ[−X] ≤ αρµ (Q)+ γf (µ) ∀µ ∈V∀ Q ∈ Q} ∞ = {X ∈ L : EQ[−X] ≤ αρµ (Q)+ γf (µ) ∀ Q ∈ Q} µ \∈V ∞ µ = {X ∈ L : ρ (X) ≤ γf (µ)} = {Aρµ − γf (µ)}. µ µ \∈V \∈V

(ii) This is directly obtained from (i) since from Lemma 4.1 in this case we have γf (µ)=0 if

µ ∈Vf and γf (µ)= ∞ otherwise. Hence, we get

Aρ = {Aρµ − γf (µ)} = {Aρµ − γf (µ)} = Aρµ . µ µ µ \∈V \∈Vf \∈Vf

Remark 5.5. It becomes clear that the pivotal role player by ρµ for dual representations of section 4 is also present for acceptance sets. A financial interpretation is that in order for a position X be acceptable for the combination f(ρI ) it must be acceptable for all possible weighting schemes µ adjusted by a correction, represented by γf . Without such adjustment, the set would be too restrictive. In fact, for f with Positive Homogeneity we can reduce the restriction to weight schemes over Vf . Remark 5.6. The results in last Theorem are in consonance to the four cases of Theorem 4.11.

More precisely, if ρI is composed as coherent risk measures, then

∞ Aρ = X ∈ L : EQ[−X] ≤ inf {αρµ (Q)+ γf (µ)} ∀ Q ∈∪µ∈V cl(Qρµ ) . µ∈V  

Similar deductions as those for the general convex case lead to Aρ = µ∈V {Aρµ − γf (µ)}. Fur- thermore, when ρI is composed as coherent risk measures and f possessesT Positive Homogeneity we get

∞ µ Aρ = X ∈ L : EQ[−X] ≤ 0 ∀ Q ∈ clconv(∪µ∈Vf cl(Qρ )) ∞ µ = X ∈ L : EQ[−X] ≤ 0 ∀ Q ∈∪µ∈Vf cl(Qρ ) ∞ =  {X ∈ L : EQ[−X] ≤ 0 ∀ Q ∈ cl(Qρµ )} µ \∈Vf ∞ µ = {X ∈ L : ρ (X) ≤ 0} = Aρµ . µ µ \∈Vf \∈Vf Remark 5.7. It is worthy explore as examples from last Theorem the particular cases of ρµ W C µ and ρ . For ρ , note that f(R) = I Rdµ leads to γf assuming value 0 in µ and ∞ in W C V\{µ}. Thus, Vf = {µ} and we indeed have Aρµ = Aρν = Aρµ . Concerning to ρ , R ν∈Vf f R R γ WC i µ ( ) = sup leads to f = 0. Thus, we must have thatT Aρ = i∈I Aρ = µ∈V Aρ . In T T 28 W C i fact, it is straightforward that ρ (X) ≤ 0 if and only if I ρ (X)dµ ≤ 0 for any µ ∈V, which corroborates to the claim. R

6 Probability based risk measurement

6.1 Preliminaries

From now on we work on L∞ = L∞(Ω, F) the space of point-wise bounded random variables on (Ω, F) by replacing P-a.s. concepts from previous sections by their point-wise counterparts. ∞ ∞ ∞ Note that L ⊆ L (Q), ∀ Q ∈ P. We denote F = {FX,Q : X ∈ L , Q ∈ P}. In this section, uncertainty is linked to probabilities in the sense that I ⊆ P. We assume that each L∞(Q), Q ∈ I, is atom-less. Extreme choices for I are a singleton or the whole P. Other possible choices are closed balls around a reference probability measure based on distance, metric, divergence, or relation, as in Shapiro (2017), for instance. We do not pursue such details in order to conduct our approach in a more general fashion. We define risk measurement under the intuitive idea that we obtain the same functional from distinct probabilities that represent scenarios. These may have different interpretations, such as models, economic situations, heterogeneous beliefs etc.

Q Definition 6.1. A probability-based risk measurement is a family of risk measures ρI = {ρ : ∞ Q ρ ∞ ρ L → R, Q ∈ I} such that ρ (X)= R (FX,Q), ∀ X ∈ L , ∀ Q ∈I, where R : F → R is called risk functional.

Remark 6.2. Such risk functional approach is inspired in the two step risk measurement of Cont et al. (2010). This definition implies that each ρQ has Q-Law Invariance or is Q-based in Q Q ∞ the sense that if FX,Q = FY,Q, then ρ (X)= ρ (Y ), ∀X,Y ∈ L . In fact, we have the stronger Q1 Q2 ∞ Cross Law Invariance in the sense that if FX,Q1 = FY,Q2 , then ρ (X)= ρ (Y ), ∀ X,Y ∈ L ,

∀ Q1, Q2 ∈I. It is direct that if X ∈AρQ1 and FX,Q1 = FY,Q2 , then Y ∈AρQ2 . For more details on Cross Law Invariance, see Laeven and Stadje (2013).

Example 6.3. Cross Law Invariance is respected in all cases exposed in Example 2.5. We have indeed:

Q EL 1 −1 (i) Expected Loss (EL): EL (X)= R (FX,Q)= −EQ[X]= − 0 FX,Q(s)ds.

Q V aRα −1 R (ii) Value at Risk (VaR): V aRα (X)= R (FX,Q)= −FX,Q(α), α ∈ [0, 1].

Q ESα 1 α Q (iii) Expected Shortfall (ES): ESα (X) = R (FX,Q) = α 0 V aRs (X)ds, α ∈ (0, 1] and Q ES0 Q ES0 (X)= R (FX,Q)= V aR0 (X)= − ess infQ X. R

Q ML −1 (iv) Maximum loss (ML): ML (X)= R (FX,Q)= −ess infQ X = −FX,Q(0).

We could consider risk measurement composed by Q-based risk measures ρQ without a common link Rρ. However, we would be very close to the standard combination theory of previous sections, but without the modeling intuition. We then take an interest in risk measures which consider the whole set I in the sense that they are a combination of probability-based risk measurements as ρ(X)= f(ρI(X)), where f : X → R is a combination function. If FX,Q =

29 FY,Q, ∀ Q ∈I, then ρ(X)= f(ρI(X)) = f(ρI(Y )) = ρ(Y ). This kind of risk measure is called I-based functional in Wang and Ziegel (2018).

µ µ Remark 6.4. Let Q be defined as Q (A)= I Q(A)dµ, ∀ A ∈ F, µ ∈V. We do not necessarily µ have that Q ∈I. However, Assumption 4.4Rassures that the integral is well defined when each µ ρQ is convex. Under this framework one has a temptation to write ρQ (X)= ρµ(X). However, this in general is not the case. From proposition 5 of Acciaio and Svindland (2013), we have µ for finite I, that Rρ is concave in F when each ρQ is convex, i.e. ρQ (X) ≥ ρµ(X). There is equality if, and only if ρQ(X)= ELQ(X). Such result is a consequence of linearity for the map ∞ Q → EQ[X], ∀ X ∈ L and Corollary 2 of Acciaio and Svindland (2013), which claims EL is the only law invariant convex risk measure that is convex in F.

6.2 Representation

It is not possible to obtain a representation like those in Theorem 2.7 for combinations over a probability-based risk measurement since these risk measures are not determined by a single probability measure. However, it is nonetheless possible to adapt such representations. To that, we need a Lemma regarding measurability issues.

Q ∞ Lemma 6.5. Let ρI = {ρ : L → R, Q ∈ I} be a probability-based risk measurement composed of convex risk measures. Then the following maps are G-measurable:

Q ∞ (i) Q → V aRα (X), ∀ X ∈ L , ∀ α ∈ [0, 1].

Q ∞ (ii) Q → ESα (X), ∀ X ∈ L , ∀ α ∈ [0, 1].

Q Q ∞ Q (iii) Q → (0,1] ESα (X)dm , ∀ X ∈ L , ∀ α ∈ [0, 1] and for any {m ∈ M: Q ∈ I}. R Proof. (i) From Assumption 4.4, we have that Q → FX,Q(x) = Q(X ≤ x) is G-measurable ∀ X ∈ L∞, ∀ x ∈ R. We then have for a fixed pair (X, α) ∈ L∞ × [0, 1] that for any k ∈ R

Q −1 Q ∈I : V aRα (X) ≥ k = Q ∈I : FX,Q(α) ≤−k = {Q ∈I : FX,Q(−k) ≥ α} ∈ G. n o n o Q ∞ Thus, the maps Q → V aRα (X) are G-measurable for any X ∈ L and any α ∈ [0, 1].

(ii) We have for fixed pair (X, α) ∈ L∞ × [0, 1] that

α Q 1 Q ESα (X)= V aRs (X)ds = sup EQα [−Y ] , α 0 Q Z Y ∼X where −1 1s≤α F dQα (1 − s)= , ∀ s ∈ [0, 1]. dQ ,Q α It is straightforward to note that Qα is a probability measure absolutely continuous in

relation to Q. From Assumption 4.4 we have that Q → EQα [−Y ] is G-measurable for any Q Y ∼ X. By repeating the deduction in item (i) of Lemma 4.15, we conclude that the Q ∞ maps Q → ESα (X) are G-measurable for any X ∈ L and any α ∈ [0, 1].

30 (iii) Following similar steps as in item (ii), fix a pair (X, α) ∈ L∞ × [0, 1] and {mQ ∈ M: Q ∈ I}. We have for each Q ∈I that

Q Q ESα (X)dm = sup EQ′ [−Y ] , (0,1] Q Z Y ∼X

where −1 1 Q F ′ (1 − s)= dm (v), ∀ s ∈ [0, 1]. dQ ,Q v dQ Z(s,1] By replacing the argument in item (ii), which comes to be a special case of item (iii), with ′ α Q Q Q instead of Q we prove that the maps Q → (0,1] ESα (X)dm are G-measurable for ∞ Q any X ∈ L , any α ∈ [0, 1] and for any {m ∈ MR : Q ∈ I}.

µ µ In the following Proposition, we establish a direct representation between ρ and ESα, or µ µ even V aRα. This fact links ρ to spectral risk measures.

Q ∞ Proposition 6.6. Let ρI = {ρ : L → R, Q ∈ I} be a probability-based risk measurement composed of convex risk measures and ρµ : L∞ → R defined as in (2.7). Then:

(i) ρµ can be represented as:

µ µ ∞ ρ (X)= sup ESα(X)dm − βρµ (m) , ∀ X ∈ L , (6.1) m∈M (Z(0,1] )

with βρµ : M→ R+ ∪ {∞} defined as in (4.9).

(ii) If in addition ρQ fulfills, for every Q ∈I, Positive Homogeneity, then the representation is µ µ ∞ ρ (X)= sup ESα(X)dm, ∀ X ∈ L , (6.2) m∈cl(Mρµ ) Z(0,1]

where Mρµ is defined as in (4.10).

(iii) If ρQ also is, for every Q ∈I, comonotone, then the representation is

µ µ ∞ ρ (X)= ESα(X)dm, ∀ X ∈ L , (6.3) Z(0,1]

where m ∈ cl(Mρµ ).

Proof. Since L∞ ⊆ L∞(Q), ∀ Q ∈I, we have that each ρQ can be considered as the restriction of a functional over L∞(Q). Thus, from Theorems 2.6 and 2.7, ρQ can be represented as Q combinations of ESα over probabilities m ∈ M. The proof follows similar steps of Theorem 4.7 by noticing that

Q Q Q Q µ ESα (X)dm dµ = ESα (X)dµdm dµ = ESα(X)dm, ZI Z(0,1] ZI Z(0,1] ZI Z(0,1] Q where m = I m dµ ∈ M. R 31 Remark 6.7. The representation in item (iii) have a spectral analogous as

1 µ µ ρ (X)= V aRα(X)φ(α)dα, Z0 µ where φ is as in Example 2.12. This is directly obtained by the definition of ESα. Remark 6.8. The case of ρµ is a special one, instead of a rule. For instance, this link for ρW C is frustrated since

W C W C ρ (X) ≤ sup ESα (X)dm − βρWC (m) , m∈M (Z(0,1] )

min W C where βρWC (m)= inf β Q (m). Indeed, Propositions 3.3 and 3.5 shows that ρ is subadditive Q∈I ρ for comonotone random variables because the supremum in (2.6) is not attained in general. Under the probability-based risk measurement framework with some conditions on the set I, such as being tight, Bartl et al. (2019) provide situations whereby the supremum is attained. W C Moreover, Theorem 1 of Wang and Ziegel (2018) shows that V aRα possesses Comonotonic Additivity in the case of finite cardinality for I. In Theorems 3 and 4 of Wang and Ziegel (2018) a representation for I-based risk measures for the comonotone and coherent cases is exposed when I is a finite set and restrictions regarding assumptions over the probabilities Q ∈I. In the next Corollary, we expose a representation that suits for convex cases and general I, despite being limited to those generated by a combination.

Q ∞ Corollary 6.9. Let ρI = {ρ : L → R, Q ∈ I} be a probability-based risk measurement composed of convex risk measures, f : X → R possessing Monotonicity, Translation Invariance, ∞ Convexity and Fatou continuity, and ρ: L → R defined as ρ(X)= f(ρI(X)). Then:

(i) ρ can be represented as

µ ∞ ρ(X)= sup ESα(X)dm − βρ(m) , ∀ X ∈ L , (6.4) µ∈V,m∈M (Z(0,1] )

where βρ as in (4.13).

(ii) If in addition to the initial hypotheses f possesses Positive Homogeneity, then the penalty

term becomes βρ(m)= inf βρµ (m), where Vf is as in Remark 4.2. µ∈Vf

(iii) If in addition to the initial hypotheses ρi possess, for any i ∈ I, Positive Homogeneity,

then βρ(m)= ∞, ∀ m ∈ M\∪µ∈V cl(Mρµ ).

(iv) If in addition to the initial hypotheses we have the situations in (ii) and (iii), then the representation of ρ becomes

µ ∞ ρ(X)= sup ESα(X)dm, ∀ X ∈ L , (6.5) Vf (0,1] µ∈Vf ,m∈Mρ Z

Vf where Mρ is as (4.14).

32 (v) If in addition to the initial hypotheses ρi possess, for any i ∈I, Comonotonic Additivity, µ then βρ(m)= ∞, ∀ m ∈ M\∪µ∈V {mcc}, where

µ µ mcc = argmax ESα(X)dm, ∀ µ ∈V. m∈cl(Mρµ ) Z(0,1]

(vi) If in addition to the initial hypotheses we have (ii) and (v), then the representation of ρ becomes µ ∞ ρ(X)= sup ESα(X)dm, ∀ X ∈ L , (6.6) Vf (0,1] µ∈Vf ,m∈Mρ,cc Z

Vf µ where Mρ,cc is the closed convex hull of ∪µ∈Vf {mcc}.

Proof. The claims are directly obtained from Theorem 4.11, Corollary 4.18 and Proposition 6.6.

6.3 Stochastic orders

It is reasonable to consider risk measures that are suitable for decision making. This is typically addressed under monotonicity with respect to stochastic orders, see B¨auerle and M¨uller (2006) for instance. However, in the presence of uncertainty regarding choice of a probability measure, adaptations must be done.

Definition 6.10. We consider the following orders for X,Y ∈ L∞:

−1 −1 (i) X 1,Q Y (Q-based stochastic dominance of first order) if and only if FX,Q(α) ≥ FY,Q(α), ∀α ∈ [0, 1].

α −1 (ii) X 2,Q Y (Q-based stochastic dominance of second order) if and only if 0 FX,Q(s)ds ≥ α F −1 s ds, α , 0 Y,Q( ) ∀ ∈ [0 1]. R R −1 −1 (iii) X 1,I Y (I-based stochastic dominance of first order) if and only if FX,Q(α) ≥ FY,Q(α), ∀α ∈ [0, 1], ∀ Q ∈I.

α −1 (iv) X 2,I Y (I-based stochastic dominance of second order) if and only if 0 FX,Q(s)ds ≥ α F −1 s ds, α , , Q 0 Y,Q( ) ∀ ∈ [0 1] ∀ ∈I. R A riskR measure ρ: L∞ → R is said to respect any of the previous orders if X  Y implies ρ(X) ≤ ρ(Y ), ∀ X,Y ∈ L∞.

Q Remark 6.11. Of course, if ρ respects 1,Q or 2,Q for any Q ∈ I and f is monotone, then

ρ = f(ρI) respects 1,I or 2,I, respectively. When I = {P} and risk measures are functionals over L∞(Ω, F, P), the standard case, some well known results arise: P-Law Invariance and

Monotonicity are equivalent to respect regarding 1,P; and any P-law invariant convex risk measure respects 2,P. See the reference books cited in the introduction for more details. The facts in Remark 6.11 are not always true under general I, as we show in the next Proposition. Such results highlight the discussion that considering a framework robust to the choice of probability measures may lead to paradoxes regarding decision making based on risk

33 measures. Nonetheless, such situation is avoided when one considers risk measures generated by adequate composition of probability-based risk measurements. To that, we need the following auxiliary definition.

Definition 6.12. A risk measure ρ: L∞ → R has a (ES, I) representation if

µ ∞ ρ(X)= sup ESα(X)dm − β(m) , ∀ X ∈ L , µ∈V,m∈M (Z(0,1] ) where β : M→ R+ ∪ {∞} is a penalty term.

Remark 6.13. It is straightforward to note that any functional of this kind is a I-based Fatou continuous convex risk measure. From Proposition 6.6 and Corollary 6.9 we have that this is the case for risk measures composed by probability-based risk measurements with proper combinations f. It is also the case for I-based coherent risk measures for finite and mutually singular I, as in Theorem 4 of Wang and Ziegel (2018).

Proposition 6.14. Let ρ: L∞ → R be a risk measure. Then:

(i) If ρ respects 1,I, then ρ is I-based and monotone. The converse is true when I = P.

(ii) If ρ has a (ES, I) representation (and thus it is a I-based Fatou continuous convex risk

measure), then it respects 2,I. The converse is true under the additional hypotheses of ρ monetary and convex for comonotone pairs, plus I be a singleton.

∞ Proof. (i) Let FX,Q = FY,Q ∀ Q ∈ I for arbitrary X,Y ∈ L . Then by hypothesis we have

ρ(X) = ρ(Y ). Let X ≥ Y . Then X 1,I Y and hence ρ(X) ≤ ρ(Y ). For the converse,

note that when I = P we have that ρ is always I-based since FX,Q = FY,Q for any Q ∈ P

implies X = Y . Let X 1,I Y . Then X ≥ Y and thus Monotonicity implies ρ(X) ≤ ρ(Y ).

Q Q (ii) Let X 2,I Y . Thus ESα (X) ≤ ESα (Y ), ∀ α ∈ [0, 1], ∀ Q ∈ I. From monotonicity of integral and supremum, as well non negativity of penalty terms, we have that ρ(X) ≤ ρ(Y ) for any ρ with a (ES, I) representation. Regarding the converse claim, it is a known result from literature since we have the standard case for a singleton I, see Proposition 5.1 in F¨ollmer and Knispel (2013) for instance.

Remark 6.15. In (i), the difficulty in stating a converse claim is because X 1,I Y does not imply X ≥ Y point-wisely. Even when we assume that there is a reference measure P and we work on L∞(Ω, F, P) under P almost surely sense, the best we can get is to replace P by Q ⊆ P. Since not every I-based convex risk measure possess a (ES, I) representation, it is not possible to obtain the result in (ii) by only assuming Convexity, as occurs in standard case. The converse statement in (ii) needs to be restrictive due to the impossibility of existence for any pair ∞ ∞ X,Y ∈ L , a comonotone pair ZX ,ZY ∈ L such that for any Q ∈I we have FX,Q = FZX ,Q and FY,Q = FZY ,Q, which is essential to the proof when I is a singleton.

34 6.4 Elicitability and worst case risk measure

A recently highlighted statistical property is Elicitability, which enables the comparison of competing models in risk forecasting. See Ziegel (2016) and the references therein for more details. In this subsection we adapt it to our framework.

2 Definition 6.16. A map S : R → R+ is called scoring function if it has the following proper- ties:

(i) S(x,y) = 0 if and only if x = y;

(ii) y → S(x,y) is non-decreasing for y>x and non-increasing for y

(iii) S(x,y) is continuous in y, for any x ∈ R.

Q ∞ A probability-based risk measurement ρI = {ρ : L → R, Q ∈ I} is elicitable if exists a scoring 2 function S : R → R+ such that

Q ∞ ρ (X)= − arg min EQ [S(X,y)] , ∀ X ∈ L , ∀ Q ∈I. (6.7) y∈R

Remark 6.17. Elicitability can be restrictive because depending on the demanded financial properties at hand we may end up with only one example of risk functional which satisfies the requisites, see Theorem 4.9 of Bellini and Bignozzi (2015) and Theorem 1 in Kou and Peng (2016). For instance, EL and VaR are elicitable under scores (x − y)2 and α(x − y)+ + (1 − α)(x − y)−, respectively, while ES and ML are not. In our framework, when Q = Q′, ∀ Q ∈I, and f possesses Translation Invariance it is easy to observe that ρ = f(ρI) inherits elicitability from ρI since RX is a constant. However, we can express a non-elicitable risk measure in the case it is a worst case combination (f W C) of ELQ and the supremum over I is attained for any X ∈ L∞. This is very useful for coherent risk measures from the nature of their dual representations.

Q ∞ Proposition 6.18. Let ρI = {ρ : L → R, Q ∈ I} be a probability-based risk measurement 2 W C ∞ which is elicitable under the scoring function S : R → R+ and ρ : L → R defined as in ∞ (2.6) such that the supremum in I is attained in QX for each X ∈ L . Then we have that

W C ∞ ρ (X)= − arg min EQX [S(X,y)], ∀ X ∈ L . (6.8) y∈R

Proof. For any X ∈ L∞ we have that

W C QX ρ (X)= ρ (X)= − arg min EQX [S(X,y)]. y∈R

Q Q Remark 6.19. For instance, let I = Q P and ρ = EL , : ∀ Q ∈I. Thus we obtain ESα

W C P 2 ρ (X)= ESα(X)= − arg min EQX (X − y) , y∈R  

35 dQX where Q = argmax{E [−X]: Q ∈ Q P }, which we know to have relative density = X Q ESα dP 1 1 −1 . Similar reasoning can be developed for the concepts of Backtestability and Iden- α X≤FX,P(α) tifiability, as in Acerbi and Szekely (2017), but we do not pursue them in this paper.

7 Model uncertainty

7.1 Preliminaries

In this section we use our developed framework in order to quantify risk uncertainty. Our approach here is the one focused on variability of the risk measurement process instead of accuracy of some particular model. See Mller and Righi (2020) for a recent discussion in such distinctions. We now define such quantification as a functional over ρI .

Q ∞ Definition 7.1. Let ρI = {ρ : L → R, Q ∈ I} be a probability-based risk measurement and ∞ ∞ f : X∩ K → R+. Then a model uncertainty measure is a functional M : L → R+ defined as

M(X)= f(ρI(X)). (7.1)

Remark 7.2. It is straightforward to see that in this case combinations can be as f = f1 − f2, ∞ where f1,f2 : X ∩ K → R and f1 ≥ f2. We would then get M(X) = f1(ρI (X)) − f2(ρI (X)). For simplicity in order to have well definiteness we have restricted the domain of combinations to K∞. Adaptations for the whole X can be done.

Example 7.3. Fix m ∈V. We have the following examples of model uncertainty measures:

(i) A very aggressive example is the worst case one that is related to the range, explored in

Cont (2006) for instance, where we have f1(R) = supi∈I R(i) and f2(R) = infi∈I R(i). In this case we get f = f W C − f BC and

M W C(X) = sup ρi(X) − inf ρi(X)= ρW C (X) − ρBC (X), (7.2) i∈I i∈I

where BC stands for best case.

W C µ (ii) Variants of the worst case approach are half ranges with the pairs f1 = f and f2 = f , µ BC as in Kerkhof et al. (2010), or f1 = f and f2 = f , as in Breuer and Csisz´ar (2016). We thus get the upper and lower range model uncertainty measures as

M UR(X) = sup ρi(X) − ρi(X)dµ = ρW C (X) − ρµ(X), (7.3) i∈I ZI M LR(X)= ρi(X)dµ − inf ρi(X)= ρµ(X) − ρBC (X). (7.4) i∈I ZI (iii) Standardization of the combinations f from previous examples are plenty available on the literature. See Barrieu and Scandolo (2015), Bernard and Vanduffel (2015) and Danielsson et al. (2016) for the examples we present here. In this case we can get formulations for f as

f W C − f µ f µ − f BC f W C − f µ f µ − f BC , , , , f W C − f BC f W C − f BC f W C f W C

36 f W C − f µ f µ − f BC f W C − f µ f µ − f BC , , , . f BC f BC f µ f µ

(iv) Usual p-norms, p ∈ [1, ∞], can also be considered, as in Mller and Righi (2020), as well as their semi negative and semi positive counterparts. In this case we have f p(R) = 1 1 µ p p p µ − p p I |R − f (R)| dµ and the semi counterparts as f−(R) = I | (R − f (R)) | dµ 1 p µ + p p andR f+(R)= I | (
R − f (R)) | dµ . Note that they are allR well defined since we are
∞ dealing in X ∩RK . Thus, we get the
model uncertainty measures

1 p p µ p M (X)= |ρI(X) − ρ (X)| dµ , (7.5) I Z  1 p p µ − p M−(X)= (ρI (X) − ρ (X)) dµ , (7.6) I Z  1 p p µ − p M+(X)= (ρI (X) − ρ (X)) dµ . (7.7) ZI 

Remark 7.4. Note that in all such examples if I is a singleton, then M = 0, which reflects the absence model uncertainty. Also, it would be desirable that the value for M(X) be bigger as I gets larger. This occurs with all functionals in items (i), (ii) and (iii) in the last Example. Remark 7.5. In the previous examples, ρµ plays a role for benchmark. It is typical in the µ literature to have a dominant or reference measure Q0 ∈ I. In this case, we replace f by Q0 µ Q0 f (R)= R(Q0) and, consequently, ρ by ρ in the formulations. This avoids the uncertainty from choosing a specific µ ∈ V. However, on the other side we must to assume that there is a reference Q0. Remark 7.6. Intuitively, f plays the role for a dispersion. In fact, such concept is linked to the notion of deviation measures as proposed in Rockafellar et al. (2006). See also Grechuk et al. (2009), Rockafellar and Uryasev (2013) for more details. The crucial property to be added in relation the ones we have already studied for risk measures is Translation Insensitivity, i.e., M(X + C)= M(X), ∀ C ∈ R, ∀ X ∈ L∞.

Remark 7.7. The inheritance by M of properties from ρI and f are similar to the ones exposed Q ∞ in section 3. In addition, if ρI = {ρ : L → R, Q ∈ I} is composed as risk measures with ∞ Translation Invariance and f fulfills Translation Insensitivity. Then M : L → R+ also posses Translation Insensitivity. It is straightforward that Translation Insensitivity is the case when f = f1 − f2 with both f1 and f2 possessing Translation Invariance. Moreover, if f1 is convex and f2 concave, we have f = f1 − f2 convex. Furthermore, since f ≥ 0, there is no relevance for studying usual acceptance sets.

7.2 Proposed approach

We now provide some generalization for the quantification by proposing a new class of model uncertainty measures. Our goal in here is to introduce such concept, leaving numerical or empirical analysis for future studies.

37 ∞ Definition 7.8. Let u: X∩ K → R be as in (2.9) of Example 2.13. Also let g : R → R+ some Borel function. We assume both u and g are normalized to u(0) = g(0) = 0. Then we define f u,g(R)= u(g(R − f µ(R))). In this case we obtain the model uncertainty measure

u,g µ M (X)= u(g(ρI (X) − ρ (X))). (7.8)

Remark 7.9. By varying u and g we have determined a whole class. Note that most of measures in Example 7.3 lie in this class. In Jokhadze and Schmidt (2020) the special case of u being a spectral risk measure and g(x)= x is investigated. We now explore some properties for the proposed M u,g. We emphasize that is not our intention to discuss which set of properties such kind of measure must fulfill since this is an open gap for the literature. We then precisely focus on those that characterize deviations and dispersion. In what follows we are fixing a probability µ ∈V.

Proposition 7.10. Let f u,g : X ∩ K∞ → R be as in Definition 7.8. Then we have:

(i) f u,g fulfills Translation Insensitivity.

(ii) If u(R) > f µ(R) for non-constant R and g convex, then f u,g has Non-negativity, i.e. f u,g(R) = 0 for any constant R ∈X∩ K∞ and f u,g(R) > 0 for any non-constant R ∈ X ∩ K∞.

(iii) If both u and g fulfill any property among Convexity, Positive Homogeneity and Additivity, then also does f u,g.

(iv) If g(x) ≤ |x|, ∀ x ∈ R, then f u,g fulfills Upper Range Dominance, i.e. f u,g(R) ≤ sup R − ∞ I Rdµ for any R ∈X∩ K .

(v)R If g is continuous and u is Lebesgue continuous, then f u,g fulfills Lebesgue continuity.

Proof. (i) For any C ∈ R we have that (R + C) − I (R + C)dµ = R − I Rdµ is true for any ∞ R ∈X∩ K . R R (ii) That f u,g(C) = 0, ∀C ∈ R is a consequence of item (i). Let R ∈X∩K∞ be non-constant. Then

u g R − Rdµ > g R − Rdµ dµ ≥ g R − Rdµ dµ = 0.   ZI  ZI  ZI  ZI  ZI   The second step is due to the Jensen inequality.

(iii) Since u is monotone and convex, its composition with any convex function is also convex. For Positive Homogeneity and Additivity, the result is obvious.

(iv) Since u is monetary utility we have for any R ∈X∩ K∞ that

u g R − Rdµ ≤ u R − Rdµ ≤ u sup R − Rdµ = sup R − Rdµ.   ZI   ZI   ZI  ZI

38 ∞ ∞ (v) Let {Rn}⊂X∩ K be bounded and such that lim Rn = R ∈X∩ K . Then n→∞

u,g µ µ u,g f (R)= u g lim (Rn − f (Rn)) = u lim g (Rn − f (Rn)) = lim f (Rn). n→∞ n→∞ n→∞      We have use Dominated convergence in the first equality. The others follow from continuity of g and Lebesgue continuity of u, respectively.

Remark 7.11. Such properties characterize f u,g as a deviation measure on X ∩ K∞. This rein- forces the interpretation that we are looking for the variability or dispersion among “estimates” Q ρ ∞ {ρ (X)= R (FX,Q), Q ∈ I} for given X ∈ L . Of course, such reasoning can be extrapolated to other probabilistic functionalist beyond risk measures. From the properties of f u,g we directly derive those for M u,g. The nest Corollary exposes these results.

Q ∞ Corollary 7.12. Let ρI = {ρ : L → R, Q ∈ I} be a probability-based risk measurement and u,g ∞ M : L → R+ be defined as in (7.8). Then we have the following:

(i) If I is a singleton, then M u,g(X) = 0, ∀ X ∈ L∞.

u,g (ii) If ρI is composed as risk measures that posses Translation Invariance, then M possesses Translation insensitivity.

µ (iii) If ρI is composed as risk measures that posses Translation Invariance, u(R) >f (R) for non-constant R, g convex and injective, then M u,g(C) = 0, ∀ C ∈ R and M u,g(X) > 0 for any X ∈ L∞ with ρW C(X) − ρBC (X) > 0.

u,g (iv) If the members of ρI , u and g posses Positive Homogeneity, then also does M .

u,g (v) If ρI is composed as law invariant risk measures, then M possesses Law Invariance.

(vi) If ρI is composed as comonotone risk measures together to g and u be additive, then M possesses Comomonotonic additivity.

(vii) If g(x) ≤ |x|, ∀ x ∈ R, then M u,g(X) ≤ ρW C (X) − ρµ(X), ∀ X ∈ L∞.

(viii) If ρI is composed by point-wise bounded risk measures that posses any property among continuity from above, below, Fatou or Lebesgue, g is continuous and u is Lebesgue con- tinuous, then also does M u,g.

Proof. The claims are directly obtained from Propositions 3.5, 3.9 and 7.10.

Remark 7.13. The value for M u,g(X) is bigger as I gets larger for suitable choices of g and u. The property in item (ii) is closely related to Non-negativity for deviation measures, but in the usual for it must be valid for any non-constant X. The difference here is that even

a non-constant X can have a constant ρI (X). In item (vi), if in addition ρI is composed

as monetary risk measures and ρI (X) ≥ EQ[−X] for some Q ∈ P, then M is lower range u,g ∞ u,g dominated, i.e. M (X) ≤ EQ[X] − inf X, ∀ X ∈ L . This because in this case M (X) ≤

39 W C µ ρ (X) − ρ (X) ≤ − inf X + EQ[X]. This property plays a key role for dual representations of deviation measures. Regarding norm continuity, since f u,g is not monotone, we dot not have even Lipschitz continuity preservation, as in item (i) of Proposition 3.9. Remark 7.14. Since f u,g is not monotone, we can no longer guarantee that M u,g is convex, even when both u and g have this property. Thus, there is no reason to deal with dual representations and other convex related concepts. Nonetheless, there is still space for a discussion if model uncertainty measures must or must not be convex (or even quasi-convex) functionals. In this case, a financial position such as λX + (1 − λ)Y can in fact exhibit more model uncertainty than the individual counterparts due to the joint distribution of the pair X,Y .

References

Acciaio, B., Svindland, G., 2013. Are law-invariant risk functions concave on distributions? Dependence Modeling 1, 54–64.

Acerbi, C., 2002. Spectral measures of risk: A coherent representation of subjective risk aversion. Journal of Banking & Finance 26, 1505–1518.

Acerbi, C., Szekely, B., 2017. General properties of backtestable statistics. Working Paper .

Aliprantis, C.D., Border, K.C., 2006. Infinite Dimensional Analysis. 3 ed., Springer, Berlin.

Ang, M., Sun, J., Yao, Q., 2018. On the dual representation of coherent risk measures. Annals of Operations Research 262, 29–46.

Artzner, P., Delbaen, F., Eber, J., Heath, D., 1999. Coherent measures of risk. Mathematical Finance 9, 203–228.

Barrieu, P., El Karoui, N., 2005. Inf-convolution of risk measures and optimal risk transfer. Finance and Stochastics 9, 269–298.

Barrieu, P., Scandolo, G., 2015. Assessing financial model risk. European Journal of Operational Research 242, 546 – 556.

Bartl, D., Drapeau, S., Tangpi, L., 2019. Computational aspects of robust optimized certainty equivalents and option pricing. Mathematical Finance in press.

B¨auerle, N., M¨uller, A., 2006. Stochastic orders and risk measures: Consistency and bounds. Insurance: Mathematics and Economics 38, 132–148.

Bellini, F., Bignozzi, V., 2015. On elicitable risk measures. Quantitative Finance 15, 725–733.

Bellini, F., Laeven, R.J.A., Rosazza Gianin, E., 2018. Robust return risk measures. Mathematics and Financial Economics 12, 5–32.

Bernard, C., Vanduffel, S., 2015. A new approach to assessing model risk in high dimensions. Journal of Banking & Finance 58, 166 – 178.

40 Breuer, T., Csisz´ar, I., 2016. Measuring distribution model risk. Mathematical Finance 26, 395–411.

Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M., Montrucchio, L., 2011. Risk measures: Rationality and diversification. Mathematical Finance 21, 743–774.

Cont, R., 2006. Model uncertainty and its impact on the pricing of derivative instruments. Mathematical Finance 16, 519–547.

Cont, R., Deguest, R., Scandolo, G., 2010. Robustness and sensitivity analysis of risk measure- ment procedures. Quantitative Finance 10, 593–606.

Danielsson, J., James, K.R., Valenzuela, M., Zer, I., 2016. Model risk of risk models. Journal of Financial Stability 23, 79 – 91.

Delbaen, F., 2002. Coherent risk measures on general probability spaces, in: Sandmann, K., Sch¨onbucher, P.J. (Eds.), Advances in Finance and Stochastics: Essays in Honour of Dieter Sondermann. Springer Berlin Heidelberg, pp. 1–37.

Delbaen, F., 2012. Monetary Utility Functions. Lecture Notes: University of Osaka.

El Karoui, N., Ravanelli, C., 2009. Cash subadditive risk measures and iterest rate ambiguity. Mathematical Finance 19, 561–590.

Emmer, S., Kratz, M., Tasche, D., 2015. What is the best risk measure in practice? a comparison of standard measures. Journal of Risk 18, 31–60.

F¨ollmer, H., Knispel, T., 2013. Convex risk measures: Basic facts, law-invariance and beyond, asymptotics for large portfolios, in: MacLean, L., Ziemba, W. (Eds.), Handbook of the Fundamentals of Financial Decision Making. World Scientific, pp. 507–554.

F¨ollmer, H., Schied, A., 2002. Convex measures of risk and trading constraints. Finance and stochastics 6, 429–447.

F¨ollmer, H., Schied, A., 2016. Stochastic Finance: An Introduction in Discrete Time. 4 ed., de Gruyter.

Fritelli, M., Rosazza Gianin, E., 2005. Law invariant convex risk measures. Advances in mathematical economics 7, 33–46.

Frittelli, M., Maggis, M., 2018. Disentangling price, risk and model risk: V&r measures. Math- ematics and Financial Economics 12, 219–247.

Grechuk, B., Molyboha, A., Zabarankin, M., 2009. Maximum Entropy Principle with General Deviation Measures. Mathematics of Operations Research 34, 445–467.

Guo, S., Xu, H., 2019. Distributionally robust shortfall risk optimization model and its approx- imation. Mathematical Programming 174, 473498.

41 Jokhadze, V., Schmidt, W., 2020. Measuring model risk in financial risk management and pricing. International Journal of Theoretical and Applied Finance in press.

Jouini, E., Schachermayer, W., Touzi, N., 2006. Law invariant risk measures have the Fatou property. Advances in Mathematical Economics 9, 49–71.

Jouini, E., Schachermayer, W., Touzi, N., 2008. Optimal risk sharing for law invariant monetary utility functions. Mathematical Finance 18, 269–292.

Kellner, R., Rosch, D., 2016. Quantifying market risk with value-at-risk or expected shortfall? consequences for capital requirements and model risk. Journal of Economic Dynamics and Control 68, 45 – 63.

Kerkhof, J., Melenberg, B., Schumacher, H., 2010. Model risk and capital reserves. Journal of Banking & Finance 34, 267 – 279.

Kiesel, R., R¨uhlicke, R., Stahl, G., Zheng, J., 2016. The wasserstein metric and robustness in risk management. Risks 4, 32.

Koch-Medina, P., Munari, C., Siki´c,ˇ M., 2017. Diversification, protection of liability holders and regulatory arbitrage. Mathematics and Financial Economics 11, 63–83.

Kou, S., Peng, X., 2016. On the measurement of economic tail risk. Operations Research 64, 1056–1072.

Kou, S., Peng, X., Heyde, C., 2013. External risk measures and basel accords. Mathematics of Operations Research 38, 393417.

Kratschmer, V., Schied, A., Zahle, H., 2014. Comparative and qualitative robustness for law- invariant risk measures. Finance and Stochastics 18, 271–295.

Kusuoka, S., 2001. On law invariant coherent risk measures. Advances in mathematical eco- nomics 3, 158–168.

Laeven, R.J.A., Stadje, M., 2013. Entropy coherent and entropy convex measures of risk. Mathematics of Operations Research 38, 265–293.

Maggis, M., Meyer-Brandis, T., Svindland, G., 2018. Fatou closedness under model uncertainty. Positivity 22, 1325–1343.

Mller, F., Righi, M., 2020. Model risk measures: A review and new proposals on risk forecasting. Working Paper .

Pflug, G., R¨omisch, W., 2007. Modeling, Measuring and Managing Risk. 1 ed., World Scientific.

Qian, P.Y., Wang, Z.Z., Wen, Z.W., 2019. A composite risk measure framework for decision making under uncertainty. Journal of the Operations Research Society of China 7, 43–68.

Righi, M., 2019. A composition between risk and deviation measures. Annals of Operations Research 282, 299–313.

42 Righi, M., 2020. Inf-convolution and optimal risk sharing with arbitrary sets of risk measures. Working Paper .

Righi, M., Ceretta, P., 2016. Shortfall Deviation Risk: an alternative to risk measurement. Journal of Risk 19, 81–116.

Righi, M., M¨uller, F., Moresco, M., 2019. On a robust risk measurement approach for capital determination errors minimization. Working Paper .

Rockafellar, R., Uryasev, S., 2013. The fundamental risk quadrangle in risk management, optimization and statistical estimation. Surveys in Operations Research and Management Science 18, 33–53.

Rockafellar, R., Uryasev, S., Zabarankin, M., 2006. Generalized deviations in risk analysis. Finance and Stochastics 10, 51–74.

R¨uschendorf, L., 2013. Mathematical Risk Analysis. Springer.

Shapiro, A., 2017. Distributionally robust stochastic programming. SIAM Journal on Opti- mization 27, 2258–2275.

Svindland, G., 2010. Continuity properties of law-invariant (quasi-)convex risk functions on L∞. Mathematics and Financial Economics 3, 39–43.

Wang, R., Ziegel, J., 2018. Scenario-based risk evaluation. Working Paper .

Ziegel, J., 2016. Coherence and Elicitability. Mathematical Finance 26, 901–918.

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