On Banach Spaces of Vector-Valued Random Variables and Their Duals

arXiv:1703.10367v2 [math.FA] 16 Jun 2017 hsppritoue aahsae o etrvle rand vector-valued for spaces Banach introduces paper This h pcscnb soitdwt ikfntoasadw dem we and functionals risk with associated be can spaces The pc-aud admvariables. random space-valued) xesosi h eodpr fteppr eflydescribe fully from We essentially differs paper. spaces the L these of for part theory second duality the in extensions iestain.A xml sgvnb osdrn portfo a considering by given is vec example portfolio An or situations. functions life Vector-valued finance. ematical i nice particularly is property. theory Radon–Nikodým duality the The enjoys reflexive. are they rbblt pc n eod eetn hmt vector-val to them extend we Firs second, ways. and two space in probability functions a for spaces rearrangement extend r slrea osbesc htteascae ikfuncti risk associated the that such possible as B large generally as more are consider we Here, vector-valued. be to in is said outcome random The currencies. Introduction 1 hmskle@ahmtkt-hmizd;alois.pichl [email protected]; ∞ ∗ nBnc pcso etrvle admvralsand variables random vector-valued of spaces Banach On ti aua oadesdffrne/smlrte between similarities differences/ address to natural is It nipratmtvto o osdrn hs pcsderiv spaces these considering for motivation important An u o an not but , aut fMteais hmizUiest fTechnology of University Chemnitz Mathematics, of Faculty fterda pcsi em fvco esrswt ausi values with measures vector of terms Bana in these spaces of dual c properties their which key of variables elaborate random We natu of continu presented. spaces a Lipschitz spaces larger are in strictly functionals defined on these defined that are shown functionals is risk it and So-called finance. ical etivratsae,da ersnain ikmeasures risk representation, dual spaces, invariant ment Keywords: variabl random vector-valued of spaces Banach introduce We 00MteaisSbetClassification: Subject Mathematics 2010 L p hi ul oiae yrs measures risk by motivated duals their pc ngnrladfrhr hi uli o vnsmlrt similar even not is dual their further, and general in space etrvle admvrals aahsae frno va random of spaces Banach variables, random Vector-valued hmsKle li Pichler Alois Kalmes Thomas R d ue1,2017 19, June nti oiaigeape h eae admvral is variable random related the example, motivating this in Abstract [email protected] 1 rmr:4E0 64.Scnay 62P05 Secondary: 46E40. 46E30, Primary: L p mvralsi rtpr.Teespaces These part. first a in variables om pcs h e pcsaelre than larger are spaces new The spaces. tcatcdominance stochastic , osaentrlypeeti ayreal many in present naturally are tors nl eancontinuous. remain onals aeta h ulo h tt space state the of dual the that case n i ihivsmnsin investments with lio L u.Ters ucinl antbe cannot functionals risk The ous. ,rno aibe r osdrdon considered are variables random t, h ulsaeo h tt space. state the of space dual the n e (i.e., ued nc pc-audrno variables. random space-valued anach hsae n ierepresentations give and spaces ch 1 etsapriua neetfrthe for interest particular a reates sfo eetdvlpet nmath- in developments recent from es a a nteeBnc spaces Banach these on way ral ntaeta h pcsintroduced spaces the that onstrate and h ul ftenwsae.The spaces. new the of duals the smtvtdfo mathemat- from motivated es L emn bt authors) (both Germany , p R pcsadw lbrt on elaborate we and spaces ∗ d rmr eea Banach general more or , ibe,rearrange- riables, o L p pcs However, spaces. d a,different say, , Rüschendorf [29] introduces and considers vector-valued risk functionals first. Svindland [30], Filipović and Svindland [14], Kupper and Svindland [18] and many further authors consider and discuss different domain spaces for risk measures on portfolio vectors, for example Orlicz spaces (as done in Cheridito and Li [6] and Bellini and Rosazza Gianin [4]). Ekeland and Schachermayer [12] consider the domain space L∞ for these risk measures. Ekeland et al. [13] provide the first multivariate generalization of a Kusuoka representation for risk measures on vector-valued random variables on L2. In contrast, the present paper extends these spaces and presents the largest possible Banach spaces for which those functionals remain continuous. The resulting spaces are neither Orlicz nor Lebesgue spaces, as considered in the earlier literature. The spaces, which we consider, are in a way related to function spaces (rearrangement spaces) introduced by Lorentz [20, 21], following earlier results obtained by Halperin [16]. For unexplained notions from the theory of vector measures we would like to refer the reader to the book by Diestel and Uhl [11].

Outline of the paper. The following section (Section 2) provides the mathematical setting in- p cluding the relation to mathematical ﬁnance. The Banach spaces Lσ(P,X) of X-valued random variables, introduced in Section 3, constitute the natural domains of risk functionals. We demon- strate that risk functionals are continuous with respect to the norm of the space introduced. In Section 4 we give a representation of the dual spaces of these Banach spaces in the scalar-valued case. This representation is used in Section 5 to derive representations of the duals in the general vector-valued case.

2 Mathematical setting and motivation

We consider a probability space (Ω, , P ) and denote the distribution function of an R-valued random variable Y by F F (q) := P (Y q)= P ( ω : Y (ω) q ) . Y ≤ { ≤ } The generalized inverse is the nondecreasing and left-continuous function

F −1(α) := inf q : P (Y q) α , Y { ≤ ≥ } also called the quantile or Value-at-Risk. With X = (X, ) we denote a Banach space and by X∗ its continuous dual space. We use the notation ϕ, x fork·kϕ(x), ϕ X∗ and x X. As usual we denote for p [1, ) by Lp(P,X) the Bochner-Lebesgueh i space of p∈-Bochner integrable∈ X-valued random variables∈ Y∞on (Ω, , P ) whose norm we denote by . Recall that for Y Lp(P,X) F k·kp ∈ 1 1/p ∞ 1/p Y = F −1 (u)pdu = ptp−1 1 F (t) dt . (1) k kp kY k − kY k Z0 Z0 In this paper Banach spaces of vector-valued, strongly measurable random variables are introduced by weighting the quantiles in a diﬀerent way than (1). The present results extend and generalize characterizations obtained in Pichler [26], where only real valued random variables and p =1 are considered (and elaborated in a context of insurance).

2 Remark 1. We shall assume throughout the paper that the probability space (Ω, , P ) is rich enough to carry a [0, 1]-valued, uniform distribution.1 If this is not the case, then oneF may replace Ω by Ω˜ := Ω [0, 1] with the product measure P˜(A B) := P (A) Lebesgue measure(B). Every random variable×Y on Ω extends to Ω˜ by Y˜ (ω,u) := Y×(ω) and U(ω,u· ) := u is a uniform random variable, as P˜ (U u)= P˜ (Ω [0,u]) = u. We denote the set of [0, 1]-valued uniform random variables on (Ω, , P )≤by U (0, 1). × FWith an R-valued random variable Y one may further associate its generalized quantile transform

′ F (y,u) := (1 u) lim FY (y )+ u FY (y). − · y′↑y ·

The random variable F (Y,U) is uniformly distributed again and F (Y,U) is coupled in a comonotone way with Y , i.e., the inequality F (Y,U)(ω) F (Y,U)(ω′) Y (ω) Y (ω′) 0 holds P P almost everywhere (see, e.g., Pflug and Römisch [25−, Proposition 1.3]). − ≥ ⊗ Relation to mathematical finance: risk measures and their continuity properties. Risk measures on R-valued random variables have been introduced in the pioneering paper Artzner et al. [3]. An R-valued random variable is typically associated with the total, or accumulated return of a portfolio in mathematical finance. (The prevalent interpretation in insurance is the size of a claim, which happens with a probability specified by the probability measure P .) The aggregated portfolio is composed of individual components, as stocks. From the perspective of comprehensive risk management it is desirable to understand not only the risk of the accumulated portfolio, but also its components. These more general risk measures on Rd-valued random variables have been considered first in Burgert and Rüschendorf [5] and further progress was made, for example, by Rüschendorf [29], Ekeland et al. [13] and Ekeland and Schachermayer [12]. Ekeland and Schachermayer [12, Theorem 1.7] obtain a Kusuoka representation (cf. Kusuoka [19]) for risk measures based on Rd-valued random variables. The risk functional identified there in the “regular case” for the homogeneous risk functional on random vectors is

ρ (Y ) := sup E Z, Y ′ : Y ′ Y , (2) Z { h i ∼ } ′ ′ d 2 where Y Y indicates that Y and Y enjoy the same law in R . ρZ is called the maximal correlation∼ risk measure in direction Z. The rearrangement inequality (see e.g. McNeil et al. [22, Theorem 5.25(2)], also known as Chebyshev’s sum inequality, cf. Hardy et al. [17, Section 2.17]) provides an upper bound for the natural linear form in (2) by

1 E E ∗ E −1 −1 Z, Y Z Y K Z ℓd Y K FkZk (u) FkY k(u)du, (3) | h i| ≤ k k ·k k≤ ·k k 1 ·k k≤ · ℓd · Z0 1 where the norms and ∗ are dual to each other on Rd (here, K > 0 is the constant linking the ∗ k·k k·k d norms by K d on (the dual of) R ). k·k ≤ · k·kℓ1 1U is uniform, if P (U ≤ u)= u for all u ∈ [0, 1]. 2 ′ ′ d That is, P (Y1 ≤ y1,...Yd, ≤ yd)= P Y1 ≤ y1,...,Yd ≤ yd for all (y1,...yd) ∈ R .

3 The maximal correlation risk measure (2) employs the linear form E Z, Y , which satisﬁes the bounds (3). This motivates ﬁxing the function h i

σ( ) := F −1 ( ) (4) kZk d · ℓ1 · and to consider an appropriate vector space of random variables endowed with

1 Y := σ(u) F −1 (u)du, k kσ · kY k Z0

(cf. Pichler [28, 27], Ahmadi-Javid and Pichler [2]). It turns out that σ is a norm (Theorem 4 below) on this vector space of random variables and that the maximalk·k correlation risk measure is continuous with respect to the norm (Proposition 7).

p 3 The vector-valued Banach spaces Lσ(P, X) Motivated by the observations made in the previous section we introduce the following notions. Definition 2. A nondecreasing, nonnegative function σ : [0, 1) [0, ), which is continuous 1 → ∞ from the left and normalized by 0 σ(u)du = 1, is called a distortion function (in the literature occasionally also spectrum function, cf. Acerbi [1]). R Definition 3. For a distortion function σ, a Banach space (X, ) and a probability space (Ω, , P ) we define for p [1, ) and a strongly measurable X-valued randomk·k variable Y on (Ω, , P )F ∈ ∞ F p E p p Y σ,p := sup σ(U) Y = sup σ(U(ω)) Y (ω) dP (ω), k k uniform k k uniform k k U U ZΩ where the supremum is taken over all U U (0, 1), i.e., over all [0, 1]-valued, uniformly distributed random variables U on (Ω, , P ). Moreover,∈ we set F Lp (P,X) := Y :Ω X strongly measurable and Y p < , σ { → k kσ,p ∞} where as usual we identify X-valued random variables which coincide P -almost everywhere. Obviously, for σ = 1 one obtains the classical Bochner-Lebesgue spaces Lp(P,X) which are well-known to be Banach spaces.

p p Theorem 4. Lσ(P,X) is a vector space and p,σ is a norm on Lσ(P,X) turning it into a Banach space which embeds contractively into Lp(P,Xk·k). Moreover, for each X-valued, strongly measurable Y on (Ω, , P ) and every U U (0, 1) which is coupled in comonotone way with Y it follows that F ∈ k k 1 Y p = E (σ(U) Y p)= σ(u)F −1 (u)pdu. (5) k kσ,p k k kY k Z0 Proof. We denote the probability measure on (Ω, ) with P -density σ U for some U U (0, 1) by F ◦ ∈ σ(U)P and the expectation of a non-negative random variable Z on (Ω, , σ(U)P ) by EU (Z). We obviously have F p E p Y σ,p = sup U Y , k k U∈U (0,1) k k

4 p p which implies that Lσ(P,X) is a subspace of the intersection of Banach spaces U∈U (0,1) L (σ(U)P,X) and that is a seminorm on Lp (P,X). σ,p σ T By thek·k rearrangement inequality (see, e.g., McNeil et al. [22, Theorem 5.25(2)]), the well-known fact that F −1 = σ and (F −1 )p = F −1 it follows for every U U (0, 1) and each X-valued, σ(U) kY k kY kp ∈ strongly measurable Y on (Ω, , P ), that F 1 E (σ(U) Y p) σ(u)F −1 (u)pdu k k ≤ kY k Z0 so that 1 Y p σ(u)F −1 (u)pdu. (6) k kσ,p ≤ kY k Z0 Moreover, if we ﬁx for an X-valued, strongly measurable Y on (Ω, , P ) some U U (0, 1) such that U and Y are coupled in a comonotone way (such U exists dueF to our general∈ assumption on (Ω, , P ) madek k in Remark 1) then (Kusuoka [19]) F 1 E σ(U) Y p = σ(u)F −1 (u)pdu. k k kY k Z0 Together with (6) we obtain for each X-valued, strongly measurable Y on (Ω, , P ) that there is U U (0, 1) such that F ∈ 1 Y p = E (σ(U) Y p)= σ(u)F −1 (u)pdu, k kσ,p k k kY k Z0 proving (5). p In order to see that the seminorm σ,p on Lσ(P,X) is in fact a norm we apply the continuous version of Chebychev’s inequalityk·k (see, e.g., Gradshteyn and Ryzhik [15, Eq. 12.314]) to the −1 p nonnegative, nondecreasing functions σ and (FkY k) on [0, 1) to obtain

1 1 1 1 σ(u)F −1 (u)pdu σ(u)du F −1 (u)pdu = F −1 (u)pdu = E ( Y p), kY k ≥ · kY k kY k k k Z0 Z0 Z0 Z0 −1 p −1 where the last equality follows from (FkY k) = FkY kp . In particular, together with (5) we obtain for every X-valued, strongly measurable Y

E ( Y p) Y p , k k ≤k kσ,p which proves that Lp (P,X) embeds contractively into Lp(P,X) and that Y =0 implies Y =0 σ k kσ,p so that σ,p is indeed a norm. k·k p Finally, in order to prove that Lσ(P,X) is a Banach space when equipped with the norm σ,p, p pk·k we ﬁrst note that a Cauchy sequence (Yn)n∈N in Lσ(P,X) is also a Cauchy sequence in L (P,X) so that there is Y Lp(P,X) with Y = lim Y in Lp(P,X). From this we conclude that ∈ n→∞ n Y = limk→∞ Ynk P -almost everywhere on Ω for some subsequence (Ynk )k∈N of (Yn)n∈N. Since for each ε> 0 there is N N such that for all U U (0, 1) ∈ ∈ εp > E (σ(U) Y Y p) k n − mk

5 whenever n,m N it follows with Fatou’s Lemma that for every U U (0, 1) and each n N we have ≥ ∈ ≥

p p p p E (σ(U) Y Yn )= E ( lim σ(U) Yn Yn ) lim inf E (σ(U) Yn Yn ) ε , k − k k→∞ k k − k ≤ k→∞ k k − k ≤ i.e., Y Y εp for every n N. Thus, we conclude that k − nkσ,p ≤ ≥ Y = (Y Y )+ Y Lp (P,X) − N N ∈ σ p and that (Yn)n∈N converges to Y in Lσ(P,X). p −1 Remark 5. By (5) Lσ(P,X)-membership of Y only depends on the quantile function FkY k so that p Lσ(P,X) is invariant with respect to rearrangements. From the deﬁnition of σ,p it follows p ∞ k·k immediately that Lσ(P,X) is an L (P )-module and that αY σ,p α ∞ Y σ,p for all α L∞(P ) and each Y Lp (P,X). k k ≤ k k k k ∈ ∈ σ p We next show that the Lσ(P,X)-spaces behave like the classical Bochner-Lebesgue spaces Lp(P,X) when one varies the exponent p [1, ). ∈ ∞ Proposition 6. Let p, p′ [1, ) be such that p

1 1 1 ′ p r p /p −1 p Y σ,p = σ (u)σ (u)FkY k(u) du k k 0 Z ′ 1/r p/p 1 1 ′ −1 p p σ(u)du σ(u)F (u) du = Y ′ , ≤ kY k k kσ,p Z0 Z0 which proves i) while ii) follows from (5), 1/r +1/(p′/p)=1, and Hölder’s inequality since

′ 1 1 1/r 1 ′ p/p p −1 p r −1 p p Y = σ(u)F (u) du σ (u)du F (u) p du k kσ,p kY k ≤ kY k Z0 Z0 Z0 1 1/r ′ ′ p/p = σr(u)du E Y p k k Z0 holds for each X-valued, strongly measurable Y on (Ω, , P ). F For a Banach space X with (continuous) dual space X∗ we write as usual x∗, x := x∗(x), x X, x∗ X∗. The dual norm on X∗ will also be denoted by . If Z is an X∗h-valued,i Bochner integrable∈ ∈ random variable on (Ω, , P ) such that E Z = 1k·kthen σ := F −1 is a distortion F k k Z kZk function. For two X-valued, strongly measurable Y , Y on (Ω, , P ) we write Y Y if they have 1 2 F 1 ∼ 2 the same law, i.e., if P Y1 = P Y2 .

6 Proposition 7. Let X be a real Banach space and let Z be an X∗-valued, Bochner integrable random variable on (Ω, , P ) such that E Z =1. Then, for every p [1, ) F k k ∈ ∞ p ′ ′ ρZ : L (P,X) R, Y sup E Z, Y : Y Y σZ → 7→ { h i ∼ } is a well-defined subadditive, convex functional. Moreover, for Y , Y Lp (P,X) we have 1 2 ∈ σZ ρ (Y ) ρ (Y ) Y Y . | Z 1 − Z 2 |≤k 1 − 2kσ,p ′ −1 −1 ′ p p Proof. It follows from that ′ . Hence, whenever Y Y FkY k = FkY k Y LσZ (P,X) Y LσZ (P,X) ∼ ∈ p ∈ by (5) in Theorem 4 . From the strong measurability of Z and Y LσZ (P,X) it follows immediately that ω Z(ω), Y (ω) is an R-valued random variable on (Ω, ,∈ P ). The rearrangement inequality, the definition7→ h of σ , (5i) in Theorem 4 and Proposition 6 implyF that for Y ′ Y Lp (P,X) Z ∼ ∈ σZ 1 E Z, Y ′ E( Z Y ′ ) σ (u)F −1 (u)du = Y Y , | h i| ≤ k kk k ≤ Z kY k k kσ,1 ≤k kσ,p Z0 which proves that ρZ is well-defined and that

ρ (Y ) Y . (7) | Z |≤k kσ,p

Obviously, ρZ (λY ) = λρZ (Y ) for all λ > 0. Moreover, from the deﬁnition of ρZ and strong measurability it follows immediately that ρZ is subadditive. Therefore,

ρ (Y )= ρ(Y + Y Y ) ρ (Y )+ ρ (Y Y ). Z 1 2 1 − 2 ≤ Z 2 Z 1 − 2

Interchanging the roles of Y1, Y2 in the above inequality gives

ρ (Y ) ρ (Y ) ρ (Y Y ), | Z 1 − Z 2 |≤ Z 1 − 2 which together with (7) proves ρ (Y ) ρ (Y ) Y Y . | Z 1 − Z 2 |≤k 1 − 2kσ,p p In the remainder of this section we will provide a closer look at the Banach spaces Lσ(P,X). Proposition 8. Let X = 0 . Then the following are equivalent. 6 { } i) For all p [1, ) the spaces Lp (P,X) and Lp(P,X) are isomorphic as Banach spaces. ∈ ∞ σ ii) There is p [1, ) such that Lp (P,X)= Lp(P,X) as sets. ∈ ∞ σ iii) σ is bounded.

p p Proof. Obviously, i) implies ii). By Theorem 4, Lσ(P,X) embeds contractively into L (P,X). Thus, if ii) holds, this embedding is onto so that by Banach’s Isomorphism Theorem there is C > 0 such that Y Lp(P,X) : sup σ(U(ω)) Y (ω) pdP (ω) C Y (ω) pdP (ω), ∀ ∈ U k k ≤ k k U∈ (0,1) ZΩ ZΩ

7 where U (0, 1) is deﬁned as before. Choose f L1(P, R) and x X with x = 1. Then Y (ω) := f(ω) 1/px deﬁnes an element of Lp(P,X) so that∈ for any U U∈(0, 1) we havek k | | ∈

σ(U(ω))f(ω)dP (ω) σ(U(ω)) Y (ω) pdP (ω) ≤ k k ZΩ ZΩ

C Y (ω) pdP (ω) ≤ k k ZΩ = C f(ω) dP (ω) < . | | ∞ ZΩ Since f L1(P, R) was chosen arbitrarily it follows that σ U L∞(P, R) which by U U (0, 1) and by the∈ fact that σ is nondecreasing implies boundedness◦ of ∈σ. Thus, iii) follows from∈ii). p p Finally, iii) and the fact that Lσ(P,X) embeds contractively into L (P,X) for any p [1, ) implies i) by Theorem 4. ∈ ∞

Proposition 9. We have the following:

i) For every p [1, ), L∞(P,X) embeds contractively into Lp (P,X). ∈ ∞ σ ii) Simple functions are dense in Lp (P,X) for every p [1, ). σ ∈ ∞ Proof. It follows from the deﬁnition of quantile function that 0 F −1 Y for every X-valued, ≤ kY k ≤k k∞ strongly measurable Y on (Ω, , P ) which implies by (5) in Theorem 4 F 1 Y p = σ(u)F −1 (u)pdu Y p , k kσ,p kY k ≤k k∞ Z0 proving i). In order to prove ii) let Y Lp (P,X) and ﬁx ε (0, 1). We choose u (0, 1) such that ∈ σ ∈ ε ∈ 1 σ(u)F −1 (u)pdu < εp. By the strong measurability of Y there are N with P (N)=0 and uε kY k ∈F a separable, closed subspace X of X such that 1 c Y is X -valued. Let x ; j N be a dense R 1 N 1 { j ∈ } subset of X1. Denoting the open ball about xj with radius ε in X by Bε(xj ) we choose Borel subsets Ej Bε(xj ) such that X1 j∈NEj and such that the Ej are pairwise disjoint. Then −1 ⊆ ⊆ ∪ −1 ((1N c Y ) (Ej ))j∈N is a pairwise disjoint sequence in such that P ( j∈N(1N c Y ) (Ej ))=1. Let n N be such that F ∪ ∈ n −1 P ((1N c Y ) (Ej )) >uε (8) j=1 X n 1 −1 and set E := j=1( N c Y ) (Ej ). ∪ p p 1 c p p Obviously, for t 0 we have E Y t Y t so that F1Ec kY k (t) FkY k (t). Therefore, ≥ { k k ≤ } ⊇ {k k ≤ } ≥ u [0, 1] : t 0; F1 p (t) u t 0; F p (t) u , ∀ ∈ { ≥ Ec kY k ≥ }⊇{ ≥ kY k ≥ } −1 −1 which implies F1 p F p . Furthermore, Ec kY k ≤ kY k p F1 p (0) = P (1 c Y = 0) P (E) Ec kY k E k k ≥

8 −1 −1 −1 p so that F1 p (u)=0 for every u [0, P (E)], which together with F p = (F ) and (8) Ec kY k ∈ kY k kY k yields for all u [0, 1] ∈ −1 1 −1 p 1 −1 p F1 p (u) (P (E),1](u)F (u) (u ,1](u)F (u) . (9) Ec kY k ≤ kY k ≤ ε kY k n 1 1 −1 1 c Deﬁning Yε := j=1 ( Nc Y ) (Ej ) xj it follows from the deﬁnition of E that N Y Yε ε c k − k ≤ on E while Yε =0 on E . For every U U (0, 1) we obtain P ∈

p p p σ(U) 1N c Y Yε dP = σ(U) 1E 1N c Y Yε dP + σ(U) 1Ec Y dP Ω k − k Ω k − k Ω k k Z Z 1 Z p −1 ε σ(U)dP + σ(u)F p (u)du 1Ec kY k ≤ Ω 0 Z 1 Z εp + σ(u)F −1 (u)pdu < 2εp, ≤ kY kp Zuε where we used the rearrangement inequality (see McNeil et al. [22, Theorem 5.25(2)]) in the ﬁrst inequality and (9) in the second one while the last inequality follows form the choice of uε. Thus, Y Y < √p 2ε, proving ii). k − εk Theorem 10. For X = 0 the following are equivalent. 6 { } p i) Lσ(P,X) is a Hilbert space. ii) X is a Hilbert space, p =2, and σ =1 on (0, 1). Proof. Obviously, ii) implies i). Let Eα be chosen with P (Eα)= α. For α (0, 1) and x X a straightforward calculation gives for Y ∈F= 1 x that F −1 = x 1 . Moreover,∈ for x , x∈ X with x = x =1 and Eα kY k k k (1−α,1) 1 2 ∈ k 1k k 2k 1 1 c Y1 := Eα x1, Y2 := Eα x2 we have Y1 Y2 σ,p =1. Thus, by the parallelogram identity and (5) we obtain for arbitrary α [0, 1] k ± k ∈ 1 1 1= Y Y 2 + Y + Y 2 = Y 2 + Y 2 (10) 2k 1 − 2kσ,p 2k 1 2kσ,p k 1kσ,p k 2kσ,p 1 2/p 1 2/p = σ(u)du + σ(u)du . Z1−α Zα Pick α (0, 1) so that 0 < α σ(u)du < 1. If p> 2, then ∈ 0 R 1 2/p 1 2/p 1= σ(u)du + σ(u)du Z1−α Zα 1 1 > σ(u)du + σ(u)du Z1−α Zα α 1 σ(u)du + σ(u)du =1, ≥ Z0 Zα as σ is nondecreasing. This is a contradiction and hence p 2. ≤

9 Define the function 1 2/p f(α) := σ(u)du . (11) Zα Since σ is continuous from the left f is differentiable from the left with increasing left derivative, thus f is convex. Further we have f(α)+ f(1 α)=1 by (10) so that f is concave as well. Hence, f is affine, i.e., f(α) = b + c α, and we deduce− from f(1) = 0,f(0) = 1, and (11) the particular form · p p −1 σ(u)= (1 u) 2 (12) 2 − 1 p/2 which implies 1−α σ(u)du = α . Next consider measurable sets A and B with A B. The parallelogram law (10), applied to R ⊆ the random variables Y1 := 1A x and Y2 := 1B x, reads 1 1 2/p P (B) P (A) + P (B)p/2 P (A)p/2 +2pP (A)p/2 = P (A)+ P (B), 2 − 2 − i.e., P (B)p/2 + (2p 1)P (A)p/2 = (3P (A)+ P (B))p/2 . − We may specify the sets further by P (B)=4P (A), then the latter equality reduces to

4p/2 + (2p 1)=7p/2 − so that we are left with solving the equation

log 7 2x 1= x 2 log 2 − for x =2p. log 7 The convex function x log 4 does not have more than two intersections with the line 2x 1, and these are x =1 and x =4, i.e., p =0 and p =2. − p =0 does not qualify, and the distortion function for p =2 is σ( )=1, by (12). This concludes the proof. ·

4 The dual space in the scalar valued case

p p p K K R C In this section we are going to determine the dual space of Lσ := Lσ(P ) := Lσ(P, ), , . p ∗ ∗ ∈{ } For ϕ Lσ(P ) we denote the dual norm of ϕ by ϕ σ,p. Some of the results presented in this section∈ are inspired by Lorentz [21]. k k

Definition 11. As usual we denote by L0(P ) the set of K-valued random variables on (Ω, , P ), where random variables which coincide P -almost surely are identified. We define the Köthe dualF of p Lσ(P ) as Lp (P )× := Z L0(P ); Y Lp (P ): ZY L1(P ) . σ { ∈ ∀ ∈ σ ∈ } Since L∞(P ) Lp (P ) for all p [1, ) it follows from taking Y = 1 Z that Z L1(P ) ⊆ σ ∈ ∞ {Z6=0} |Z| ∈ whenever Z Lp (P )×. ∈ σ

10 Proposition 12. For every Z Lp (P )× ∈ σ sup E(ZY ) ; Y 1 < . {| | k kσ,p ≤ } ∞ Moreover, ϕ : Lp (P ) K, ϕ (Y )= E(ZY ) Z σ → Z p ∗ belongs to Lσ(P ) and Φ: Lp (P )× Lp (P )∗,Z ϕ σ → σ 7→ Z is a linear isomorphism with

Z Lp (P )× : Φ(Z) ∗ = sup E(ZY ) ; Y 1 . (13) ∀ ∈ σ k kσ,p {| | k kσ,p ≤ } p p × Proof. Obviously, ϕZ is a well-deﬁned, linear functional on Lσ(P ) for every Z Lσ(P ) . The assumption ∈ = sup E(ZY ) ; Y 1 ∞ {| | k kσ,p ≤ } p implies the existence of a sequence (Yk)k∈N in the unit ball of Lσ(P ) such that k N : k2 E(ZY ) E( ZY ). ∀ ∈ ≤ | k |≤ | k| ˜ 1 ZYk p N Because Yk := {ZYk6=0} ˜ Yk belongs to the unit ball of Lσ(P ), k the completeness of |ZYk | ∈ n 1 p ˜ N p p × Lσ(P ) implies that ( k=1 k2 Yk)n∈ converges in Lσ(P ) to some Y . As Z Lσ(P ) it follows that ZY L1(P ). ∈ ∈ P p p But on the other hand, since Lσ(P ) embeds contractively into L (P ) by Theorem 4 it follows nl 1 ˜ N that some subsequence ( k=1 k2 Yk)l∈ also converges P -almost surely to Y . Therefore, P -almost surely we have P n n n l 1 l 1 l 1 ZY = Z lim ( Y˜k) = lim ZY˜k = lim ZYk (14) l→∞ k2 l→∞ k2 l→∞ k2 | | Xk=1 kX=1 Xk=1 and by an application of the Monotone Convergence Theorem we conclude

n n n l 1 l 1 l E(ZY )= E( lim ZYk ) = lim E( ZYk ) lim 1 l→∞ k2 | | l→∞ k2 | | ≥ l→∞ Xk=1 Xk=1 kX=1 which contradicts ZY L1(P ). Hence, ∈ > sup E(ZY ) ; Y 1 ∞ {| | k kσ,p ≤ } so that ϕ Lp (P )∗ with Z ∈ σ ϕ ∗ = sup E(ZY ) ; Y 1 . (15) k Z kσ,p {| | k kσ,p ≤ } This implies that Φ is a well-deﬁned linear mapping which satisﬁes (13). In order to show that Φ is injective choose Z Lp (P )× with Φ(Z)=0. We set Y := 1 Z . Since simple functions ∈ σ {Z6=0} |Z| belong to Lp (P ) it follows easily that Y Lp (P ). It follows σ ∈ σ 0=Φ(Z)(Y )= E( Z ), | |

11 so that Z =0. p 1 −1 In order to prove surjectivity of Φ let ϕ Lσ(P ). For E and Y = E we have F|Y | = 1 ∈ ∈ F (1−P (E),1] so that by (5)

1 ϕ(1 ) ϕ∗ 1 = ϕ ∗ ( σ(u)du)1/p. | E |≤k σ,pkk E kσ,p k kσ,p Z1−P (E) Using this inequality, it is straightforward to show that

µ : K, µ(E) := ϕ(1 ) F → E is a complex measure which is P -continuous, i.e., µ(E)=0 whenever P (E)=0. An application 1 1 E 1 of the Radon-Nikodým Theorem yields some Z L (P ) such that µ(E) = Ω E ZdP = (Z E) for all E . For simple functions Y it follows∈ϕ(Y ) = E(ZY ). As soon as we have shown that p ∈× F R Z Lσ(P ) it follows from the above and Theorem 9 that ϕ = Φ(Z). ∈ p × p In order to show Z Lσ(P ) we ﬁrst observe that αY Lσ(P ) and αY σ,p α ∞ Y σ,p p ∈ ∞ ∈ k k ≤ k k kN k for every Y Lσ(P ) and each α L (P ). Therefore, by setting En := Z n ,n , we 1 ∈ ∈ p p ∗ {| | ≤ }∗ ∈ ∗ have En Y σ,p Y σ,p for each Y Lσ(P ) which implies ϕn Lσ(P ) and ϕn σ,p ϕ σ,p k k ≤1 k k ∈ ∈ E 1 k k ≤ k k where ϕn(Y ) := ϕ( En Y ). For simple functions Y we have ϕn(Y )= (Z En Y ). Additionally, by Hölder’s inequality and Theorem 4 we obtain for arbitary Y Lp (P ) ∈ σ E( Z 1 Y ) nE( Y ) n Y , | En | ≤ | | ≤ k kσ,p 1 p × p so that Z En Lσ(P ) . Because simple functions are dense in Lσ(P ) by Theorem 9 we conclude ∈ 1 from the above Φ(Z En )= ϕn. Finally, since

ZY ZY E( Z 1 Y )= ϕ (1 Y ) ϕ ∗ 1 Y | En | | n {ZY 6=0} ZY |≤k nkσ,pk {ZY 6=0} ZY kσ,p | | | | ϕ Y ≤k kσ,pk kσ,p it follows with the aid of the Monotone Convergence Theorem that

E( ZY ) = lim E( Z 1E Y ) ϕ σ,p Y σ,p | | n→∞ | n | ≤k k k k for each Y Lp (P ) so that Z Lp (P )×. ∈ σ ∈ σ Remark 13. With the aid of the fact that for α L∞(P ) the linear mapping Y αY is well p ∈ 7→ p × deﬁned and continuous from Lσ(P ) into itself it is straightforward to see that Z Lσ(P ) whenever Z Lp (P )× and that in this case ϕ ∗ = ϕ ∗ . | | ∈ ∈ σ k Z kσ,p k |Z|kσ,p p × p We next aim at giving a representation of Lσ(P ) and thus of the dual space of Lσ(P ). For this purpose we introduce the following notion.

Deﬁnition 14. For a distortion function σ we deﬁne

1 S := S : [0, 1] R, S (α)= σ(u)du. σ → σ Zα

12 Remark 15. Obviously, S is a continuous, nonincreasing function with S(0) = 1,S(1) = 0. If we set

u0 := inf u> 0; σ(u) > 0 we have u0 < 1, S|[0,u0] = 1 and S|[u0,1] is an increasing bijection from { } −1 [u0, 1] to [0, 1]. By abuse of notation we denote the inverse of S|[u0,1] by S . For α , α [0, 1], α < α and λ (0, 1) it follows from the fact that σ is nondecreasing that 1 2 ∈ 1 2 ∈

λα1+(1−λ)α2 S(λα + (1 λ)α ) S(α )= σ(u)du 1 − 2 − 1 − Zα1 σ(λα + (1 λ)α ) (1 λ)(α α ) ≥− 1 − 2 − 2 − 1 and

α2 S(α ) S(λα + (1 λ)α )= σ(u)du 2 − 1 − 2 − Zλα1+(1−λ)α2 σ(λα + (1 λ)α ) λ(α α ) ≤− 1 − 2 2 − 1 so that S(λα + (1 λ)α ) S(α ) 1 − 2 − 1 σ(λα + (1 λ)α ) (1 λ)(α α ) ≥− 1 − 2 − 2 − 1 S(α ) S(λα + (1 λ)α ) 2 − 1 − 2 , ≥ λ(α α ) 2 − 1

which implies S(λα1 + (1 λ)α2) λS(α1)+(1 λ)S(α2), i.e., S is concave. In particular, S is differentiable from the left− and from≥ the right on (0−, 1], (on [0, 1), resp.) and since σ is continuous ′ from the left it is straightforward to show that for the left derivative we have Sl(α) = σ(u),u (0, 1]. − ∈ Recall that for a non-negative random variable Z the average value-at-risk of level α [0, 1) is AV R 1 1 −1 ∈ defined as @ α(Z)= 1−α α FZ (u)du. Definition 16. For a distortionR function σ, Z L0(P ), and α [0, 1) we define ∈ ∈ AV R 1 F −1(u)du ∗ @ α( Z ) α |Z| Z σ,∞ := sup 1 | | (= sup 1 ). (16) | | α∈[0,1) Sσ(α) α∈[0,1) 1−α R α σ(u)du

′ 0 ′R Moreover, we say that Z L (P ) σ-dominates Z (in symbols Z σ< Z) if there is a uniform random variable U U (0, 1) such∈ that ∈ AV@R σ(U) Z′ AV@R ( Z ) for all α< 1. (17) α | | ≥ α | | Further we deﬁne, for p (1, ), ∈ ∞ Z ∗ := inf Z′ : Z′ < Z (18) | |σ,q k kσ,q σ n o where q (1, ) is the conjugate exponent to p, i.e., 1/p +1/q =1 and where as usual inf := . ∈ ∞ ∗ ∅ ∞ Finally, for p [1, ) with conjugate exponent q, i.e. 1/p +1/q = 1, we set Lσ,q(P ) := Z L0(P ); Z ∗ < ∈ (and∞ we identify random variables which coincide P -almost everywhere).{ ∈ | |σ,q ∞}

13 0 From the definition of quantile functions it follows for Z1, Z2 L (P ) with Z1 Z2 that F −1 F −1 which implies Z ∗ Z ∗ . Since also F −1 = α∈F −1 for α |K it| ≤ follows | | also |Z1| |Z2| 1 σ,q 2 σ,q |αZ1| |Z1| ∗≤ ∗ AV| | R ≤ | | | | ∈ αZ1 σ,q = α Z1 σ,q. Since @ α is subadditive (cf. Pflug and Römisch [25]) it follows easily that | ∗ | | | | | 0 Lσ,q(P ) is a subspace of L (P ). ∗ Remark 17 (Stochastic dominance of second order). The definition of σ,∞ reflects the duality of risk functionals. Indeed, the supremum (16) can be restated as | · |

η 1 Z ∗ = inf η 0 : AV@R ( Z ) σ(u)du for all α< 1 . | |σ,∞ ≥ α | | ≤ 1 α · − Zα By the rearrangement inequality (cf. McNeil et al. [22, Theorem 5.25(2)]) this equivalent formulation involves the statement AV@R ( Z ) AV@R η σ(U) , (19) α | | ≤ α where U U (0, 1). Choosing U to be coupled in a comonotone way with Z it follows ∈ | | Z ∗ = inf η 0,U U (0, 1) : AV@R ( Z ) AV@R η σ(U) for all α< 1 | |σ,∞ ≥ ∈ α | | ≤ α Following Ogryczak and Ruszczyński [23], (19) is equivalent to saying that Z is dominated by Z ∗ σ(U) in second stochastic order.3 | | k kσ · Remark 18. i) By the choice α =0 in (16) it follows that

1 Z ∗ AV@R ( Z )= F −1(u)du = E Z = Z , (20) | |σ,∞ ≥ 0 | | |Z| | | k k1 Z0 so that L∗ (P ) L1(P ). σ,∞ ⊆ ii) Since for Z, Z′ L0(P ) with Z′ < Z we have for p [1, ) with Proposition 6 ∈ σ ∈ ∞ 1 1 −1 −1 ′ ′ Z = F (u)du σ(u)F ′ (u)du = Z Z k k1 |Z| ≤ |Z | k kσ,1 ≤k kσ,p Z0 Z0 it also follows that L∗ (P ) L1(P ). σ,q ⊆ Proposition 19. For Z L0(P ) we have ∈ Z ∗ = inf η 0; F : [0, 1) [0, ) nondecreasing: | |σ,∞ { ≥ ∀ → ∞ 1 1 F −1(u)F (u)du η σ(u)F (u)du , |Z| ≤ } Z0 Z0 and for p (1, ) ∈ ∞ Z ∗ = inf Z′ ; Z′ L0(P ) such that F : [0, 1) [0, ) nondecreasing: | |σ,q {k kσ,q ∈ ∀ → ∞ 1 1 −1 −1 F (u)F (u)du σ(u)F ′ (u)F (u)du , |Z| ≤ |Z | } Z0 Z0 where as usual 1/p +1/q =1. 3Cf. Dentcheva and Ruszczyński [7, 8, 9] for stochastic dominance of second order.

14 Proof. Since 1 is a nondecreasing, non-negative function for every α [0, 1) it follows [α,1] ∈ 1 1 −1 η 0; F : [0, 1) [0, ) nondecreasing: F|Z| (u)F (u)du η σ(u)F (u)du { ≥ ∀ → ∞ 0 ≤ 0 } 1 Z 1 Z η 0; α [0, 1) : F −1(u)du η σ(u)du . ⊆{ ≥ ∀ ∈ |Z| ≤ } Zα Zα On the other hand, if for some η 0 we have ≥ 1 1 α [0, 1) : F −1(u)du η σ(u)du ∀ ∈ |Z| ≤ Zα Zα it follows for all γ ,...,γ [0, ) and every choice of α <...<α [0, 1) that 1 n ∈ ∞ 1 n ∈ 1 n 1 n F −1(u) γ 1 (u)du η σ(u) γ 1 (u)du. |Z| j [αj ,1] ≤ j [αj ,1] 0 j=1 0 j=1 Z X Z X Since every non-negative, nondecreasing function F : [0, 1) [0, ) is the pointwise limit of a non- n 1 → ∞ decreasing sequence of such step functions j=1 γj [αj ,1] it follows from the Monotone Convergence Theorem that 1 P 1 F −1(u)F (u)du η σ(u)F (u)du |Z| ≤ Z0 Z0 for all such F . Hence it also holds

1 1 −1 η 0; F : [0, 1) [0, ) nondecreasing: F|Z| (u)F (u)du η σ(u)F (u)du { ≥ ∀ → ∞ 0 ≤ 0 } 1 Z1 Z η 0; α [0, 1) : F −1(u)du η σ(u)du ⊇{ ≥ ∀ ∈ |Z| ≤ } Zα Zα which proves the ﬁrst claim. The rest of the proposition is proved mutatis muntandis.

Proposition 20. For a distortion function σ and p [1, ) we have L∗ (P ) Lp (P )× and ∈ ∞ σ,q ⊆ σ sup E(ZY ) ; Y Lp (P ), Y 1 Z ∗ (21) {| | ∈ σ k kσ,p ≤ } ≤ | |σ,q for every Z L∗ (P ), where q is the conjugate exponent to p. ∈ σ,q ∗ 1 Proof. Let p = 1. For Z Lσ,∞(P ) it follows for arbitrary Y Lσ(P ) from the rearrangement inequality combined with Proposition∈ 19 ∈

1 1 E( ZY ) F −1(u)F −1(u)du Z ∗ σ(u)F −1(u)du | | ≤ |Z| |Y | ≤ | |σ,∞ |Y | Z0 Z0 = Z ∗ Y . | |σ,∞k kσ,1 1 × ∗ Hence, Z Lσ(P ) and the above inequality also implies that Z σ,∞ is an upper bound for sup E(ZY∈) ; Y L1 (P ), Y 1 . | | {| | ∈ σ k kσ,1 ≤ }

15 ∗ ′ q Next let p (1, ) and q the corresponding conjugate exponent. For Z Lσ,q(P ) let Z Lσ(P ) ′ ∈ ∞ 1 ∈ ∈ with Z σ< Z. For arbitrary Y Lσ(P ) it follows from the rearrangement inequality combined with Proposition 19 and Hölder’s inequality∈

1 1 −1 −1 −1 −1 E( ZY ) F (u)F (u)du σ(u)F ′ (u)F (u)du | | ≤ |Z| |Y | ≤ |Z | |Y | Z0 Z0 Z′ Y . ≤k kσ,qk kσ,p p × ′ q ′ Thus, Z Lσ(P ) and because Z Lσ(P ) with Z σ< Z was chosen arbitrarily, it follows that sup E(ZY∈ ) ; Y L1 (P ), Y 1∈ is bounded by Z ∗ . {| | ∈ σ k kσ,p ≤ } | |σ,q ∗ p × In order to show that in fact Lσ,q(P )= Lσ(P ) holds as well as equality in inequality (21) we have to distinguish the cases p =1 and p (1, ). We begin with the case p =1. ∈ ∞ Proposition 21. For a K-valued random variable Z on (Ω, , P ) and α [0, 1) there is Eα such that P (E )=1 α and F ∈ ∈F α − 1 AV@R ( Z )= E( Z 1 ). α | | 1 α | | Eα − Proof. Let E with ∈F Z > F −1(α) E Z F (α) {| | |Z| }⊆ ⊆ {| |≥ |Z| } be arbitrary. Denoting the positive part of an R-valued function f as usual by f+ it follows

1 1 AV@R ( Z )= F −1(u)du α | | 1 α |Z| − Zα 1 1 = F −1(α)+ (F −1(u) F −1(α)) du |Z| 1 α |Z| − |Z| + − Z0 1 = F −1(α)+ E ( Z F −1(α)) 1 (22) |Z| 1 α | |− |Z| E − 1 1 = F −1(α)+ E( Z 1 ) F −1(α)P (E) |Z| 1 α | | E − 1 α |Z| 1 − 1 − = E( Z 1 )+(1 P (E))F −1(α). 1 α | | E − 1 α |Z| − − −1 From the deﬁnition of F|Z| it follows immediately that

P ( Z < F −1(α)) α P ( Z F −1(α)) | | |Z| ≤ ≤ | |≤ |Z| so that P ( Z > F −1(α)) 1 α and P ( Z F −1(α)) 1 α. | | |Z| ≤ − | |≥ |Z| ≥ − Let U be a [0, 1]-valued, uniformly distributed random variable on (Ω, , P ). We deﬁne F E := Z > F −1(α) Z = F −1(α) U [0,β] β {| | |Z| } ∪ {| | |Z| }∩{ ∈ } for β [0, 1] and set ∈ f : [0, 1] [0, 1],f(β) := P (E ). → β

16 From the properties of a probability measure it follows easily that f is continuous as well as

f(0) = P ( Z > F −1(α)) 1 α and f(1) = P ( Z F −1(α)) 1 α. | | |Z| ≤ − | |≥ |Z| ≥ − Hence, there is β [0, 1] such that for E we have P (E )=1 α and it follows from (22) that 0 ∈ β0 β0 − Eβ0 has the desired property.

1 × For the case p =1 we can now give the desired intrinsic description of Lσ(P ) . 1 × ∗ Proposition 22. For a distortion function σ it holds Lσ(P ) = Lσ,∞(P ) and for every Z 1 × ∈ Lσ(P ) we have Z ∗ = sup E(ZY ) ; Y L1 (P ), Y 1 . | |σ,∞ {| | ∈ σ k kσ,1 ≤ } 1 × Proof. Let Z Lσ(P ) . By Proposition 21, for any α [0, 1) there is Eα such that AV R ∈ 1 E 1 ∈ ∈ F @ α( Z ) = 1−α ( Z Eα ) and P (Eα)=1 α. Employing the notation from Proposition 12 we obtain| | | | − 1 Z 1 AV@R ( Z )= ϕ ( 1 1 ) ϕ ∗ 1 α | | | Z 1 α {Z6=0} Z Eα |≤k Z kσ,1 1 αk Eα kσ,1 − 1 | | − ∗ 1 −1 = ϕZ σ,1 σ(u)F1 (u)du k k 1 α Eα − Z0 1 1 1 = ϕ ∗ σ(u)du = ϕ ∗ S (α), k Z kσ,1 1 α k Z kσ,1 1 α σ − Z1−P (Eα) − so that Z ∗ is ﬁnite and bounded above by | |σ,∞ ϕ ∗ = sup E(ZY ) ; Y L1 (P ), Y 1 . k Z kσ,1 {| | ∈ σ k kσ,1 ≤ } Proposition 20 now yields the rest of the claim. Combining Propositions 12 and 22 we immediately derive the next result.

∗ ∗ Theorem 23. Let σ be a distortion function. Then σ,∞ is a norm on Lσ,∞(P ) turning it into a Banach space. Moreover, | · |

Φ : (L∗ (P ), ∗ ) (L1 (P )∗, ∗ ),Z (Y Φ(Z)(Y ) := E(ZY )) σ,∞ | · |σ,∞ → σ k·kσ,1 7→ 7→ is an isometric isomorphism.

In order to derive an analogous representation for the case p (1, ) we need an equivalent result to 22 for this case. This requires some preparation. We begin∈ by∞ recalling a notion from Lorentz [21].

Deﬁnition 24. Let σ be a distortion function. A function H : [0, 1] R is called S -concave → σ if whenever y,b R are such that ySσ(α1)+ b = H(α1) and ySσ(α2)+ b = H(α2) for some α < α [0, 1] then∈ H(α) yS (α)+ b for each α [α , α ]. 1 2 ∈ ≥ σ ∈ 1 2 The next proposition is essentially contained in Lorentz [21, Proof of Theorem 3.6.2]. Neverthe- less, we include its proof for the reader’s convenience.

17 Proposition 25. Let σ be a distortion function and let u0 := inf u> 0; σ(u) > 0 . Moreover, let H be a set of S -concave functions such that H is constant{ for every H H}. Assume that σ |[0,u0] ∈ α [0, 1] : F (α) := inf H(α); H H > . ∀ ∈ { ∈ } −∞

Then, F is Sσ-concave.

Proof. Let y,b R and let α1 < α2 [0, 1] such that ySσ(αj )+ b = F (αj ), j = 1, 2. Let H H be arbitrary. Since∈ H is constant∈ there are y,¯ ¯b R such that yS¯ (α )+ ¯b = H(α ), j =1∈, 2. |[0,u0] ∈ σ j j In case of y y¯ 0 it follows that yS¯ σ +¯b (ySσ + b) is nonincreasing while yS¯ σ +¯b (ySσ + b) is nondecreasing− in≥ case of y y¯ 0. Therefore,− S -concavity of H together with yS −(α )+ b − ≤ σ σ j ≤ yS¯ σ(αj )+ ¯b, j =1, 2 implies α [α , α ]: yS (α)+ b yS¯ (α)+ ¯b H(α). ∀ ∈ 1 2 σ ≤ σ ≤ Since H H was arbitrary, we conclude that F yS + b on [α , α ]. ∈ ≥ σ 1 2 Proposition 26. Let σ be a distortion function, u0 := inf u> 0; σ(u) > 0 , and let y1,y2,b1,b2 R { } ∈ . Moreover, let H be Sσ-concave, continuous from the right in u0 such that H|[0,u0] is constant. If for y,b R and α1, α2 [0, 1] with α1 < α2 and u0 < α2 we have H(αj )= ySσ(αj )+ b, j = 1, 2, then it follows∈ that yS +∈b H on [0, 1] (α , α ). σ ≥ \ 1 2 Proof. It is straightforward to show that if for α, β [0, 1], α < β, it holds y S (α)+b = y S (α)+ ∈ 1 σ 1 2 σ b2 and y1Sσ(β)+ b1 >y2Sσ(β)+ b2 then β>u0 and γ (max α, u ,β]: y S (γ)+ b >y S (γ)+ b (23) ∀ ∈ { 0} 1 σ 1 2 σ 2 while for α, β [u , 1], α < β, the conditions y S (α)+ b > y S (α)+ b and y S (β)+ b = ∈ 0 1 σ 1 2 σ 2 1 σ 1 y2Sσ(β)+ b2 imply γ [α, β): y S (γ)+ b >y S (γ)+ b . (24) ∀ ∈ 1 σ 1 2 σ 2 In case of α u it follows from the hypothesis that H is constant that trivially yS+b H 1 ≤ 0 |[0,u0] ≥ on [0, α1]. Now let u0 < α1. We assume that yS(α)+ b

yS˜ σ(α)+ ˜b = H(α) and yS˜ (α2)+ ˜b = H(α2). The S -concavity of H hence implies H yS˜ + ˜b on [α, α ]. In particular σ ≥ σ 2 yS (α )+ b = H(α ) yS˜ (α )+ ˜b. (25) σ 1 1 ≥ σ 2 On the other hand

yS˜ σ(α)+ ˜b = H(α) > ySσ(α)+ b and yS˜ σ(α2)+ ˜b = H(α2)= ySσ(α2)+ b so that by (24) we obtain yS˜ +˜b>yS +b on [α, α ) which contradicts (25). Therefore yS +b H σ σ 2 ≥ on (u0, α1). Since Sσ is continuous and H is continuous from the right in u0 the same inequality holds on [u , α ). Because S and H are constant on [0,u ] we obtain yS + b H on [0, α ]. 0 1 σ 0 ≥ 1 It remains to show that yS + b H on [α2, 1] as well. Assume there is α [α2, 1] with yS (α)+ bu there≥ are again y,˜ ˜b R such that ∈ σ 2 0 ∈

yS˜ σ(α1)+ ˜b = H(α1) and yS˜ (α)+ ˜b = H(α).

18 Because H is Sσ-concave this implies

β [α , α]: H(β) yS˜ (β)+ ˜b. (26) ∀ ∈ 1 ≥ σ On the other hand

yS˜ σ(α1)+ ˜b = H(α1)= ySσ(α1)+ b and yS˜ σ(α)+ ˜b = H(α) > ySσ(α)+ b.

By (23) it follows yS˜ + ˜b>yS + b on (max α ,u , α]. In particular σ σ { 1 0}

yS˜ σ(α2)+ ˜b>ySσ(α2)+ b = H(α2)

which contradicts (26).

Deﬁnition 27. Let σ be a distortion function. For a continuous function G : [0, 1] R we deﬁne → ∗ R ∗ Gσ : [0, ) , Gσ(y) := inf ySσ(α) G(α). ∞ → α∈[0,1] −

Then G(α)+ G∗ (y) yS (α) for all α [0, 1],y 0 so that σ ≤ σ ∈ ≥ R ∗ Gσ : [0, 1] , Gσ(α) := inf ySσ(α) Gσ(y) → y≥0 − is well-deﬁned and satisﬁes G G. σ ≥ Remark 28.

i) Setting as before u0 := inf u > 0; σ(u) > 0 we have that Sσ|[u0,1] is a bijection from [u0, 1] onto [0, 1]. Denoting by abuse{ of notation its} inverse with S−1 it follows

−1 ∗ α [0, 1] : Gσ(S (α)) = inf yα Gσ(y) ∀ ∈ y≥0 − so that ˜ R ˜ ∗ Gσ : [0, 1] , Gσ(α) := inf yα Gσ(y) → y≥0 −

is well-defined. Being the infimum of nondecreasing and concave functions G˜σ is nondecreasing and concave, too. Therefore, G˜σ is differentiable from the right on [0, 1) with non-negative and nonincreasing right derivative. We obviously have Gσ = G˜σ Sσ so that the concavity ◦ dl dr of Sσ implies that Gσ is concave, too. Denoting left and right derivatives by dα and dα respectively, an appropriate adaption of your favorite proof of the chain rules yields d d d d α (0, 1] : l G (α)= r G˜ (S (α)) l S (α)= r G˜ (S (α))σ(α). ∀ ∈ dα σ dα σ σ dα σ −dα σ σ

1 Combined with G (1) G (α)= dl G (u)du we obtain σ − σ α dα σ R 1 α [0, 1] : G (α)= G (1) + H(u)σ(u)du (27) ∀ ∈ σ σ Zα for a non-negative, nondecreasing function H on [0, 1] which is continuous from the left.

19 ii) For y 0 the function yS G∗ (y) is obviously S -concave. It therefore follows from ≥ σ − σ σ Proposition 25 that Gσ is Sσ-concave. Moreover, being the inﬁmum of nonincreasing functions Gσ is nonincreasing.

iii) Because Sσ|[0,u0] =1 it follows that Gσ is constant on [0,u0]. Hence, for all α1, α2 [0, 1], α1 < α , there are y 0,b G∗ (y) with G (α )= yS (α ) b for j =1, 2. ∈ 2 ≥ ≥ σ σ j σ j − Gσ(α2)−Gσ(α1) Indeed, if α2 > u0 we choose y = which is well-deﬁned and non-negative Sσ(α2)−Sσ (α1) because Sσ is strictly decreasing on [u0, 1] and Gσ is nonincreasing. Then

b = ySσ(α2) Gσ(α2) inf ySσ(α) Gσ(α) − ≥ α∈[0,1] − ∗ ∗ = inf ySσ(α) (inf yS˜ σ(α) Gσ(˜y)) Gσ(y). α∈[0,1] − y˜≥0 − ≥ In case of α u we may choose y =0 so that 2 ≤ 0 ∗ ∗ b = Gσ(α2)= Gσ(0) = inf y Gσ(y) Gσ(0). − − − y≥0 − ≥ If additionally G is nonincreasing it holds ∗ ∗ Gσ(0) = inf y Gσ(y) Gσ(0) = inf ( G(α)) = sup G(α)= G(0) y≥0 − ≤− − α∈[0,1] − α∈[0,1] so that because G G we conclude ≤ σ b = G (0) = G(0) = G∗ (0). − σ − σ Proposition 29. Let G : [0, 1] R be continuous and nonincreasing. If α (0, 1) is such that G (α) > G(α) then there are 0 →α <α<α 1 and y 0 such that ∈ σ ≤ 1 2 ≤ ≥ β (α , α ): G (β)= yS (β) G∗ (y). ∀ ∈ 1 2 σ σ − σ Proof. By continuity of Sσ and Gσ there are 0 α1 <α<α2 1 such that Gσ(α2) > G(α1). From Remark 28 iii) we conclude the existence≤ of y 0 and b≤ G∗ (y) such that G (α ) = ≥ ≥ σ σ j ySσ(αj ) b, j =1, 2 and such that y =0 in case of α2 u0. Because− S and G are nonincreasing and y 0 it follows≤ from σ σ ≥ inf Gσ(β)= Gσ(α2) > G(α1) = sup G(β) β∈(α1,α2) β∈(α1,α2) that ySσ b G on (α1, α2). If α −u ≥, we have seen in Remark 28 iii) that without loss of generality we may assume y =0 2 ≤ 0 and b = G(0). Since G is nonincreasing it thus follows ySσ b = G(0) G on [0, 1]. If α −> u we apply Proposition 26 to G to conclude yS− b G ≥on [0, 1] (α , α ). Since 2 0 σ σ − ≥ σ \ 1 2 Gσ G and ySσ b G on (α1, α2) we obtain also in this case ySσ b G on [0, 1]. ≥ − ≥ − ≥ ∗ So in both cases ySσ b G or equivalently ySσ G b on [0, 1] so that Gσ(y) = inf α ySσ(α) − ∗≥ ∗ − ≥ − G(α) b. Since also b Gσ(y) it follows b = Gσ(y). ≥ ≥ ∗ Finally, since Gσ is Sσ-concave and Gσ(αj )= ySσ(αj ) Gσ(αj ) holds for j =1, 2 it follows for β (α , α ) − ∈ 1 2 ∗ ∗ ∗ ySσ(β) Gσ(y) Gσ(β) = inf yS˜ σ(β) Gσ(˜y) ySσ(β) Gσ(y) − ≤ y˜≥0 − ≤ − which proves the claim.

20 Proposition 30. Assume that G : [0, 1] R is continuous, nonincreasing, and that G(1) = 0. → Then Gσ(0) = G(0) and Gσ(1) = 0.

Proof. We already observed in Remark 28 iii) that Gσ(0) = G(0). Using the compactness of [0, 1], G(1) = 0, and that Sσ(α)=0 implies α =1 it follows 1 n N k N α [0, 1] : G(α) > k S (α)+ ∀ ∈ ∃ n ∈ ∀ ∈ n σ n N N ∗ which implies that for every n there is kn with 1/n Gσ(kn). Because Sσ(1) = 0 we derive ∈ ∈ − ≤ ∗ ∗ Gσ(1) = inf ( Gσ(y)) inf ( Gσ(kn)) 0= G(1) Gσ(1) y≥0 − ≤ n∈N − ≤ ≤

which gives Gσ(1) = 0. Combining Propositions 29 and 30 we immediately obtain the next result.

Proposition 31. Let G : [0, 1] R be continuous and nonincreasing such that G(1) = 0. If α (0, 1) satisﬁes G (α) > G(α) there→ are 0 α <α<α 1 and y 0 such that G = yS G∗ (y∈) σ ≤ 1 2 ≤ ≥ σ σ − σ on (α1, α2) and Gσ(αj )= G(αj ) for j =1, 2. We have now everything at hand to derive the analogue of Proposition 22.

Lemma 32. Let σ be a distortion function and p (1, ) with conjugate exponent q. Then p × ∗ p × ∈∗ ∞ ∗ p Lσ(P ) = Lσ,q(P ), for every Z Lσ(P ) it holds Z σ,q = ϕZ σ,p, and there is Y Lσ(P ) with Y =1 such that ϕ (Y )= ∈ϕ ∗ . | | k k ∈ k kσ,p Z k Z kσ,p Proof. By Proposition 20 we only have to show Lp (P )× L∗ (P ) and that σ ⊆ σ,q sup E(ZY ) ; Y Lp (P ), Y 1 { | ∈ σ k kσ,p ≤ } ∗ p × is an upper bound for Z σ,q for any Z Lσ(P ) . So we fix Z Lp (P|)×|. By Remark ∈13 we also have Z Lp (P )×. We define ∈ σ | | ∈ σ 1 G : [0, 1] [0, ), G(α) := F −1(u)du → ∞ |Z| Zα and observe that G is well-defined by Z L1(P ). G is obviously continuous, differentiable from the left, nonincreasing with G(1) =| 0|. ∈ By Proposition 31 and Remark 28 i) there is a non- negative, nondecreasing function H on [0, 1] which is continuous from that left such that Gσ(α)= 1 α H(u)σ(u)du. If there is α (0, 1) with G (α) > G(α) it follows immediately from Proposition 31 that there are R ∈ σ 0 α1 <α<α2 1 and y 0 such that H(u)= y for u (α1, α2) and Gσ(αj )= G(αj ), j =1, 2 so≤ that ≤ ≥ ∈

α2 α2 H(u)qσ(u)du = yq−1 H(u)σ(u)du = yq−1(G (α ) G (α )) σ 1 − σ 2 Zα1 Zα1 α2 = yq−1(G(α ) G(α )) = H(u)q−1F −1(u)du. 1 − 2 |Z| Zα1

21 On the other hand, if Gσ(α) = G(α) by continuity there is a maximal closed interval [α1, α2] containing α such that G and Gσ coincide on [α1, α2]. Thus, on (α1, α2) the left derivatives of G −1 and Gσ coincide, i.e., F|Z| = Hσ on (α1, α2) which implies again

α2 α2 q q−1 −1 H(u) σ(u)du = H(u) F|Z| (u)du. Zα1 Zα1 Combining these arguments gives

α , α [0, 1] : G (α )= G(α ), G (α )= G(α ) (28) ∀ 1 2 ∈ σ 1 1 σ 2 2 α2 α2 q q−1 −1 H(u) σ(u)du = H(u) F|Z| (u)du. ⇒ α α Z 1 Z 1 In order to proceed, we distinguish two cases. First we assume that there is a strictly increasing sequence (αn)n∈N in (0, 1) converging to 1 such that G(αn)= Gσ(αn). We deﬁne Y := (1 Hq−1) U, n [0,αn] ◦ where U U (0, 1) is coupled in a comonotone way with Z . From 1/p +1/q = 1 it follows that Y p = (1∈ Hq) U which implies | | | n | [0,αn] ◦ 1 αn p −1 p q Yn = F (u) σ(u)du = H (u)σ(u)du < (29) k kσ,p |Yn| ∞ Z0 Z0 1 p since H in nondecreasing and 0 σ(u)du =1 so that Yn = Yn Lσ(P ). Using the notation from Proposition 12, because Z and U are coupled in a comonotone| | ∈ way we have by (29) and (28) | | R applied to α1 =0 and α2 = αn

αn αn H(u)qσ(u)du = Hq−1(u)F −1(u)du = E( Y Z ) |Z| | n|| | Z0 Z0 αn 1/p ∗ ∗ q ϕ|Z| σ,p Yn σ,p = ϕ|Z| σ,p H(u) σ(u)du ≤k k k k k k 0 Z which gives 1 1/q 1 q ∗ [0,αn] H(u) σ(u)du ϕ|Z| σ,p 0 ≤k k Z for all n N. Using that limn→∞ αn = 1 an application of the Monotone Convergence Theorem yields ∈ 1 1/q H(u)qσ(u)du ϕ ∗ (30) ≤k |Z|kσ,p Z0 ′ q ′ ′ −1 so that Z := H U belongs to L (P ). Because Z = Z and F ′ = H it follows from ◦ σ | | |Z | 1 1 α [0, 1] : H(u)σ(u)du = G (α) G(α)= F −1(u)du ∀ ∈ σ ≥ |Z| Zα Zα that Z′ < Z which combined with (30) yields Z L∗ and σ ∈ σ,q Z ∗ ϕ ∗ = ϕ ∗ | |σ,q ≤k |Z|kσ,p k Z kσ,p

22 where we have used Remark 13 in the last equality. Since also Z ∗ Z ∗ we obtain from (30) k kσ,p ≤ | |σ,q 1 1/q ∗ q ′ ′ ϕZ σ,p = H(u) σ(u)du = inf Z σ,q; Z σ< Z . (31) k k 0 {k k } Z Now we deﬁne Z Y := 1 Hq−1(U). (32) {Z6=0} Z | | Then the same arguments used in deriving (29) combined with (30) show that Y Lp (P ) and ∈ σ 1 1/p q Y σ,p = H (u)σ(u)du . k k 0 Z Moreover, using that Z and U are coupled in a comonotone way, (28) applied to α = 0 and | | 1 α2 =1, and (31) give

1 ϕ (Y )= E(ZY )= E( ZY )= H(u)q−1F −1(u)du Z | | |Z| Z0 1 1 1/q 1 1/p = H(u)qσ(u)du = H(u)qσ(u)du H(u)qσ(u)du Z0 Z0 Z0 = ϕ ∗ Y . k Z kσ,pk kσ,p

Next, if there is no strictly increasing sequence (αn)n∈N in (0, 1) converging to 1 such that G(α ) = G (α ) there is β (0, 1) such that G(u) < G (u) for all α (β, 1) and such that n σ n ∈ σ ∈ G(β) = Gσ(β). It therefore follows from Proposition 31 that there is y 0 such that H = y on (β, 1). Because H is nondecreasing this implies that H is bounded so that≥ trivially

1 H(u)qσ(u) < . ∞ Z0 By repeating the arguments from the first part of the proof it follows for U U (0, 1) coupled ′ ′ q ′∈ in a comonotone way with Z that Z := H U satisfies Z Lσ(P ) and Z σ< Z which gives ∗ ∗ | | ∗ ′ ◦ ∈ Z Lσ,q(P ) and Z σ,q = ϕZ σ,p = Z σ,q. Defining Y as in (32) finally gives again ϕZ (Y ) = ϕ∈ ∗ Y which| | provesk thek claim.k k k Z kσ,pk kσ,p Combining Proposition 12 and Lemma 32 we immediately derive the next result.

∗ Theorem 33. Let σ be a distortion function and p (1, ) with conjugate exponent q. Then σ,q ∗ ∈ ∞ |·| is a norm on Lσ,q(P ) turning it into a Banach space. Moreover, Φ : (L∗ (P ), ∗ ) (Lp (P )∗, ∗ ),Z (Y Φ(Z)(Y ) := E(ZY )) σ,q | · |σ,q → σ k·kσ,p 7→ 7→ p ∗ p is an isometric isomorphism. Moreover, for every ϕ Lσ(P ) there is Y Lσ(P ) with Y σ,p =1 such that ϕ(Y )= ϕ ∗ . ∈ ∈ k k k kσ,p Corollary 34. For a distortion function σ and p (1, ) the Banach space Lp (P ) is reﬂexive. ∈ ∞ σ Proof. This is an immediate consequence of James’ Theorem (see, e.g. Diestel [10, Theorem I.3]) and Theorem 33.

23 Proposition 35. Simple functions (and thus L∞) are dense in L∗ (P ), whenever q < . σ,q ∞ Proof. Let F contain all finite sigma algebras for which the measure P is defined. Note that (F, ) is a filter, and the proof of Proposition 9Factually demonstrates that ⊆ E (Y ) Y 0 k |F − kσ,p −→F whenever F increases. RecallF first ∈ that AV@R E(Y ) AV@R (Y ). Indeed, it follows from the conditional Jensen α |F ≤ α inequality (cf. Williams [31, Section 34]) that E(Y ) q E (Y q) , and hence, using |F − + ≤ − +|F Pflug [24], 1 AV@Rα E(Y ) = min q + E E(Y ) q |F q∈R 1 α |F − + − 1 min q + EE (Y q)+ ≤ q∈R 1 α − |F − 1 = min q + E (Y q)+ = AV@Rα(Y ). q∈R 1 α − |F − ′ Suppose that Z σ< Z. It follows that 1 1 1 −1 −1 −1 σ(u)F ′ (u)du F (u)du F (u)du Z ≥ Z ≥ E(Z|F) Zα Zα Zα ′ E E ∗ ∗ for every α 1, that is Z σ< (Z ) and thus (Z ) σ,q Z σ,q. The assertion follows as E (Z ): ≤ F is arbitrarily close|F to Z in thek norm|F k by≤ Proposition k k 9. { |F F ∈ } k·kσ,q 1 We close this section by having a closer look at Lσ(P ) and its dual space. 1 Theorem 36. The dual space of Lσ(P ) is not separable. 1 Proof. It is enough to assume that σ is unbounded, as for bounded σ we have that Lσ(P ) is isomorphic to L1(P ) by Proposition 8 and its dual L∞(P ) is not separable. For β [0, 1] consider the random variables ∈ σ(U) if U β, Zβ := ≤ σ(1 + β U) if U >β ( − U ∗ for a (fixed) uniform random variable U (0, 1). Notice, that Zβ σ,1 = 1, since Zβ is a rearrangements of σ(U). Assume that β<γ∈ and observe that k k Z Z = σ(1 + γ U) σ(1 + β U) γ − β − − − σ(1 + γ U) σ(1 + β γ) ≥ − − − whenever U>γ. Then it holds that AV R ∗ @ α Zγ Zβ Zγ Zβ σ,1 lim sup 1 − k − k ≥ α→1 1 1−α α σ(u)du AV R @ αR σ(1 + β U) σ(1 + β γ) lim sup −1 − − . ≥ α→1 1 1−α α σ(u)du R 24 1 1 Now, as σ is unbounded, the denominator is unbounded as well (indeed, we have 1−α α σ(u)du σ(α)) and hence ≥ R AV R ∗ @ α σ(1 + β U) σ(1 + β γ) Zγ Zβ σ,1 lim sup −1 − − k − k ≥ α→1 1 1−α α σ(u)du 1 1 σ(u)du σ(1R + β γ) = lim 1−α α − − =1. (33) α→1 1 1 R 1−α α σ(u)du

R 1 ∗ Suppose ﬁnally that there is a dense sequence (Dk)k∈N Lσ(P ) . For β [0, 1] ﬁxed there N ∗ 1 ⊂∗ ∗∈ ∗ is k such that Zβ Dk σ,1 < 2 . But 1 Zβ Zγ σ,1 Zβ Dk σ,1 + Dk Zγ σ,1, ∈ k − k ∗ 1 ≤ k − k ≤ k − k k − k from which follows that Dk Zγ σ,1 > 2 whenever γ = β. Hence only countably many Zβ can be k − k 6 ∗ 1 N N approximated by the sequence (Dk)k∈ with a distance Zβ Dk σ < 2 and (Dk)k∈ thus is not dense giving the desired contradiction. k − k

5 The dual space in the vector-valued case

p K In this section we determine the dual space of Lσ(P,X) for arbitrary Banach spaces X over R, C . We denote the space of X-valued simple functions on (Ω, , P ) by (X), i.e., ∈ { } F S (X)= Y :Ω X; Y (Ω) is ﬁnite and x X : Y −1( x ) . S { → ∀ ∈ { } ∈F} Then it is straightforward to see and well-known that

ϕ : (X) K; ϕ linear and continuous with respect to { S → k·k∞} and µ : X∗; µ vector measure of bounded variation { F → } are isomorphic via the linear mapping

Φ: ϕ µ (E)(x) := ϕ(1 x), x X, E . (34) 7→ ϕ E ∈ ∈F For a vector measure µ we denote by µ its variation. | | Lemma 37. For a linear mapping ϕ : (X) K we have S → n n sup ϕ( 1 x ) ; E partition of Ω, x X, 1 x 1 {| Ej j | j ∈F j ∈ k Ej j kσ,p ≤ } j=1 j=1 X X n n = sup α µ (E ) ; E partition of Ω, α K, α 1 1 { | j |k ϕ j k j ∈F j ∈ k j Ej kσ,p ≤ } j=1 j=1 X X n n = sup α 1 d µ ; E partition of Ω, α K, α 1 1 , { | j Ej | | ϕ| j ∈F j ∈ k j Ej kσ,p ≤ } Ω j=1 j=1 Z X X where µϕ is deﬁned as in (34).

25 Proof. For a partition E1,...,En of Ω, α1,...,αn K, and z1,...,zn X with zj =1 we have ∈F ∈ ∈ k k n n n α 1 z p = sup σ(U) α p 1 dP = α , 1 p , k j Ej jkσ,p | j | Ej k j Ej kσ,p j=1 U∈U(0,1) Ω j=1 j=1 X Z X X p where the norm on the left hand side is the one on Lσ(P,X) while the norm on the right hand side p denotes the one on Lσ(P ). Therefore we conclude n n sup ϕ( 1 x ) ; E partition of Ω, x X, 1 x 1 | Ej j | j ∈F j ∈ k Ej j kσ,p ≤ j=1 j=1 n X X o n = sup ϕ( α 1 z ) ; E partition of Ω, α 0,z X, z =1 | j Ej j | j ∈F j ≥ j ∈ k jk j=1 n X n and α 1 1 k j Ej kσ,p ≤ j=1 X o n = sup ϕ(α 1 z ) ; E partition of Ω, α 0,z X, z =1 | j Ej j | j ∈F j ≥ j ∈ k jk j=1 n X n and α 1 1 k j Ej kσ,p ≤ j=1 X o n n = sup α µ (E ) ; E partition of Ω, α K, α 1 1 , | j |k ϕ j k j ∈F j ∈ k j Ej kσ,p ≤ j=1 j=1 n X X o which gives the first equality. Using the definition of µ we continue | ϕ| n n sup α µ (E ) ; E partition of Ω, α K, α 1 1 | j |k ϕ j k j ∈F j ∈ k j Ej kσ,p ≤ j=1 j=1 n X X o n = sup αi µϕ(Ej ) ; Ej partition of Ω, αj K =1 | | k k ∈F ∈ i j:αi=αj n αi pairwiseX different X n and α 1 1 k j Ej kσ,p ≤ j=1 X o n K = sup αi µϕ j:αi=αj Ej ; Ej partition of Ω, αj i=1 | || | ∪ ∈F ∈ n αi pairwiseX different n and α 1 1 k j Ej kσ,p ≤ j=1 X o n n = sup α 1 d µ ; E partition of Ω, α K, α 1 1 | j Ej | | ϕ| j ∈F j ∈ k j Ej kσ,p ≤ Ω j=1 j=1 n Z X X o which proves the second equality.

26 Deﬁnition 38. For a distortion function σ, p [1, ), and a Banach space X we deﬁne ∈ ∞ (X) := ϕ : (X) K; ϕ linear and continuous with respect to . Lσ,p S { S → k·kσ,p}

Lemma 39. Let Φbe the natural isomorphism from (34). Then Φ( σ,p (X) ) coincides with the set L S µ : X∗; µ is a σ-additive vector measure of bounded variation { F → d µ such that µ P and | | L∗ (P ) , | | ≪ dP ∈ σ,q } and d µ ϕ (X) : ϕ = | ϕ| ∗ , ∀ ∈ Lσ,p S k k | dP |σ,q where q is the conjugate exponent to p. Proof. For ϕ (X) it follows from the density of (X) in Lp (P,X) that ϕ extends to a ∈ Lσ,p S S σ unique element of Lp (P,X)× which we still denote by ϕ. For a pairwise disjoint sequence (E ) N σ j j∈ in and its union E it follows for arbitrary x X F ∈ m ∞ µϕ E (x) µϕ Ej (x) = µϕ Ej (x) = ϕ(1∪∞ E x) | − | | | | j=m+1 j | j=1 j=m+1 [ [ ∗ ϕ x 1∪∞ E σ,p ≤k kσ,p k kk j=m+1 j k 1 ∗ 1/p = ϕ σ,p x σ(u)du . k k k k 1−P∞ P (E ) Z j=m+1 j With the aid of Lebesgue’s Dominated Convergence Theorem it follows

m 1 ∗ 1/p µϕ(E) µϕ(Ej ) ϕ σ(u)du m→∞ 0. σ,p ∞ k − k≤k k 1−P P (E ) → j=1 Z j=m+1 j X Thus Φ(ϕ)= µϕ is a σ-additive vector measure. Moreover, for every ﬁnite partition E1,...,En of Ω and x ,...,x X with x 1 we have ∈F 1 n ∈ k j k≤ n n n µ E (x ) = ϕ(1 x ) = sign ϕ(1 x ) ϕ(1 x ) | ϕ j j | | Ej j | Ej j Ej j j=1 j=1 j=1 X X X n = ϕ( sign ϕ(1 x ) 1 x ) | Ej j Ej j | j=1 X n ϕ ∗ sign ϕ(1 x ) 1 x ≤k kσ,p k Ej j Ej j kσ,p j=1 X n ∗ 1 p 1 p 1/p ϕ σ,p sup sign ϕ( Ej xj ) Ej xj σ(U)dP ≤k k U k k U∈ (0,1) ZΩ j=1 X ∗ 1/p ∗ ϕ σ,p sup σ(U)dP = ϕ σ,p, ≤k k U k k U∈ (0,1) ZΩ

27 where for a complex number α as usual sign (α) = α in case α = 0, resp. sign (0) = 0. Thus, for |α| 6 arbitrary ε> 0 it follows for suitable choices xε X from the above inequality that j ∈ n n ε µ (E ) µ E (xε) + ϕ ∗ + ε, k ϕ j k≤ | ϕ j j | n ≤k kσ,p j=1 j=1 X X n ∗ ∗ i.e., j=1 µϕ(Ej ) ϕ σ,p which in turn implies µϕ (Ω) ϕ σ,p. Hence Φ(ϕ) = µϕ is of bounded variation.k k≤k k | | ≤ k k P Since µϕ is σ-additive the same holds for µϕ (see Diestel and Uhl [11, Proposition I.1.9]), i.e., µ is a (ﬁnite) measure on . If E satisﬁes| | P (E)=0 it follows for x X | ϕ| F ∈F ∈

1/p 1E x σ,p = x ( sup σ(U)dP ) =0 k k k k U u∈ (0,1) ZE and therefore µϕ(E) = 0. If E1,...,En is a partition of E it follows P (Ej )=0 and thus n µ (E )k =0 whichk implies µ (E)=0∈ F. By an application of the Radon-Nikodým Theorem j=1 k ϕ j k | ϕ| we obtain g L1(P ),g 0 such that P ϕ ∈ ϕ ≥

E : g dP = µ (E). ∀ ∈F ϕ | ϕ| ZE From the fact that (X) is dense in Lp (P,X) and (K) is dense in Lp (P ) it follows with Lemma 39 S σ S σ n n ϕ ∗ = sup ϕ( 1 x ) ; E partition of Ω, x X, 1 x 1 k kσ,p | Ej j | j ∈F j ∈ k Ej j kσ,p ≤ j=1 j=1 n X X o n = sup α 1 d µ ; E partition of Ω, α K | j Ej | | ϕ| j ∈F j ∈ Ω j=1 n Z X n and α 1 1 k j Ej kσ,p ≤ } j=1 X n = sup α 1 g dP ; E partition of Ω, α K | j Ej | ϕ j ∈F j ∈ Ω j=1 n Z X n and α 1 1 k j Ej kσ,p ≤ j=1 X o p = sup fgϕ dP ; f Lσ(P ), f σ,p 1 , Ω | | ∈ k k ≤ n Z o p × ∗ ∗ ∗ so that in particular gϕ Lσ(P ) = Lσ,q(P ) and ϕ σ,p = gϕ σ,q. Since ϕ σ,p( (X)) was ∈ k k | | ∈ L ∗S chosen arbitrarily this finally shows that Φ( σ,p( (X)) is contained in the set of X -valued, σ- additive vector measures of bounded variationL suchS that their bounded variation measure admits a ∗ P -density in Lσ,q(P ). Next let µ be such a measure and set ϕ := Φ−1(µ). We have to show that ϕ belongs to (X) . But from the density of (K) in Lp (P ) it follows immediately together with Lemma 37 Lσ,p S S σ 28 that n n sup ϕ( 1 x ) ; E partition of Ω, x X, 1 x 1 {| Ej j | j ∈F j ∈ k Ej j kσ,p ≤ } j=1 j=1 X X n n d µ = sup α 1 | | dP ; E partition of Ω, α K, α 1 1 { | j Ej | dP j ∈F j ∈ k j Ej kσ,p ≤ } Ω j=1 j=1 Z X X d µ = | | ∗ < , | dP |σ,p ∞ which shows ϕ (X) . ∈ Lσ,p S Definition 40. Let X be a Banach space, σ a distortion function, and p [1, ) with conjugate exponent q. Then we define ∈ ∞

L∗ (P,X∗) := µ : X∗; µ is a σ-additive vector measure of bounded σ,q { F → d µ variation such that µ P and | | L∗ (P ) , | | ≪ dP ∈ σ,q } which is obviously a subspace of the space of all X∗-valued vector measures on . Moreover, for F µ L∗ (P,X∗) we set µ ∗ := d|µ| ∗ . Then, ∗ is obviously a norm on L∗ (P,X∗). ∈ σ,q | |σ,q | dP |σ,q | · |σ,q σ,q ∗ ∗ p Remark 41. For µ Lσ,q(P,X ) it follows from Lemma 39 and the density of (X) in Lσ(P,X) −1 ∈ S p that Φ (µ) can be extended in a unique way to a continuous linear functional on Lσ(P,X) which we again denote by Φ−1(µ). For Y Lp (P,X) we also write for obvious reasons ∈ σ Y dµ := Φ−1(µ)(Y ). ZΩ With this notation the following theorem is an immediate consequence of Lemma 39, Proposition 22, and Lemma 32. Theorem 42. Let X be a Banach space, σ a distortion function, and p [1, ) with conjugate exponent q. Then (L∗ (P,X∗), ∗ ) is a Banach space and the mapping ∈ ∞ σ,q | · |σ,q

Ψ : (L∗ (P,X∗), ∗ ) (Lp (P,X)∗, ∗ ), µ (Y Y dµ) σ,q | · |σ,q 7→ σ k·kσ,p 7→ 7→ ZΩ is an isometric isomorphism. Deﬁnition 43. For a Banach space X, p [1, ) with conjugate exponent q we deﬁne ∈ ∞ Lq ∗(P,X) := Z :Ω X; Z strongly measurable, Z L∗ (P ) σ { → k k ∈ σ,q } q ∗ q,∗ ∗ and for Z Lσ (P,X) we set Z σ := Z σ,q, where as usual we identify random variables which coincide P∈-almost everywhere.| It| follows|k easilyk | that Lq ∗(P,X) is a vector space and q,∗ a norm. σ | · |σ Remark 44. For Z Lq ∗(P,X∗) it follows from Z L∗ (P ) L1(P ) that ∈ σ k k ∈ σ,q ⊆

µ : X∗,µ (E) := Z dP Z F → Z ZE

29 is a well-deﬁned, σ-additive vector measure of bounded variation with µZ (E) = E Z dP (see, e.g., [11, Theorem II.2.4]). A straightforward calculation gives for Y |(X|) k k ∈ S R Y dµ = Z(ω), Y (ω) dP (ω). Z h i ZΩ ZΩ Moreover, for Z Lq ∗(P,X∗) and Y Lp (P,X) it follows from Z L∗ (P ) and Y Lp (P ) ∈ σ ∈ σ k k ∈ σ,q k k ∈ σ

Z(ω), Y (ω) dP (ω) Z(ω) Y (ω) dP (ω) Z ∗ Y |h i| ≤ k kk k ≤|k k |σ,q |k k |σ,p ZΩ ZΩ = Z q ∗ Y | |σ k kσ,p which implies that ψ : Lp (P,X) K, Y E( Z, Y ) is a well-deﬁned continuous linear functional Z σ → 7→ h i which coincides on the dense subspace (X) with Ψ(µZ ). Together with Theorem 42 this shows that S ι : (Lq ∗(P,X∗), q,∗) Lp (P,X)∗,Z Y E( Z, Y ) σ | · |σ → σ 7→ 7→ h i is an isometry. As in the case of Bochner-Lebesgue spaces we have the following result. Theorem 45. For a Banach space X, a distortion function σ, and p [1, ) with conjugate exponent q the isometry ∈ ∞

ι : (Lq ∗(P,X∗), q,∗) Lp (P,X)∗,Z Y E( Z, Y ) σ | · |σ → σ 7→ 7→ h i is an isomorphism if and only if X∗ has the Radon-Nikodým property with respect to (Ω, , P ). F Proof. Assume first, that X∗ has the Radon-Nikodým property with respect to (Ω, , P ). By p ∗ F Remark 44 we only have to show surjectivity of ι. For an arbitrary ϕ Lσ(P,X) there is by Theorem 42 a σ-additive X∗-valued vector measure of bounded variation∈ such that µ P and d|µ| ∗ d|µ| ∗ ∗ | | ≪ dP Lσ,q(P ) with ϕ = dP σ,q. By the Radon-Nikodým property of X it follows that there ∈ 1 ∗ k k | | is Z L (P,X ) such that µ(E)= E Z dP for all E . Since µ (E)= E Z dP (see e.g. [11, ∈ q ∗ ∗ ∈F | | k k Theorem II.2.4]) it follows that Z Lσ (P,X ) and ι(Z)= µ showing the surjectivity of ι. Now, let ι be an isometric isomorphism.∈R The proof that X∗ has the Radon-NikodýmR property is along the same lines as the proof of the corresponding implication of [11, Theorem IV.1.1]. However, we include the proof for the reader’s convenience. So, let µ : X∗ be a P -continuous vector F → measure of bounded variation and fix E0 such that P (E0) > 0. By the Hahn Decomposition Theorem applied to the signed measure kP∈ F µ for large enough k > 0 gives the existence of − | | B ,B E0, P (B) > 0 such that µ (E) kP (E) for all E , E B. For Y (X), Y = n∈ F1 x⊆ with pairwise disjoint E | | and≤ x X we define∈ F ⊆ ∈ S j=1 Ej j j ∈F j ∈ P n ϕ(Y )= µ(E B)(x ). j ∩ j j=1 X Denoting the norm in L1(P,X) as usual by Theorem 4 then gives k·k1 n ϕ(Y ) k µ(E B)(x ) k Y k Y | |≤ k j ∩ j k≤ k k1 ≤ k kσ,p j=1 X

30 so that the obviously linear mapping ϕ on (X) is continuous with respect to σ,p. By Proposi- S p ∗ k·k tion 9 ϕ extends (in a unique way) to an element of Lσ(P,X) which we still denote by ϕ. Since ι is supposed to be surjective there is Z Lq ∗(P,X∗) L1(P,X∗) such that ∈ σ ⊆ Y Lp (P,X): ϕ(Y )= E( Z, Y ). ∀ ∈ σ h i 1 Since µ(E B)(x) = ϕ( E x) = E Z(ω), x dP (ω) = E Z(ω) dP (ω), x for all E , x X it follows that∩µ(E B)= Z dP . h i h i ∈ F ∈ ∩ E R R Because E0 with P (E0) > 0 was chosen arbitrarily, it follows from [11, Corollary III.2.5] that 1∈F ∗ R there is Z L (P,X ) such that µ(E)= E Z dP for all E which proves the Radon-Nikodým property of∈X∗ with respect to (Ω, , P ). ∈F F R 6 Summary

This paper introduces Banach spaces, which naturally carry risk measures for vector-valued returns. Risk measures are continuous on these spaces, and the spaces are as large as possible. The spaces are built based on duality, and in this sense are natural for risk measures involving vector-valued returns. We provide a complete characterization of the topological dual, which essentially simpliﬁes if the dual of the state space enjoys the Radon–Nikodým property. It is a key property of these spaces that the corresponding risk functional is continuous (in fact, Lipschitz continuous) with respect to any of the associated norms introduced, such that they all qualify as a domain space for the risk measure.

References

[1] C. Acerbi. Spectral measures of risk: A coherent representation of subjective risk aversion. Journal of Banking & Finance, 26:1505–1518, 2002. doi: 10.1016/S0378-4266(02)00281-9. [2] A. Ahmadi-Javid and A. Pichler. Norms and Banach spaces induced by the Entropic Value- at-Risk. 2016. [3] P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath. Coherent Measures of Risk. Mathematical Finance, 9:203–228, 1999. doi: 10.1111/1467-9965.00068. [4] F. Bellini and E. Rosazza Gianin. Haezendonck–Goovaerts risk measures and Orlicz quantiles. Insurance: Mathematics and Economics, 51(1):107–114, 2012. doi: 10.1016/j.insmatheco.2012. 03.005. [5] C. Burgert and L. Rüschendorf. Consistent risk measures for portfolio vectors. Insurance: Mathematics and Economics, 38(2):289–297, 2006. ISSN 0167-6687. doi: 10.1016/j.insmatheco. 2005.08.008. [6] P. Cheridito and T. Li. Risk measures on Orlicz hearts. Mathematical Finance, 19(2):189–214, 2009. doi: 10.1111/j.1467-9965.2009.00364.x. [7] D. Dentcheva and A. Ruszczyński. Optimization with stochastic dominance constraints. SIAM Journal on Optimization, 14(2):548–566, 2003. doi: 10.1137/S1052623402420528.

31 [8] D. Dentcheva and A. Ruszczyński. Convexification of stochastic ordering. C. R. Acad. Bulgare Sci., 57(3):5–10, 2004. [9] D. Dentcheva and A. Ruszczyński. Portfolio optimization with stochastic dominance constraints. Journal of Banking & Finance, 30:433–451, 2006. doi: 10.1016/j.jbankfin.2005.04.024. [10] J. Diestel. Geometry of Banach spaces—selected topics. Lecture Notes in Mathematics, Vol. 485. Springer-Verlag, Berlin-New York, 1975. [11] J. Diestel and J. J. Uhl, Jr. Vector Measures. Number 15 in Mathematical Surveys. American Mathematical Society, 1977. URL http://books.google.com/books?id=EQFjD90fXWAC. [12] I. Ekeland and W. Schachermayer. Law invariant risk measures on L∞(Rd). Statistics & Risk Modeling, 28(3):195–225, 2011. doi: 10.1524/stnd.2011.1099. [13] I. Ekeland, A. Galichon, and M. Henry. Comonotonic measures of multivariate risk. Mathemat- ical Finance, 22(1):109–132, 2012. ISSN 1467-9965. doi: 10.1111/j.1467-9965.2010.00453.x. [14] D. Filipović and G. Svindland. The canonical model space for law-invariant convex risk measures is L1. Mathematical Finance, 22(3):585–589, 2012. doi: 10.1111/j.1467-9965.2012.00534. x. [15] I. S. Gradshteyn and J. M. Ryzhik. Table of integrals, series, and products. San Diego, CA: Academic Press. xlvii, 1163 p., 7th edition, 2000. [16] I. Halperin. Function spaces. Canadian Journal of Mathematics, 5:273–288, 1953. doi: 10. 4153/CJM-1953-031-3. [17] G. H. Hardy, J. E. Littlewood, and G. Pólya. Inequalities. Cambridge University Press, 1988. [18] M. Kupper and G. Svindland. Dual reopresentation of monotone convex functions on L0. Proceedings of the American Mathematical Society, 139(11):4073–4086, 11 2011. doi: 10.1090/ S0002-9939-2011-10835-9. [19] S. Kusuoka. On law invariant coherent risk measures. In Advances in mathematical economics, volume 3, chapter 4, pages 83–95. Springer, 2001. doi: 10.1007/978-4-431-67891-5. [20] G. G. Lorentz. On the theory of spaces Λ. Pacific Journal of Mathematics, 1(3):411–429, 1951. [21] G. G. Lorentz. Bernstein Polynomials. Chelsea Publ., 1986. [22] A. J. McNeil, R. Frey, and P. Embrechts. Quantitative Risk Management. Princeton University Press, 2005. [23] W. Ogryczak and A. Ruszczyński. Dual stochastic dominance and related mean-risk models. SIAM Journal on Optimization, 13(1):60–78, 2002. doi: 10.1137/S1052623400375075. [24] G. Ch. Pflug. Some remarks on the Value-at-Risk and the Conditional Value-at-Risk. In S. Uryasev, editor, Probabilistic Constrained Optimization, volume 49, chapter 15, pages 272– 281. Springer US, 2000. doi: 10.1007/978-1-4757-3150-7. [25] G. Ch. Pflug and W. Römisch. Modeling, Measuring and Managing Risk. World Scientific, River Edge, NJ, 2007. doi: 10.1142/9789812708724.

32 [26] A. Pichler. The natural Banach space for version independent risk measures. Insurance: Mathematics and Economics, 53(2):405–415, 2013. doi: 10.1016/j.insmatheco.2013.07.005. [27] A. Pichler. Insurance pricing under ambiguity. European Actuarial Journal, 4(2):335–364, 2014. doi: 10.1007/s13385-014-0099-7. [28] A. Pichler. A quantitative comparison of risk measures. Annals of Operations Research, 2017. doi: 10.1007/s10479-017-2397-3. [29] L. Rüschendorf. Law invariant convex risk measures for portfolio vectors. Statistics & Decisions, pages 97–108, 2009. doi: 10.1524/stnd.2006.24.1.97.

[30] G. Svindland. Subgradients of law-invariant convex risk measures on L1. Statistics & Decisions, 27:169–199, 2009. doi: 10.1524/stnd.2009.1040. [31] D. Williams. Probability with Martingales. Cambridge University Press, Cambridge, 1991. URL http://books.google.com/books?id=RnOJeRpk0SEC.