Risk measures for vector-valued returns

Risk Measures for Vector-Valued Returns

Alois Pichler

June 1, 2015

Abstract

Portfolios, which are exposed to different currencies, have separate and different returns in each individual currency and are thus vector-valued in a natural way.

This paper investigates the natural domain of these risk measures. A Banach space is pre- sented, for which the risk measure is continuous, and which reflects the vector-valued outcomes of the corresponding risk measures from mathematical finance. We develop its key properties and describe the corresponding duality theory. We finally outline extensions of this space, which are along classicalL^pspaces.

Keywords: Risk Measures, Rearrangement Inequalities, Stochastic Dominance, Dual Rep- resentation

Classification: 90C15, 60B05, 62P05

1 Introduction

Classical risk measures have been extended in several directions. Rudloff et al., Molchanov and other authors extend the concept toset-valued risk functionals in the papers [11,25,2,16]. Jouini at al. [19] and Burgert and Rüschendorf [4] follow a different approach by extending the concept of risk measures to vector-valued random variables (cf. also Kabanov [20]). Risk measures on R^d-valued random variables are naturally present in many real life situations. An example is given by considering a portfolio, which has exposures in d (say) different currencies, where each individual portfolio is exposed to uncertainty and subject to individual considerations on risk. A further example is a consolidated financial statement of an internationally operating company, for example an insurance company. Ekeland and Schachermayer [10] consider the domain spaceL^∞ for these risk measures. The first multivariate generalization of a Kusuoka representation for risk measures on vector-valued random variables is provided by Ekeland et al. [9] on L² (notice the difference to multi-valued risk objectives, cf. [14]).

Svindland et al. [36, 12, 21], but many other authors as well mention and consider different domain spaces for risk measures (for example Orlicz spaces, cf. Cheridito [5] or Bellini [3]). Here we consider domain spaces, for which the risk measure is continuous, and extend this basic setting in two different directions.

In a first extension we employ elementary representations of risk measures and use them to define a Banach space to carry the investment strategies and risk measures in a natural way. This

∗Norwegian University of Science and Technology. The author acknowledges support from the Research Council of Norway (RCN) through project 207690.

Contact:aloisp@ntnu.no

setting allows a generalization to vector-valued (R^d, or more general Banach space-valued) random variables. The corresponding space has properties similar to L¹ and thus generalizes the before mentioned domain spaces. Its dual is fully described in case that the state space enjoys the Radon–

Nikodým property.

The second part of the paper extends the new space by picking up differences or similarities betweenL¹ and L^p spaces. The duality theory for these spaces essentially differs from the initial space. Again, these spaces are large enough to carry risk measure in a natural way, and—above all—the risk measure is continuous on the spaces described.

Outline. The following section (Section 2) provides the mathematical setting and elaborates initial details on the relation between the space and the continuity properties of the risk functional.

The Banach space L^p_σ to carry a risk measures for vector-valued random variables is introduced in Section 3. It is demonstrated that risk functionals are continuous with respect to the norm of the space introduced. The following Section4 elaborates the dual of L¹_σ (p= 1), while Section 5 establishes the dual ofL^p_σ forp >1.

The new space is larger than L^∞, but not an L^p space in general. The spaces are related to rearrangement spaces introduced by G. Lorentz in [24, 23] (following earlier results obtained by Halperin [15], cf. Pick et al. [31]).

2 Mathematical setting and motivation

We consider a probability space (Ω,F, P) and denote thedistribution function (cdf) of aR-valued random variableY by

F_Y(q) :=P(Y ≤q) =P({ω: Y(ω)≤q}).

Thegeneralized inverse is the nondecreasing and lower semi-continuous function F_Y⁻¹(α) := inf{q: P(Y ≤q)≥α},

also called the quantile or conditional Value-at-Risk. The Average Value-at-Risk is AV@R_α(Y) := 1

1−α ˆ 1

α

F_Y⁻¹(α) = min

q∈R q+ 1

1−αE(Y −q)₊ (1)

(cf. Rockafellar and Uryasev [32] and Pflug [27] for the latter equality).

With (X,k·k) we denote a separable Banach space. LetY: Ω→(X,k·k) be a strongly measur- able random variable. We writekYk for the [0,∞)-valued random variable

kYk:ω7→ kY(ω)k.

L^p(X) stands for the Banach space (the Lebesgue–Bochner space) of all (equivalence classes of) X-valued, Bochner integrable random variablesY with finitep-mean norm,

kYk_p:= EkYk^p1/p

= ˆ

Ω

kY(ω)k^pP(dω) ^1/p

<∞ (1≤p <∞).

Thep-mean norm can be expressed by the quantile and its generalized inverse by kYk_p=

ˆ 1 0

F_kY⁻¹_k(u)^pdu ^1/p

= ˆ 1

0

p t^p−1 1−F_kYk(t) dt

^1/p

(1≤p <∞). (2)

ForR-valued random variables, i.e., (X,k·k) = (R,|·|), the spaceL^p(X) =:L^p represents the usual Lebesgue space.

This paper introduces a new Banach space on vector-valued, strongly measurable random vari- ables by weighting the quantiles in a different way than (2). The results obtained extend and generalize characterizations obtained in Pichler [29], where only real valued random variables and p= 1 are considered in a context of insurance. Here, we characterize the dual space and prove that the new space does not have a pre-dual.

Remark 1. We shall assume throughout the paper that the probability space is rich enough to carry a uniform distribution.¹ If this is not the case, then one 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.

With anR-valued random variableY one may further associate itsgeneralized quantiletransform F(y, u) := (1−u)·lim

y⁰↑yF_Y⁻¹(y⁰) +u·F_Y⁻¹(y).

The random variableF(Y, U) is uniformly distributed again, andF(Y, U) is coupled in a comonotone way withY (cf. Pflug and Römisch [28]).

The relation to risk measures and their continuity properties. A Kusuoka representation (cf. Kusuoka [22]) for risk measures based on R^d-valued random variables is extracted in Ekeland and Schachermayer [10, Theorem 1.7]. The risk functional identified in the “regular case” in [10]

for the homogeneous risk functional on random vectors is

ρZ(Y) := sup{EhZ, Y⁰i:Y⁰∼Y}, (3) where Y ∼ Y⁰ indicates that Y and Y⁰ enjoy the same law in R^d.² ρ_Z is called the maximal correlation risk measure in directionZ.

The linear form in (3) isEhZ, Yi, wherehZ, Yi(ω) =Pd

i=1Zi(ω)Yi(ω) is the inner product and Z≥0 is normalized to satisfy

E

" _d X

i=1

|Zi|

#

= 1, that is,

1 =EkZk_`d 1 =

ˆ 1 0

F_kZk⁻¹

`d1

(u)du.

The rearrangement inequality (cf. Hardy et al. [17]) provides an upper bound for the linear form by

|EhZ, Yi| ≤EkZk^∗· kYk ≤EK· kZk_`d

1· kYk ≤K· ˆ 1

0

F_kZk⁻¹

`d1

(u)F_kY⁻¹_k(u)du, (4) where the norms k·k on R^d and k·k^∗ are dual to each other. K >0 is the constant linking the norms byk·k^∗≤K· k·k_`d

1 on (the dual of)R^d. Without loss of generality we may (and will) assume thatK= 1 (otherwise, consider the equivalent normk·k⁰:=Kk·kinstead ofk·konR^d).

1Uis uniform, ifP(U≤u) =ufor allu∈[0,1].

2That is,P(Y1≤y1, . . . Yd≤yd) =P Y₁⁰≤y1, . . . Y_d⁰≤yd

for all (y1, . . . yd)∈R^d.

The maximal correlation risk measure (3) employs the linear formEhZ, Yi, which satisfies the bounds (4). This motivates fixing the function

σ(·) :=F_kZk⁻¹

`d1

(·) (5)

and to endow a space with the form kYk_σ:=

ˆ 1 0

σ(u)F_kY⁻¹_k(u)du.

It turns out thatk·k_σ is a norm (Proposition6below) and the maximal correlation risk measure is continuous with respect to the norm. We collect these observations in the following definition and proposition.

Definition 2 (Weighting function). A nondecreasing, nonnegative function σ: [0,1) → [0,∞), which is normalized by´1

0 σ(u)du= 1, is called adistortion function(in the literature occasionally alsospectrum, cf. Acerbi [1]).

The following proposition establishes the continuity relation.

Proposition 3. ρZ, the maximal correlation risk measure in directionZ defined in(3)is Lipschitz continuous with respect to the normk·k_σ, that is,

|ρZ(Y)−ρZ(Y⁰)| ≤ kY −Y⁰k_σ,

whereσis defined in (5)and provided that ρ_Z is finite valued onY andY⁰. Proof. Note first thatρZ is subadditive (convex), and thus

ρ_Z(Y) =ρ_Z(Y⁰+Y −Y⁰)≤ρ_Z(Y⁰) +ρ_Z(Y −Y⁰).

ChoseZ⁰ ∼Zsuch thatρZ(Y−Y⁰) =EhZ⁰, Y −Y⁰i. Repeating the sequence of inequalities in (4) reveals that

ρZ(Y) ≤ ρZ(Y⁰) +EhZ⁰, Y −Y⁰i

≤ ρZ(Y⁰) + ˆ 1

0

σ(u)F_kY⁻¹_−Y0k(u)du≤ρZ(Y⁰) +kY −Y⁰k_σ. The assertion is immediate by interchanging the roles ofY and Y⁰.

3 The vector-valued Banach space L

(X )

The risk measureρZ introduced in the introduction is continuous with respect tok·k_σ. However, the proper space has not been specified. This section introduces the space and the norm in a more general setting. Basic properties of the space are elaborated.

Definition 4. For a distortion functionσand a random variableY with outcomes in the Banach space (X,k·k) define

kYk_σ,p:= sup

Uuniform Eσ(U)kYk^p1/p

, (6)

where the supremum is among all uniform random variablesU. In line withp-measurable random variables we denote byL^p_σ(X) the space of equivalence classes from

nY: Ω→X strongly measurable and kYk_σ,p<∞o ,

where random variables are identified which cannot be distinguished by the probability measureP. Remark 5. We shall use the abbreviationskYk_σ:=kYk_σ,1 as well asLσ(X) :=L¹_σ(X).

Basic characterization ofL^p_σ(X). The spaceL^p_σ(X), equipped with the normk·k_σ,p, is indeed a Banach space (see Theorem11below). We elaborate important relations and comparisons with the norm ofL^p(X) spaces first.

Proposition 6. k·k_σ,p is a seminorm onL^p_σ(X)whenever1≤p <∞and it holds that

kYk_σ,p = ˆ 1

0

σ(u)F_kY⁻¹_k(u)^pdu ^1/p

. (7)

Moreover, the supremum in (6)is attained.

Proof. It is evident thatkYk_σ,p, as defined in (6), is positively homogeneous. As for the triangle inequality notice that

(Eσ(U)kY1+Y2k^p)^1/p ≤ Eσ(U) (kY1k+kY2k)^p1/p

=

E

σ(U)^1/pkY1k+σ(U)^1/pkY2kp^1/p

=

σ(U)^1/pkY1k+σ(U)^1/pkY2k p

≤

σ(U)^1/pkY1k _p+

σ(U)^1/pkY2k _p

=

E σ(U)^1/pkY1k^p1/p +

E σ(U)^1/pkY2k^p1/p by Minkowski’s inequality. By passing to the supremum it follows that

kY1+Y2k_σ,p≤ kY1k_σ,p+kY2k_σ,p, the triangle inequality.

To accept (7) let U be coupled in a comonotone way with kYk (which exists according to Remark1). It follows from the rearrangement inequality that the supremum in (6) is attained for U and further that

Eσ(U)kYk^p= ˆ 1

0

σ(u)F_kY⁻¹_k(u)^pdu, the assertion.

Remark 7. Clearly,kYk_σ,p=kYk_pandL^p_σ(X) =L^p(X) for the (trivial) distortion functionσ(·) = 1.

Example 8. It follows directly from (7) that the norm of an indicator function of a setE is k1Ek_σ,p=

ˆ 1 1−P(E)

σ(u)du

!¹/p

=S 1−P(E)^1/p ,

where

S(α) :=

ˆ 1 α

σ(u)du. (8)

Notice thatP(E)≤P(E)^1/p≤ k1^Ek_σ,p≤1.

Equation (2) in the introduction provides a possibility to compute the norm directly, without involving its generalized inverse. The following corollary generalizes the formula for the normk·k_σ,p. Corollary 9. The seminorm k·k_σ,p can be expressed in terms of the cdf F_kYk directly (without involving its generalized inverseF_kY⁻¹_k) as

kYk^p_σ,p= ˆ ∞

0

p y^p−1·S F_kYk(y) dy.

Proof. By Riemann–Stieltjes integration by parts and change of variables it holds that kYk^p_σ,p =

ˆ 1 0

σ(u)F_kY⁻¹_k(u)^pdu=− ˆ 1

0

F_kY⁻¹_k(u)^pdS(u)

= −F_kY⁻¹_k(u)^p·S(u)

1 u=0+

ˆ 1 0

S(u) dF_kY⁻¹_k(u)^p

= 0 + ˆ _∞

0

S F_kYk(u) du^p=

ˆ _∞

0

pu^p−1S F_kYk(u) du, askYk ≥0. This is the assertion as announded.

Comparison of norms. The following inequalities relate the norms ofL^p_σ(X) andL^p(X). They turn out to be useful to establish completeness of the linear spaceL^p_σ(X).

Proposition 10. Forp < p⁰ it holds that

kYk₁≤ kYk_p≤ kYk_σ,p≤ kYk_σ,p0 (9) and henceL^p(X)⊇L^p_σ(X)⊇L^p_σ⁰(X). Further

kYk_σ,p≤ kσk¹_q^/p· kYk_p0,

whereq= _p0^p−p⁰ andkσk_q:=´1

0 σ(u)^qdu^1/q .

Proof. The functionσ(·) and the function F_kY⁻¹_k(·) are nondecreasing and nonnegative. It follows from the continuous version of Chebyshev’s sum inequality (cf. Hardy et al. [17]) that

kYk^p_σ,p = ˆ 1

0

σ(u)F_kY⁻¹_k(u)^pdu≥ ˆ 1

0

σ(u)du· ˆ 1

0

F_kY⁻¹_k(u)^pdu=kYk^p_p,

which is the second inequality. Further it holds by Hölder’s inequality (note that ¹_q+_p0¹/p = 1) that kYk^p_σ,p =

ˆ 1 0

σ(u)^1qσ(u)^p01/pF_kY⁻¹_k(u)^pdu

≤ ˆ 1

0

σ(u)du ^1/q

· ˆ 1

0

σ(u)F_kY⁻¹_k(u)^p0du ^p/p0

=kYk^p_σ,p0

and

kYk^p_σ,p = ˆ 1

0

σ(u)F_kY⁻¹_k(u)^pdu

≤ ˆ 1

0

σ(u)^qdu ^1/q

· ˆ 1

0

F_kY⁻¹_k(u)^pp

0 pdu

^p/p0

=kσk_q· kYk^p_p0, from which the assertions are immediate.

Theorem 11. The pair L^p_σ(X),k·k_σ,p

is a Banach space.

Proof. Suppose that the sequenceYk,k= 1,2, . . . is a Cauchy sequence. Then it follows from (9) thatYk is Cauchy with respect toL¹(X). As L¹(X) is complete there is a limitY ∈L¹(X) with kYk−Yk₁→0, ask→ ∞. It remains to be shown thatY ∈L^p_σ(X).

From convergence inL¹(X) it follows further thatkYkkconverges in distribution, and thus that F_kY⁻¹

kk(α)→F_kY⁻¹_k(α) at every point of continuity (cf. van der Vaart [37, Lemma 21.2]). As Yk is Cauchy with respect to k·k_σ,p there is k^∗ ∈ Nsuch that kY_k^∗−Y_kk_σ,p < ε for all k > k^∗, and it follows thatkYkk_σ,p≤ kYk^∗−Ykk_σ,p+kYk^∗k_σ,p <kYk^∗k_σ,p+ε. Now

kYk^p_σ,p = ˆ 1

0

σ(u)F_kY⁻¹_k(u)^pdu= ˆ 1

0

σ(u) lim inf

k→∞ F_kY⁻¹

kk(u)^pdu

≤ lim inf

k→∞

ˆ 1 0

σ(u)F_kY⁻¹

kk(u)^pdu= lim inf

k→∞ kYkk^p_σ,p<

kYk^∗k_σ,p+ε^p

<∞

by Fatou’s inequality. Hence,Y ∈L^p_σ(X) andL^p_σ(X) thus is complete.

Proposition 12. Simple functions, andL^∞(X)are dense inL^p_σ(X). Even more,L^∞(X)⊆L^p_σ(X) and

kYk_σ,p≤ kYk_∞ wheneverY ∈L^∞(X)andp <∞.

Proof. As for the second assertion note that 0≤F_kY⁻¹_k(·)≤ kYk_∞, and thus kYk^p_σ,p=

ˆ 1 0

σ(u)F_kY⁻¹_k(u)^pdu≤ ˆ 1

0

σ(u)kYk^p_∞du=kYk^p_∞. ForY ∈L^p_σ(X) chooseu_ε<1 such that´1

uεσ(u)F_kY⁻¹_k(u)^pdu < ε^p. Further, asX is separable, one may find a sequence {xi: i= 1, . . .} ⊂ X such that S

i=1B_ε/2(xi) ⊇X. Let Ei ⊆B_ε/2(xi) be disjoint sets with X = S∞

i=1Ei, and find n such that Pn

i=1P(Ei) > 1−ε. Define xi :=

E(Y|Y ∈Ei). By rearranging the enumeration, one may assume thatkxik ≤ kxi+1kfor alli≤n.

Finally defineYε:=Pn

i=11^Eixi. Note, that kY −Yεk ≤εonSn

i=1Ei, and P(kY −Yεk ≥ε)≤ε.

Hence

kY −Yεk^p_σ,p≤ ˆ 1

0

F_kY⁻¹_−Y

εk(u)^pσ(u)du≤ ˆ u_ε

0

ε^pσ(u)du+ ˆ 1

uε

F_kY⁻¹_k(u)^pσ(u)du <2ε^p. (10) The assertion follows, asYε is a simple function and the right side of (10) can be made arbitrarily small.

The essential relation to the risk measures introduced in the introductory Section 2 is the following proposition on continuity. This is an immediate consequence of Proposition 3 and (9) (withp= 1).

Proposition 13. The risk measure ρ_Z, considered on the spaceL^p_σ(X), is (Lipschitz) continuous

|ρZ(Y)−ρZ(Y⁰)| ≤ kY −Y⁰k_σ,p for everyp≥1.

For the sake of completeness we mention the following statement.

Theorem 14. The space L^p_σ(X)is not a Hilbert space, unless p= 2andσ(·) = 1.

Proof. It is straight forward to verify that the random variablesY1=c·1^E andY2=c⁰·1E^{ with 0< P(E)<1 violate the parallelogram law.

4 Duality theory for L

(X ) (p = 1)

We shall establish first that the spaceLσ does not have a pre-dual space and hence is not reflexive.

This is the same result as forL¹ (although the same proof does not apply forLσ).

The second part of this section introduces Bochner integrable random variables. This is essential to establish the dual ofL_σ(X), which involves the Radon–Nikodým property of the state spaceX.

4.1 A pre-dual does not exist

The following statement establishes non-existence of a pre-dual space. Perhaps it is interesting to note that the following proof works without particular knowledge about the dual space ofLσ(X).

Theorem 15. The Banach space L¹_σ(X),k·k_σ

does not have a pre-dual: there does not exist a Banach space(E,k·k), say, such that its topological dual space is(E,k·k)^∗= L¹_σ(X),k·k_σ

. Proof. Suppose that L¹_σ,k·k_σ

were the dual of (E,k·k). Then, by Alaoglu’s Theorem (cf. Woj- taszczyk [39]), the unit ball ofL¹_σ(X) is weakly* compact. Further, by the Krein–Milman Theorem, the closed unit ball equals the closure of the convex hull of its extreme points. However, we shall demonstrate now that there is a random variable in the unit ball ofL¹_σ(X) which is not contained in the closure of its extreme points.

Suppose thatY ∈L¹_σ(X) withkYk_σ= 1. Define Y⁰(ω) :=

(_kY(ω)k−1

kY(ω)k Y(ω) if kY(ω)k ≥1,

0 else

and set λ := kY⁰k_σ. As kY⁰(ω)k < kY(ω)k (except on {kY(·)k= 0}) it follows that 0 ≤ λ = kY⁰k_σ<kYk_σ= 1. Note, that

F_kY⁻¹0k(·) = maxn

F_kY⁻¹_k(·)−1,0o

andF_kY⁻¹_−Y0k(·) = minn

F_kY⁻¹_k(·),1o ,

and both functions are nondecreasing. Further it holds thatF_kY⁻¹_k=F_kY⁻¹0k+F_kY⁻¹_−Y0k, such that 1 =kYk_σ =

ˆ 1 0

σ(u)F_kY⁻¹_k(u)du= ˆ 1

0

σ(u)

F_kY⁻¹0k(u) +F_kY⁻¹_−Y0k(u) du

= ˆ 1

0

σ(u)F_kY⁻¹0k(u)du+ ˆ 1

0

σ(u)F_kY⁻¹_−Y0k(u)du

= kY⁰k_σ+kY −Y⁰k_σ=λ 1 λY⁰

σ

+ (1−λ)

1

1−λ(Y −Y⁰) σ

,

provided that λ > 0. As ¹_λY⁰

_σ = 1, it follows that

1

1−λ(Y −Y⁰)

_σ = 1 and further that Y =λ·_λ¹Y⁰+ (1−λ)·_1−λ¹ (Y −Y⁰) is the convex combination of two different random variables with norm 1, such thatY is not an extreme point in the unit ball of Lσ.

Every extreme point in the unit sphere ofLσ(X) thus satisfiesλ= 0, i.e.,kY(ω)k ≤1 for almost allω∈Ω. Further, any convex combination of extremal points satisfieskY(ω)k ≤1 for almost all ω∈Ω, and every pointY from the closure of the convex hull of extreme points satisfieskY(ω)k ≤1 as well.

Chooseu∈(0,1) such thatσ(u)>0, a measurable setA∈ Fwith 1−u < P(A)<1 and define the random variable Y :=1^A· ¹

kxk·´₁

1−P(A)σ(u)dux(where 0 6=x∈X). ThenkYk_σ = 1, but Y is not in the closure of the convex combination of extreme points, askY(ω)k>1 forω∈A. Hence, the unit ball ofLσ is strictly larger than the closure of the convex hull of its extreme points. This completes the proof.

The following is an immediate consequence.

Corollary 16. The Banach space L¹_σ(X),k·k_σ

is not reflexive.

We shall outline below that the dual of L¹_σ(X),k·k_σ

is not separable.

4.2 The dual of L

(X)

The duality theory ofLσ(X) involves the dual of the state spaceX. Here we establish the relevant space first and relate it toLσ(X) in a second step.

Definition 17. For a random variable Z with values in (X,k·k) define kZk^∗_σ:= sup

α<1

AV@R_α(kZk)

1 1−α

´1 ασ(u)du

(11)

and the set of all (equivalence classes of)X-valued Bochner integrable functions L^∗_σ(X) :=

Z : Ω→X: kZk^∗_σ<∞ .

Remark 18 (Stochastic dominance of second order). Risk functionals are convex functionals and hence enjoy a representation involving the convex conjugate according the Fenchel–Moreau theorem (cf. Ruszczyński et al. [35]). The convex conjugate of risk functionals on various spaces involves second order stochastic dominance relations. This is elaborated in the literature, cf. for example Shapiro [34], Föllmer and Schied [13]or Pichler [30].

The definition ofk·k^∗_σ reflects the duality of risk functionals. Indeed, the supremum (11) can be restated as

kZk^∗_σ= inf

η >0 :AV@Rα(kZk)≤η· ˆ 1

α

σ(u)dufor allα <1

. This equivalent formulation involves the statement

AV@Rα(kZk)≤AV@Rα η σ(U)

, (12)

whereU is coupled in a comonotone way withkZk. Following Ogryczak and Ruszczyński [26], (12) is equivalent to saying thatkZkis dominated bykZk^∗_σ·σ(U) in second stochastic order.³

Remark 19. By the choiceα= 0 in (11) it follows that kZk^∗_σ≥AV@R₀(kZk) =

ˆ 1 0

F_kZk⁻¹(u)du=EkZk=kZk₁, (13) such thatL¹(X)⊇L^∗_σ(X) for everyσ.

Theorem 20. L^∗_σ(X),k·k^∗_σ

is a Banach space.

Proof. The normk·k^∗_σis positively homogeneous, as the Average Value-at-Risk,AV@R, is positively homogeneous. The triangle inequality is satisfied, because of the triangle inequality in the state space (X,k·k), and as

AV@Rα(kZ1+Z2k)≤AV@Rα(kZ1k+kZ2k)≤AV@Rα(kZ1k) +AV@Rα(kZ2k) by monotonicity and subadditivity of the Average Value-at-Risk.

It remains to be shown thatL^∗_σ(X) is complete with respect to the normk·k^∗_σ. To this end let Zk be a Cauchy sequence. Then there is an indexk^∗ such that kZkk^∗_σ ≤ kZk^∗k^∗_σ+kZk−Zk^∗k^∗_σ <

kZk^∗k^∗_σ+ε. Hence,AV@Rα(kZkk)≤ ^kZk_1−α^∗k∗σ+ε´1

ασ(u)du.

By (13), the sequenceZk is a Cauchy sequence for the norm k·k₁as well, and by completeness ofL¹(X) it follows that there is a limit Z ∈L¹(X). As in the proof of Theorem 11the sequence Zk converges in distribution, and thusF_kZ⁻¹

kk(α)→F_kZk⁻¹(α).

By Fatou’s inequality again, AV@Rα(kZk) = 1

1−α ˆ 1

α

F_kZk⁻¹(u)σ(u)du= 1 1−α

ˆ 1 α

lim inf

k→∞ F_kZ⁻¹

kk(u)σ(u)du

= lim inf

k→∞

1 1−α

ˆ 1 α

F_kZ⁻¹

kk(u)σ(u)du

= lim inf

k→∞ AV@Rα(kZkk)≤ kZk^∗k^∗_σ+ε 1−α

ˆ 1 α

σ(u)du.

3Cf. Dentcheva et al. [6,7,33] for stochastic dominance of second order.

It follows thatkZk^∗_σ≤ kZk^∗k^∗_σ+ε <∞andZ ∈L^∗_σ(X), thusL^∗_σ(X) is complete.

Bochner integrable functions. We consider vector measures µ : F → X which are finitely additive (µ(E1∪E2) = µ(E1) +µ(E2) whenever E1 and E2 are disjoint members of F). The variationof a (finitely additive) vector measureµis

kµk(E) := sup

π

X

Ei∈π

kµ(Ei)k,

where the supremum is among all partitionsπ={E₁, . . . E_n}ofEinto afinitenumber of pairwise disjoints members of F. For measures µ with bounded variation (kµk(Ω) < ∞) the variation kµk:F →Ris a (finitely additive) measure.

Before we proceed to identify the dual let us recall the following fact for Bochner integrable functions. Consider f ∈ L¹(X), which is Bochner integrable by definition with norm kfk₁ =

´kfkdP. The vector-valued measure induced byf is

µ_f(E) :=

ˆ

E

fdP.

For the Bochner integrable function f it holds further that ´

EkfkdP = kµ_fk(E), such that in particularkfk₁=kµ_fk(Ω) (cf. Diestel et al. [8, Theorem 4, p. 46]).

Further, by the Hahn–Banach theorem, it holds that kµ_f(E⁰)k = sup_kx∗k≤1x^∗ µ_f(E⁰) , such

that ˆ

E

kfkdP =kµ_fk(E) = sup

π,k^x∗_ik^≤1

n

X

i=1

x^∗_i µ_f(E_i) ,

wherex^∗_i ∈X^∗ andπ={E1, . . . En}is a finite partition ofE again.

Recall now thatx^∗_i µ_f(E_i)

=x^∗_i´

EifdP

=´

Eix^∗_i(f)dP by Hille’s theorem, such that ˆ

E

kfkdP = sup

π,k^x∗_ik≤1

ˆ * _n X

i=1

1^Eix^∗_i, f +

dP.

If the state space off is a dual space itself,f ∈L¹(X^∗), then it is obvious by the same reasoning

that ˆ

E

kfkdP = sup

π,kx_ik≤1

ˆ * f,

n

X

i=1

1^Eixi

+

dP, (14)

wherexi are chosen in the unit ball of the space X.

We are ready to prove the following embedding.

Theorem 21. Let (X,k·k) be a Banach space with separable dual (X,k·k)^∗ =: X^∗,k·k^∗ . Then L^∗_σ(X^∗),k·k^∗_σ

is isometric to a subspace of Lσ(X),k·k_σ^∗ .

Proof. LetZ ∈L^∗_σ(X^∗) be a random variable with valuesZ(ω)∈X^∗ and consider the linear form

`_Z(Y) :=EhZ, Yi= ˆ

Ω

hZ, YidP = ˆ

Ω

hZ(ω), Y(ω)iP(dω).

We demonstrate first thatk`Zk=kZk^∗_σ.

From the Hardy–Littlewood inequality (cf. [17] and (4)) it follows that

`Z(Y) =EhZ, Yi ≤EkZk kYk ≤ ˆ 1

0

F_kZk⁻¹(u)F_kY⁻¹_k(u)du.

DefineG(α) :=´1

αF_kZk⁻¹(u)du, observe thatG(0) =EkZk=kZk₁, then

`Z(Y) ≤ − ˆ 1

0

F_kY⁻¹_k(u) dG(u) =−F_kY⁻¹_k(u)G(u)

1 u=0

+ ˆ 1

0

G(u) dF_kY⁻¹_k(u)

= F_kY⁻¹_k(0)kZk₁+ ˆ 1

0

G(u) dF_kY⁻¹_k(u).

Note now thatG(α) =´1

αF_kZk⁻¹(u)du≤ kZk^∗_σ·´1

ασ(u)du=kZk^∗_σ·S(α) and the integratorF_kY⁻¹_k(·) is nondecreasing (S(α) =´1

ασ(u)du, cf. (8)). Hence,

`_Z(Y)≤F_kY⁻¹_k(0)kZk₁+kZk^∗_σ ˆ 1

0

S(u) dF_kY⁻¹_k(u).

By Riemann–Stieltjes integration by parts thus,

`Z(Y) ≤ F_kY⁻¹_k(0)kZk₁+kZk^∗_σF_kY⁻¹_k(u)S(u)

1

u=0− kZk^∗_σ ˆ 1

0

F_kY⁻¹_k(u) dS(u)

= F_kY⁻¹_k(0) kZk₁− kZk^∗_σ

+kZk^∗_σ· ˆ 1

0

σ(u)F_kY⁻¹_k(u)du

≤ kZk^∗_σ· kYk_σ,

askZk₁− kZk^∗_σ ≤0 by (13), and asF_kY⁻¹_k(0) = ess infkYk ≥0. It follows thatk`Zk ≤ kZk^∗_σ. For the converse inequality find α <1 such that ^AV@R₁ ^α(kZk)

1−α

´1

ασ(u)du ≥ kZk^∗_σ−ε. Find a setE such

that n

kZk> F_kZk⁻¹(α)o

⊆E⊆n

kZk ≥F_kZk⁻¹(α)o

andP(E) = 1−α. (15) It holds thatAV@Rα(kZk) = _1−α¹ ´

EkZkdP. Letπ ={E1, . . . En} be a partition of E and xi in the unit sphere ofX be chosen such that

ˆ

E

kZkdP <

ˆ * Z,

n

X

i=1

1Eix_i +

dP+ε, (16)

which is possible by (14).

DefineY :=Pn

i=11Eix_i and observe that kYk_σ =

ˆ 1 0

σ(u)F_kY⁻¹_k(u)du= ˆ 1

α

σ(u)du≤1.

From (16) and (15) it follows that

EhZ, Yi+ε ≥ (1−α)AV@R_α(kZk) = AV@R_α(kZk)

1 1−α

´1 ασ(u)du

kYk_σ

≥ kZk^∗_σ−ε

kYk_σ≥ kZk^∗_σkYk_σ−ε.

As ε > 0 is chosen arbitrarily it follows that k`Zk ≥ kZk<sup>∗</sup><sub>σ</sub>, that is k`Zk = kZk^∗_σ, which is the assertion.

Theorem 22. LetX^∗ have the Radon–Nikodým property. Then the dualL_σ(X)^∗ can be identified withL^∗_σ(X^∗).

Proof. Let`∈Lσ(X)<sup>∗</sup> be fixed. Define theX<sup>∗</sup>-valued measure µ(E)(x) :=`(1E·x) forE∈ F.

Letxi be in the closed unit ball ofX andEi be a finite partition. Then

n

X

i=1

µ(Ei)(xi)

=

`

n

X

i=1

xi1^Ei

!

≤ k`k ·

n

X

i=1

xi1^Ei

σ

≤ k`k.

Taking the supremum with respect to all tessellations Ei and xi in the unit ball shows that the X^∗-valued measureµ is of bounded variation. By the Radon–Nikodým property of X^∗ there is a measurableZ: Ω→X^∗, such that

µ(E) = ˆ

E

ZdP = ˆ

1^EZdP.

Explicitly,`(Y) =Pn i=1

´

EiZ(x_i)dP =´

hZ, YidP for simple functionsY =Pn

i=11Eix_i.

It remains to verify thatZ ∈Lσ(X^∗) and`(·) =EhZ,·i. To this end defineEn :={kZk ≤n}.

The functional

`_n(Y) :=

ˆ

E_n

hZ, YidP

is a bounded linear functional, as Z ·1^En is bounded. As `n(·) = `(·) for all simple functions supported byEn, it follows that`n(·) =`(·) forall random variablesY ∈Lσ(X) supported byEn. That is,

`(Y ·1En) = ˆ

hZ·1En, YidP

and kZ·1Enk^∗_σ ≤ k`k. The monotone convergence theorem implies that kZk<sup>∗</sup><sub>σ</sub> ≤ k`k, such that Z∈L^∗_σ(X^∗). Finally it follows that

`(Y) = lim

n→∞

ˆ

hZ·1En, YidP = ˆ

hZ, YidP

for all Y ∈ Lσ(X) by continuity and the inequalities already established, which establishes that

`(·) =EhZ,·ifor allY ∈Lσ(X). This completes the proof.

Theorem 23. The space L^∗_σ(X^∗),k·k^∗_σ

is not separable.

check this!

Proof. Consider the random variablesY_A :=σ(U)1Axfor someA with 0< P(A)<1 and Y_B :=

σ(U)1BxwithA∩B=∅and some fixedx∈X. ThenkY_A−Y_Bk=kY_A+Y_Bk>kY_Ak=:r.

Suppose thatC is a countable, dense subset ofL^∗_σ. all sets functions

5 Duality of the spaces L

for p > 1

The dual spaceL^p_σ,p >1, is of different nature than forp= 1, which was outlined in the previous section (Section 4). However, similar to L^p, the spaces L^p_σ are reflexive. But although reflexive, it turns out that the duals ofL^p_σ are not L^q_σ spaces, except for the classical (Lebesgue) case with σ(·) = 1.

We develop the duality theory for the state space R. The results extend to the general state spaces (X,k·k), but this extension is in line with the previous section.

5.1 The dual of L

Definition 24. We say thatZ⁰ σ-dominatesZ (in symbolsZ⁰σ<Z) if there is a uniform random variableU such that

AV@Rα σ(U)Z⁰

≥AV@Rα(Z) for allα <1. (17) Further we define the mapping

kZk^∗_σ,q:= infn

kZ⁰k_σ,q:Z⁰_σ<|Z|o

(18) and the set (of equivalence classes of)L^q∗_σ :=n

Z :kZk^∗_σ,q<∞o .

Remark 25. By Hoeffding’s Lemma (cf. [18]) one may assume that Z⁰ and U are coupled in a comonotone way in (17). This establishes the equivalence

Z⁰_σ<Z, iff ˆ 1

α

σ(u)F_Z⁻¹0 (u)du≥ ˆ 1

α

F_|Z|⁻¹(u)du for all α <1.

Proposition 26. It holds that

kZk₁≤ kZk^∗_σ,q≤ kZk_σ,q and thusL¹⊇L^q∗_σ ⊇L^q_σ.

Proof. Suppose thatZ⁰ is feasible for (18),Z⁰_σ<|Z|. Then kZk₁=

ˆ 1 0

F_|Z|⁻¹(u)du≤ ˆ 1

0

σ(u)F_Z⁻¹0 (u)du=kZ⁰k_σ,1≤ kZ⁰k_σ,q by choosingα= 1 in (18) and by (9), from which the first inequality is immediate.

The second inequality follows from Z σ<Z. Indeed, by Chebyshev’s sum inequality it holds that

AV@R_α σ(U)Z

= 1

1−α ˆ 1

α

σ(u)F_Z⁻¹(u)du

= ˆ 1

0

σ α+u(1−α)

F_Z⁻¹ α+u(1−α) du

≥ ˆ 1

0

σ α+u(1−α) du·

ˆ 1 0

F_Z⁻¹ α+u(1−α) du

= 1

1−α ˆ 1

α

σ(u)du· 1 1−α

ˆ 1 α

F_Z⁻¹(u)du≥AV@R_α(Z).