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Robustness in the Optimization of Risk Measures

In this paper, we study issues of robustness in the context of Quantitative Risk Management. Depending on the underlying objectives, we develop a general methodology for determining whether a given risk measurement related optimization problem is robust. Motivated by practical issues from financial regulation, we give special attention to the two most widely used risk measures in the industry, Value-at-Risk (VaR) and Expected Shortfall (ES). We discover that for many simple representative optimization problems, VaR generally leads to non-robust optimizers whereas ES generally leads to robust ones. Our results thus shed light from a new angle on the ongoing discussion about the comparative advantages of VaR and ES in banking and insurance regulation. Our notion of robustness is conceptually different from the field of robust optimization, to which some interesting links are discovered.


Introduction
The main focus of this paper is the study of robustness properties of optimization procedures within Quantitative Risk Management (QRM). For this, we introduce a novel general framework, which at the same time is conceptually intuitive and mathematically sophisticated.
A key question concerns the influence of the choice of the underlying objective on the resulting robustness properties in risk optimization. In particular, we are interested in the two most popular regulatory risk measures, the Value-at-Risk (VaR) and the Expected Shortfall (ES), and their robustness properties in the context of risk optimization.
In QRM, the concept of robustness for risk measures is traditionally studied at the level of objective functionals without involving optimization problems; see Cont et al. (2010), Kou et al. (2013), Krätschmer et al. (2014Krätschmer et al. ( , 2017, , and the references therein. In the literature on robust optimization (see e.g. Ben-Tal et al. (2009)), model uncertainty is typically incorporated through modifying the objective functional or the constraints.
Our setup is different from the above perspectives; it builds on the simple observation that a risk measure as a standalone function may be robust, but fail to have desirable robustness properties when this measure is used within an optimization context. To briefly illustrate our ideas, suppose that a risk factor is represented by a random variable X arising from a stochastic model. An investor has to optimize her position according to the best of her knowledge, and hence we shall refer to X as the best-of-knowledge model, and the true model, denoted by Z, is typically unknowable. Ideally, a good model X is close to Z in a sense to be made clear later.
Based on the best-of-knowledge model X and an objective functional ρ, an optimized position is chosen as a function f (X) of X. Whereas the position f (X) may have a desirable objective value ρ(f (X)), this does not guarantee that ρ(f (Z)) is also desirable if Z is "slightly" different from X. In the absence of a perfect model, which almost always is the case in financial applications, this issue becomes crucially important. In the present paper, we put this observation into a rigorous quantitative framework.
The paper Cont et al. (2010) compares the qualitative robustness of VaR and coherent risk measures and concludes with the claim that VaR is better here. Some later papers, e.g.  and Krätschmer et al. (2014Krätschmer et al. ( , 2017, put the corresponding arguments into a different perspective, showing that ES also has certain desirable robustness properties. Both streams of research assumed that both VaR and, say, ES are applied to the same financial position. In reality, however, the regulatory choice of one risk measure creates certain incentives, just like any other aspect of regulation. These incentives become effective even before the risk measure has ever been applied in measuring the risk. That is, once a specific risk measure has been chosen, portfolios will be optimized with respect to that risk measure (at least to some extend). Thus, in reality, VaR and ES will never be applied to the same position. Therefore, one cannot decouple the technical properties of a risk measure from the incentives it creates. In our current paper, we make a first attempt of taking the effects created by the incentives into account and then arrive at a completely different conclusion than the previous literature.
The contribution and the structure of the paper are outlined below. In Section 2, we introduce the theoretical framework of robustness in optimization. The framework is quite general and it includes many practical problems in various fields of applications, not necessarily confined to finance and insurance. In Section 3, some preliminary results connecting robustness and continuity are presented. Keeping the problem of risk measures in mind, several representative and interesting optimization problems are listed in Section 4. As one of our contributions, we provide analytical solutions to these optimization problems in Sections 5 and 6 for VaR and ES, respectively. From our results, we shall see that, indeed, for the case of VaR which is argued by many as a robust risk measure, its corresponding optimization is highly non-robust and a small model uncertainty would ruin the optimality of the optimized positions, whereas for the case of ES the optimized positions are generally robust. In Section 7, we present some discussions on the implications of our results for the desirability of specific regulatory risk measures, an on-going debate in the financial industry (BCBS (2016), IAIS (2014)). Our results yield a (further) strong argument against using VaR as a risk measure within banking and insurance regulation; for a related discussion on robustness in the realm of risk sharing, see Embrechts et al. (2018). In the last section, Section 8, we discuss our notion of robustness in the context of distributionally robust optimization (e.g. Natarajan et al. (2008), Goh and Sim (2010)). The proofs of all results are put in the appendix.
2 Theoretical framework

Notation
We work with an atomless probability space (Ω, F , P). Let L q be the set of all random variables in (Ω, F , P) with finite q-th moment, q ∈ (0, ∞), L 0 be the space of all P-a.s. finite random variables, and L ∞ the set of all essentially bounded elements of L 0 . For a positive integer n, write L q n = (L q ) n . For a vector x ∈ R n , |x| is its Euclidean norm. Throughout, for any X ∈ L 0 , F X represents the distribution function of X. The mappings ess-inf(·) and ess-sup(·) on L 0 stand for the essential infimum and the essential supremum of a random variable, respectively. We write X d = Y if the random variables X and Y have the same distribution under P. For x ∈ R, denote by δ x the point-mass probability measure at x. Write R = (∞, ∞].

Optimization of risks without uncertainty
Let X be an n-dimensional random vector, where n is a positive integer. The random vector X is called an economic vector, which includes all random sources in an economic model under study, such as potential losses, traded securities, hedging instruments, insurance contracts, macro economic factors, or pricing densities.
Let G n be the set of measurable functions mapping R n to R. A random variable g(X) where g ∈ G n represents a risky position of an investor, in which positive values represent a loss and negative values represent a gain. The investor's problem is to choose among admissible positions g(X) for some functions g in an admissible set G ⊂ G n .
For a set G ⊂ G n , we formulate the problem to minimize: ρ(g(X)) subject to g ∈ G, where ρ is an objective functional mapping a set containing {g(X) : g ∈ G} to R. Here one prefers a smaller value of the objective functional over a larger value. Objective functionals considered may be general; examples include (up to a sign change) mean-variance functionals, expected utilities, rank-dependent utility functionals, functionals in cumulative prospect theory, and various risk measures as discussed in Artzner et al. (1999) and Föllmer and Schied (2016).
Our main interest will be, however, in the risk measures Value-at-Risk (VaR) and Expected Shortfall (ES 1 ).
The elements in the optimization problem (1) can be summarized by an objective functional ρ and a pair (X, G). We always assume that the domain of the objective functional ρ contains {g(X) : g ∈ G}, otherwise (1) is meaningless.
Example 1. An illustrative example is the classic problem of hedging in a financial market.
Suppose that an investor currently faces a risk W and would like to hedge against W . She has access to hedging instruments in a set {g(Y ) : g ∈ G ′ } where Y is an (n − 1)-dimensional economic vector and G ′ ⊂ G n−1 , n 2. Typically, the set G ′ involves a budget constraint.
Equivalently, she chooses risky positions in the set {W − g(Y ) : g ∈ G ′ }, which represents all possible hedged positions she may attain. Note that is an n-dimensional random vector and 1 A formal definition of VaR and ES is given in Section 4. ES is also known as CVaR, CTE, AVaR and TVaR, depending on the context (see e.g. Pflug and Werner (2007), McNeil et al. (2015) and Föllmer and Schied (2016)). We use the term ES to be consistent with the Basel Committee on Banking Supervision (BCBS (2016)), because our study is motivated by the comparative advantages of VaR and ES in regulation; see discussions in Section 7.
therefore the hedging problem is a special case of our general setting. Here we allow W to be arbitrarily dependent on Y : If W is a financial risk and Y is the vector of asset prices in a complete financial market, then it may be that W is a function of Y . On the other hand, if W represents a property and casualty insurance risk and Y is the vector of asset prices in a financial market, then it may be reasonable to assume that W and Y are independent.
In the discussion of this paper, we keep the choice of (X, G) as generic as possible. Some special cases and examples are studied in Sections 4-6.
Remark 1. One can also consider continuous-time models where optimizers are chosen over a set of stochastic processes (e.g. admissible trading strategies). Our discussions apply to such problems, as long as the optimizers are functions of the random source X, be it finite dimensional or infinite dimensional. In fact, in many classic financial models, the continuous-time optimization problem (such as the hedging example above) can be translated into a single-period optimization problem via the martingale approach; see Föllmer et al. (2009). For the sake of simplicity, we will focus in this paper on the case of optimization over functions of finite dimensional vectors.

Uncertainty and robustness in optimization
We proceed to put uncertainty into the optimization problems described above. For X ∈ L 0 n , G ⊂ G n and an objective functional ρ, denote by ρ(X; G) the minimum possible value of ρ, namely, ρ(X; G) = inf{ρ(g(X)) : g ∈ G}, and by G * (X, ρ) the set of optimizing functions, that is, G * (X, ρ) = {g ∈ G : ρ(g(X)) = ρ(X; G)}.
Note that G * (X, ρ) might be an empty set. For g ∈ G * (X, ρ), we refer to g(X) as an optimized position.
The optimization problem (1) is often subject to severe model uncertainty resulting from the stochastic assumptions made on X. Let Z be a set of possible economic vectors including X; Z may be interpreted as the set of alternative models 2 . Suppose that the real economic vector Z ∈ Z is different from the perceived economic vector X. The information we have at hand is about X rather than Z, and we shall refer to X as the best-of-knowledge model and Z as the true model, which is unknowable. We have to make decisions according to the best of our knowledge, that is, choosing g ∈ G * (X, ρ) optimizing our objective ρ. The real optimized position g(Z) may be different from the perceived optimized position g(X). If Z and X are close to each other according to some (pseudo-)metric π (e.g. L ∞ -metric on the space of bounded random vectors), we would like ρ(g(Z)) to be close to ρ(g(X)) in order to make sense of the optimized position g(Z), which may no longer be optimal. In other words, we naturally would desire some continuity of the mapping Z → ρ(g(Z)) at Z = X. Note that we are not interested in the optimizers for the problem of optimizing g(Z) over g ∈ G; it is indeed hopeless to make any decisions according to information that we do not have 3 .
Putting this into the context of the above hedging example, suppose that the real economic vector Z ∈ Z is different from X and an investor has the real risk f (Z) to hedge where f ∈ G n .
The information she has is about X, and she hedges from a set of instruments {g(Z) : g ∈ G ′ } where G ′ ⊂ G. In this case, f (Z) − g(Z) is the remaining risk she actually faces after hedging.
Under this setting, she naturally desires some continuity of the function Z → ρ(f (Z) − g(Z)) at Z = X.
We also note that, in our framework, the admissible set G is not subject to model uncertainty; the investor knows which positions she can choose in the optimization problem. For instance, in the above hedging example, a budget constraint that determines G is not affected by the model assumptions made for X; it is simply the observed prices for the hedging instruments.
In light of the above consideration, we endow the set Z of all possible economic vectors with a pseudo-metric π. Below we give three prominent examples of π, which will appear throughout the paper.
(i) For a subset Z of L ∞ n , the L ∞ -metric π ∞ n is defined as (ii) For q ∈ [1, ∞) and a subset Z of L q n , the L q -metric π q n is defined as (iii) For a subset Z of L 0 n , the pseudo-metric π W n is defined as where π P is the Prokhorov metric over the set of probability distribution measures 4 . In this case, convergence in π W n is equivalent to convergence in distribution (or weak convergence).
The reason of considering a pseudo-metric instead of a metric is to be able to incorporate cases such as (4) for objective functionals based on the distributions of risks, for instance, law-invariant risk measures.
We call (G, Z, π) an uncertainty triplet if G ⊂ G n and (Z, π) is a pseudo-metric space of n-random vectors. For a given uncertainty triplet, we say that an objective functional ρ is compatible if ρ maps G(Z) = {g(Z) : Z ∈ Z, g ∈ G} to R, and ρ(g(Y )) = ρ(g(Z)) for all g ∈ G and Y, Z ∈ Z with π(Y, Z) = 0, i.e. Y and Z are indistinguishable to the problem.
Definition 1. Let (G, Z, π) be an uncertainty triplet. A compatible objective functional ρ is robust at X ∈ Z relative to (G, Z, π) if there exists g ∈ G * (X, ρ) such that the function In this paper, the concept of robustness is always in the sense of Definition 1 and it should not be confused with the classic qualitative robustness as studied in e.g. Cont et al. (2010), Kou et al. (2013), Krätschmer et al. (2014) and . On the other hand, in contrast to the robust optimization literature (e.g. Goh and Sim (2010), Wiesemann et al.
(2014)), our focus is the robustness of objective functionals in optimization, instead of how to solve particular optimization problems. For this, we do not assume specific properties (e.g. convexity) of the objective functionals. As such, our setup and methodology are also different from classic ones in the optimization literature.
We make a few immediate observations about Definition 1. Robustness is a joint property of (ρ, X, G, Z, π), and only a π-neighbourhood of X in Z matters in the definition. If ρ is robust at X relative to (G, Z, π), then ρ is also robust at X relative to (G, Z ′ , π) if X ∈ Z ′ ⊂ Z, and the same holds true for (G, Z, π ′ ) if π ′ is a stronger pseudo-metric than π. On the other hand, if the optimization problem does not admit a solution, that is, G * (X, ρ) = ∅, then ρ is not robust at X.
Remark 2. Each component of the vector X may have a different economic meaning. Some of them may be subject to more severe model uncertainty whereas others may be free of model uncertainty. This can be reflected in the choice of Z which may be contained in a low-dimensional subset of the set of n-random vectors.
Remark 3. There are some alternative ways to formulate the notion of robustness in Definition 1. For instance, one may use topologies instead of metrics, or use uncertainty on the set of probability measures instead of that on the set of random vectors. By choosing appropriate metrics and topologies, these formulations become equivalent. To keep the paper accessible to risk management practice, we choose to work with metrics on the set of random vectors.

General relations between robustness and continuity
Clearly, the robustness of ρ relies on both some continuity of ρ on G(Z) and some continuity of functions in G * (X, ρ). In particular, if G * (X, ρ) = ∅, then ρ is not robust by definition. In what follows, we give a few general results where G * (X, ρ) contains a continuous function. However, later we will see that in many practical problems, G * (X, ρ) does not have any continuous elements for commonly used risk measures such as the Value-at-Risk and the Expected Shortfall; their robustness properties will be investigated in detail in Sections 5 and 6.
For a bijection g ∈ G n and a pseudo-metric space (Z, π) of n-dimensional random vectors, let (g(Z), π g ) be another pseudo-metric space defined as Proposition 1. Suppose that for an uncertainty triplet (G, Z, π), X ∈ Z and a compatible objective functional ρ, G * (X, ρ) contains a bijection g, and ρ is π g -continuous on g(Z). Then ρ is robust at X relative to (G, Z, π).
Next we look at the basic settings of Z = L ∞ n , Z = L q n , and Z = L 0 n , equipped with the L ∞ metric π ∞ n in (2), the L q metric π q n in (3), and the pseudo-metric π W n in (4), respectively.
Proposition 2. Let ρ be a compatible objective functional for the uncertainty triplet (G, Z, π) and X ∈ Z.
Proposition 2 provides simple criteria for verifying robustness of some objective functionals based on continuity of the optimizing functions in G * (X, ρ). As we shall see in Sections 4-6, for the popular risk measures VaR and ES, such criteria may not be very useful, as typically the optimizing functions lack the corresponding continuity. More detailed analyses are needed to draw meaningful conclusions for such objectives.
5 A function g : R n → R is linearly growing if for some C > 0, |g(y)| C|y| for all y ∈ R n with |y| > 1. This property is satisfied by, for instance, Lipschitz-continuous functions.
As mentioned before, the robustness in optimization is a property of a functional, depending on an uncertainty triplet. Therefore, in order to study the robustness of certain functionals, we need to specify the functional as well as the optimization problem and the corresponding uncertainty model.
Our main interest is in the risk measures Value-at-Risk (VaR) and Expected Shortfall (ES).
Here, a positive value of Y represents a loss and a negative value represents a gain. The VaR at confidence level p ∈ (0, 1) is defined as For any p ∈ (0, 1), we summarize some well known robustness properties of VaR p and ES p below.
1.) VaR p is continuous with respect to convergence in distribution (and hence, (Wasserstein) L q -convergence for q ∈ [1, ∞]) at X if and only if the inverse cdf of X is continuous at p; see, e.g., Proposition 7.3.1 in Shorack (2000).
2.) It follows immediately from 1.) and (5) that ES p is continuous with respect to convergence in distribution on every uniformly integrable subset of L 1 . In particular, ES p is continuous On the other hand, ES p is not continuous with respect to convergence in distribution on any set containing L ∞ + .
We concentrate on some simple optimization problems, which highlight the fundamental differences between the two risk measures above. Let n = 1 and X be a random loss. Suppose that random losses can be traded on a financial market with pricing density γ. By holding a risky position g(X), one receives the monetary amount E[γg(X)]. Let x 0 represent the budget of the investor, then the optimization problem is taken over the functions g satisfying the budget We consider the following three classic setups of optimization problems in a financial market.
Recall that our optimization problem is to minimize ρ(g(X)) subject to g ∈ G, for some G ⊂ G n .
(a) An optimal investment or hedging problem in the case of a complete market: The problem of optimal investing or hedging with a no short-selling or over-hedging constraint: In this problem, we assume 0 x 0 < E[γX] to avoid triviality.
(c) The problem of optimal investing or hedging with a bounded constraint: for some m > 0, In this problem, we assume 0 x 0 < m to avoid triviality.
Note that Problem (c) cannot be treated as a special case of Problem (b) as X in Problem (b) not only serves as the constraint but also as the source of randomness in the optimized position g(X).
We make the following assumptions on X, γ and our choice of the uncertainty (Z, π).
Assumption 1 (Assumptions on optimization problems). 1. X 0 and the distribution function of X is continuous and strictly increasing on (ess-infX, ess-supX).
The assumptions for X and γ are standard; for instance, they are satisfied by the classic Black-Scholes model of a single asset with terminal price X. Recall that X represents the source of randomness and hence γ is naturally a measurable function of X. We will use the notation γ for both the random variable γ(X) and the function γ : R → R. Assumption 1 will be imposed throughout the rest of the paper, and we do not specify X and (Z, π) unless necessary.

Optimization of Value-at-Risk
In this section we discuss the robustness properties of VaR p , p ∈ (0, 1), with respect to the three optimizations problems (a)-(c). From the results obtained, we shall see that VaR is generally not robust for these optimization problems. Recall that by the definition of robustness (Definition 1), in order to show such non-robustness, we have to identify all possible forms of the optimizing positions. We present the solutions to the optimization problems in Section 5.1 and then our conclusions on the robustness properties in Section 5.2.

Solutions to optimization problems
We first look at Problem (a) for for VaR p , that is, It turns out that a VaR is not robust for any choice of (Z, π) since the optimization problem (6) does not have a solution.
Next, we turn toward solving Problem (b) for VaR p , that is, where 0 . The optimization problem (7) is non-trivial to solve but fortunately it admits explicit solutions as we will see below. In what follows, let q = VaR p (X; G ns ) = inf VaR p (g(X)) : g ∈ G ns .
For this problem, an optimal solution always exists (see Lemma 1 in Appendix A.2). We summarize the solutions in the following proposition. Two cases are distinguished. If q = 0, the risk can be completely hedged under the criterion of VaR p . In this case, the solution of Problem (7) is generally not unique. This case is arguably less relevant, since there is no risk left according to VaR p after optimization. In the more interesting case q > 0, we identify the form of all possible solutions to Problem (7).
Before we proceed, we introduce the notion of a uniform tranform U of a random variable For an atomless probability space, the existence of the uniform transform of any Y is guaranteed. See e.g. Lemma A.32 of Föllmer and Schied (2016).
(ii) If q = 0, a solution of Problem (7) is given by (iii) If q > 0, any solution to Problem (7) has the form The random variable U in the preceding proposition is defined on the probability space (Ω, σ(X), P). It can therefore be represented as a function of X, and so the random variables on the right-hand sides of (8) and (9) are indeed functions of X. To represent these functions in a more explicit form, we require the following (very weak) continuity assumption, which holds in particular if (X − q)γ has a continuous distribution.
Under Assumption 2, we immediately obtain a.s.; we thus obtain the following corollary.
Corollary 1. Under Assumption 2, Problem (7) admits a P-a.s. unique solution that is of the where c = VaR p ((X − q)γ).
Remark 4. Using (10) and the budget condition E[γg * (X)] = x 0 , one can (numerically) solve for the value of q.
Finally, we turn to Problem (c) for VaR p , that is, where 0 x 0 < m. The optimization problem (11) is studied by He and Zhou (2011) and it admits a simple solution, which we briefly present below. In what follows, similarly to Section 5.2, let q ′ = VaR p (X; G bd ) = inf VaR p (g(X)) : g ∈ G bd .
Similarly to Problem (b), we present solutions of Problem (c) in the cases q ′ = 0 and q ′ > 0 separately.
Proposition 5. Let U be a uniform transform of γ on the probability space (Ω, σ(X), P). (ii) If q ′ = 0, a solution of Problem (11) is given by (iii) If q ′ > 0, any solution to Problem (11) has the form In order to establish the unique form of the optimizing functions for Problem (11), we make the following assumption which is analogous to Assumption 2.
Corollary 2. Under Assumption 3, Problem (11) admits a P-a.s. unique solution that is of the where c = VaR p (γ).

Robustness of VaR in optimization
Now, we have gathered enough ingredients to establish the (non-)robustness of VaR. In the following theorem, (Z, π) is any one of the choices specified in Assumption 1. The proof of this theorem involves some additional results which are given in Appendix A.2.
Theorem 1. For p ∈ (0, 1) and X ∈ Z, (i) VaR p is not robust relative to (G cm , Z, π); (ii) under Assumption 2, VaR p is not robust at X relative to (G ns , Z, π); (iii) under Assumption 3, VaR p is not robust at X relative to (G bd , Z, π).
Theorem 1 implies that, for all three settings of optimization problems and all choices of commonly used (pseudo-)metrics, VaR p is not robust, and this result holds for a general continuously distributed random variable X. As a consequence, VaR p has the poorest possible robustness in the context of optimization. This can already be seen from the form of its optimizing functions: the optimizing functions g * always have a jump at a p-quantile of g * (X), which makes it most vulnerable to model uncertainty.
In this section we discuss the robustness properties of ES p , p ∈ (0, 1), with respect to the three optimizations problems (a)-(c). From the results obtained, we shall see that ES is generally robust for these optimization problems, in sharp contrast to the case of VaR. Problems (a)-(c) are technically completely different for the two choices of risk measures. In order to show robustness of ES, we only need to specify a specific optimizing function and then show its corresponding continuity. Similarly to the organization of the previous section, we present the solutions to the optimization problems in Section 6.1 and then our conclusion on robustness in Section 6.2.
Recall again that Assumption 1 is imposed throughout this section.

Solutions to optimization problems
We first study Problem (a) for ES p , where 0 < p < 1, that is, Our first result states that Problem (a) either does not have a solution or has a trivial solution.
The following condition on the essential supremum γ turns out to be crucial for Problem (a).

Proposition 6. Problem (13) admits a solution if and only if Assumption 4 holds, and if
Assumption 4 holds, a solution is given by g * (·) = x 0 .
If Assumption 4 holds, the solution to Problem (13) is a constant function g * (·) = x 0 , which corresponds to perfect hedging of the original financial position X (e.g. through purchase of an insurance cover). Since the risk is perfectly hedged, there is no model uncertainty in the optimized position anymore.
Next, we turn toward solving Problem (b) for ES p , where 0 < p < 1, that is, where 0 x 0 < E[γX]. Based on Theorem 8.26 of Föllmer and Schied (2016), which is a slight generalization of a result by Sekine (2004), the form of an optimized position is obtained explicitly in the following proposition.
Finally, we turn to Problem (c) for ES p , where 0 < p < 1, that is, to minimize ES p (g(X)) subject to g ∈ G 1 : E[γg(X)] x 0 , 0 g(X) m, where 0 x 0 < m. The solution to this problem is similar to the case of Problem (14).

Robustness of ES in optimization
To establish the robustness of ES in optimization, we make the following regularity assumption, which is satisfied by essentially all practical models of financial markets with continuous model space.
Assumption 5. Either γ is a constant, or γ is a continuous function and γ(X) is continuously distributed.
Comparing Theorems 1 and 2, ES has advantages over VaR in terms of robustness for all the settings of Problems (a)-(c). In particular, the optimized positions for ES in the settings of Problems (b)-(c) are non-trivial but they have much better continuity than the corresponding optimized positions for VaR. Note that Assumptions 2, 3 and 5 impose some continuity properties on the underlying random variables. These are realistic for most practical models.
Remark 5. Generally, ES p is not robust at X relative to (G ns , Z, π W 1 ) or (G bd , Z, π W 1 ) for any set Z that is not uniformly integrable, and this is due to the fact that ES p is not continuous with respect to π W 1 , as mentioned in Section 4. On the other hand, if Z is uniformly integrable, then ES p is robust at X relative to (G ns , Z, π W 1 ) or (G bd , Z, π W 1 ) under Assumption 5, using the same proof as in Theorem 2. For recent academic discussions on various issues related to the desirability of VaR and ES in banking and insurance regulation, we refer to Kou and Peng (2016), Fissler and Ziegel (2016), Embrechts et al. (2018), Armstrong and Brigo (2018) and the references therein.
Our framework can certainly be applied to other settings such as other risk measures or expected utility/loss functions, and other settings of optimization problems. However, due to our main interest in the comparative properties of risk measures, we confine our present study to the case of VaR and ES, and several representative optimization problems. Certainly, there are many future directions to explore on robustness issues in optimization.

Is distributionally robust optimization robust?
We conclude this paper by discussing our notion of robustness in the context of distributionally robust optimization. The results in Sections 5-6 show that VaR is generally not robust for the optimization problems (a)-(c). In a classic setting of distributionally robust optimization (e.g. Quaranta and Zaffaroni (2008), Zhu and Fukushima (2009), Blanchet and Murthy (2018)), the objective functional itself is evaluated under the worst-case value over a set of possible mod-els representing uncertainty. By taking the worst-case value of the objective, model uncertainty is naturally incorporated into the optimization problem. One naturally wonders whether the robustness properties of risk measures would be improved with such an approach.
To formulate this consideration mathematically, let ρ be a compatible objective functional for an uncertainty triplet (G, Z, π) and X ∈ Z. We look at the following optimization problem, which is a robust version of (1), where ε > 0. Denote by G * (X, ρ, ε) the set of functions g ∈ G minimizing (18). Clearly, if we allow ε = 0 in (18), then G * (X, ρ, 0) = G * (X, ρ) and we are back in the setting of Section 2. In the problem (18), an investor is interested in the risk measure value ρ(g(Z)) of the risky position g(Z), in which Z is the unknowable true model. Therefore, similarly to Definition 1, we say that the objective functional ρ is robust for the setting (18) if there exists g ∈ G * (X, ρ, ε) such that the function Z → ρ(g(Z)) is π-continuous at Z = X.
Unfortunately, the minimax problem (18) is not easy to solve analytically, even for the representative settings in Section 4 and in the cases of VaR and ES. Typically, a linear programming approach has to be applied for such problems. As ES p is already shown to be generally robust in Section 6, we focus on the question of whether VaR p becomes more robust in this context.
In order to obtain analytic results, we look at a simple setting of Problem (c). Letting (G, Z, π) = (G bd , L ∞ , π ∞ 1 ), we can formulate the problem to minimize: sup where 0 x 0 < m. Similarly to Section 5, let In addition to Assumption 1 in Section 4, we make the following assumption, which can be seen as a stronger version of Assumption 3.
Fortunately, with Assumption 6, we are able to obtain an explicit form of the solution to Problem (19), allowing us to compare the corresponding robustness property with what we obtain in Section 5.
7 The monotonicity of γ as a function of X has a simple economic meaning. Recall that X represent the loss of an asset. Hence, Assumption 6 requires that the pricing density is larger when the asset has a larger loss. This requirement is satisfied by classic equilibrium models in the notion of Arrow-Debreu (Arrow and Debreu (1954)).

Proposition 9. Under Assumption 6, Problem (19) admits a solution of the form
With the solution g * in Proposition 9, the mapping Z → VaR p (g * (Z)) is π ∞ 1 -continuous at Z = X, implied by the continuity of VaR mentioned in Section 4. As a consequence, VaR p is robust for the setting (18). This observation is in sharp contrast with Theorem 1, where we see that VaR p is not robust relative to (G bd , L ∞ , π ∞ 1 ) under some weak assumptions. Therefore, at least for the above illustrative example, the modified optimization problem (19) improves the robustness of VaR. It is unclear how this result can be generalized to other optimization problems, as analytic results for (18) are rarely available.
Although VaR p becomes robust in the setting (19), its optimizing function takes a similar form as in Proposition 5, that is, a bi-atomic distribution which results in a big loss with small probability. Note that if ε approaches to zero, then the above solution g * converges to the solution in Proposition 5. This type of optimizing functions is highly undesirable and is subject to considerable model uncertainty if ε is small; see the discussions in Section 7.

A.1 Proofs in Section 3
Proof of Proposition 1. It suffices to show that the function Z → ρ(g(Z)) is π-continuous. By definition, for any X, Y ∈ Z, π g(Z) (g(X), g(Y )) = π(X, Y ). Thus the π g(Z) -continuity of ρ is equivalent to the π-continuity of the function Z → ρ(g(Z)).
Proof of Proposition 2. (i) It suffices to show that, as k → ∞, X k → X in π ∞ n implies that g(X k ) → g(X) in π ∞ 1 . This is a direct consequence of the Heine-Cantor Theorem (see Theorem 4.19 of Rudin (1976)).
(ii) X k → X w.r.t. π q n implies that {|X k | q } k∈N is uniformly integrable and that X k → X in probability. It follows from the Continuous Mapping Theorem that g(X k ) → g(X) in probability. Moreover, for sufficiently large c, Therefore, (|g(X k )| q ) is uniformly integrable and, in turn, g(X k ) → g(X) w.r.t. π q 1 .
(iii) It suffices to show that, as k → ∞, X k → X in π W n implies that g(X k ) → g(X) in π W 1 . This is a direct consequence of the Continuous Mapping Theorem.
Proof of Proposition 3. Note that we assumed that X is continuously distributed. Let y = VaR p (X). We have P(X and hence (6) does not have an optimizer.
We first verify that Problem (7) always has a solution in the following lemma. This lemma will be used later in the proofs of Proposition 4 and Theorem 1.
Proof of Lemma 1. Define Q through dQ/dP = γ and let µ = Q • X −1 . The set G ns is then a uniformly integrable subset of L 1 (µ). Let {g n } n∈N be a minimizing sequence for VaR p in G ns . By the Dunford-Pettis and Eberlein-Šmulian theorems (Theorems IV.8.9 and V.6.1 of Dunford and Schwartz (1958)), there exists a subsequence {g n k } k∈N that converges weakly in L 1 (µ) to some function g * ∈ L 1 (µ). Since G ns is convex and closed in L 1 (µ), we get g * ∈ G ns . Moreover, weak convergence in L 1 (µ) implies clearly that the laws of g n k (X) converge weakly to the one of g * (X). But VaR p is a left-hand quantile and hence lower semicontinuous with respect to weak convergence (see, e.g., Exercise A.6.1 in Föllmer and Schied (2016)). This proves that g * is optimal.
(ii) This has been seen in (i).
(iii) We give the complete proof in Lemma 2, where the desired result is stated as part (v).
Lemma 2. Let g * be a solution to Problem (7). Assuming q > 0, the following hold.
(ii) If g ∈ G ns has the form g(X) = X1 A + (X ∧ q)1 A c for some A ∈ F with P(A) = 1 − p, then g is a solution to Problem (7).
(iv) The solutions to the problem are A * = {U > p} a.s. where U is a uniform transform of (X − q)γ on the probability space (Ω, σ(X), P).
(iii) First, note that for any random variable Y , one can write Y = Y 1 A + Y 1 A c for some A ∈ F with P(A) = 1 − p and {Y > VaR p (Y )} ⊂ A. For instance, A can be chosen as From there, we can write g * (X) = g * (X)1 A + g * (X)1 A c , with P(A) = 1 − p and g * (X) VaR p (g * (X)) = q on A c . Since X is continuously distributed, A can be chosen as a set in σ(X). Let g(X) = X1 A + (X ∧ q)1 A c .
(iv) Note that we can rewrite the objective in (7) as Therefore, to solve Problem (7), it is equivalent to maximize E[(X − q) + γ1 A ] over A ∈ F with P(A) = 1 − p. Under Assumption 2, and by the Hardy-Littlewood inequality in the form of Embrechts and Wang (2015, Lemma 3.1), the solutions to this problem are Hence, g ∈ G ns , and using part (ii), we know that g is a solution to (7). Furthermore, using part (i), , by part (iv) again, A maximizes the objective in Problem (21).
Proof of Proposition 5. (i) This follows from the same proof of Proposition 4 (i).
(ii) It can be easily checked that VaR p (g * (X)) = 0 and g * ∈ G bd .
(iii) The proof is similar to that of Proposition 4 (iii).
To show Theorem 1, we need the following lemma.
Lemma 3. Let φ : R → R be a measurable function and a ∈ R such that P(φ(X) a) = p ∈ (0, 1). There exists a sequence {Z n } n∈N ⊂ Z such that the distribution measure of Z n is absolutely continuous with respect to that of X, Z n → X in π, and P(φ(Z n ) a) < p.
(i) Assume that there exists a uniform random variable U ε on [−ε, ε] and independent of X. Let By Assumption 1, X is has positive density over R, and X ε also has density over R. Hence, the distribution measure of X ε is absolutely continuous with respect to that of X. It is easy to see that |X ε − X| ε, and P(φ(X ε ) > a) = P(X ε ∈ B) P(X ∈ B) + P (X + U ε ∈ B, X ∈ B) > 1 − p.
(ii) If there does not exist a uniform random variable independent of X, we extend the original probability space to allow for such a random variable, and define X ε via (22). Then, we Xε is the quantile function of X ε . NowX ε is well defined in the probability space (Ω, F , P). By the facts that, || · || ∞ is monotone in convex order, andX ε − X is smaller than X ε − X in convex order (see e.g. Corollary 3.28 of Rüschendorf (2013)), we have In either cases (i) or (ii), taking Z n = X 1/n , n ∈ N, we complete the proof of the lemma for the metric π = π ∞ 1 , which is the strongest metric among choices specified in Assumption 1. Therefore, the lemma holds for all choices of (Z, π) in Assumption 1.
Proof of Theorem 1. (i) This statement follows directly from Proposition 3.
(ii) Let µ X be the distribution measure of X and c = VaR p ((X − q)γ). By Lemma 3, there exists a non-negative sequence of random variables in Z, Z n → X in π, such that Z n has a distribution measure absolutely continuous with respect to µ X , and P(γ(Z n )(Z n − q) > c) > 1 − p. Note that this implies VaR p (Z n 1 {γ(Zn)(Zn−q)>c} ) = VaR p (Z n ). For g * ∈ G ns (X, VaR p ), from Corollary 1 we know We have g * (Z n ) = Z n 1 {γ(Zn)(Zn−q)>c} + (Z n ∧ q)1 {γ(Zn)(Zn−q) c} a.s.

It follows that
Further, by definition of VaR p as a left-quantile, for any of the convergence types π we consider, where the last inequality follows from Lemma 1. Therefore, noting that VaR p (g * (X)) = q, lim n→∞ VaR p (g * (Z n )) > VaR p (g * (X)) so that the mapping Z → VaR p (g * (Z)) is not continuous at Z = X with respect to π ∞ 1 .
Noting that VaR p (g * (X)) = q < m, we have so that the mapping Z → VaR p (g * (Z)) is not continuous at Z = X with respect to π ∞ 1 .

A.3 Proofs in Section 5
Proof of Proposition 6. By the dual representation of ES p (e.g. Theorem 8.14 of McNeil et al. (2015)), we have If ess-supγ 1 1−p , then γ ∈ B p , and hence for any g ∈ G cm , ES p (g(X)) E[γg(X)] x 0 . Clearly, taking the constant function g * (·) = x 0 we have g * ∈ G cm and ES p (g * (X)) = x 0 . Therefore, g * is a solution to Problem (13).
Proof of Proposition 7. The result follows from Theorem 8.26 of Föllmer and Schied (2016). The fact that r is a p-quantile of g * (X) is stated in the proof of Theorem 8.26 in Föllmer and Schied (2016).
Proof of Proposition 8. Similarly to Proposition 7, the result follows from Theorem 8.26 of Föllmer and Schied (2016).
Proof of Theorem 2. (i) Since an optimal allocation is a constant function, it is not subject to any model uncertainty, that is, g(Z) = g(X) for all Z ∈ Z.
(ii) It suffices to show the statement for (Z, π) = (L 1 , π 1 1 ) since the statement for (Z, π) = (L q , π q 1 ) for q ∈ (1, ∞] is weaker. Let {Z n } n∈N ⊂ L 1 be a sequence of random variable such that Z n → X in L 1 . We will show g * (Z n ) → g * (X) in L 1 for some g * which optimizes (14).
Because 0 g * (Z n ) Z n , {g * (Z n )} n∈N is also uniformly integrable, and hence it converges to g * (X) in L 1 .
Finally, noting that ES p is continuous with respect to π 1 1 (see Section 4), we have g * (Z n ) → g * (X) as n → ∞. Thus, ES p is robust at X relative to (G ns , L 1 , π 1 1 ). As explained before, this also implies that ES p is robust at X relative to (G ns , L q , π q 1 ), q ∈ (1, ∞]. (iii) This follows by the same proof of (ii).

A.4 Proofs in Section 8
To prove Proposition 9, we need the following two lemmas. In what follows, we denote by A ε = {x ∈ R : |x − y| ε for some y ∈ A} for a set A ⊂ R and ε > 0.
Lemma 4. If A ⊂ R is either compact or an interval, then A ε ∈ B(R) and Proof. If A is a compact set or an interval, then so is A ε , which proves A ε ∈ B(R). Next, for any Y ∈ L ∞ with π ∞ 1 (Y, X) ε, the condition Y ∈ A implies X ∈ A ε a.s. Therefore, P(Y ∈ A) P(X ∈ A ε ), leading to sup π ∞ 1 (Y,X) ε P(Y ∈ A) P(X ∈ A ε ). To show the opposite direction of the inequality, it suffices to take Y = f A (X)1 {X∈Aε} + X1 {X ∈Aε} , where, for a compact set A, f A (x) is a nearest point of x in A (to be precise, there can be two such nearest points; by taking f A (x) to be the lower of the two, f A becomes lower semicontinuous and, hence, measurable). In the case in which A is a nondegenerate interval, we fix a point a in the interior of the interval and let otherwise.
In both cases, |Y − X| ε, and P(Y ∈ A) = P(X ∈ A ε ), leading to the desired result.
We conclude that, in both cases, P(X ∈ A * ) P(X ∈ A).
As a consequence, it suffices to search for optimizers h ∈ G bd of the form (26). Moreover, for such an h, we have E[γh(X)] = mQ(g(X) > a) + aQ(g(X) a), where Q is given by dQ/dP = γ.
Due to the inner regularity of the law Q • X −1 , we can find, for any a ′ > a, a compact set Since P(Y ∈ K) P(g(Y ) > a) 1 − p for all Y ∈ L ∞ with π ∞ 1 (Y, X) ε, we have sup π ∞ 1 (Y,X) ε VaR p (h ′ (Y )) = a ′ .
Let us denote by K the class of all compact set K ⊂ R satisfying P(Y ∈ K) 1 − p for all Y ∈ L ∞ with π ∞ 1 (Y, X) ε. The above argument shows that K is not empty. Define a function where a K ∈ R is such that E[γh K (X)] = x 0 . The existence of a K is guaranteed by P(X ∈ K c ) p > 0. Note that 0 < a K < m since x 0 < m and q > 0.
The preceding argument shows that it is sufficient to construct a function h * such that E[γh * (X)] = x 0 and sup π ∞ 1 (Y,X) ε for all K ∈ K . We define h * by where a * m is such that E[γh * (X)] = x 0 . Now let K ∈ K be given and h K of the form (27).