Rama Cont

Romain Deguest

Xue Dong He

Loss-based risk measures

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A Lebesgue Continuity

A main focus of quantitative modeling in finance has been to measure the risk of financial portfolios. In connection with the widespread use of Value-at-Risk and related risk measurement methodologies, a considerable theoretical literature [START_REF] Acerbi | Spectral measures of risk: a coherent representation of subjective risk aversion[END_REF][START_REF] Acerbi | Coherent measures of risk in everyday market practice[END_REF]; [START_REF] Artzner | Coherent Measures of Risk[END_REF]; [START_REF] Cont | Robustness and sensitivity analysis of risk meausurement procedures[END_REF]; [START_REF] Föllmer | Convex measures of risk and trading constraints[END_REF]; [START_REF] Föllmer | Stochastic Finance: An Introduction in Discrete Time[END_REF]; [START_REF] Frittelli | Putting order in risk measures[END_REF]; [START_REF] Heyde | External Risk Measures and Basel Accords[END_REF]; [START_REF] Mcneil | Quantitative risk management: concepts, techniques, and tools[END_REF] has focused on the design of appropriate risk measures for financial portfolios. In this approach, a risk measure is represented as a map ρ : L ∞ → R assigning to each (bounded) random variable X ∈ L ∞ (Ω, F , P)-representing the gain of a portfolio-a number which measures the risk of this portfolio. A framework often used as a starting point is the axiomatic setting of [START_REF] Artzner | Coherent Measures of Risk[END_REF], which defines a coherent risk measure as a map ρ : L ∞ → R that is 1. Monotone (decreasing): ρ(X) ≤ ρ(Y ) provided X ≥ Y .

2. Cash-additive (additive with respect to cash reserves): ρ(X + c) = ρ(X) -c for any c ∈ R.

3.

Positive homogeneous: ρ(λX) = λρ(X) for any λ ≥ 0.

4. Sub-additive: ρ(X + Y ) ≤ ρ(X) + ρ(Y ). [START_REF] Artzner | Coherent Measures of Risk[END_REF] argue that these axioms correspond to desirable properties of a risk measure, such as the reduction of risk under diversification. However, some of these axioms have unintended consequences and have been revised in various ways. Replacing positive homogeneity property by the more general convexity property leads to the class of convex risk measures [START_REF] Föllmer | Convex measures of risk and trading constraints[END_REF][START_REF] Föllmer | Stochastic Finance: An Introduction in Discrete Time[END_REF][START_REF] Frittelli | Putting order in risk measures[END_REF]; replacing subadditivity by co-monotonic sub-additivity [START_REF] Heyde | External Risk Measures and Basel Accords[END_REF][START_REF] Song | Risk measures with comonotonic subadditivity or convexity and respecting stochastic orders[END_REF] allows to restore the robustness of risk estimators [START_REF] Cont | Robustness and sensitivity analysis of risk meausurement procedures[END_REF].

Even more basic issues arise when considering the above axioms as the sole characterization of a "risk measure". Intuitively, the risk of a portfolio is associated with the magnitude of its losses, not its gains. In the case of a statistical risk measure, this means that the risk measure should be defined in terms of the left tail of the portfolio gain (distribution). Indeed, the most popular risk measures-Value at Risk and expected shortfall-are based on the left tail of the loss distribution. However, this natural property is not implied by any of the axioms of coherent risk measures as outlined in [START_REF] Artzner | Coherent Measures of Risk[END_REF].

If one further distinguishes, as in [START_REF] Heyde | External Risk Measures and Basel Accords[END_REF], 'external' risk measures used for determining capital requirements from 'internal' risk measures used for a firm's risk management -determining position limits, allocation of capital, etc.-then the seemingly innocent axiom of cash-additivity-that the risk of a portfolio decreases by c if combined with a cash position of size c-also comes under question. This axiom makes sense if ρ(X) is interpreted as a capital requirement for portfolio X, i.e., for external risk measures. But if ρ(X) is used to gauge the risk of the portfolio X, this axiom implies that if one combines a risky position X with a cash position c = ρ(X) then the combined position X + c has a "zero risk": ρ(X + c) = 0, which is meaningless in a risk management context. Also, as noted already by [START_REF] Jarrow | Put option premiums and Coherent Risk Measures[END_REF], the axiom of cash-additivity excludes some natural examples of risk measures, such as the value of the put option on a portfolio's net value [START_REF] Jarrow | Put option premiums and Coherent Risk Measures[END_REF].

El [START_REF] Karoui | Cash subadditive risk measures and interest rate ambiguity[END_REF] challenge the axiom of cash additivity from a different angle. They show that, when considering risk measures defined on future (instead of discounted) value of portfolio gains in order to take into account interest rate risk , cash additivity is inevitably violated and needs to be replaced by cash subadditivity.

Another issue, very important in practice but somewhat neglected in the literature on risk measures, is the issue of statistical estimation of risk, and the design of robust estimators for risk measures: lack of robustness of a risk estimation procedure makes it difficult, if not impossible, to use in practice. Unfortunately, as shown in [START_REF] Cont | Robustness and sensitivity analysis of risk meausurement procedures[END_REF], another unintended consequence of the axioms of coherent risk measures is that lead to non-robust risk estimators: the subadditivity property requires dependence on extreme tail events, which leads to high sensitivity to outliers [START_REF] Cont | Robustness and sensitivity analysis of risk meausurement procedures[END_REF].

We consider in this paper an alternative approach to defining risk measures which addresses these concerns. Starting from the requirement that risk measures of financial portfolios should be based on their losses, not their gains, we define the notion of loss-based risk measure and study the properties of this class of risk measures.

We claim that our theoretical framework is a more natural one for the purpose of risk measurement and management of financial portfolios, and englobes various natural examples of risk measures used in practice. Examples of such loss-based risk measures have been considered by [START_REF] Jarrow | Put option premiums and Coherent Risk Measures[END_REF], who studied the put option premium as a risk measure, and [START_REF] Staum | Excess invariance and shortfall risk measures[END_REF], in the context of a finite dimensional probability space. Our analysis extends these results to a more general setting: we use a slightly different set of axioms and obtain a characterization of these risk measures on a general probability space.

Section 1 defines loss-based risk measures and provides a representation theorem in terms of a penalty function (Theorem 1), which adapts the results of [START_REF] Föllmer | Stochastic Finance: An Introduction in Discrete Time[END_REF] to our setting. This representation result is made more explicit in the case of distribution-based risk measures (Theorem 2) and examples are given in Section 1.5. We then study the statistical robustness, in the sense of [START_REF] Cont | Robustness and sensitivity analysis of risk meausurement procedures[END_REF], of estimators for loss-based risk measures. In the case of statistical (i.e. distribution-based) risk measures we present a general criterion for qualitative robustness of such risk estimators (Section 2). These results extend results of [START_REF] Cont | Robustness and sensitivity analysis of risk meausurement procedures[END_REF] to loss-based risk measures. In Section 2.3 we provide statistically robust versions of these risk estimators. The theoretical results on the qualitative robustness of these estimators are confirmed by a quantitative sensitivity analysis based on influence functions (Section 3).

1 Loss-Based Risk Measures 1.1 Notations be the space of bounded random variables.

Let

P(Ω, F , P) := {X ∈ L 1 (Ω, F , P) : X ≥ 0, ||X|| 1 = 1},
and

M(Ω, F , P) := {X ∈ L 1 (Ω, F , P) : X ≥ 0, ||X|| 1 ≤ 1}.
Then P(Ω, F , P) can also be regarded as the set of P-absolutely continuous probability measures on (Ω, F , P), and M(Ω, F , P) as the set of P-absolutely continuous measures µ such that µ(Ω) ≤ 1.

Denote by P((0, 1)) the set of probability measures on the open unit interval (0, 1), and M((0, 1)) the set of positive measures µ on (0, 1) such that µ((0, 1)) ≤ 1. Let Ψ((0, 1)) : = φ : (0, 1) → R + | φ(•) is decreasing on (0,1) and

1 0 φ(z)dz ≤ 1 , (1) Φ((0, 1)) : = φ : (0, 1) → R + | φ(•) is decreasing on (0,1) and 1 0 φ(z)dz = 1 , (2)
both of which can be identified as subsets of M((0, 1)). Finally, for any random variables X, let X ∧ 0 := min(X, 0).

Definition

Consider a map ρ : L ∞ (Ω, F , P) → R which associates to each random variable X, representing the future payoff of a portfolio, a number ρ(X) representing its risk. [START_REF] Artzner | Coherent Measures of Risk[END_REF] have argued that, in order for ρ to qualify as a monetary risk measure, it has to satisfy at least the two following properties:

(i) Cash additivity: adding cash to a portfolio reduces the risk by the same amount.

∀X ∈ L ∞ (Ω, F , P), ∀α ∈ R, ρ(α + X) = ρ(X) -α
(ii) Monotonicity: if a portfolio X has higher payoff than a portfolio Y in all scenarios, then it has lower risk.

∀X, Y ∈ L ∞ (Ω, F , P), X ≤ Y ⇒ ρ(X) ≥ ρ(Y ).
The cash-additivity property (i) is necessary to interpret ρ(X) in terms of capital requirements for X: the capital requirement for a portfolio X may be satisfied by combining it with a position in cash equal to ρ(X) since ρ(X + ρ(X)) = ρ(X)ρ(X) = 0. However if one uses ρ(X) as a measure of risk, one might be tempted to think that the position X + c has "no residual risk", which is counterintuitive and, indeed, nonsensical if taken literally. This observation suggests that the cashinvariance property may not be a reasonable requirement for a good risk measure. Also, as noted by [START_REF] Jarrow | Put option premiums and Coherent Risk Measures[END_REF], the cash-additivity requirement excludes a natural risk measure, the premium of a put on the loss:

ρ(X) = E[max(-X, 0)]
Moreover, the initial motivation of risk measures is to quantify the risk associated with portfolio's losses. Thus, a risk measure should focus on the loss of a portfolio, and two portfolios which have the same losses should lead to the same risk measure.

With these concerns in mind, we define a class of risk measures, which we call loss-based risk measures, as follows.

Definition 1 (Loss-based risk measures) A mapping ρ :

L ∞ (Ω, F , P) → R + is called a loss-based risk measure if it satisfies (a) Cash Loss: for any α ∈ R + , ρ(-α) = α; (b) Monotonicity: for any X, Y ∈ L ∞ (Ω, F , P), if X ≤ Y, then ρ(X) ≥ ρ(Y ).
(c) Convexity: for any X, Y ∈ L ∞ (Ω, F , P) and 0 < α < 1,

ρ(αX + (1 -α)Y ) ≤ αρ(X) + (1 -α)ρ(Y ).
(d) Loss-dependency: the risk of a portfolio only depends on its losses, i.e., ∀X ∈ L ∞ (Ω, F , P), ρ(X) = ρ(X ∧ 0)

. Property (a) is a "normalization" property which states that the risk of a (nonrandom) cash liability is its face value. Properties (b) and (c) have well-known interpretations in terms of increasing loss diversification, see e.g. [START_REF] Föllmer | Convex measures of risk and trading constraints[END_REF]. Property (d) entails that portfolios with the same losses have the same risk. This implies that in fact the monotonicity property (b) may be replaced by the weaker

X ∧ 0 ≤ Y ∧ 0 ⇒ ρ(X) ≥ ρ(Y ).
Note that property (d) prevents the risk measure from satisfying the cash-additivity property (i): a loss-based risk measure is not "coherent" in the sense of [START_REF] Artzner | Coherent Measures of Risk[END_REF].

Loss-based risk measures are related to the class of cash subadditive risk measures defined by El [START_REF] Karoui | Cash subadditive risk measures and interest rate ambiguity[END_REF]. Indeed, properties (a) and (d), together with a weak semi-continuity condition, imply the cash subadditivity property defined in El Karoui and Ravanelli (2009, Def 3.1). To see this, let us first assume the following semi-continuity condition: for any X n → X in L ∞ (Ω, F , P), ρ(X) ≤ liminf n→∞ ρ(X n ). This condition is weaker than the Fatou property commonly assumed in the risk measure literature. Now, consider any X ∈ L ∞ (Ω, F , P) and α ≥ 0. For any ǫ ∈ (0, 1), we have

ρ((1 -ǫ)X -α) = ρ((1 -ǫ)X + ǫ(- α ǫ )) ≤ (1 -ǫ)ρ(X) + ǫρ(- α ǫ )) = (1 -ǫ)ρ(X) + α,
where the inequality is due to the convexity property (c) and the last equality is due to the Cash Loss property (a). Letting ǫ ↓ 0 and using semi-continuity, we have

ρ(X -α) ≤ ρ(X) + α, ∀X ∈ L ∞ (Ω, F , P), α ≥ 0,
which also implies

ρ(X + α) ≥ ρ(X) -α, ∀X ∈ L ∞ (Ω, F , P), α ≥ 0.
Thus, loss-based risk measures are cash-subadditive. On the other hand, it is easy to see that cash subadditivity in the sense of El [START_REF] Karoui | Cash subadditive risk measures and interest rate ambiguity[END_REF] does not imply the Cash Loss property (a). However, our focus in this study is not only on cash-subadditivity: the Lossdependency property (d) is at least as important as replacing cash additivity by cash subadditivity in a risk management context. Thus, loss-based risk measures deserve to be highlighted and treated separately although they are special cases of cash subadditive risk measures El [START_REF] Karoui | Cash subadditive risk measures and interest rate ambiguity[END_REF].

The following useful lemma shows that that Convexity (c) and Monotonicity (b) together imply L ∞ (Ω, F , P) continuity for a risk measure.

Lemma 1 Any map ρ : L ∞ (Ω, F , P) → R which is monotone and convex is continuous on L ∞ (Ω, F , P).

Proof Consider a sequence (X n ) n≥1 that converges to X in L ∞ (Ω, F , P). Then the sequence ε n := ||X n -X|| ∞ , n ≥ 1, converges to zero and we can assume without loss of generality that

ε n ≤ 1, n ≥ 1. Let α n := √ ε n , β n := εn αn + ||X|| ∞ + 1, then (1 -α n )X n -α n β n = X n -ε n -α n (||X|| ∞ + 1 + X n ) ≤ X where the inequality is because X n -X ≤ ||X n -X|| ∞ and ||X|| ∞ + 1 + X n ≥ 0.
By the monotonicity and convexity of ρ, we derive

ρ(X) ≤ ρ((1 -α n )X n -α n β n ) ≤ (1 -α n )ρ(X n ) + α n ρ(-β n ) ≤ (1 -α n )ρ(X n ) + α n ρ(-(||X|| ∞ + 2)), which leads to ρ(X n ) ≥ ρ(X) -α n ρ(-(||X|| ∞ + 2)) 1 -α n .
Letting n → ∞ and using the fact that α n converges to zero, we immediately have lim inf n→∞ ρ(X n ) ≥ ρ(X), i.e., ρ is lower semi-continuous in L ∞ (Ω, F , P). A similar argument leads to the upper semi-continuity of ρ in L ∞ (Ω, F , P).

Representation Theorem

We give now a characterization of loss-based risk measures that have a semicontinuity property, the Fatou property. A (loss-based) risk measure ρ satisfies the Fatou property if, for any sequence (X n ) n≥1 uniformly bounded in L ∞ (Ω, F , P) such that X n → X almost surely, we have

ρ(X) ≤ lim inf n→∞ ρ(X n ).
(3)

Theorem 1 The following are equivalent 1. ρ is a loss-based risk measure satisfying Fatou property (3).

2. There exists a convex function

V : M(Ω, F , P) → [0, ∞] satisfying inf Y 1 ≥1-ǫ V (Y ) = 0 for any ǫ ∈ (0, 1), (4) such that ρ(X) = - inf Y ∈M(Ω,F ,P) {E[(X ∧ 0)Y ] + V (Y )}, ∀X ∈ L ∞ (Ω, F , P). (5)
Proof Assume ρ is a loss-based risk measure satisfying the Fatou property. First, following Theorem 2.1 in [START_REF] Delbaen | A general version of the fundamental theorem of asset pricing[END_REF] or Theorem 3.2 in [START_REF] Delbaen | Coherent risk measures on general probability spaces[END_REF], ρ is lower semi-continuous under weak* topology if and only if it satisfies the Fatou property. By Frittelli and Rosazza Gianin (2002, Theorem 6) there exists

V : L 1 (Ω, F , P) → (-∞, ∞] such that ρ(X) = - inf Y ∈L 1 (Ω,F ,P) {E[XY ] + V (Y )} = - inf Y ∈L 1 (Ω,F ,P) {E[(X ∧ 0)Y ] + V (Y )}, ∀X ∈ L ∞ (Ω, F , P),
where the second equality is due to the Loss-dependency property (d) of the loss-based risk measure. Furthermore, we have the dual relation

V (Y ) = sup X∈L ∞ (Ω,F ,P) {-ρ(X) -E[XY ]}, ∀Y ∈ L 1 (Ω, F , P). For any Y ∈ L 1 (Ω, F , P), let A := {Y < 0}. If P(A) > 0, V (Y ) ≥ sup n≥1 {-ρ(n1 A ) -E[n1 A Y ]} = sup n≥1 {-E[n1 A Y ]} = +∞,
where the first equality holds because n1 A ≥ 0 and ρ is monotone with ρ(0

) = 0. Now, if Y ≥ 0 and Y 1 > 1, we have V (Y ) ≥ sup α≥0 {-ρ(-α) -E[(-α)Y ]} = sup α≥0 {(E[Y ] -1)α} = +∞.
Thus the domain of V lies in M(Ω, F , P). Next, it is easy to see that V (Y ) ≥ 0 for any Y ∈ L 1 (Ω, F , P), because ρ(0) = 0. Finally, for any ǫ ∈ (0, 1) and any α > 0,

α = ρ(-α) = sup Y ∈M(Ω,F ,P) {αE[Y ] -V (Y )} = max sup Y ∈M(Ω,F ,P), Y 1 <1-ǫ {αE[Y ] -V (Y )}, sup Y ∈M(Ω,F ,P), Y 1 ≥1-ǫ {αE[Y ] -V (Y )} ≤ max α(1 -ǫ), sup Y ∈M(Ω,F ,P), Y 1 ≥1-ǫ {αE[Y ] -V (Y )} ≤ max α(1 -ǫ), α - inf Y ∈M(Ω,F ,P), Y 1 ≥1-ǫ {V (Y )} .
Thus we conclude that V (•) must satisfy (4). On the other hand, one can check that ρ represented in ( 5) is a loss-based risk measure satisfying the lower-semi-continuity under weak* topology and thus the Fatou property.

We can see in the representation theorem that the domain of the penalty function V (•) is a subset of M(Ω, F , P), the set of all measures with total mass less than one. This property is also observed in El [START_REF] Karoui | Cash subadditive risk measures and interest rate ambiguity[END_REF] ). Compared with the representation theorem in El [START_REF] Karoui | Cash subadditive risk measures and interest rate ambiguity[END_REF], we have an additional condition (4) due to Cash Loss property. Moreover, the dual representation formula (5) only depends on the negative part of X due to the Lossdependency property (d).

Loss-based statistical risk measures

Most of the risk measures used in finance are statistical, or distribution-based risk measures, i.e. they depend on X only through its distribution F X (•):

F X (•) = F Y (•) ⇒ ρ(X) = ρ(Y ).
Following ideas from [START_REF] Kusuoka | On law invariant coherent risk measures[END_REF], [START_REF] Frittelli | Law invariant convex risk measures[END_REF], and [START_REF] Jouini | Law invariant risk measures have the Fatou property[END_REF], we derive a representation theorem for loss-based statistical risk measures.

Theorem 2 Let ρ be a loss-based statistical risk measure. There exists a convex function

v : Ψ((0, 1)) → [0, ∞] satisfying inf 1 0 φ(z)dz≥1-ǫ v(φ) = 0 for any ǫ ∈ (0, 1), (6) such that ρ(X) = -inf φ∈Ψ 1 0 (G X (z) ∧ 0)φ(z)dz + v(φ) , ∀X ∈ L ∞ (Ω, F , P). (7)
Proof First, we remark that a statistical risk measure necessarily satisfies Fatou property. Indeed, as noted in Theorem 2.2, [START_REF] Jouini | Law invariant risk measures have the Fatou property[END_REF], distributionbased convex functionals in L ∞ (Ω, F , P) satisfy Fatou property if and only if they are lower semi-continuous in L ∞ (Ω, F , P) norm. By Lemma 1, any loss-based risk measures is L ∞ (Ω, F , P)-continuous, so automatically satisfies the Fatou property if it is distribution-based. We then need to build a connection between ( 5) and ( 7) when the risk measure is distribution-based, which can be done following the lines of Jouini et al. (2006, Theorem 2.1).

Motivated by the representation (7), we sometimes abuse the notation by writing

ρ(G(•)) = -inf φ∈Ψ 1 0 (G(z) ∧ 0)φ(z)dz + v(φ) (8)
for any bounded quantile functions, if we are considering a loss-based statistical risk measure.

Remark 1 If, in addition, the loss-based risk measure ρ satisfies (e) Cash-loss Invariance: for any X ∈ L ∞ (Ω, F , P), X ≤ 0, and

α ∈ R + , ρ(X-α) = ρ(X) + α,
then one can select V in the representation (5) and v in the representation (7) such that the domain of V lies in P(Ω, F , P) and the domain of v lies in Φ((0, 1)). In other words, ρ(X) = ρ(X ∧ 0) for some convex monetary risk measure ρ. Indeed, for any X ∈ L ∞ (Ω, F , P), one can define

ρ(X) = ρ(X -α X ) -α X ,
where α X is any upper-bound of X. By the Cash-loss Invariance property of ρ, ρ is well-defined. Furthermore, it is easy to check that ρ is a cash-invariant risk measure and ρ(X) = ρ(X ∧ 0) = ρ(X ∧ 0). However, even though ρ satisfies the Cash-loss Invariance property, it is still not cash invariant. Indeed, in general, we only have

ρ(X + α) = ρ((X + α) ∧ 0) ≥ ρ(X ∧ 0 + α) = ρ(X ∧ 0) -α = ρ(X) -α,
for any X ∈ L ∞ (Ω, F , P) and α ∈ R.

Examples

Example 1 (Put option premium) [START_REF] Jarrow | Put option premiums and Coherent Risk Measures[END_REF] argues that a natural measure of a firm's insolvency risk is the premium of a put option on the firms equity, which is given by the positive part of its net value X (assets minus liabilities), i.e.

E Q [-min(X, 0)] where Q is an appropriately chosen pricing model. One can generalize this to any portfolio whose net value is represented by a random variable X: the downside risk of the portfolio can be measured by

ρ(X) := E[-min(X, 0)] (9)
This example satisfies all the properties of Definition 1: it is a loss-based risk measure.

In particular, as noted by [START_REF] Jarrow | Put option premiums and Coherent Risk Measures[END_REF], it is not cash-additive.

Example 2 (Scenario-based margin requirements) When determining margin requirements for derivative transactions, the objective of a central clearing facility is to compute the margin requirement of each participant in the clearinghouse in order to cover losses incurred by the clearing participants portfolio over a liquidation horizon T (typically, a few days). A popular method for computing such margin requirements -used for example by various futures and options exchanges-is to select a certain number of "stress scenarios" for risk factors affecting the portfolio and compute the margin requirement as the maximum loss over these scenarios. If one denotes the portfolio P&L over the horizon T by X, then the margin requirement ρ(X) is given by

ρ(X) = max{-min(X(ω 1 ), 0), ..., -min(X(ω n ), 0)} (10)
where ω 1 , ..., ω n are the stress scenarios. Naturally, the clearinghouse only consider the losses of the portfolio when computing the margin: as a result, such margin requirements may be viewed as a loss-based risk measure. This example satisfies all the properties of Definition 1: it is a loss-based risk measure. This is the main idea behind the SPAN method used by the Chicago Merchantile Exchange (CME). Interestingly, the SPAN method was considered as an initial motivation for the definition of coherent risk measures in [START_REF] Artzner | Coherent Measures of Risk[END_REF]. Yet it is easy to check that (10) is loss-dependent and therefore not cash-additive, so is not a coherent risk measure.

Example 3 (Expected tail-loss) This popular risk measure is defined as

ρ(X) := - 1 β β 0 (G X (z) ∧ 0)dz, X ∈ L ∞ (Ω, F , P) (11)
Note that, by construction, this risk measure focuses on the left tail of the P&L distribution, since it only involves the quantile function on (0, β). Nonetheless, the classical definition of expected shortfall -1 β β 0 G X (z)dz does not satisfy Loss-dependency property (d) of Definition 1 since G X (β) might be greater that 0 for some X. Therefore, we insert G X (z) ∧ 0 in its definition to turn it into a loss-based risk measure. We notice that the put option premium is an expected tail-loss by taking β = 1.

Example 4 (Spectral tail measures) A large class of loss-based statistical risk measures is obtained by taking weighted averages of quantiles with various weight functions φ ∈ Φ:

ρ(X) := - 1 0 (G X (z) ∧ 0)φ(z)dz, X ∈ L ∞ (Ω, F , P) (12)
We call such a risk measures a spectral tail measure : it is a loss-based version of the spectral risk measures defined by [START_REF] Acerbi | Spectral measures of risk: a coherent representation of subjective risk aversion[END_REF]. This example satisfies the Cash-loss Invariance property (e) and is also positively homogeneous: for any λ > 0, ρ(λX) = λρ(X). Notice that the expected tail loss is a spectral tail measure, with φ(z) =

1 β 1 (0,β) (z).
Example 5 (Loss certainty equivalent) Consider u(•) ∈ C 4 (R + ) which is strictly increasing and strictly convex. Assume u ′ /u ′′ is concave. Consider the following map:

ρ(X) := u -1 (Eu(|X ∧ 0|)) , X ∈ L ∞ (Ω, F , P). ( 13
)
It is clear that ρ satisfies Cash Loss, Monotonicity, and Loss-dependency in Definition 1. By Hardy et al. (1959, Theorem 106), ρ is also convex. Thus, ρ is a loss-based convex risk measure. We call ρ a loss certainty equivalent . By definition, ρ is distributional-based and

ρ(X) = u -1 1 0 u(|G X (z) ∧ 0|)dz , X ∈ L ∞ (Ω, F , P). ( 14
)
If u(x) = x p , x ≥ 0 for certain p ≥ 1, we will speak of the L p loss certainty equivalent . Here, when p = 1, u(•) is not strictly convex. However, in this case, ( 13) is still well-defined and the risk measure is actually the put option premium, which is also a loss-based risk measure. Thus, we include the case p = 1 here and identify the put option premium as a special case of L p loss certainty equivalent in the following. One can show that the L p loss certainty equivalent is the only loss certainty equivalent satisfying positive homogeneity, i.e., for any λ > 0, ρ(λX) = λρ(X). The L p loss certainty equivalent has the following dual representation

ρ(X) = - inf Y ∈M q (Ω,F ,P) E [(X ∧ 0)Y ] ( 15
)
where 1 < q ≤ ∞ is the conjugate of p, i.e., 1 p + 1 q = 1, and M q (Ω, F , P) is the set of all nonnegative random variables with L q norms less than or equal to one. Moreover, it has the distribution-based representation

ρ(X) = -inf φ∈Ψ q ((0,1)) 1 0 (G X (z) ∧ 0)φ(z)dz. ( 16
)
where Ψ q ((0, 1)) is the set of φ ∈ Ψ((0, 1)) such that 1 0 φ(z) q dz ≤ 1. Let u(x) = e βx , x ≥ 0 for certain β > 0, then the loss certainty equivalent becomes the entropic loss certainty equivalent , a loss-based version of the entropic risk measure studied in [START_REF] Frittelli | Putting order in risk measures[END_REF]; [START_REF] Föllmer | Convex measures of risk and trading constraints[END_REF]; [START_REF] Föllmer | Stochastic Finance: An Introduction in Discrete Time[END_REF]. One can show that the entropic loss certainty equivalent and L 1 loss certainty equivalent are the loss certainty equivalent satisfying Cash-loss Invariance property. It is well known in the literature that the entropic loss certainty equivalent has the dual representation

ρ(X) = - inf Y ∈P(Ω,F ,P) {E[(X ∧ 0)Y ] + V (Y )}, (17)
where

V (Y ) = E [Y ln Y ] - inf Y ∈P(Ω,F ,P) E [Y ln Y ] , (18)
and the distributional-based representation

ρ(X) = -inf φ∈Φ((0,1)) 1 0 (G X (z) ∧ 0)φ(z)dz + v(φ), (19)
where

v(φ) = 1 0 φ(z) ln φ(z)dz -inf φ∈Φ((0,1)) 1 0 φ(z) ln φ(z)dz. ( 20
)
See e.g. [START_REF] Frittelli | Putting order in risk measures[END_REF]; [START_REF] Föllmer | Convex measures of risk and trading constraints[END_REF]; [START_REF] Föllmer | Stochastic Finance: An Introduction in Discrete Time[END_REF]; [START_REF] Frittelli | Law invariant convex risk measures[END_REF].

Robustness of risk estimators

In practice, for measuring the risk of a portfolio, aside from the theoretical choice of a risk measure, a key issue is the estimation of the risk measure, which requires the choice of a risk estimator [START_REF] Cont | Robustness and sensitivity analysis of risk meausurement procedures[END_REF]. In this section we study the robustness property of empirical risk estimators built from some distribution-based risk measure. We follow the ideas in Cont et al. ( 2010) but we will view risk measures as functionals on the set of quantile functions, rather than the set of distribution functions. As we will see, this makes the study of continuity properties of loss-based risk measures easier.

Moreover, from Theorem 2, a loss-based statistical risk measure can be represented by ( 7), which suggests that it is more natural to work directly on quantile functions.

Denote by Q the set of all quantile functions and by D the set of all distribution functions. The Lévy-Prokhorov metric between two distribution functions F 1 ∈ D and F 2 ∈ D is defined as

d P (F 1 , F 2 ) inf{ǫ > 0 : F 1 (x -ǫ) -ǫ ≤ F 2 (x) ≤ F 1 (x + ǫ) + ǫ, ∀x ∈ R}.
This metric appears to be the most tractable one on D and it induces the same topology as the usual weak topology on D.

The quantile set Q and the distribution set D are connected by the following one-on-one correspondence

D → Q F (•) → F -1 (•)
where

F -1 (t) = inf {x ∈ R | F (x) ≥ t} , t ∈ (0, 1)
is the left continuous inverse of F (•). Such a correspondence, together with the Lévy-Prokhorov metric on D induces a metric on Q which we denote by d. The convergence under this metric can be characterized by the following: for any G n , G ∈ Q, G n → G if and only if G n (z) → G(z) at any continuity points of G.

Most of the time, we work with quantile functions that are continuous on (0, 1) in order to avoid irregularities due to the presence of atoms. In practice, it is not restrictive to focus on continuous quantile functions. Indeed, people do assume the continuity of quantile functions in many applications, e.g., when computing the Value at Risk. The study of discontinuous quantile functions is more technical and of little interest, so we choose not to pursue in this direction. In the following, we denote by Q c the set of all continuous quantile functions. We also denote by Q ∞ the set of all bounded quantile functions, and Q ∞ c the set of all bounded continuous quantile functions.

A Family of Distribution-based Risk Measures

Motivated by the representation (7), we consider the following Fenchel-Legendre transform ρ : Q → [0, ∞]:

ρ(G(•)) := -inf m∈dom(v) (0,1) (G(z) ∧ 0)m(dz) + v(m) , (21)
where v :

M((0, 1)) → [0, ∞] is the penalty function satisfying inf m((0,1))≥1-ǫ v(m) = 0 for any ǫ ∈ (0, 1), (22)
and dom(v) is the domain of v, i.e., dom(v) := {m ∈ M((0, 1)

) | v(m) < ∞}. ( 23
)
It is easy to see that ρ(G(•)) is well-defined for any G(•) ∈ Q, and dom(ρ), the domain of ρ, contains Q ∞ . It is clear that ( 21) is a generalization of (7). Such a generalization in the context of risk measures has been considered by [START_REF] Cont | Robustness and sensitivity analysis of risk meausurement procedures[END_REF] in the case of spectral risk measures, by [START_REF] Heyde | External Risk Measures and Basel Accords[END_REF] for risk measures on finite-dimensional probability spaces, and by [START_REF] Song | Risk measures with comonotonic subadditivity or convexity and respecting stochastic orders[END_REF] in a general setting. The generalization (21) sacrifices the convexity of ρ viewed as a map on L ∞ (Ω, F , P). However, if ρ is a map defined on Q, it satisfies the following properties 1. Cash Loss:

for any α ∈ R + , ρ(-α) = α; 2. Monotonicity: for any G 1 (•), G 2 (•) ∈ Q such that G 1 (z) ≤ G 2 (z), 0 < z < 1, ρ(G 1 (•)) ≥ ρ(G 2 (•)); 3. Quantile Convexity: for any G 1 (•), G 2 (•) ∈ Q and 0 < α < 1, ρ(αG 1 (•) + (1 - α)G 2 (•)) ≤ αρ(G 1 (•)) + (1 -α)ρ(G 2 (•)).
4. Loss-dependency: for any G(•) ∈ Q, ρ(G(.)) = ρ(G(.) ∧ 0).

Qualitative Robustness

In practice, in order to compute the risk measure ρ(G(•)) of portfolio whose P&L has quantile G(•), one has first to estimate the distribution or quantile of the P&L distribution of the portfolio from data, and then apply the risk measure ρ to the estimated distribution. One of the popular ways is to apply the risk measure to the empirical distribution.

A sequence of samples of G(•) ∈ Q is a sequence of random variables X 1 , X 2 , • • • which are i.i.d and follow the distribution G -1 . We denote by X this sequence of samples and X n its first n samples. For each sample size n, the empirical distribution is defined as

F emp X n (x) = 1 n n i=1 1 X i ≤x , x ∈ R, (24)
and the empirical quantile is defined as

G emp X n (z) := (F emp X n ) -1 (z) = X (⌊nz⌋+1) , z ∈ (0, 1), ( 25
)
where ⌊a⌋ denotes the integer part of a and

X (1) ≤ • • • ≤ X (n) . In practice, the quantity ρ(X n ) := ρ(G emp X n (•)
) is computed as the estimated risk measure ρ(G(•)) given the n samples X n . The risk estimator ρ is defined on ∪ n≥1 R n , the set of all possible samples (X n ) n≥1 , and has values in R + . Since the samples can be regarded as random variables, so is the risk estimator ρ(X n ). We denote by L n ( ρ, G) the distribution function of ρ(X n ).

ρ is said to be consistent at

G = F -1 ∈ dom(ρ) if lim n→∞ ρ(X n ) = ρ(F emp X n ) -1 ) almost surely. ( 26
)
Because the true risk measure ρ(G(•)) is estimated by ρ(X n ), the consistency is the minimal requirement for a meaningful risk measure. In the following we denote by Q ρ the set of quantiles G at which ρ is well defined and consistent, and

Q ρ c the continuous quantiles in Q ρ .
The following definition of robust risk estimator is considered in Cont et al. (2010, Definition 4):

Definition 2 [START_REF] Cont | Robustness and sensitivity analysis of risk meausurement procedures[END_REF]) Let ρ be defined by ( 21) and C ⊂ Q ρ be a a set of plausible loss quantiles. ρ is C-robust at G ∈ C if for any ε > 0 there exist δ > 0 and n 0 ≥ 1 such that, for all G ∈ C,

d( G, G) ≤ δ =⇒ d P (L n ( ρ, G), L n ( ρ, G)) < ε, ∀n ≥ n 0 . A risk estimator ρ is called C-robust if it is C-robust at any G ∈ C.
The choice of the Lévy-Prokhorov distance is natural in robust statistics (see [START_REF] Huber | Robust Statistics[END_REF]). It is worth mentioning, as pointed out by [START_REF] Huber | Robust Statistics[END_REF], that the use of a different metric, even if it also metrizes the weak topology, may lead to a different class of robust estimators.

The following proposition which is cited from [START_REF] Cont | Robustness and sensitivity analysis of risk meausurement procedures[END_REF] shows that the robustness of the risk estimator ρ is equivalent to the continuity of the risk measure ρ.

where the second inequality is due to Fatou's lemma. On the other hand, for each η < 1, by Lemma 2, G n (z) → G(z) uniformly for z ∈ (0, η]. Thus we have

lim inf n→∞ U(G n (•)) = lim inf n→∞ inf m∈dom(v) (0,1) (G n (z) ∧ 0)m(dz) + v(m) ≥ lim inf n→∞ inf m∈dom(v) (0,1) (G n (z ∧ η) ∧ 0)m(dz) + v(m) = inf m∈dom(v) (0,1) (G(z ∧ η) ∧ 0)m(dz) + v(m) = U(G(• ∧ η)),
where the second equality holds because G n (z ∧η) → G(z ∧η) for z ∈ (0, 1) uniformly. Finally,

U(G(•)) ≥ U(G(• ∧ η)) = inf m∈dom(v) (0,1) (G(z ∧ η) ∧ 0)m(dz) + v(m) = inf m∈dom(v) (0,1) (G(z) ∧ 0)m(dz) + v(m) - (0,1) (G(z) ∧ 0 -(G(z ∧ η) ∧ 0))m(dz) ≥ U(G(•)) -lim z↑1 G(z) ∧ 0 -G(η) ∧ 0 → U(G(•)) as η ↑ 1. Therefore we conclude lim inf n→∞ U(G n (•)) ≥ U(G(•)).
The following result characterizes risk measures ρ : Q → R + defined by ( 21) which are continuous on Q.

Theorem 3 Let ρ be defined by (21) and C be any subset of Q c such that C ⊇ Q ∞ c . The following are equivalent:

(i) ρ(•), when restricted to C, is continuous at any G(•) ∈ C. (ii) There exists 0 < δ < 1 such that sup m∈dom(v) m((0, δ)) = 0. (27) Furthermore, if (ii) holds, Q ρ c = Q c . Proof (ii) ⇒ (i)
The proof is analogous to the proof of Lemma 3. Let (ii) hold for some 0 < δ < 1.

For simplicity, we work with U(.) -ρ(.) and first show that U is lower semi-continuous. For each η < 1, we have

lim inf n→∞ U(G n (•)) = lim inf n→∞ inf m∈dom(v) [δ,1) (G n (z) ∧ 0)m(dz) + v(m) ≥ lim inf n→∞ inf m∈dom(v) [δ,1) (G n (z ∧ η) ∧ 0)m(dz) + v(m) = inf m∈dom(v) [δ,1) (G(z ∧ η) ∧ 0)m(dz) + v(m) = U(G(• ∧ η)),
where the second equality holds because ) where Lemma 2 applies. Then, by monotonicity, we have

G n (z ∧ η) converges to G(z ∧ η) uni- formly for z ∈ [δ, 1
0 ≤ U(G(•)) -U(G(• ∧ η)). Finally, U(G(•)) ≥ U(G(• ∧ η)) = inf m∈dom(v) [δ,1) (G(z ∧ η) ∧ 0)m(dz) + v(m) = inf m∈dom(v) [δ,1) (G(z) ∧ 0)m(dz) + v(m) - (0,1) (G(z) ∧ 0 -(G(z ∧ η) ∧ 0))m(dz) ≤ U(G(•)) -lim z↑1 G(z) ∧ 0 -G(η) ∧ 0 → U(G(•))
as η ↑ 1. Thus we have lim inf n→∞ U(G n (•)) ≥ U(G(•)), and can conclude that U is lower semi-continuous.

Next, we show that U is also upper-semi-continuous.

lim sup n→∞ U(G n (•)) ≤ inf m∈dom(v) lim sup n→∞ (0,1) (G n ∧ 0)(z)m(dz) + v(m) ≤ inf m∈dom(v) (0,1) (G(z) ∧ 0)m(dz) + v(m) = U(G(•)),
where the second inequality is due to Fatou's lemma. Together with the lowersemi-continuity, U(•) is continuous at G, and so is ρ.

(i) ⇒ (ii) We prove it by contradiction. If (ii) is not true, there exists δ n > 0, and

m n ∈ dom(v), such that δ n → 0 and m n ((0, δ n )) > 0. Define G n (z) :=      -β n 0 < z ≤ δ n , βn δn (z -2δ n ) δ n < z ≤ 2δ n 0 2δ n < z < 1, n ≥ 1,
where

β n := v(m n ) + 1 m n ((0, δ n )) , n ≥ 1. It is obvious that G n ∈ Q ∞ c ⊂ C, n ≥ 1. Because δ n ↓ 0, G n (•) → 0 in C. On the other hand, U(G n (•)) ≤ (0,1) G n (z)m n (dz) + v(m n ) ≤ -1,
and U(0) = 0. Thus, U(•) is not continuous at 0 which is a contradiction.

Finally, if (ii) holds, for any G(•) ∈ Q c , we have ρ(G(•)) = ρ(G(• ∨ δ)). Because G(• ∨ δ) is bounded from below, by Lemma 3, ρ is consistent at G(• ∨ δ), and therefore consistent at G(•). In other words, Q ρ c = Q c .
Remark 2 [START_REF] Jouini | Law invariant risk measures have the Fatou property[END_REF] consider the Lebesgue property of a convex monetary risk measure and give an equivalent dual characterization. Actually, the Lebesgue property is a weaker continuity property of ρ compared with the continuity we consider here. To compare these two continuity properties, we mimic the result in [START_REF] Jouini | Law invariant risk measures have the Fatou property[END_REF] to give a dual characterization in Appendix A.

Combining Proposition 1 and Theorem 3, we have

Corollary 1 Let ρ be defined by (21) and C be any subset of

Q ρ c such that C ⊇ Q ∞ c . Then the following are equivalent 1. ρ is C-robust 2. There exists 0 < δ < 1 such that sup m∈dom(v) m((0, δ)) = 0. (28) Furthermore, if ρ is C-robust, then Q ρ c = Q c .
As the following corollary shows, loss-based statistical risk measures do not lead to robust risk estimators.

Corollary 2 Let ρ be a loss-based statistical risk measure and C be any subset of

Q ρ c such that C ⊇ Q ∞ c . Then ρ is not C-robust.
Proof By Theorem 2, ρ can be represented as ( 7) where the penalty function v satisfies (6). As a result, there exists a φ ∈ Ψ((0, 1))∩dom(v) such that 1 0 φ(z)dz ≥ 1 2 . Because φ(•) is decreasing on (0,1), we must have δ 0 φ(z)dz > 0 for any δ > 0. By Corollary 1, ρ is not C-robust.

Statistical robustness of risk estimators

Corollary 1 gives a sufficient and necessary condition for a loss-based risk measure ρ represented by equation ( 21) to be continuous and therefore lead to a robust empirical risk estimator. In particular, loss-based statistical risk measures lead to non robust risk estimators. In the following, we provide one way to robustify risk estimator computed from loss-based statistical risk measures. Fixed some δ ∈ (0, 1), for any loss-based statistical risk measure ρ, consider its δ-truncation

ρ δ (G(•)) := ρ(G(• ∨ δ)), G ∈ Q. ( 29
)
Note that we could also consider risk measures given by representation ( 21) and propose a modified version of this larger class of risk measures leading to robust risk estimators. The restriction to loss-based statistical risk measures is only made to illustrate further the conflict between robustness and convexity.

From the representation (7), we can find the representation of ρ δ , which in turns shows that ρ δ is robust. Define the map π : Ψ → M((0, 1)) which associates to any density function φ in Ψ a measure m defined by

m(dz) :=      φ(z)dz, δ < z < 1, (0,δ] φ(t)dt δ z , z = δ, 0, 0 < z < δ,
where δ z is the Dirac measure at z. Noticing that π is not a bijective map leads to an additional definition since the penalty function v(m) for m ∈ π(Ψ((0, 1))) := {m | m = π(φ) for some φ ∈ Ψ((0, 1))} cannot be derived uniquely from v(φ) where m = π(φ). Therefore by denoting π

-1 (m) = {φ ∈ Ψ((0, 1)) | π(φ) = m}, we define for m ∈ π(Ψ), v δ (m) := inf φ∈Ψ∩π -1 (m) v(φ).
From the representation (7), we have

ρ δ (G) = -inf φ∈Ψ (G(δ) ∧ 0) (0,δ] φ(z)dz + (δ,1) (G(z) ∧ 0)φ(z)dz + v(φ) = -inf m∈π(Ψ) (0,1) (G(z) ∧ 0)m(dz) + v δ (m) . (30)
Finally, from (6), it is easy to see that v δ satisfies (22).

From the representation (30), it is immediate to see that the δ-truncation ρ δ is no longer convex since measures m ∈ π(Ψ((0, 1))) have a point mass at δ and therefore do not admit a density on (0, 1). On the other hand, it is also straightforward to see that ρ δ is Q c -robust because each m ∈ π(Ψ((0, 1))) satisfies m((0, δ)) = 0.

Example 6 The δ-truncation of the spectral tail measure ( 12) is given by

ρ δ (G) = 1 δ (G(z) ∧ 0)φ(z)dz + G(δ) δ 0 φ(z)dz, G ∈ Q. ( 31
)
Example 7 The δ-truncation of the loss certainty equivalent is given by

ρ δ (G(•)) = u -1 1 0 u(|G(t ∨ δ) ∧ 0|)dt , G ∈ Q. ( 32
)
In particular, the δ-truncation of the L p risk measure is given by

ρ δ (G) = 1 0 |G(z ∨ δ) ∧ 0| p dz 1 p , G ∈ Q, (33)
and its representation is expressed as

ρ δ (G) = - inf m∈π(Ψ q ((0,1))) 1 0 (G(z) ∧ 0)m(dz), G ∈ Q. ( 34
)
The δ-truncation of the entropic risk measure is given by

ρ δ (G) = 1 β log (0,1) e -β G(z∨δ)∧0 dz , G ∈ Q, ( 35
)
and its representation is expressed as

ρ δ (G) = - inf m∈π(Φ((0,1))) (0,1) (G(z) ∧ 0)m(dz) + v δ (m) , G ∈ Q (36)
where for each m ∈ π(Φ((0, 1)))

v δ (m) = inf φ∈Φ∩π -1 (m) v(φ) = 1 β (δ,1) m ′ (z) log (m ′ (z)) dz + 1 β lim z↓δ (m ′ (z) log (m ′ (z)) z) -inf φ∈Φ((0,1)) 1 0 φ(z) ln φ(z)dz.
Here m ′ (z), δ < z < 1 denotes the density of m on (δ, 1) w.r.t to Lebesgue measure. This density is well-defined because m ∈ π(Φ((0, 1))).

Remark 3 In the case of the Expected Shortfall, [START_REF] Cont | Robustness and sensitivity analysis of risk meausurement procedures[END_REF] have proposed another truncation for ES α

ES δ,α (G) = 1 α -δ (δ,α) G(z) dz.
Hence, applying this idea to the larger class of loss-based statistical risk measures would lead to defining the following δ-truncation

ρ δ (G) = -inf φ∈ π(Ψ) (0,1) (G(z) ∧ 0) φ(z)dz + v δ ( φ) ,
where the map π : Ψ → M((0, 1)) associates to any function φ ∈ Ψ another function φ ∈ M((0, 1)) defined by

φ(z) := φ(z)1 (δ,1) (z) (δ,1) φ(z)dz ∈ Ψ,
and where the penalty function is given by

v δ ( φ) := inf φ∈Ψ∩ π -1 ( φ) v(φ).
In this context, such truncations ρ δ are less tractable than ρ δ since they cannot be computed from the initial risk measure ρ as in (29).

Sensitivity analysis of risk estimators

In the previous section, we have studied the robustness of risk estimators in a qualitative sense. One may argue that the above results rely on the choice of a topology (weak topology) together with a distance (Lévy-Prokhorov distance) on the space of probability distributions. As illustrated in Appendix A, the choice of a weaker topology could lead to different robustness properties. Nonetheless, as noted in [START_REF] Huber | Robust Statistics[END_REF], the choice of the weak continuity to study robustness is natural in statistics. To further illustrate this statement in this section, we study sensitivity properties of risk estimators without relying on any topology and show that it leads to the same conclusions as before, singling out the same class of risk estimators as non-robust.

This may be done by quantifying the sensitivity of risk estimators using influence functions [START_REF] Cont | Robustness and sensitivity analysis of risk meausurement procedures[END_REF][START_REF] Hampel | The influence curve and its role in robust estimation[END_REF]. Fix a risk estimator ρ for which the estimation is based on applying the risk measure ρ to empirical quantile functions. Then, its sensitivity function at the quantile function G of a distribution F , in the direction of the Dirac mass at z is equal to

S(z; G) lim ǫ→0 + ρ(ǫδ z + (1 -ǫ)F ) -ρ(F ) ǫ ,
for any z ∈ R such that the limit exists. Note that S(z; G) is nothing but the directional derivative of the risk measure ρ at F in the direction δ z ∈ D. S(z, G) measures the sensitivity of the risk estimator based on a large sample to the addition of a new data point. It has already been used in the field of risk measure by [START_REF] Heyde | External Risk Measures and Basel Accords[END_REF]; [START_REF] Cont | Robustness and sensitivity analysis of risk meausurement procedures[END_REF]. In the latter, the authors consider different estimation methods, using both empirical and parametric distributions. In that case, the definition of the sensitivity function should be considered with more attention since the risk measure ρ would have to be replaced with an effective risk measure incorporating both the choice of the risk measure and the estimation method as explained in [START_REF] Cont | Robustness and sensitivity analysis of risk meausurement procedures[END_REF].

Unbounded sensitivity functions

In this section, we compute the sensitivity function of the loss certainty equivalent. We find that this risk measure has unbounded sensitivity function which is consistent with our findings of Section 2. Note that, unlike the setting of Section 2, this result makes no reference to any topology on the set of loss distributions.

Proposition 2 The sensitivity function of the loss certainty equivalent (13) is given by

S(z; G) = u(|z ∧ 0|) -u(ρ(G)) u ′ (ρ(G)) . (37)
It is easy to compute that

δ δ-ǫ 1-ǫ u(|G (t) ∧ 0|)1 {t>F (z)} dt = 0, F (z) ≥ δ, (1 -δ)u(|G(δ) ∧ 0|)ǫ + o(ǫ), F (z) < δ, δ 1-ǫ δ u(|G (t) ∧ 0|)1 {t≤F (z-)} dt = δu(|G(δ) ∧ 0|)ǫ + o(ǫ), F (z-) > δ, 0, F (z-) ≤ δ.
Noticing (42), we can compute that

1 δ u(|G (t) ∧ 0|)1 {F (z-)<t≤F (z)} dt =      0, F (z) < δ, (F (z) -δ)u(|z ∧ 0|), F (z-) ≤ δ ≤ F (z), (F (z) -F (z-))u(|z ∧ 0|), δ < F (z-).
Similarly, we can compute that

1 δ u(|z ∧ 0|)1 {(1-ǫ)F (z-)<t≤ǫ+(1-ǫ)F (z)} dt =      o(ǫ), F (z) < δ, [(1 -ǫ)(F (z) -δ) + ǫ(1 -δ)] u(|z ∧ 0|), F (z-) ≤ δ ≤ F (z), [(1 -ǫ)(F (z) -F (z-)) + ǫ] u(|z ∧ 0|), δ < F (z-). Therefore 1 δ u(|G ǫ (t) ∧ 0|)dt = (1 -ǫ) 1 δ u(|G (t) ∧ 0|)dt +      [u(|z ∧ 0|) -δu(|G(δ) ∧ 0|)] ǫ + o(ǫ), δ < F (z-), (1 -δ)u(|z ∧ 0|)ǫ + o(ǫ), F (z-) ≤ δ ≤ F (z), (1 -δ)u(|G(δ) ∧ 0|)ǫ + o(ǫ), δ > F (z).
On the other hand

G ǫ (δ) ∧ 0 -G(δ) ∧ 0 =      G δ-ǫ 1-ǫ ∧ 0 -G(δ) ∧ 0, δ > ǫ + (1 -ǫ)F (z), G δ 1-ǫ ∧ 0 -G(δ) ∧ 0, δ ≤ (1 -ǫ)F (z-), z ∧ 0 -G(δ) ∧ 0, (1 -ǫ)F (z-) < δ ≤ ǫ + (1 -ǫ)F (z).
We discuss case by case.

1. G(δ) > 0. Because G(•) is continuous at δ, it is easy to show that G ǫ (δ) ∧ 0 - G(δ) ∧ 0 = 0 when ǫ is sufficiently small. 2. G(δ) = 0. In this case, we have G ǫ (δ) ∧ 0 -G(δ) ∧ 0 =      G δ-ǫ 1-ǫ , δ > ǫ + (1 -ǫ)F (z), 0, δ ≤ (1 -ǫ)F (z-), z ∧ 0, (1 -ǫ)F (z-) < δ ≤ ǫ + (1 -ǫ)F (z)
when ǫ is sufficiently small. Recalling (43), we conclude that

G ǫ (δ) ∧ 0 -G(δ) ∧ 0 =      0, δ < F (z-), 0, F (z-) ≤ δ ≤ F (z), -(1 -δ)G ′ (δ)ǫ + o(ǫ), δ > F (z).
3. G(δ) < 0. In this case, we have

G ǫ (δ) ∧ 0 -G(δ) ∧ 0 =      G δ-ǫ 1-ǫ -G(δ), δ > ǫ + (1 -ǫ)F (z), G δ 1-ǫ -G(δ), δ ≤ (1 -ǫ)F (z-), z -G(δ), (1 -ǫ)F (z-) < δ ≤ ǫ + (1 -ǫ)F (z).
when ǫ is small enough, leading to

G ǫ (δ) ∧ 0 -G(δ) ∧ 0 =      δG ′ (δ)ǫ + o(ǫ), δ < F (z-), 0, F (z-) ≤ δ ≤ F (z), -(1 -δ)G ′ (δ)ǫ + o(ǫ), δ > F (z). Notice that 1 0 u(|G ǫ (t ∨ δ) ∧ 0|)dt - 1 0 u(|G(t ∨ δ) ∧ 0|)dt = 1 δ u(|G ǫ (t) ∧ 0|)dt - 1 δ u(|G(t) ∧ 0|)dt +δ (u(|G ǫ (δ) ∧ 0|) -u(|G(δ) ∧ 0|)) . Then, when G(δ) > 0, lim ǫ↓0 1 0 u(|G ǫ (t ∨ δ) ∧ 0|)dt - 1 0 u(|G(t ∨ δ) ∧ 0|)dt ǫ = lim ǫ↓0 1 δ u(|G ǫ (t) ∧ 0|)dt -

A Lebesgue Continuity

Following [START_REF] Jouini | Law invariant risk measures have the Fatou property[END_REF], we consider a weaker continuity property for ρ:

Definition 3 (Lebesgue continuity) Let ρ be defined in (21). ρ is Lebesgue continuous at G ∈ Q if for any G n ∈ Q such that G n is uniformly bounded and

G n → G, ρ(G n ) → ρ(G).
The following result gives the dual characterization of the Lebesgue continuity and shows how Theorem 3 is modified when weak continuity is replaced by Lebesgue continuity.

Theorem 4 Let ρ be defined as in (21). Then, the following are equivalent

1

  δ u(|G(t) ∧ 0|)dt + δ (u(|G ǫ (δ) ∧ 0|) -u(|G(δ) ∧ 0|)) |G ǫ (t) ∧ 0|)dt -1 δ u(|G(t) ∧ 0|)dt -δu ′ (|G(δ) ∧ 0|) (G ǫ (δ) ∧ 0 -G(δ) |G ǫ (t ∨ δ) ∧ 0|)dt -1 0 u(|G(t ∨ δ) ∧ 0|)dt ǫ = -) ≤ δ ≤ F (z), u ′ (0)δ(1δ)G ′ (δ), δ > F (z). |G ǫ (t ∨ δ) ∧ 0|)dt -∧ 0|) -δu(|G(δ)|) -δ 2 u ′ (|G(δ)|)G ′ (δ), δ < F (z-), (1 -δ)u(|z ∧ 0|), F (z-) ≤ δ ≤ F (z), (1 -δ)u(|G(δ)|) + δ(1 -δ)u ′ (|G(δ)|)G ′ (δ), δ > F (z(t ∨ δ) ∧ 0|)dt + ∧ 0|) -δ 2 u ′ (|G(δ)|)G ′ (δ), δ < F (z-), u(|z ∧ 0|), F (z-) ≤ δ ≤ F (z), u(|G(δ)|) + δ(1 -δ)u ′ (|G(δ)|)G ′ (δ), δ > F (z).

Finally

  , applying chain rule and (43), we immediately have (38)-(40).From Proposition 3, it is easy to compute that supz∈R δ)G ′ (δ), G(δ) = 0, -u(ρ δ (G)) + u(|G(δ)|) + δ(1 -δ)u ′ (|G(δ)|)G ′ (δ), G(δ) < 0.Thus the truncated loss certainty equivalent has bounded sensitivity functions, which is consistent with the robustness properties of risk estimators derived in Section 2.Remark 4 If F has a continuous positive density f in the neighborhood of G(δ), then G(•) is differentiable at δ andG ′ (δ) = 1 f (G(δ)) .Moreover, denoting by SVaR δ (z; G) the sensitivity function of VaR δ at G, which, from the proof of Proposition 3, isSVaR δ (z; G) δ)) , z < G(δ),we can rewrite, for G(δ) < 0, the sensitivity of the δ-truncation of the loss certainty equivalent as:S(z; G) = 1 u ′ (ρ δ (G)) [-u(ρ δ (G)) + u(|(z ∨ G(δ)) ∧ 0|) -δu ′ (|G(δ)|)SVaR δ (z; G)] .

  (i) ρ(•) is Lebesgue continuous at any G ∈ Q ∞ c . (ii) For each c > 0, lim δ↓0 sup v(m)≤c m((0, δ)) = 0.Proof We work with U(•) := -ρ(•).

  (i) ⇒ (ii) If (ii) does not hold, there exists c > 0, ǫ 0 > 0, δ n ↓ 0, and m n ∈ P((0, 1)) withv(m n ) ≤ c, such that m n ((0, δ n )) ≥ ǫ 0 . Define -2δ n ), δ n < z ≤ 2δ n , (z) ∧ 0)m n (dz) + v(m n ) ≤ -2c + c = -c < 0.On the other hand, G n (•) is uniformly bounded, and becauseδ n ↓ 0, G n (•) → G(•) ≡ 0. Because U(0) = 0, U(•) is not Lebesgue continuous at 0, which is a contradiction. (ii) ⇒ (i) Suppose G, G n ∈ Q ∞ c are uniformly bounded and G n → G. On the one z) ∧ 0)m(dz) + v(m) = U(G(•)),where the first equality is due to bounded convergence theorem.On the other hand, let M > 0 be a uniform bound of G n , i.e., sup n≥1,z∈(0,1) |G n (z)| ≤ M. For each fixed 0 < ǫ < 1, and n ≥ 1, we can find m n ∈ dom(v) such thatU(G n (•)) ≥ (0,1) (G n (z) ∧ 0)m n (dz) + v(m n ) -ǫ. (44) Let c n := v(m n ) < ∞, then c n ≤ U(G n (•)) -(z) ∧ 0)m(dz) + v(m) -ǫ, n ≥ 1.By monotonicity of U, we can find 0< η < 1 such that U(G(• ∧ η)) ≥ U(G(•))ǫ.Because lim δ↓0 sup v(m)≤c m((0, δ)) = 0, we can find 0 < δ < η such that sup v(m)≤c m((0, δ)) < ǫ/M. By Lemma 2, G n (z) → G(z) uniformly on [δ, η].

  Representation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Loss-based statistical risk measures . . . . . . . . . . . . . . . . . . . 1.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Family of Distribution-based Risk Measures . . . . . . . . . . . . . 2.2 Qualitative Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Statistical robustness of risk estimators . . . . . . . . . . . . . . . . .

	2 Robustness of risk estimators
	2.1 A

Contents 1 Loss-Based Risk Measures 1.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 3 Sensitivity analysis of risk estimators 3.1 Unbounded sensitivity functions . . . . . . . . . . . . . . . . . . . . . 3.2 Boundedness of sensitivity functions for robust risk estimators . . . .

Proposition 1 [START_REF] Cont | Robustness and sensitivity analysis of risk meausurement procedures[END_REF]) Let ρ be a risk measure and G ∈ C ⊂ Q ρ . The following are equivalent:

1. ρ, when restricted to C, is continuous at G; 2. ρ is C-robust at G.

In the following, we are going to investigate the continuity of ρ, which finally clarifies whether ρ is robust or not. The following lemma is useful.

Lemma 2 Let -∞ < a < b < +∞, and G n , G be increasing functions on [a, b]. Suppose G(z) is continuous on [a, b] and

The following lemma shows that any risk measure ρ in (21

Lemma 3 Let ρ be given in (21). Then ρ is consistent at any G ∈ Q c which is bounded from below. In particular,

Proof Let G ∈ Q c which is bounded from below, and X 1 , X 2 , . . . be its samples. By Glivenko-Cantelli theorem, G emp

In the following, for each fixed ω, let G n := G emp X n . For simplicity, we work with

Proof By denoting G ǫ (•) the quantile function corresponding to the distribution function

Now, simple calculus leads to (37).

From Proposition 2, if lim x→∞ u(x) = ∞, lim z↓-∞ S(z; G) = +∞, showing that the sensitivity function is unbounded. In particular, for the L p and entropic risk measures, the sensitivity functions are unbounded.

Boundedness of sensitivity functions for robust risk estimators

In this section, we compute the sensitivity functions of the δ-truncated versions of the loss certainty equivalents. These truncated versions were introduced in Section 2.3 in order to obtain robust risk estimators. The conclusion of the following proposition is that by truncating these risk measures, their sensitivity functions become bounded. Therefore, the robustness properties of risk estimators derived in Section 2 are consistent with the sensitivity functions computed in this section.

Proposition 3 Consider the δ-truncation of the certainty equivalent (32) with 0 < δ < 1. Assume F (z) < 1 and G(•) is differentiable at δ. Then the sensitivity S(z; G) can be computed as follows:

Proof First let us recall some properties of left-continuous inverse functions. For any distribution function F , denote by F -1 its left-continuous inverse. Then for any t ∈ [0, 1], x ∈ R,

and thus

In the following, we denote by F (•) the distribution function associated with G(•). By assumption, G(•) is differentiable and thus continuous at δ. We claim that

Indeed, from (41), we deduce that

. Thus, to prove (43), we only need to show that if

On the one hand, from (41), z ≥ G(δ). On the other hand, for any ε > 0 small enough, F (z -ε) < δ +ε, which again by ( 41) leads to z -ǫ < G(δ + ε). Letting ε ↓ 0, by the continuity of G(•) at δ, we conclude that z ≤ G(δ).

Denote by G ǫ (•) the quantile function corresponding to the distribution function F ǫ (•) = (1 -ǫ)F (•) + ǫδ z (•). We only need to compute

and S(z; G) follows from the chain rule.

Straightforward computation shows that

Thus Thus we have lim inf n→∞ U(G n (•)) ≥ U(G(•)). In summary, U is Lebesgue continuous at G and so is ρ.


J Pištora

M Les Ák

E Lišková

Š Viš Ovský

I Harward

P Maslankiewicz

K Balin

Z Celinski

J Mistrík

R Lopusnik

J Vl Ek

Effect of FeF 2 on magneto-optic response in FeF 2 /Fe/FeF 2 sandwiches

Keywords: 78, 20, Ci

The room temperature optical constants n and k of MBE grown FeF 2 films are reported. Because of poor chemical stability, FeF 2 had to be coated with a protective Au layer. Reflection spectral ellipsometry in the photon energy range between 1.3 and 5.2 eV was performed on structures with a typical profile Au(0.5 nm)/FeF 2 (120 nm)/Au(30 nm) /Ag(20 nm) /Fe(0.6 nm) grown on GaAs(001) substrate. The spectra of n and k in FeF 2 were subsequently employed in the design of FeF 2 /Fe/FeF 2 sandwiches considered as magneto-optic (MO) sensors for weak microwave currents. Their MO response was evaluated using reflection MO (Kerr) spectroscopy at polar magnetization. The present results may be of interest in MO studies of magnetic nanostructures with Fe/FeF 2 /Fe, including MO magnetometry and MO magnetic domain imaging.

Introduction

Interest in FeF 2 is motivated by exchange bias studies [START_REF] Noqués | Exchange bias[END_REF][START_REF] Stamps | Mechanisms for exchange bias[END_REF][START_REF] Fitzsimmons | Pinned magnetization in the antiferromagnet and ferromagnet of an exchange bias system[END_REF][START_REF] Widuch | Variation in ferromagnetic domain density and domain asymmetry in Fe/FeF 2 exchange bias structures[END_REF] since anti-ferromagnet FeF 2 has a strong uniaxial anisotropy that makes modeling of the exchange bias Fe/FeF 2 /Fe nanostructures easier. The effect of exchange bias on the magnetic characteristics of Fe/FeF 2 /Fe can be studied using fixed wavelength magneto-optic (MO) magnetometry in reflection. Chemical compatibility with Fe was found by Lopusnik et al. [START_REF] Lopusnik | Optimization of magneto-optical response of FeF 2 /Fe/FeF 2 sandwiches for microwave field detection[END_REF] to be favorable for magneto-optic (MO) reflection sensors based on FeF 2 /Fe/FeF 2 sandwiches operating at a Ti: sapphire laser wavelength of 810 nm (1.53 eV). The sensors, which exploit strong MO activity in reflection of Fe in the infrared spectral region, were considered for the detection of weak magnetic fluxes up to microwave frequencies with submicron resolution. In the sandwich nanostructure, the MO response can be tuned to its maximum by a proper choice of the Fe layer thickness and that of the sandwiching FeF 2 layers. In the practical design, an Au layer protects the FeF 2 /Fe/FeF 2 sandwich which is deposited on a reflecting Ag or Au layer. The entire structure is grown on a GaAs substrate with a Fe seed layer to form an epitaxial Ag or Au reflector.

An evaluation of the MO response in the Fe/FeF 2 /Fe exchange bias and FeF 2 /Fe/FeF 2 sensor structures requires the optical and MO constants for all layer media. These are available in the literature except those for ultrathin FeF 2 films. Their bulk counterpart, FeF 2 single crystals have rutile-type structure (D 4h ) and become anti-ferromagnetic at temperatures T<T N =78.4 K [START_REF] Ohlmann | Antiferromagnetic Resonance in FeF 2 at Far-Infrared Frequencies[END_REF]. Infrared optical absorption, magnetic circular dichroism (MCD), and light scattering studies of FeF 2 have focused on antiferromagnetic resonance [START_REF] Ohlmann | Antiferromagnetic Resonance in FeF 2 at Far-Infrared Frequencies[END_REF], optical transitions with spin change [START_REF] Russell | Zeeman Splitting of Narrow Lines in Crystalline MnF 2 , FeF 2 , and CoF 2 . Evidence for Spin-Wave Excitations[END_REF][START_REF] Mcclure | Electronic Transitions with Spin Change in Several Antiferromagmetic Crystals[END_REF], light scattering by magnons [START_REF] Cheng | Magnetic Circular Dichroism of Sharp Optical Transitions in Antiferromagnetic FeF 2[END_REF], Raman optical activity [START_REF] Lockwood | Magnetic Raman optical activity in FeF 2[END_REF], etc. Giordano et al. measured the infrared reflectivity spectrum at photon energies between 0.05 and 0.6 eV on an optically clear light yellow 4 mm thick FeF 2 single crystal [START_REF] Giordano | Infrared spectru[END_REF]. McClure et al. [START_REF] Mcclure | Electronic Transitions with Spin Change in Several Antiferromagmetic Crystals[END_REF] and Cheng et al. [START_REF] Cheng | Magnetic Circular Dichroism of Sharp Optical Transitions in Antiferromagnetic FeF 2[END_REF] reported two bands in the absorption spectrum: One centered near 2.67 eV and another stronger one centered near 3.20 eV. However, above the photon energy of 0.08 eV, no systematic information on optical spectra in FeF 2 is available.

The present work reports on the room temperature optical constants n and k of submicron FeF 2 films in the photon energy range between 1.3 and 5.2 eV (240-950 nm), determined from reflection spectroscopic ellipsometry (SE). The films were MBE grown on noble metal reflectors separated from GaAs (001) substrates by a seed Fe layer. A thin protecting Au layer was added on the top of the structure since the FeF 2 films were found chemically unstable when exposed to ambient humidity. The n and k extracted from the ex situ SE data were then used for the design of FeF 2 /Fe/FeF 2 structures with an optimized MO response, in reflection, at polar magnetization, and normal light incidence, for a wavelength of 810 nm. The FeF 2 /Fe/FeF 2 structures were characterized using MO reflection (Kerr) spectroscopy at polar magnetization. In the spectral range from 1.4 to 4.5 eV, the best theoretical fit to the MO experimental data was achieved by replacing the sharp Fe/FeF 2 interfaces with 2 nm thick transition layers. These transition layers were characterized by the optical and MO constants of an effective medium consisting of Fe (50%) and FeF 2 (50%).

Samples

The optical constants n and k in FeF 2 were computed from SE experiments using two structures denoted as Sample A and Sample B (fig. 1). Their composition profiles are listed in table 1 The n and k of FeF 2 extracted from the SE data on samples A and B were used in the design of Au/FeF 2 /Fe/FeF 2 /Ag/Fe/GaAs multilayers, samples C and D, with an optimized MO response at the desired wavelengths. The structures of samples C and D are shown in figure 2, while individual layer thicknesses for samples C and D are listed in table 2. The details of the sample preparation were given in the previous paper [START_REF] Lopusnik | Optimization of magneto-optical response of FeF 2 /Fe/FeF 2 sandwiches for microwave field detection[END_REF]. We would like to point out that fluorides on metallic templates frequently grow in columnar fashion, and may form rough interfaces between Fe/FeF 2 [START_REF] Malkinski | Correlation between microstructural and magnetic properties in Fe/KCoF 3 bilayers[END_REF][START_REF] Malkinski | Microstructure and magnetic properties in Fe/KCoF 3 thin films[END_REF].

Spectroscopic ellipsometry of FeF 2

The Au/FeF 2 /Au/Ag/Fe/GaAs sandwiches were characterized using a UVISEL spectroscopic ellipsometer (Horiba Jobin Yvon) with the optical element sequence PSMA (Polarizer-Sample-Modulator-Analyzer). The phase of the probing polarized monochromatic beam is modulated using a photoelastic modulator (PEM), and the resulting AC photoelectronic signal at the photodetector output is converted to DC using a lock-in amplifier. The spectra were measured at photon energies ranging from 1.3 to 5.2 eV (from 240 to 960 nm) in 0.02 eV steps, and at the incidence angles of 60°, 70°and 75°. Two azimuth setting for the PEM, M = 0°and M = 45°, had to be used to acquire unique values of the ellipsometric angles and . Figure 3 shows the experimental and modeled and for Sample A at the incidence angle of 70°. Sample B produced similar results, deviating by less than 1% from Sample A, and thus only Sample A is shown and discussed for simplicity. The modeling procedure is described next.

The analytic complex relative scalar permittivity in the polycrystalline FeF 2 layer, ε(E), was computed using SCOUT software for optical spectroscopy [14]. The Kim extended oscillator model [15], developed from the work by Kim et al. [START_REF] Kim | Modeling the optical dielectric function of semiconductors: Extension of the critical-point parabolic-band approximations[END_REF], was chosen as the most appropriate for FeF 2 . In the model, ε(E), in the energy range of 1-6 eV, can conveniently be expressed in terms of a sum of three oscillators and a constant term (ε ∞ ), i.e.

= ∞ + = 3 1 ) ( ) ( ) ( i i e E E χ ε ε (1)
where the i-th component of electric susceptibility is given by

E E E E E E i i i P i e ) ( i ) ( ) ( 2 2 ) ( 0 2 ) ( ) ( - - = χ (2)
- + - = Γ 2 ) ( ) ( 0 2 ) ( ) ( ) ( 1 1 exp ) ( i i i i i E E E (3)
where (i) is the damping constant. The parameter , ) (i the so-called Gauss-Lorentz switch, is allowed to vary between 0 and . The value 0 been successfully applied to a broad class of metals [START_REF] Raki | Optical properties of metallic films for vertical-cavity optoelectronic devices[END_REF], semiconductors, and dielectrics [START_REF] Djuriši | Progress in the room temperature optical functions of semiconductors[END_REF].

For CaF 2 and SrF 2 , both transparent fluorides, ε(E) between 1 and 6 eV can be approximated with a single Lorentz oscillator term with ≈

) ( 0 i E
15.5 eV assigned to electric dipole allowed inter-band transitions [START_REF] Wemple | Behavior of the electronic dielectric constant in covalent and ionic materials[END_REF][START_REF] Palik | Calcium Fluoride (CaF 2 )[END_REF]. In FeF 2 , there are also contributions to ε(E) due to 3d electrons of Fe 2+ . The modeling process started with six fitting parameters, i.e., the optical constants of FeF 2 and the thicknesses of the four topmost layers Au/FeF 2 /Au/Ag with a fixed thickness of the Fe seed layer.

Fixed parameters included the Lorentz oscillator at ≈ In the transparent fluorides CaF 2 and SrF 2 , k 0, and n varies between 1.42 (1.5 eV) and 1.48 (5.0 eV) [START_REF] Wemple | Behavior of the electronic dielectric constant in covalent and ionic materials[END_REF][START_REF] Palik | Calcium Fluoride (CaF 2 )[END_REF]. The higher values of n, non-zero k and in FeF 2 (figures 4 and 5), originate from electron transitions involving 3d levels of Fe 2+ ions in octahedral coordination [START_REF] Leslie | An Introduction to Transition Metal Chemistry, Ligand Field Theory[END_REF][START_REF] Sugano | Multiplets of Transition-Metal Ions in Crystals[END_REF].

) ( 0 i E [eV] ) i ( p E [eV] (i) [eV] ) i ( ) 1 ( e χ 3

MO spectroscopy of FeF 2 /Fe/FeF 2

The use of reflection MO (Kerr) spectroscopy for the characterization of FeF 2 /Fe/FeF 2 multilayers is illustrated on Samples C and D. MO Kerr rotation and ellipticity spectra at polar magnetization and nearly normal incidence were measured in the photon energy range from 1.3 to 4.5 eV (275-950 nm).

The optical element sequence in the set-up consisted of 450W Xe arc lamp -monochromatorpolarizer (0°) -calibrated DC Faraday rotator -AC Faraday rotator -calibrated phase plate (0°or 90°) -sample -analyzer (90°) -photomultiplier. Samples were subjected to a magnetic field of ±0.3 Tesla perpendicular to their surface (polar magnetization, ±M). This field is about seven times smaller than the field required to saturate thin iron films (2.1 Tesla) out of plane. The rotation spectra were determined with the phase plate removed, i.e. at zero phase retardation, from the current difference in the DC Faraday rotator at ±M. The procedure was repeated with the phase plate of a known retardation, , inserted into the sequence. The ellipticity was deduced from the compensation currents in the DC Faraday rotator at zero and retardations. Figure 6 shows the MO polar Kerr rotation and ellipticity spectra in Samples C and D. The spectra display maxima in the MO response near 1.5 eV, as required for the sensor operation. The structure centered near 3.9 eV, more distinct in Sample D, is due to the shifted plasma resonance in Ag [START_REF] Wooten | OpticalProperties of Solids[END_REF]. The spectra were simulated with the help of a multilayer model with abrupt interfaces [START_REF] Liskova | Magnetooptical AlN/Fe/AlN structures optimized for the operation in the violet spectral region[END_REF], schematically shown in figure 2a, using the optical constants of FeF 2 from table 4. For the other layer media, the literature values were used [21-23, 28, 29]. The thicknesses of the four topmost layers (Au/FeF 2 /Fe/FeF 2 ) were allowed to vary. The results are included in table 2 as entries C1 and D1. The model correctly reproduced both the position of the maxima in MO response at 1.53 eV, trends in the spectra, and provided refined thicknesses for the layers. However, the thickness values for the bottom FeF 2 layer were significantly higher compared to those evaluated during the growth. The abrupt interface model, however, ignores the roughness at FeF 2 /Fe interfaces. To account for this effect, a more realistic model with transition layers at the Fe/FeF 2 interfaces, schematically shown in figure 2b, was employed. The addition of transition layers at both Fe/FeF 2 interfaces, with a total thickness of 4 nm, and characterized with the optical and MO constants of an effective medium [START_REF] Aspnes | Investigation of effective medium models of microscopic roughness by spectroscopic ellipsometry[END_REF] consisting of Fe (50%) and FeF 2 (50%), improved agreement between the thickness values evaluated during the growth (entries C and D in table 2) and the modeling (entries C2 and D2 in table 2). In the model with transition layers, the MO maxima near 1.53 eV are shifted by +0.03 eV with respect to those obtained from the model with abrupt interfaces. Up to this small shift, the MO spectra computed with these models are practically the same. Only the spectra for the effective medium model with transition layers are therefore included in figure 6. Two transition layers sandwiching Fe of equal thickness (2 nm) were considered but the model was found insensitive to thickness redistribution between the two Fe/FeF 2 interfaces, provided that the total transition layer thickness was kept at 4 nm.

Summary

The paper reports the room temperature permittivity, ε(E), of polycrystalline submicron FeF 2 films deduced from spectroscopic ellipsometry on sandwiches at photon energies, E, between 1.3 and 5.2

eV. The ε(E) spectra display features typical for a dielectric 3d compound with a high density of magnetic cations, and can be conveniently expressed analytically as a sum of three physically reasonable oscillators. The two lower oscillators correspond to 3d electron transitions involving Fe 2+ pairs coupled via F -, the mechanism typical for other compounds with high density magnetic cations coupled by superexchange [START_REF] Fontijn | A consistent interpretationof the magneto-optical spectra of spinel type ferrites (invited)[END_REF]. Their contributions to ε(E) are superimposed on a wing of an allowed electric dipole transition with a well separated resonance around 15 eV. A lower degree of covalency in FeF 2 results not only in much lower Néel temperature (T N ), but also in lower oscillator strengths of 3d electron transitions. This, in turn, results in a two orders in magnitude lower absorption coefficient (figure 5) compared to antiferromagnetic and ferrimagnetic oxides.

The ε(E) spectra enabled a more precise evaluation of the magneto-optic (MO) response in FeF 2 /Fe/FeF 2 sandwiches considered for MO sensing of microwave current with submicron resolution.

The measured maximum MO rotation at a 15% saturation of Fe reached 1 degree. MO studies of FeF 2 /Fe/FeF 2 structures supported the expected presence of intermediate mix layers at the Fe/FeF 2 interfaces. Their contribution to MO response was approximated using effective medium theory. The information on ε(E) of FeF 2 (not available in the literature so far) will be useful in interface studies of layered systems with FeF 2 by means of MO magnetometry, MO magnetic domains imaging, MO spectroscopy, and spectroscopic ellipsometry. In particular, it may be used for selective sensing of Fe layers in exchange coupled Fe/FeF 2 /Fe sandwiches. The optical constants of FeF 2 submicron films may serve as a guide to optical constants of bulk FeF 2 .

Figure 1 .

 1 Figure 1. Structure of FeF 2 sandwiches.

Figure 2 .

 2 Figure 2. Structure of FeF 2 /Fe/FeF 2 multilayers: abrupt interfaces (a) and effective medium model with transition interface layers (b).

Figure 3 .

 3 Figure 3. Measured and modeled spectra of the ellipsometric parameters, and , in Sample A at an incidence angle of 70°.

  is considered Lorentzian. The contribution to ε(E) originating from the spectral range

E 3 ( 0 E

 30 15 eV, and literature values for the optical constants of Au, Ag, Fe and GaAs[START_REF] Johnson | Optical constants of the noble metals[END_REF][START_REF] Johnson | Optical constants of the transition metals[END_REF][START_REF] Palik | Gallium Arsenide (GaAs)[END_REF]. The FeF 2 layer thickness, t=118 nm, deduced from modeling, was reasonably close to t=120 nm measured with a thickness monitor during growth. Using the modeled t and ε(E), SCOUT software provided and consistent with the experiment, as shown in figure3. The parameters entering Eq. (1) for ε(E) are collected in table 3. The procedure provided ) i ( at the resonance photon energies for the first ( 119 and 0.841, respectively, corresponding to Gaussian profiles. The exact position of ) has a negligible effect on the parameters of the first and second oscillators, and consequently on the permittivity in the spectral region of interest (E<6 eV). For example, a shift of

0 EE

 0 modified the permittivity by less than 0.1%. The large separation between the strong allowed transition centered at ) 3 ( and the transitions involving 3d electrons, centered at ) is typical for fluorides[START_REF] Leslie | An Introduction to Transition Metal Chemistry, Ligand Field Theory[END_REF].

Figure 4 .Table 3 .

 43 Figure 4. Real (a) and imaginary (b) parts of the complex relative permittivity in FeF 2 as a function of photon energy, E, determined from spectral ellipsometry on Sample A.

Figure 5 .

 5 Figure 5. Absorption coefficient, λ π α / 4 k = , of FeF 2 as a function of photon energy, E, determined from ellipsometry measurements on Sample A.

Figure 6 .

 6 Figure 6. Magneto-optical Kerr rotation (circles) and ellipticity (triangles) spectra as a function of photon energy, E, measured at polar magnetization ±0.3 Tesla in Samples C and D. The computed spectra are represented with solid lines.

Table 1 .

 1 . Samples for spectral ellipsometry.

		Au [nm]	FeF 2 [nm]	Au[nm] Ag[nm] Fe[nm]	GaAs
	A	0.5	120	30	20	0.6	Substrate
	B	0.5	120	60	6.5	0.6	Substrate

Table 2 .

 2 Multilayers with optimized MO response.

		Au	FeF 2	Effective	Fe	Effective	FeF 2	Ag	Fe	GaAs
		[nm]	[nm]	medium	[nm]	medium	[nm]	[nm]	[nm]
				[nm]		[nm]
	C	0.5	2.5	-	8	-	50	50	0.6	Substrate
	C1	0.3	3	-	8.5	-	63	50	0.6	Substrate
	C2	0.3	1	2	6.8	2	58	50	0.6	Substrate
	D	0.5	2.5	-	8	-	70	50	0.6	Substrate
	D1	0.3	3	-	9	-	98	50	0.6	Substrate
	D2	0.3	1	2	7.3	2	90	50	0.6	Substrate

Table 4 .

 4 Optical constants n and k of polycrystalline FeF 2 as a function of wavelength, .

	[nm]	n	k	[nm]	n	k
	235 1.84375 0.15733	400	1.97849 0.08298
	245	1.88023 0.17467	425	1.98678 0.06481
	255	1.91734 0.17021	450	1.98721 0.04764
	265	1.94265 0.15116	475	1.98325 0.03436
	275	1.95361 0.12881	500	1.97771 0.02494
	285	1.95434 0.11045	525	1.97206 0.01842
	295	1.95006 0.09844	550	1.96686 0.0139
	305	1.94467 0.09238	575	1.96229 0.01073
	315	1.94053 0.09074	600	1.95832 0.00846
	325	1.93876 0.09183	700	1.94712 0.00392
	335	1.93965 0.09411	800	1.94051 0.00227
	345	1.94301 0.09636	900	1.93629 0.00153
	350	1.94547 0.09719	1000 1.93341 0.00114
	375	1.9623 0.0955

Acknowledgments

Experimental assistance of Martin Veis is acknowledged. This project was supported by NSF (DMR 0605629 and DMR 0907053), Research Corporation (CC6471), CU-NIST Joint Project, Czech Science Foundation grants (202-06-0531 and 102/09/H074), Grant Agency of Charles University (3146/2008) and by Ministry of Education, Youth and Sport of the Czech Republic (MSM 6198910016, 0021520834 and ME09045).


M Gladysiewicz
email: marta.gladysiewicz@pwr.wroc.pl

R Kudrawiec

J Misiewicz

G Cywi Ski

M Siekacz

C Skierbiszewski

Broadening of intersubband and interband transitions in InGaN/AlInN multi quantum wells

The authors have applied modulation spectroscopy to study intersubband (IS) and interband (IB) transitions in InGaN/AlInN multi quantum well (QW) structures with the QW width varying from 1.3 to 1.8 nm, and proposed a simple method to analyze the broadening of IS and IB transitions. Within this method the measured broadenings of IS and IB transitions have been compared with theoretical calculations performed within the effective mass approximation for a QW system with various QW width fluctuations. In the framework of this comparison it has been found that the QW width fluctuation in the investigated InGaN/AlInN QW system is close to 2 monolayers.

In order to achieve intersubband (IS) absorption at fiber-optics telecommunication wavelengths (i.e., 1.3 and 1.55 µm) in GaN-based quantum well (QW) structures, the QW width has to be smaller than 10 monolayers (MLs) [1][2][3][4][5][6][7][8]. In this regime of QW width, its small fluctuation (even 1 ML) significantly influences the energies of optical transitions.

Recently, it has been shown that individual absorption peaks corresponding to an integer number of MLs can be observed for GaN/AlN quantum wells [7,8]. The homogeneous broadening of IS absorption for this system (i.e., GaN/AlN QW with no QW width fluctuation) has been found to be ~67 meV [7]. In the case of indium containing QWs (i.e., InGaN/AlInN QWs), the broadening of optical transitions is expected to be higher because of the alloying effect which leads to higher inhomogeneities in this system, including indium segregation effects and, very likely, higher fluctuations of the QW width. However, the incorporation of indium atoms into GaN/AlN QWs is needed or/and profitable in many applications and therefore InGaN/AlInN QWs are the subject of many studies and applications in optoelectronic devices.

In this work we have applied modulation spectroscopy [9] to study energies and broadenings of IS and interband (IB) transitions in InGaN/AlInN multi QWs (MQWs) of various QW widths. So far the IS absorption in GaN/AlN QW structures was investigated by the transmission and/or photo induced absorption spectroscopy. In this paper we have applied photomodulated transmission (PT) [START_REF] Kudrawiec | [END_REF] to investigate the energy and broadening of IS absorption instead of the standard transmission measurement. The broadening of IB transitions has been investigated by contactless electroreflectance (CER) [11,12]. In order to evaluate the broadening of IS and IB transitions and determine the QW width fluctuation a simple theoretical model has been proposed.

For this study we grew InGaN(d)/AlInN(3nm) multi QWs (MQWs) samples of various QW widths (d=1.3, 1.6, and 1.8nm) by plasma-assisted molecular-beam epitaxy on Ga polar GaN/Al 2 O 3 templates. 40 period InGaN/AlInN MQWs were grown after a 20 nm In 0.1 Ga 0.9 N buffer layer and capped with an undoped 100 nm In 0.1 Ga 0.9 N layer. The InGaN wells were uniformly doped with Si at a level of•10 20 cm -3 . The structural quality of these samples were studied by high resolution transmission electron microscopy (TEM) and X-ray diffraction (XRD) mapping techniques. Both XRD and TEM results confirm good structural quality [START_REF] Cywi Ski | North America Molecular Beam Epitaxy conference[END_REF]. The indium concentration in InGaN QWs and AlInN barriers has been determined to be ~5 and ~0.5%, respectively.

A conventional experimental set-up with a tungsten halogen lamp (250 W) as a probe light source, a 0.55 m monochromator and an appropriate detector (PbS photodiode and photomultiplier for infrared and ultraviolet spectral range, respectively) was applied for obtaining photomodulated transmittance and contactless electroreflectance spectra. For optical measurements, samples were prepared in a way to form a multipass waveguide as shown in Ref. 6. For PT measurements the probe beam was sent through this waveguide and the photomodulation was provided by a laser beam (266 nm line from YAG laser) at the normal incidence on the sample surface. For CER experiments, samples were mounted in a capacitor with one semitransparent electrode. This electrode was kept at a distance of ~0.5 mm from the sample surface while the sample itself was fixed on the bottom copper electrode. A maximum peak-to-peak alternating voltage of ~3.0 kV with the frequency of 285 Hz was applied. CER spectra were measured at near normal incidence of the probing beam on the sample surface.

Other relevant details of CER measurements were discussed in Ref. 12. The broadening of 1e-2e transition is a few times larger than the homogeneous broadening of IS absorption reported for indium free systems (~350 meV for InGaN/AlInN QWs versus 67 meV for GaN/AlN MQWs). In addition, it is visible that this broadening increases when the QW width decreases. In order to extract energies and broadenings for the intersubband absorption in the three samples, the standard fitting procedure assuming Lorentzian lineshape [START_REF] Aspnes | [END_REF] has been applied. According to this model PT spectrum can be fitted using the following formula

( ) ( ) m i i E E e C E T T - ⋅ Γ ⋅ + - ⋅ = ∆ 0 ϑ , (1)
where E 0 and Γ are the energy and the broadening parameter of the transition, and C and ϑ are the amplitude and phase of the line, respectively. The term m refers to the type of critical point for optical transitions. In our case, it has been assumed that m = 3 since according to our experience such a line is very similar to the third derivative Gaussian lineshape, i.e. the most appropriate lineshape for inhomogeneous system. The fitting curves are shown by thick grey lines together with the moduli of individual resonances (dashed lines) obtained according to Eq. 2

( ) ( ) [ ] 2 2 2 0 m E E C E Γ + - = ∆ρ (2)
with parameters taken from the fit. In order to analyze broadenings of intersubband absorption and to compare them with literature data for GaN/AlN systems, we have determined the full width at half maximum (FWHM) for the moduli peak.

Figure 1 (d) shows energies of 1e-2e transition obtained from PT measurements for the three MQW samples together with theoretical calculations, which were performed in the framework of the electron effective mass approximation with material parameters taken from Ref. [START_REF] Aspnes | [END_REF]. The built-in electric field inside the InGaN/AlInN MQW has been calculated from piezoelectric and spontaneous polarizations in this system [15] with appropriate periodic boundary conditions [16] assuming 20% screening of the electric field due to the Si doping.

In order to calculate the broadening of intersubband absorption, we made an assumption that it entirely originates from the QW width fluctuations. This is a "border case" neglecting the influence of In-segregation effects, which is expected to be small in these samples because of the low indium content. This broadening is defined as γ is a contribution which is associated with QW width fluctuations. This contribution can be calculated according to Eq. ( 3)

IS IS IS 1 0 γ γ γ + = ,
w w E e e IS ∆ ∂ ∂ = 2 1 1 γ , ( 3
)
where w is the QW width and w ∆ is its fluctuation. In order to determine the magnitude of QW width fluctuations, this parameter is treated as a free parameter.

Figure 1(e) shows theoretical calculations of the broadening parameter of 1e-2e transition for w ∆ = 1, 2, and 3 ML together with experimental data, i.e., the FWHM obtained from PT measurements. A good agreement between experimental data and theoretical calculation is observed for w ∆ ≈ 2. This result is consistent with TEM measurements for these samples [START_REF] Cywi Ski | North America Molecular Beam Epitaxy conference[END_REF].

The broadening of the fundamental IB transition in these MQW has been obtained from CER measurements accompanied with the standard fitting procedure [12], see Fig. γ is associated with the QW width fluctuations. The γ 1 component is calculated according to Eq. ( 4)

w w E C H IB ∆ ∂ ∂ = 1 1 1 γ , ( 4
)
where w is the QW width and w ∆ is its fluctuation. We expect that the magnitude of QW width fluctuation depends on the growth conditions (i.e., mostly on the growth temperature). In our case, the three MQW samples were obtained at exactly the same growth conditions and they differ between themselves by the nominal QW width only. In Fig. 3 we have compared our calculations with experimental data for InGaN/AlInN MQW samples obtained at very similar growth conditions [6,[START_REF] Cywi Ski | North America Molecular Beam Epitaxy conference[END_REF]. Like in the previous case [Fig. 1(e)] we have found that a good agreement between experimental data and theoretical calculations is observed when the QW width fluctuation is close to 2 ML. In addition, such a comparison has been performed for binary GaN/AlN MQWs from Ref. [7] where the authors have identified that the homogenous broadening of IS absorption for GaN/AlN system equals 67meV. Our analysis suggests that for some GaN/AlN MQWs the larger broadening of IS absorption appears because of a QW width fluctuation. This fluctuation has been estimated to be below 1 ML. It means that 1ML thick terraces can exist in these MQWs but the size of these terraces is rather small. For large terraces (>1 µm) individual peaks, which are related to the integer number of ML, are expected in absorption spectra. Such peaks were observed for some GaN/AlN MQWs in Ref. 7. In the case of Incontaining MQWs, such peaks were not resolved in absorption spectra since the QW width fluctuations are more smeared in this system due to the alloying effect. In addition, the incorporation of indium atoms into GaN/AlN QW enhances the QW width fluctuation (compare solid and open points in Fig. 3). Indium also leads to the energy gap fluctuations but this effect is small in our InGaN/AlInN MQWs because of low indium concentration.

In conclusion, it has been clearly shown that energies and broadenings of IS and IB transitions in InGaN/AlInN MQW structures can be studied by modulation spectroscopy. In addition, a simple theoretical model to determine the QW width fluctuation from the broadening of IS and IB transitions has been presented in this work. Within this model the fluctuation of the QW width has been found to be 2 monolayers. It is worth noting that this model can be applied to the analysis of broadening parameters obtained from PT and CER (i.e. absorptionlike measurements). In the case of photoluminescence spectroscopy, the fluctuation of QW thickness cannot be determined within this theoretical model because of emissionlike character of photoluminescence, i.e., its high sensitivity to localized states. It means that the modulation spectroscopy is a very useful method to evaluate the quantum well width homogeneity in narrow QW systems. This finding is very important since optical measurements are much cheaper and easier than TEM measurements which need the time consuming sample preparation and are destructive for samples. This work was supported by the Polish Ministry of Science and Higher Education as scientific projects Nos. N507 011 32/0500 and N202 41090 33. In addition, R.K. acknowledges for financial support from the Foundation for Polish Science. [6] and transmission [7,[START_REF] Cywi Ski | North America Molecular Beam Epitaxy conference[END_REF]18]).

Figure 1 (

 1 Figure 1(a), 1(b), and 1(c) show room temperature PT spectra in the infrared spectral

where IS 0 γ

 0 corresponds to a system without QW width fluctuations (we assume that IS 0 γ = 67meV after Ref. 7) and IS 1

  2 (a), (b) and (c). The fitting curves are shown by thick grey lines together with the moduli of individual resonances (dashed lines). The strongest CER resonance at 2.9 eV is attributed to the absorption in the InGaN cap layer[12]. Remaining CER resonances are associated with the absorption in the MQW region. The notation lHkC denotes the transition between l-th hole subband and k-th conduction subband. It is worth noting that the 1H1C transition is observed in these MQW structures in CER spectroscopy (absorptionlike technique) despite the silicon doping in the QW region. It means that electron concentration in some InGaN QWs can be smaller. Such a situation is expected close to the sample surface in the surface depletion region. On the other hand it is worth noting that the IB absorption at higher k states is possible for QWs with the occupied k = 0, but this absorption is much weaker than absorption at k = 0, especially in modulation spectroscopy.

Figure 2 (γ

 2 Figure 2(d) shows the results of theoretical calculations of energy of 1H1C transition

Figure 2 (

 2 Figure 2(e) shows theoretical calculations of the broadening of 1H1C transition for w ∆ = 1, 2, and 3 ML together with experimental data, i.e., the FWHM for 1H1C transition

Figure Captions : FIG. 1 .

 :1 Figure Captions:

  FIG. 1.