Law-invariant functionals that collapse to the mean
Fabio Bellini, Pablo Koch-Medina, Cosimo Munari, Gregor Svindland
aa r X i v : . [ q -f i n . M F ] S e p Law-invariant functionals that collapse to the mean
Fabio Bellini
Department of Statistics and Quantitative MethodsUniversity of Milano-Bicocca, Italy [email protected]
Pablo Koch-Medina, Cosimo Munari
Center for Finance and Insurance and Swiss Finance InstituteUniversity of Zurich, Switzerland [email protected] , [email protected] Gregor Svindland
Institute of Probability and Statistics and House of InsuranceLeibniz University Hannover, Germany [email protected]
September 10, 2020
Abstract
We discuss when law-invariant convex functionals “collapse to the mean”. More precisely, we showthat, in a large class of spaces of random variables and under mild semicontinuity assumptions, theexpectation functional is, up to an affine transformation, the only law-invariant convex functionalthat is linear along the direction of a nonconstant random variable with nonzero expectation. Thisextends results obtained in the literature in a bounded setting and under additional assumptions onthe functionals. We illustrate the implications of our general results for pricing rules and risk measures.
Keywords : law invariance, affinity, translation invariance, pricing functionals, risk measures
In the influential paper Wang et al. [31], the authors describe an axiomatic approach to insurance pricingand provide a representation of admissible pricing rules in terms of Choquet integrals. One of the keyaxioms put forward is law invariance, stipulating that prices depend on the contracts’ payoffs only throughtheir probability distribution with respect to the “physical” probability measure. At the end of thatpaper, it is pointed out that law-invariant pricing rules based on Choquet integrals could also be usedto harmonize the pricing of insurance products and financial derivatives. It is, however, not difficult tosee that law invariance of the pricing functional cannot be expected to hold in general. For instance,the Fundamental Theorem of Asset Pricing asserts that, under suitable conditions, in a financial marketthat is frictionless and free of arbitrage opportunities, prices can be essentially expressed as expectationswith respect to a “risk-neutral” probability measure. It is with respect to this probability measure thatprices in this market are law invariant. Hence, for financial market prices to exhibit law invariance with1espect to the “physical” probability measure, the “physical” and the “risk-neutral” measures would haveto coincide. This is, however, never the case with the sole exception of a market in which the expectedreturns under the “physical” measure is the same for all assets.Prompted by the attempts in Wang [29] and Wang [30] to carry out the harmonization suggested inWang et al. [31] by means of law-invariant pricing rules, Castagnoli et al. [9] show that postulatingthe law invariance of pricing functionals is questionable also in a more general setting than that offrictionless financial markets. This was accomplished by proving that the expectation under the physicalprobability measure is the only functional defined on the space of bounded payoffs that is law invariant,sublinear, increasing, comonotonic, linear on the space of constant random variables (properties satisfiedby the pricing rules considered in Wang [29] and Wang [30]), and linear on a vector space containinga nonconstant random variable (a property satisfied by the pricing functional in any market in whichat least one of the risky traded assets is frictionless), i.e. any such functional “collapses to the mean”.This result was improved in Frittelli and Rosazza Gianin [16] by replacing sublinearity by convexity anddropping comonotonicity. We note that the more recent literature on market-consistent valuation, see e.g.Malamud et al. [25], Pelsser and Stadje [26], Dhaene et al. [11], is aware of this limitation and requiresonly partial law invariance, e.g. for payoffs that are driven by pure insurance risk only. We also refer tothe economic premium principles in B¨uhlmann [7] and B¨uhlmann [8] as an early example of premiumprinciples that are not law invariant on the entire reference payoff space.Beyond the implications for pricing in financial markets, the results in Castagnoli et al. [9] and Frittelliand Rosazza Gianin [16] raise the question of whether the “collapse to the mean” occurs for a largerclass of law-invariant functionals defined on a space X of random variables. We allow X to belong to afairly general class of locally-convex spaces consisting of integrable random variables and containing allessentially bounded random variables. In this note, we show that the collapse to the mean is the result of aninherent tension between law invariance and linearity. We prove that, under suitable lower semicontinuityproperties (which are always satisfied in the setting of [9] and [16]), the expectation functional is, upto an affine transformation, the only law-invariant convex functional ϕ : X → ( −∞ , ∞ ] that is linearalong a nonconstant random variable Z with nonzero expectation. The strategy we follow is simple. Thekey observation is that the set of random variables that have the same distribution as Z spans a densesubspace of X . As a result, linearity along Z together with law invariance forces linearity on a densesubspace. The lower semicontinuity assumption then implies that ϕ is linear on the entire space. Theresult follows by noting that the only continuous linear functionals that are law invariant are multiples ofthe expectation functional.The note is organized as follows. In Section 2 we introduce the setting together with the necessary notationand terminology. In Section 3 we show that convex functionals that are lower semicontinuous and linearalong a given direction enjoy the stronger property of being translation invariant along the same direction.In Section 4 we establish our main result on the “collapse to the mean”. Some applications of our resultare discussed in Section 5. Let (Ω , F , P ) be a nonatomic probability space. We denote by L the set of equivalence classes of randomvariables, i.e. Borel measurable functions X : Ω → R , with respect to almost-sure equality under P . Inline with standard practice, we do not distinguish explicitly between an element of L and any of itsrepresentatives. In particular, the elements of R are naturally identified with random variables that arealmost-surely constant. For two random variables X, Y ∈ L we write X ∼ Y whenever X and Y havethe same probability law with respect to P . The expectation under P is denoted by E . The standardLebesgue spaces are denoted by L p for p ∈ [1 , ∞ ]. We say that a set X ⊂ L is law invariant if X ∈ X for every X ∈ L such that X ∼ Y for some Y ∈ X .2 ssumption 2.1. We denote by ( X , X ∗ ) a pair of law-invariant vector subspaces of L containing L ∞ .We denote by σ ( X , X ∗ ) the weakest linear topology on X with respect to which, for every Y ∈ X ∗ , thelinear functional on X given by X E [ XY ] is continuous. Remark 2.2. (i) Note that, under our assumptions, σ ( X , X ∗ ) is not metrizable. As a result, in general,one needs to work with nets instead of sequences. Recall that a net ( X α ) ⊂ X converges to an element X ∈ X with respect to the topology σ ( X , X ∗ ) if and only if E [ X α Y ] → E [ XY ] for every Y ∈ X ∗ .(ii) Note that for every nonzero X ∈ X there exists Y ∈ X ∗ , namely either Y = { X> } or Y = { X< } (which belong to X ∗ because they are bounded), such that E [ XY ] = 0. Similarly, for every nonzero Y ∈ X ∗ there exists X ∈ X such that E [ XY ] = 0. Hence, ( X , X ∗ ) is a dual pair. In particular, Theorem5.93 in Aliprantis and Border [1] implies that, endowed with σ ( X , X ∗ ), the space X is a locally-convexHausdorff topological vector space whose topological dual can be identified with X ∗ . Example 2.3 ( Orlicz Spaces ) . Let
Φ : [0 , ∞ ) → [0 , ∞ ] be an Orlicz function, i.e. a convex, left-continuous, increasing function which is finite on a right neighborhood of zero and satisfies Φ(0) = 0 . Theconjugate of Φ is the function Φ ∗ : [0 , ∞ ) → [0 , ∞ ] defined by Φ ∗ ( u ) := sup t ∈ [0 , ∞ ) { tu − Φ( t ) } . Note that Φ ∗ is also an Orlicz function. For every X ∈ L define the Luxemburg norm by k X k Φ := inf (cid:26) λ ∈ (0 , ∞ ) ; E (cid:20) Φ (cid:18) | X | λ (cid:19)(cid:21) ≤ (cid:27) . The corresponding Orlicz space is given by L Φ := { X ∈ L ; k X k Φ < ∞} . The heart of L Φ is the space H Φ := (cid:26) X ∈ L Φ ; ∀ λ ∈ (0 , ∞ ) : E (cid:20) Φ (cid:18) | X | λ (cid:19)(cid:21) < ∞ (cid:27) . The classical Lebesgue spaces are special examples of Orlicz spaces. Indeed, if Φ( t ) = t p for p ∈ [1 , ∞ ) and t ∈ [0 , ∞ ) , then L Φ = H Φ = L p and the Luxemburg norm coincides with the usual p norm. Moreover, ifwe set Φ( t ) = 0 for t ∈ [0 , and Φ( t ) = ∞ otherwise, then we have L Φ = L ∞ and the Luxemburg normcoincides with the usual L ∞ -norm. Note that, in this case, H Φ = { } .In our nonatomic setting, L Φ = H Φ if and only if Φ satisfies the ∆ condition, i.e. there exist s ∈ (0 , ∞ ) and k ∈ (0 , ∞ ) such that Φ(2 t ) < k Φ( t ) for every t ∈ [ s, ∞ ) . A well-known example of a nontrivial H Φ with H Φ = L φ is obtained by setting Φ( t ) = exp( t ) − for t ∈ [0 , ∞ ) .In general, the norm dual of L Φ cannot be identified with a subspace of L . However, if Φ is finitevalued (so that H Φ = { } ), the norm dual of H Φ can always be identified with L Φ ∗ . For the case L p , for p ∈ [1 , ∞ ) , this is simply the well-known identification of the norm dual of L p with L pp − (with the usualconvention := ∞ ). For more details on Orlicz spaces we refer to Edgar and Sucheston [12].The pair ( X , X ∗ ) with X = L Φ and X ∗ ∈ { L Φ ∗ , H Φ ∗ , L ∞ } satisfies Assumption 2.1. In the following definition we introduce the necessary terminology for functionals.
Definition 2.4.
Let ϕ : X → ( −∞ , ∞ ] be a functional. The domain of ϕ is the setdom( ϕ ) := { X ∈ X ; ϕ ( X ) < ∞} . We say that the functional ϕ is: 31) proper if dom( ϕ ) is nonempty.(2) convex if ϕ ( λX + (1 − λ ) Y ) ≤ λϕ ( X ) + (1 − λ ) ϕ ( Y ) for all X, Y ∈ X and λ ∈ [0 , positively homogeneous if ϕ (0) = 0 and ϕ ( λX ) = λϕ ( X ) for all X ∈ X and λ ∈ (0 , ∞ ).(4) sublinear if it is both convex and positively homogeneous.(5) monotone if ϕ ( X ) ≥ ϕ ( Y ) for all X, Y ∈ X such that X ≥ Y .(6) law invariant if ϕ ( X ) = ϕ ( Y ) for all X, Y ∈ X such that X ∼ Y .(7) σ ( X , X ∗ ) -lower semicontinuous if for all nets ( X α ) ⊂ X and X ∈ X we have X α σ ( X , X ∗ ) −−−−−→ X = ⇒ ϕ ( X ) ≤ lim inf α ϕ ( X α ) . (8) norm-lower semicontinuous if for all sequences ( X n ) ⊂ X and X ∈ X we have X n k·k −−→ X = ⇒ ϕ ( X ) ≤ lim inf n →∞ ϕ ( X n )provided that X is equipped with a norm k · k .Finally, we say that the functional ϕ satisfies:(9) the Fatou property if for all sequences ( X n ) ⊂ X and X ∈ X we have X n a.s. −−→ X, sup n ∈ N | X n | ∈ X = ⇒ ϕ ( X ) ≤ lim inf n →∞ ϕ ( X n ) . To a proper functional ϕ : X → ( −∞ , ∞ ] we associate the dual functional ϕ ∗ : X ∗ → ( −∞ , ∞ ] defined by ϕ ∗ ( Y ) := sup X ∈X { E [ XY ] − ϕ ( X ) } . Note that ϕ ∗ is well defined and does not attain the value −∞ because ϕ is proper. The next propositionrecords the well-known dual representation of convex and lower semicontinuous functionals; see, e.g.,Theorem 2.3.3 in Z˘alinescu [32]. Proposition 2.5.
Let ϕ : X → ( −∞ , ∞ ] be proper, convex, and σ ( X , X ∗ ) -lower semicontinuous. Then,for every X ∈ X we have ϕ ( X ) = sup Y ∈X ∗ { E [ XY ] − ϕ ∗ ( Y ) } = sup Y ∈ dom( ϕ ∗ ) { E [ XY ] − ϕ ∗ ( Y ) } . The next example serves to highlight that requiring σ ( X , X ∗ )-lower semicontinuity for convex and law-invariant functionals is not as restrictive as it may seem at first sight. Example 2.6 ( Orlicz Spaces ) . The following results can be found in Proposition 2.5 in Bellini et al[4], which merely summarizes results from the literature (Jouini et al. [21], Svindland [28], and Gao etal. [17]. We also refer to Leung and Tantrawan [23] for abstract results beyond the Orlicz setting).If X is a general Orlicz space L Φ and ϕ : X → ( −∞ , ∞ ] is a proper, convex, and law invariant functional,then the following statements are equivalent:(a) ϕ is σ ( X , L ∞ ) -lower semicontinuous.(b) ϕ satisfies the Fatou property. f X is either L ∞ or an Orlicz heart H Φ for a finite Orlicz function Φ (in particular any L p with p ∈ [1 , ∞ ) ), then (a) is also equivalent to:(c) ϕ is norm lower semicontinuous.The example given in Remark 5.6 in Gao et al. [17] shows that, for a general Orlicz space, norm lowersemicontinuity does not always imply σ ( X , L ∞ ) lower semicontinuity. If ϕ is additionally monotone, then(a) is also equivalent to:(d) ϕ is continuous from below, i.e. for every increasing sequence ( X n ) ⊂ X and every X ∈ X we have X n a.s. −−→ X = ⇒ ϕ ( X n ) → ϕ ( X ) . Clearly, in all these cases, ϕ is also σ ( X , X ∗ ) -lower semicontinuous. The goal of this short section is to show the link between two properties of functionals that will play akey role in our main result in the next section, namely affinity and translation invariance. The functionalsconsidered in this section are not required to be law invariant. Throughout we assume that ( X , X ∗ ) is apair satisfying Assumption 2.1. For a set S ⊂ X we denote by span( S ) the smallest linear subspace of X containing S . If S = { Z } for some Z ∈ X , then we simply write span( Z ). Definition 3.1.
Let M be a linear subspace of X . We say that a functional ϕ : X → ( −∞ , ∞ ] is:(1) affine along M if M ⊂ dom( ϕ ) and the functional on M given by Z ϕ ( Z ) − ϕ (0) is linear. If M = span( Z ) for some Z ∈ X , then we simply say that ϕ is affine along Z . In this case, there exists a ∈ R such that for all X ∈ X and m ∈ R ϕ ( mZ ) = ϕ (0) + am. (2) translation invariant along M if ϕ is affine along M and for all X ∈ X and Z ∈ M ϕ ( X + Z ) = ϕ ( X ) + ϕ ( Z ) − ϕ (0) . If M = span( Z ) for some Z ∈ X , then we simply say that ϕ is translation invariant along Z . In thiscase, there exists a ∈ R such that for all X ∈ X and m ∈ R ϕ ( X + mZ ) = ϕ ( X ) + am. In both cases we have a = ϕ ( Z ) − ϕ (0). Remark 3.2. (i) Let
S ⊂ X and assume that ϕ : X → ( −∞ , ∞ ] is translation invariant along everyelement of S . Then, ϕ is translation invariant along span( S ). In particular, ϕ is affine on span( S ).However, note that ϕ need not be affine along span( S ) if it is affine along every element of S .(ii) Clearly, the only functionals that are translation invariant along X are those that are affine on X .By definition, translation invariance implies affinity. As shown by the next example, the converse impli-cation does not hold in general even if we assume that ϕ is convex.5 xample 3.3. Assume
W, Z ∈ L are linearly independent and define a functional ϕ : L → ( −∞ , ∞ ] by ϕ ( X ) = if X = αW + βZ for some α, β ∈ R with α < ,β if X = W + βZ for some β ∈ R , ∞ otherwise . It is not difficult to verify that ϕ is convex and also affine along Z . However, ϕ is not translation invariantalong Z because there exists no a ∈ R such that m = ϕ ( W + mZ ) = ϕ ( W ) + am = am for every m ∈ R . There are two notable classes of functionals for which affinity does imply translation invariance. The firstis the class of sublinear functionals.
Proposition 3.4.
Let ϕ : X → ( −∞ , ∞ ] be sublinear and S ⊂ X . If ϕ is affine along every element of S , then it is translation invariant along span( S ) .Proof. Recall that ϕ (0) = 0 by sublinearity and note that for every fixed Z ∈ S the functional ϕ is linearon span( Z ) by affinity. Hence, for every X ∈ X we have ϕ ( X + Z ) ≤ ϕ ( X ) + ϕ ( Z )= ϕ ( X + Z − Z ) + ϕ ( Z ) ≤ ϕ ( X + Z ) + ϕ ( − Z ) + ϕ ( Z )= ϕ ( X + Z )by sublinearity. This shows that ϕ is translation invariant along every element of S . Remark 3.2 nowimplies that ϕ is translation invariant along span( S ).We saw in Example 3.3 that in the preceding result we cannot replace sublinearity by convexity. How-ever, we may replace sublinearity by σ ( X , X ∗ )-lower semicontinuity and convexity. In this case, lowersemicontinuity forces translation invariance along the σ ( X , X ∗ )-closure of span( S ) and delivers a dualrepresentation that will be exploited in the context of law-invariant functionals in the next section. Theorem 3.5.
Let ϕ : X → ( −∞ , ∞ ] be proper, convex, and σ ( X , X ∗ ) -lower semicontinuous and S ⊂ X .If ϕ is affine along every element of S , then ϕ is translation invariant along M , where M is the σ ( X , X ∗ ) -closure of span( S ) . Moreover, for all Z ∈ M and Y ∈ dom( ϕ ∗ ) ϕ ( Z ) = E [ ZY ] + ϕ (0) . (3.1) Proof. Step 1 . Take arbitrary Z ∈ S and Y ∈ dom( ϕ ∗ ). Since mZ ∈ dom( ϕ ) for every m ∈ R by affinity,it follows from Proposition 2.5 that for every m ∈ R we havesup m ∈ R { m ( E [ ZY ] − ϕ ( Z ) + ϕ (0)) } − ϕ (0) = sup m ∈ R { E [ mZY ] − ϕ ( mZ ) } ≤ sup X ∈X { E [ XY ] − ϕ ( X ) } < ∞ . Clearly, this is only possible if ϕ ( Z ) = E [ ZY ] + ϕ (0). This establishes (3.1) when Z ∈ S . Step 2 . Take now arbitrary Z ∈ S and Y ∈ dom( ϕ ∗ ). It follows from Step 1 that E [ ZY ] = ϕ ( Z ) − ϕ (0) = E [ ZY ′ ] for every Y ′ ∈ dom( ϕ ∗ ). Hence, we infer from Proposition 2.5 that for every X ∈ X ϕ ( X + Z ) = sup Y ′ ∈ dom( ϕ ∗ ) { E [( X + Z ) Y ′ ] − ϕ ∗ ( Y ′ ) } = sup Y ′ ∈ dom( ϕ ∗ ) { E [ XY ′ ] − ϕ ∗ ( Y ′ ) } + E [ ZY ]= ϕ ( X ) + E [ ZY ]= ϕ ( X ) + ϕ ( Z ) − ϕ (0) . ϕ is translation invariant along every element of S . By Remark 3.2, it follows that ϕ istranslation invariant along span( S ). In particular, (3.1) holds also for every Z ∈ span( S ).Take now Z ∈ M and let ( Z α ) be a net in span( S ) converging to Z and Y ∈ dom( ϕ ∗ ). Then, ϕ ( Z ) ≤ lim inf α ϕ ( Z α ) = lim inf α E [ Z α Y ] + ϕ (0) = E [ ZY ] + ϕ (0)by lower semicontinuity at Z . Using translation invariance along span( S ) we have for every αϕ ( Z ) = ϕ ( Z − Z α ) + ϕ ( Z α ) − ϕ (0) = ϕ ( Z − Z α ) + E [ Z α Y ] . Hence, by lower semicontinuity at 0, we easily obtain ϕ ( Z ) = lim inf α E [ Z α Y ] + lim inf α ϕ ( Z − Z α ) ≥ E [ ZY ] + ϕ (0) . It follows that ϕ ( Z ) = E [ ZY ] + ϕ (0) for every Z ∈ M . In particular, ϕ is affine on M . To conclude theproof we may apply what we have showed so far to M instead of S .A direct consequence of the preceding result is that when the functional is affine on a set whose linearspan is σ ( X , X ∗ )-dense in X , it must be affine on the entire space. Its linear part is thus represented bya unique dual element in X ∗ . Corollary 3.6.
Let ϕ : X → ( −∞ , ∞ ] be proper, convex, and σ ( X , X ∗ ) -lower semicontinuous and S ⊂ X such that span( S ) is σ ( X , X ∗ ) -dense in X . If ϕ is affine along every element of S , then ϕ is affine on X and there exists a unique Y ∈ X ∗ such that for every X ∈ X ϕ ( X ) = E [ XY ] + ϕ (0) . Throughout this section, we assume that ( X , X ∗ ) is a pair satisfying Assumption 2.1. We establish ourmain result on the “collapse to the mean” of convex law-invariant functionals. We start by recalling awell-known result about “law-invariance equivalence classes”. Here, for every random variable X ∈ L wedenote by q X a fixed quantile function of X , i.e. a function q X : (0 , → R satisfying for every α ∈ (0 , { m ∈ R ; P ( X ≤ m ) ≥ α } ≤ q X ( α ) ≤ inf { m ∈ R ; P ( X ≤ m ) > α } . Lemma 4.1.
For all X ∈ X and Y ∈ X ∗ the set E ( X, Y ) = { E [ X ′ Y ] ; X ′ ∈ X , X ′ ∼ X } is a closedinterval such that:(i) inf E ( X, Y ) = R q X ( α ) q Y (1 − α ) dα .(ii) sup E ( X, Y ) = R q X ( α ) q Y ( α ) dα .(iii) E ( X, Y ) = { E [ XY ′ ] ; Y ′ ∈ X ∗ , Y ′ ∼ Y } .Moreover, E ( X, Y ) is reduced to a singleton if and only if either X or Y is constant.Proof. It can be proved along the lines of Theorem 9.1 in Luxemburg [24] that E ( X, Y ) is a closed intervalsatisfying assertions (i) to (iii) . We refer to Bellini et al. [4] for a detailed proof. The “if” implication7n the last assertion is clear. To establish the “only if” implication, assume that E ( X, Y ) is reduced to asingleton. In this case, we must have0 = Z q X ( α ) q Y ( α ) dα − Z q X ( α ) q Y (1 − α ) dα = Z / q X ( α )[ q Y ( α ) − q Y (1 − α )] dα + Z / q X ( α )[ q Y ( α ) − q Y (1 − α )] dα = Z / [ q X ( α ) − q X (1 − α )][ q Y ( α ) − q Y (1 − α )] dα. Now, assume that either X or Y is not constant. Upon exchanging their roles, we can assume withoutloss of generality that X is not constant. Then, we find β ∈ (0 , /
2) such that q X ( α ) − q X (1 − α ) < α ∈ (0 , β ]. Hence, the above identity can only hold if q Y ( α ) = q Y (1 − α ) for almostevery α ∈ (0 , β ]. Being nondecreasing, q Y must therefore be almost-surely constant so that Y has to beconstant. This delivers the desired implication.It is an immediate consequence of the preceding lemma that any linear and σ ( X , X ∗ )-continuous functionalthat is law invariant must “collapse to the mean”. Of course, this simple fact could also be proved directly. Proposition 4.2.
Let M be a law-invariant linear subspace of X containing a nonconstant randomvariable. Let Y ∈ X ∗ and consider the linear functional ϕ : M → R given by ϕ ( X ) = E [ XY ] . Thefollowing statements are equivalent:(a) ϕ is law invariant.(b) Y is constant. We now use Lemma 4.1 to prove that the linear space generated by all the random variables having thesame distribution as a given nonconstant random variable with nonzero expectation is σ ( X , X ∗ )-dense inthe space X . For any random variable X ∈ X set L X := { X ′ ∈ X ; X ′ ∼ X } . Lemma 4.3.
For every nonconstant Z ∈ X the following statements hold:(i) If E [ Z ] = 0 , then span( L Z ) is σ ( X , X ∗ ) -dense in X .(ii) If E [ Z ] = 0 , then the σ ( X , X ∗ ) -closure of span( L Z ) coincides with the set { X ∈ X ; E [ X ] = 0 } .Proof. Let M be the σ ( X , X ∗ )-closure of span( L Z ). The annihilator of the set M is defined by M ⊥ := { Y ∈ X ∗ ; ∀ X ∈ M , E [ XY ] = 0 } . Similarly, the annihilator of the set M ⊥ is given by M ⊥⊥ := { X ∈ X ; ∀ Y ∈ M ⊥ , E [ XY ] = 0 } . Take an arbitrary Y ∈ M ⊥ . Since Z is not constant and { E [ ZY ′ ] ; Y ′ ∈ L Y } = { E [ Z ′ Y ] ; Z ′ ∈ L Z } = { } by Lemma 4.1, it follows from the same result that Y must be constant. If E [ Z ] = 0, then we must have Y = 0. In this case, M ⊥ = { } and it follows from Corollary 5.108 in Aliprantis and Border [1] that (i) holds. If E [ Z ] = 0, then we must have M ⊥ = R . This implies that M ⊥⊥ = { X ∈ X ; E [ X ] = 0 } . Since M = M ⊥⊥ by Theorem 5.107 in Aliprantis and Border [1], we infer that (ii) holds.8 ffinity along a nondeterministic random variable with nonzero expectation By combining the previous results we can now easily establish our main result.
Theorem 4.4.
For a proper, convex, σ ( X , X ∗ ) -lower semicontinuous, law-invariant functional ϕ : X → ( −∞ , ∞ ] the following statements are equivalent:(a) The functional ϕ is affine along a nonconstant Z ∈ X with E [ Z ] = 0 .(b) The functional ϕ is translation invariant along a nonconstant Z ∈ X with E [ Z ] = 0 .(c) There exists a ∈ R such that ϕ ( X ) = a E [ X ] + ϕ (0) for every X ∈ X .Proof. It follows from Theorem 3.5 that (a) and (b) are equivalent. To conclude, we only have to showthat (a) implies (c) . To this effect, assume that ϕ is affine along a nonconstant random variable Z ∈ X with E [ Z ] = 0. Note that, by Lemma 4.3, the σ ( X , X ∗ )-closure of span( L Z ) is X . Noting that, by lawinvariance, ϕ is affine along each element of L Z , we can apply Corollary 3.6 to obtain that ϕ ( X ) = E [ XY ] + ϕ (0)for all X ∈ X and Y ∈ dom( ϕ ∗ ). It now suffices to apply Proposition 4.2 to the functional ϕ − ϕ (0) toinfer that Y must be constant and conclude the proof. Remark 4.5.
We show that lower semicontinuity is necessary for the above “collapse to the mean” tohold. Let A = { X ∈ L ; X has a discrete distribution } and define ϕ : L → ( −∞ , ∞ ] by ϕ ( X ) = ( X ∈ A , ∞ otherwise . It is clear that ϕ is convex and law invariant. Moreover, for every event E ∈ F with P ( E ) ∈ (0 ,
1) we havethat ϕ is linear (in fact, null) on the vector space spanned by the nonconstant random variable Z = E .However, ϕ fails to be σ ( L , L ∞ )-lower semicontinuous. To see this, take a positive random variable X ∈ X \ A . Then, we can always find an increasing sequence ( X n ) ⊂ A such that X n → X almost surely.It follows from the Dominated Convergence Theorem that X n → X with respect to σ ( L , L ∞ ) but ϕ ( X ) = ∞ > n →∞ ϕ ( X n ) , showing that ϕ is not σ ( L , L ∞ )-lower semicontinuous. Affinity along a nondeterministic random variable with zero expectation
If the random variable along which a functional is affine has zero expectation, then the functional is simplythe composition of a convex real function and the expectation functional.
Theorem 4.6.
For a proper, convex, σ ( X , X ∗ ) -lower semicontinuous, law-invariant functional ϕ : X → ( −∞ , ∞ ] the following statements are equivalent:(a) The functional ϕ is affine along a nonconstant Z ∈ X with E [ Z ] = 0 .(b) The functional ϕ is translation invariant along a nonconstant Z ∈ X with E [ Z ] = 0 .(c) ϕ ( X ) = ϕ ( E [ X ]) for every X ∈ X . roof. It follows from Theorem 3.5 that (a) and (b) are equivalent. To conclude, we only have to showthat (a) implies (c) . Hence, assume that ϕ is affine along a nonconstant Z ∈ X with E [ Z ] = 0. Let M = { X ∈ X ; E [ X ] = 0 } , which by Lemma 4.3 is the σ ( X , X ∗ )-closure of span( L Z ). By Theorem 3.5, ϕ ( X ) = E [ XY ] + ϕ (0)for all X ∈ M and Y ∈ dom( ϕ ∗ ). It follows from Proposition 4.2 that Y must be constant. Hence, ϕ ( X ) = ϕ ( E [ X ]) + ϕ ( X − E [ X ]) − ϕ (0) = ϕ ( E [ X ]) + ϕ (0) − ϕ (0) = ϕ ( E [ X ])by translation invariance along M . This delivers the desired implication.Although, in general, there is no full “collapse to the mean” if the functional is affine along a direction withzero expectation, we do obtain a full “collapse to the mean” as soon as we additionally have translationinvariant along constant random variables. This is a situation that is often encountered in applications. Corollary 4.7.
For a proper, convex, σ ( X , X ∗ ) -lower semicontinuous, law-invariant functional ϕ : X → ( −∞ , ∞ ] that is translation invariant along the following statements are equivalent:(a) The functional ϕ is affine along a nonconstant Z ∈ X .(b) The functional ϕ is translation invariant along a nonconstant Z ∈ X .(c) There exists a ∈ R such that ϕ ( X ) = a E [ X ] + ϕ (0) for every X ∈ X .Proof. If E [ Z ] = 0, then the equivalences follow from Theorem 4.4. If E [ Z ] = 0, it suffices to showthat (a) implies (c) due to Theorem 4.6. In this case, the same result implies that ϕ ( X ) = ϕ ( E [ X ]) forevery X ∈ X whenever (a) holds. Then, by translation invariance along 1, there exists a ∈ R such that ϕ ( X ) = ϕ (0) + a E [ X ] for every X ∈ X . In this final section we illustrate some implications of the above “collapse to the mean” in the context ofpricing functionals and risk measures. We will also point out connections to other works in the literaturewhere a “collapse to the mean” was established. Throughout the entire section we continue to denote by( X , X ∗ ) a pair satisfying Assumption 2.1. Law-invariant pricing rules
The pricing of insurance contracts is one of the key topics in actuarial science. The classical approachbased on expected utility theory is thoroughly presented in standard textbooks such as B¨uhlmann [6],Borch [5], Gerber [18]. Since the pioneering contributions of these authors, it has become customary inthe theoretical literature to address the pricing problem in an “axiomatic” way by prescribing a set ofeconomically plausible requirements that a “good” pricing rule should satisfy. An early survey of theaxiomatic approach to insurance pricing can be found in Goovaerts et al. [19] and Deprez and Gerber[10]. An updated picture is presented in Laeven and Goovaerts [22]. In a pricing setting, the elements of X are interpreted as the payoffs of financial contracts at a given future date. A payoff is called risk free whenever it is constant and risky otherwise. A “pricing rule” assigns to each payoff its (buying) price. Definition 5.1.
A pricing rule is a functional π : X → ( −∞ , ∞ ] satisfying π (0) = 0. A payoff X ∈ X is frictionless (under π ) if it satisfies the following conditions:(1) π ( − X ) = − π ( X ). 102) π ( λX ) = λπ ( X ) for every λ ∈ (0 , ∞ ).For every X ∈ X the quantity π ( X ) − ( − π ( − X )) can be interpreted as the difference between the buyingand the selling price of X , i.e. as the “bid-ask spread” of X ; see e.g. Jouini [20]. A payoff is frictionlessprecisely when its bid-ask spread is zero and the price per unit does not depend on the transacted volume.A “collapse to the mean” in a Choquet pricing framework was obtained in Theorem 1 in Castagnoliet al. [9]. That result can be equivalently formulated as follows: The expectation (under the physicalprobability measure P ) is the only law-invariant, sublinear, monotone, comonotonic pricing functional on L ∞ under which every risk-free payoff and some risky payoff are frictionless . As a direct consequence ofTheorem 4.4 we obtain the following generalization of this result. Proposition 5.2.
Let π be a proper, convex, σ ( X , X ∗ ) -lower semicontinuous, law-invariant pricing rule.If π admits a frictionless risky payoff Z ∈ X with E [ Z ] = 0 , then there exists a ∈ R such that π ( X ) = a E [ X ] for every X ∈ X . In particular, every payoff is frictionless. (The condition E [ Z ] = 0 can be removed ifthe risk-free payoff is frictionless).Proof. Take any payoff Z ∈ X and note first that Z is frictionless if π is linear along it. The conversealso holds. Indeed, if Z is frictionless, then for every m ∈ R we have π ( mZ ) = mπ ( Z ) whenever m ≥ π (0) = 0 by our initial assumption on π ) and π ( mZ ) = π ( − ( − m ) Z ) = − mπ ( − Z ) = mπ ( Z )whenever m <
0. The desired statements now follow directly from Theorem 4.4 and Corollary 4.7.
Remark 5.3.
The above proposition provides a considerable extension of Theorem 1 in Castagnoli etal. [9], which is obtained for special pricing rules in the setting ( X , X ∗ ) = ( L ∞ , L ). Note that the pricingrules considered there are automatically σ ( L ∞ , L )-lower semicontinuous. Indeed, since the risk-freepayoff 1 is assumed to frictionless, those pricing rules are translation invariant along 1 by sublinearity (seeProposition 3.4) and are therefore Lipschitz continuous with respect to the L ∞ -norm (see, e.g., Lemma 4.3in F¨ollmer and Schied [13]). This implies that they are σ ( L ∞ , L )-lower semicontinuous by Example 2.6.Hence, all the requirements in Proposition 5.2 are met in the aforementioned result. Law-invariant risk measures based on general eligible assets
The paper by Artzner et al. [2] has been a landmark contribution in the theory of risk measures. Ina regulatory context, a risk measures assign the minimal amount of capital that has to be raised andinvested in a fixed financial asset, called the eligible asset , to ensure an acceptable profit-and-loss profile.The acceptability criterion is pre-specified by the regulator. In the literature, it is standard to assumethat the eligible asset is frictionless in the sense that it is available in arbitrary quantities and its priceper unit does not depend on the transacted volume. In this case, the corresponding risk measures arenaturally translation invariant as recalled below. In the context of risk measures, the elements of X areinterpreted as (net) capital positions of financial firms at a fixed future date. Definition 5.4. A (frictionless) eligible asset is a couple S = ( S , S ) with strictly-positive price S ∈ R and nonzero positive payoff S ∈ X . We say that S is risk free if S is constant and risky otherwise. Wesay that S is cash if S = (1 , ρ : X → ( −∞ , ∞ ] is said to be an S -additive risk measure ifit satisfies the following properties:(1) ρ ( X + mS ) = ρ ( X ) − mS for all X ∈ X and m ∈ R .112) ρ ( X ) ≤ ρ ( Y ) for all X, Y ∈ X such that X ≥ Y .When S is cash, we speak of cash-additivity instead of S -additivity.It is well known that, for every X ∈ X , an S -additive risk measure can always be expressed as ρ ( X ) = inf (cid:26) m ∈ R ; X + mS S ∈ A ρ (cid:27) , where A ρ = { X ∈ X ; ρ ( X ) ≤ } . The set A ρ consists of all the capital positions that are deemedacceptable from a regulatory perspective. Hence, for every position X ∈ X , the quantity ρ ( X ) can beinterpreted as the minimum amount of capital that has to be raised and invested in the eligible asset toensure acceptability. This type of risk measures has been thoroughly investigated in the case of a casheligible asset; see e.g. F¨ollmer and Schied [13]. The case of a general eligible asset has been studied, e.g.,in Artzner et al. [3] and Farkas et al. [14, 15].There are many examples of law-invariant risk measures when the eligible asset is risk free. One questionis whether law invariance can hold when the eligible asset is risky. This question was taken up in abounded setting in Frittelli and Rosazza Gianin [16]. A slight reformulation of Proposition 9 in [16]reads as follows: The expectation (under the physical probability measure P ) is, up to a sign, the onlylaw-invariant, convex, cash-additive risk measure on L ∞ that is S -additive for a risky eligible asset S andassigns the value to the zero position . As an application of our general “collapse to the mean” we obtainthe following generalization of this result. Proposition 5.5.
Let ρ be a proper, convex, σ ( X , X ∗ ) -lower semicontinuous, law-invariant, S -additiverisk measure such that ρ (0) < ∞ . If the eligible asset S is risky, then for every X ∈ X ρ ( X ) = S E [ S ] E [ − X ] + ρ (0) . (If ρ is cash-additive, then E [ S ] = S ).Proof. Since ρ (0) ∈ R and ρ is an S -additive risk measure, we have that ρ is translation invariant and,hence, affine along the payoff S . As S is nonconstant and satisfies E [ S ] >
0, it follows from Theorem 4.4that there exist a, b ∈ R such that ρ ( X ) = a E [ X ] + b for every X ∈ X . We infer that b = ρ (0) and a = ρ ( S ) − ρ (0) E [ S ] = − S E [ S ] . If ρ is also cash-additive, then a + b = ρ (1) = ρ (0) − b −
1, showing that E [ S ] = S . Remark 5.6.
The above proposition considerably extends Proposition 9 in Frittelli and Rosazza Gianin[16], which is obtained for special risk measures in the setting ( X , X ∗ ) = ( L ∞ , L ). Note that the riskmeasures considered there are automatically σ ( L ∞ , L )-lower semicontinuous. Indeed, by cash-additivity,they are Lipschitz continuous with respect to the L ∞ -norm (see, e.g., Lemma 4.3 in F¨ollmer and Schied[13]) and therefore σ ( L ∞ , L )-lower semicontinuous by Example 2.6. Hence, the requirements in Propo-sition 5.5 are met in the aforementioned result. Relevant cash-based risk measures
In this final section we maintain the preceding framework and provide an interesting result on cash-additiverisk measures satisfying the following relevance properties.12 efinition 5.7.
We say that ρ : X → ( −∞ , ∞ ] is relevant if for every X ∈ X we have X ≥ , P ( X > > ⇒ ρ ( − X ) > strongly relevant if for every X ∈ X we have X = 0 , ρ ( X ) ≤ ⇒ ρ ( − X ) > . Note that a strongly-relevant functional that satisfies ρ (0) ≤ sensitivity , has beenstudied, e.g., in Stoica [27] and F¨ollmer and Schied [13] in connection with generalized no-arbitrageconditions.Our “collapse to the mean” can be used to show that, with the exception of the negative of the expectation,every cash-additive risk measure that is sublinear, σ ( X , X ∗ )-lower semicontinuous, and law invariant isautomatically strongly relevant. In particular, this implies that every risk measure of the above type isalways relevant. Proposition 5.8.
Let ρ be a sublinear, σ ( X , X ∗ ) -lower semicontinuous, law-invariant, cash-additive riskmeasure. Then, one of the following two alternatives holds:(i) ρ ( X ) = E [ − X ] for every X ∈ X .(ii) ρ is strongly relevant.In particular, ρ is always relevant.Proof. Assume that ρ is not strongly relevant. Then, we must find a nonzero Z ∈ X such that ρ ( Z ) ≤ ρ ( − Z ) ≤
0. As ρ is sublinear, we also have0 = ρ (0) = ρ ( Z − Z ) ≤ ρ ( Z ) + ρ ( − Z ) . This implies that ρ ( − Z ) = − ρ ( Z ). But then ρ ( mZ ) = mρ ( Z ) for every m ∈ R again by sublinearity,showing that ρ is linear along Z . Note that Z cannot be constant for otherwise0 ≤ − ρ ( Z ) = Z = ρ ( − Z ) ≤ Z = 0. As a result of Corollary 4.7, there must exist a, b ∈ R such that ρ ( X ) = a E [ X ]+ b for every X ∈ X . To conclude, it suffices to note that b = ρ (0) = 0 and a = ρ (1) = − Remark 5.9.
The preceding result does not generally hold if ρ is only assumed to be convex. To seethis, define ρ : L → ( −∞ , ∞ ] by setting ρ ( X ) = inf { m ∈ R ; E [min( X + m, ≥ − } . It is immediate to verify that ρ is a convex, σ ( L , L ∞ )-lower semicontinuous, law-invariant, cash-additiverisk measure. However, we have ρ ( −
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