Multi-utility representations of incomplete preferences induced by set-valued risk measures
aa r X i v : . [ q -f i n . M F ] S e p Multi-utility representations of incomplete preferences induced byset-valued risk measures
Cosimo Munari
Center for Finance and Insurance and Swiss Finance InstituteDepartment of Banking and Finance, University of Zurich, Switzerland [email protected]
September 10, 2020
Abstract
We establish a variety of numerical representations of preference relations induced by set-valued riskmeasures. Because of the general incompleteness of such preferences, we have to deal with multi-utilityrepresentations. We look for representations that are both parsimonious (the family of representingfunctionals is indexed by a tractable set of parameters) and well behaved (the representing functionalssatisfy nice regularity properties with respect to the structure of the underlying space of alternatives).The key to our results is a general dual representation of set-valued risk measures that unifies theexisting dual representations in the literature and highlights their link with duality results for scalarrisk measures.
Keywords : risk measures, dual representations, incomplete preferences, multi-utility representations
JEL classification : C60, G11
MSC : 91B06, 91G80
This note is concerned with the numerical representation of preference relations induced by a special classof set-valued maps. Recall that a preference (relation) over the elements of a set L is a reflexive andtransitive binary relation on L . A preference is said to be complete if any two elements x, y ∈ L arecomparable in the sense that it is always possible to determine whether x is preferred to y or viceversa.Following the terminology of Dubra et al. [14], a family U of maps u : L → [ −∞ , ∞ ] is a multi-utilityrepresentation of a preference (cid:23) if for all x, y ∈ L we have x (cid:23) y ⇐⇒ u ( x ) ≥ u ( y ) for every u ∈ U . In words, a multi-utility representation provides a numerical representation for the given preference rela-tion via a family of “utility functionals”. In view of their greater tractability, multi-utility representationsplay a fundamental role in applications. A standard problem in this context is to find representationsthat are at the same time parsimonious (the family of representing functionals is indexed by a small set ofparameters) and well behaved (the representing functionals satisfy nice regularity properties with respectto the structure of the underlying set). This is especially important for incomplete preferences, whichcannot be represented by a unique functional. 1n the broad field of economics and finance, incomplete preferences arise naturally in the presence ofmulti-criteria decision making. We refer to Aumann [5] and Bewley [8] for two classical references andto Ok [39], Dubra et al. [14], Mandler [36], Eliaz and Ok [15], Kaminski [34], Evren [16, 17], Evren andOk [18], Bosi and Herden [9], Galaabaatar and Karni [26], Nishimura and Ok [38], Bosi et al. [10], andBevilacqua et al. [7] for an overview of contributions to the theory of incomplete preferences and theirmulti-utility representations in the last twenty years.The goal of this note is to establish numerical representations of preference relations induced by a specialclass of set-valued maps that have been the subject of intense research in the recent mathematical financeliterature. To introduce the underlying economic problem, consider an economic agent who is confrontedwith the problem of ranking a number of different alternatives represented by the elements of a set L .The agent has specified a target set of acceptable or attractive alternatives A ⊆ L . We assume that, ifan alternative is not acceptable, it can be made acceptable upon implementation of a suitable admissibleaction. We represent the results of admissible actions by the elements of a set M ⊆ L and assume thata given alternative x ∈ L can be transformed through a given m ∈ M into the new alternative x + m .The objective of the agent is then to identify, for each alternative, all the admissible actions that can beimplemented to move said alternative inside the target set by way of translations. This naturally leadsto the set-valued map R : L ⇒ M defined by R ( x ) := { m ∈ M : x + m ∈ A } = M ∩ ( A − x ) . The map R can be seen as a generalization of the set-valued risk measures studied by Jouini et al. [32],Kulikov [35], Hamel and Heyde [28], Hamel et al. [29], and Molchanov and Cascos [37] in the context ofmarkets with transaction costs; by Haier et al. [27] in the context of intragroup transfers; by Feinsteinet al. [22], Armenti et al. [3], and Ararat and Rudloff [1] in the context of systemic risk. We refer tothese contributions for a discussion about the financial interpretation of set-valued risk measures in therespective fields of application and to Section 5 for some concrete examples in the context of multi-currencymarkets with transaction costs and systemic risk.The set-valued map R defined above induces a preference relation on L by setting x (cid:23) R y : ⇐⇒ R ( x ) ⊇ R ( y ) . According to this preference, the agent prefers x to y if every admissible action through which we canmove y into the target set will also allow us to transport x there. In other terms, x is preferred to y if it is easier to make x acceptable compared to y . The goal of this note is to establish numericalrepresentations of the preference (cid:23) R . Since this preference, as shown below, is not complete in general,we have to deal with multi-utility representations. In particular, we look for representations consistingof (semi)continuous utility functionals. We achieve this by establishing suitable (dual) representations ofthe set-valued map R .Our results provide a unifying perspective on the existing dual representations of set-valued risk measuresand on the corresponding multi-utility representations, which, to be best of our knowledge, have neverbeen explicitly investigated in the literature. We illustrate the advantages of such a unifying approach bydiscussing applications to multi-currency markets with transaction costs and systemic risk. In addition,we highlight where our strategy to establishing dual representations differs from the standard argumentsused in the literature. The note is structured as follows. The necessary mathematical background iscollected in Section 2. The standing assumptions on the space of alternatives and the main properties ofthe set-valued map under investigation are presented in Section 3. The main results on dual and multi-utility representations are established in Section 4 and are applied to a number of concrete situations inSection 5. 2 Mathematical background
In this section we collect the necessary mathematical background and fix the notation and terminologyused throughout the paper. We refer to Rockafellar [40] and Z˘alinescu [43] for a thorough presentation ofduality for topological vector spaces. Moreover, we refer to Aubin and Ekeland [4] for a variety of resultson support functions and barrier cones.Let L be a real locally convex Hausdorff topological vector space. The topological dual of L is denotedby L ′ . Any linear subspace M ⊆ L is canonically equipped with the relative topology inherited from L .The corresponding dual space is denoted by M ′ . For every set A ⊆ L we denote by int( A ) and cl( A )the interior and the closure of A , respectively. We say that A is convex if λA + (1 − λ ) A ⊆ A for every λ ∈ [0 ,
1] and that A is a cone if λA ⊆ A for every λ ∈ [0 , ∞ ). The (lower) support function of A is themap σ A : L ′ → [ −∞ , ∞ ] defined by σ A ( ψ ) := inf x ∈ A ψ ( x ) . The effective domain of σ A is called the barrier cone of A and is denoted bybar( A ) := { ψ ∈ L ′ : σ A ( ψ ) > −∞} . It follows from the Hahn-Banach Theorem that, if A is closed and convex, then it can be represented asthe intersection of all the halfspaces containing it or equivalently A = \ ψ ∈ L ′ { x ∈ L : ψ ( x ) ≥ σ A ( ψ ) } = \ ψ ∈ bar( A ) { x ∈ L : ψ ( x ) ≥ σ A ( ψ ) } . (2.1)If A is a cone, then bar( A ) coincides with the polar or dual cone of A , i.e.bar( A ) = A + := { ψ ∈ L ′ : ψ ( x ) ≥ , ∀ x ∈ A } . If A is a vector space, then bar( A ) coincides with the annihilator of A , i.e.bar( A ) = A ⊥ := { ψ ∈ L ′ : ψ ( x ) = 0 , ∀ x ∈ A } . Finally, if A + K ⊆ A for some cone K ⊆ L , then bar( A ) ⊆ K + . Throughout the remainder of the note, we assume that L is a real locally convex Hausdorff topologicalvector space. We also fix a closed convex cone K ⊆ L satisfying K − K = L and consider the inducedpartial order defined by x (cid:23) K y : ⇐⇒ x − y ∈ K. The above partial order is meant to capture an “objective” preference relation shared by all agents. Thisis akin to the “better for sure” preference in Drapeau and Kupper [13].
Assumption 3.1.
We stipulate the following assumptions on A and M :(A1) A is closed, convex, and satisfies A + K ⊆ A .(A2) M is a closed linear subspace of L such that M ∩ K = { } .(A3) R ( x ) / ∈ {∅ , M } for some x ∈ L .The next proposition collects a number of basic properties of the set-valued map R and its associatedpreference (cid:23) R . The properties of R are aligned with those discussed in Hamel and Heyde [28] and Hamelet al. [29]. 3 roposition 3.2. (i) (cid:23) R is monotone with respect to K , i.e. for all x, y ∈ Lx (cid:23) K y = ⇒ x (cid:23) R y. (ii) (cid:23) R is convex, i.e. for all x, y ∈ L and λ ∈ [0 , x (cid:23) R y = ⇒ λx + (1 − λ ) y (cid:23) R y. (iii) R ( x ) + K ∩ M ⊆ R ( x ) for every x ∈ L .(iv) R ( x + m ) = R ( x ) − m for all x ∈ L and m ∈ M .(v) R ( λx + (1 − λ ) y ) ⊇ λR ( x ) + (1 − λ ) R ( y ) for all x, y ∈ L and λ ∈ [0 , .(vi) R ( x ) is convex and closed for every x ∈ L .(vii) R ( x ) = M for every x ∈ L .Proof. To establish (i), assume that x (cid:23) K y for x, y ∈ L . For every m ∈ R ( y ) we have x + m = y + m + x − y ∈ A + K ⊆ A. This shows that m ∈ R ( x ) as well, so that x (cid:23) R y . To establish (ii), take λ ∈ [0 ,
1] and assume that x (cid:23) R y . For every m ∈ R ( y ) we have y + m ∈ A and, hence, x + m ∈ A . This yields λx + (1 − λ ) y + m = λ ( x + m ) + (1 − λ )( y + m ) ∈ λA + (1 − λ ) A ⊆ A, showing that m ∈ R ( λx + (1 − λ ) y ). In sum, λx + (1 − λ ) y (cid:23) R y . To see that properties (iii) to (vi)hold, it suffices to recall that R ( x ) = M ∩ ( A − x ) for every x ∈ L . Finally, to establish (vii), assume that R ( x ) = M for some x ∈ L . Take any y ∈ L and assume that R ( y ) is nonempty so that y + m ∈ A forsome m ∈ M . For all n ∈ M and λ ∈ (0 ,
1] we have λ (cid:18) x + 1 λ ( n − m ) (cid:19) + (1 − λ )( y + m ) ∈ λA + (1 − λ ) A ⊆ A by convexity. Hence, letting λ →
0, we obtain y + n ∈ A by closedness. Since n was arbitrary, we infer that R ( y ) = M . This contradicts assumption (A3), showing that R ( x ) = M must hold for every x ∈ L . Remark 3.3. (i) If M is spanned by a single element, then (cid:23) R is complete. Indeed, in this case, we canalways assume that M is spanned by a nonzero element m ∈ M ∩ K by our standing assumption. Then,for every x ∈ L such that R ( x ) = ∅ we see that R ( x ) = { λm : λ ∈ [ λ x , ∞ ) } for a suitable λ x ∈ R . This shows that (cid:23) R is complete.(ii) In general, the preference (cid:23) R is not complete when M is spanned by more than one element. Forinstance, let L = R and assume that K = A = R and M = R × { } . For x = 0 and y = (1 , − ,
0) werespectively have R ( x ) = { m ∈ M : m ≥ , m ≥ } and R ( y ) = { m ∈ M : m ≥ − , m ≥ } . Clearly, neither x (cid:23) R y nor y (cid:23) R x holds, showing that (cid:23) R is not complete.(iii) Sometimes the preference (cid:23) R is complete even if M is spanned by more than one element. Forinstance, let L = R and assume that K = A = R × R and M = { } × R . For every x ∈ L such that R ( x ) = ∅ we have R ( x ) = { m ∈ M : m ≥ − x } . This shows that (cid:23) R is complete. 4 Multi-utility representations
In this section we establish a variety of multi-utility representations of the preference induced by R , whichare derived from suitable representations of the sets R ( x ). As highlighted below, both representationshave a strong link with (scalar) risk measures and their dual representations. We refer to the appendixfor the necessary mathematical background and notation.The first multi-utility representation is based on the following scalarizations of R . Here, we set K + M := { π ∈ M ′ : π ( m ) ≥ , ∀ m ∈ K ∩ M } . Definition 4.1.
For every π ∈ K + M we define a map ρ π : L → [ −∞ , ∞ ] by setting ρ π ( x ) := inf { π ( m ) : m ∈ M, x + m ∈ A } . Moreover, we define a map u π : L → [ −∞ , ∞ ] by setting u π ( x ) := − ρ π ( x ) . The functionals ρ π are examples of the risk measures introduced in F¨ollmer and Schied [23] and generalizedin Frittelli and Scandolo [25]. We refer to Farkas et al. [19, 20] for a thorough investigation of suchfunctionals at our level of generality. The next proposition features some of their standard properties,which follow immediately from Proposition 3.2. Since the announced multi-utility representation will beexpressed in terms of the negatives of the functionals ρ π , the proposition is stated in terms of the utilityfunctionals u π . Proposition 4.2.
For every π ∈ K + M the functional u π satisfies the following properties:(i) u π is translative along M , i.e. for all x ∈ L and m ∈ Mu π ( x + m ) = u π ( x ) + π ( m ) . (ii) u π is nondecreasing with respect to (cid:23) K , i.e. for all x, y ∈ Lx (cid:23) K y = ⇒ u π ( x ) ≥ u π ( y ) . (iii) u π is concave, i.e. for all x, y ∈ L and λ ∈ [0 , u π ( λx + (1 − λ ) y ) ≥ λu π ( x ) + (1 − λ ) u π ( y ) . Remark 4.3.
Note that, unless M is spanned by one element, the closedness of the set A is not sufficientto ensure that the functionals ρ π are lower semicontinuous; see Example 1 in Farkas et al. [20]. We referto Hamel et al. [30] for a discussion on general sufficient conditions ensuring the lower semicontinuity ofscalarizations of set-valued maps and to Farkas et al. [20] and Baes et al. [6] for a variety of sufficientconditions in a risk measure setting.The first multi-utility representation of the preference induced by R rests on the intimate link betweenthe risk measures ρ π and the support functions corresponding to R . Lemma 4.4.
For every x ∈ L the set R ( x ) can be represented as R ( x ) = \ π ∈ K + M \{ } { m ∈ M : π ( m ) ≥ ρ π ( x ) } . roof. The result is clear if R ( x ) = ∅ . Otherwise, recall that R ( x ) is closed and convex by Proposition 3.2and observe that ρ π ( x ) = σ R ( x ) ( π ) for every π ∈ M ′ . We can apply the dual representation (2.1) in thecontext of the space M to obtain R ( x ) = \ π ∈ bar( R ( x )) \{ } { m ∈ M : π ( m ) ≥ ρ π ( x ) } . As R ( x ) + K ∩ M ⊆ R ( x ) again by Proposition 3.2, we conclude by noting that the barrier cone of R ( x )must be contained in K + M . Theorem 4.5.
The preference (cid:23) R can be represented by the multi-utility family U = { u π : π ∈ K + M \ { }} . Proof.
We rely on Lemma 4.4. Take any x, y ∈ L . If x (cid:23) R y , then R ( x ) ⊇ R ( y ) and ρ π ( x ) = σ R ( x ) ( π ) ≤ σ R ( y ) ( π ) = ρ π ( y )for every π ∈ K + M \ { } . Conversely, if ρ π ( x ) ≤ ρ π ( y ) for every π ∈ K + M \ { } , then for each m ∈ R ( y ) wehave π ( m ) ≥ ρ π ( y ) ≥ ρ π ( x ) for every π ∈ K + M \ { } , so that m ∈ R ( x ). This yields x (cid:23) R y and concludesthe proof. Remark 4.6.
The simple representation in Lemma 4.4 shows that the set-valued map R is completelycharacterized by the family of functionals ρ π . In the context of risk measures, one could say that a set-valued risk measure is completely characterized by the corresponding family of scalar risk measures. Thiscorresponds to the “setification” formula in Section 4.2 in Hamel et al. [30].We aim to improve the above representation in a twofold way. First, we want to find a multi-utilityrepresentation consisting of a smaller number of representing functionals. This is important to ensure amore parsimonious, hence tractable, representation. Second, we want to establish a multi-utility repre-sentation consisting of (semi)continuous representing functionals. This is important in applications, e.g.in optimization problems where the preference appears in the optimization domain.The second multi-utility representation will be expressed in terms of the following utility functionals.Here, for any functional π ∈ M ′ we denote by ext( π ) the set of all linear continuous extensions of π tothe whole space L , i.e. ext( π ) := { ψ ∈ L ′ : ψ ( m ) = π ( m ) , ∀ m ∈ M } . Definition 4.7.
For every π ∈ K + M we define a map ρ ∗ π : L → [ −∞ , ∞ ] by setting ρ ∗ π ( x ) = sup ψ ∈ ext( π ) { σ A ( ψ ) − ψ ( x ) } = sup ψ ∈ ext( π ) ∩ bar( A ) { σ A ( ψ ) − ψ ( x ) } . Moreover, we define a map u ∗ π : L → [ −∞ , ∞ ] by setting u ∗ π ( x ) = − ρ ∗ π ( x ) . (If A is a cone, then σ A = 0 on bar( A ) and the above maps simplify accordingly).The functionals ρ ∗ π are inspired by the dual representation of the risk measures ρ π , see e.g. Frittelli andScandolo [25] or Farkas et al. [20]. The precise link is shown in Proposition 4.14 below. For the timebeing, we are interested in highlighting some properties of the functionals ρ ∗ π , or equivalently u ∗ π , andproceeding to our desired multi-utility representation.6 roposition 4.8. For every π ∈ K + M the functional u ∗ π satisfies the following properties:(i) u ∗ π is translative along M , i.e. for all x ∈ L and m ∈ Mu ∗ π ( x + m ) = u ∗ π ( x ) + π ( m ) . (ii) u ∗ π is nondecreasing with respect to (cid:23) K , i.e. for all x, y ∈ Lx (cid:23) K y = ⇒ u ∗ π ( x ) ≥ u ∗ π ( y ) . (iii) u ∗ π is concave, i.e. for all x, y ∈ L and λ ∈ [0 , u ∗ π ( λx + (1 − λ ) y ) ≥ λu ∗ π ( x ) + (1 − λ ) u ∗ π ( y ) . (iv) u ∗ π is upper semicontinuous, i.e. for every net ( x α ) ⊆ L and every x ∈ Lx α → x = ⇒ lim sup u ∗ π ( x α ) ≥ u ∗ π ( x ) . Proof.
Translativity follows from the definition of ρ ∗ π . Being a supremum of affine maps, it is clear that ρ ∗ π is convex and lower semicontinuous. To show monotonicity, it suffices to observe that bar( A ) ⊆ K + by (A1) and therefore ρ ∗ π ( x ) = sup ψ ∈ ext( π ) ∩ K + { σ A ( ψ ) − ψ ( x ) } ≤ sup ψ ∈ ext( π ) ∩ K + { σ A ( ψ ) − ψ ( y ) } = ρ ∗ π ( y )for all x, y ∈ L with x (cid:23) K y , where we used that ψ ( x − y ) ≥ ψ ∈ K + .To streamline the proof of the announced multi-utility representation, we start with the following lemma.We denote by ker( π ) the kernel of π ∈ M ′ , i.e.ker( π ) := { m ∈ M : π ( m ) = 0 } . In the sequel, we will repeatedly use the fact that ker( π ) has codimension 1 in M (provided π is nonzero). Lemma 4.9.
The set A can be represented as A = \ π ∈ K + M \{ } cl( A + ker( π )) . (4.1) Moreover, for every π ∈ K + M we have bar(cl( A + ker( π ))) = bar( A ) ∩ ker( π ) ⊥ and σ cl( A +ker( π )) ( ψ ) = ( σ A ( ψ ) if ψ ∈ ker( π ) ⊥ , −∞ otherwise . Proof.
We only prove the inclusion “ ⊇ ” in (4.1) because the other assertions are clear. Assume that x ∈ cl( A + ker( π )) for every nonzero π ∈ K + M and take any ψ ∈ bar( A ). In light of (2.1), to conclude theproof we have to show that ψ ( x ) ≥ σ A ( ψ ). To this effect, let π ψ be the restriction of ψ to the space M .Since bar( A ) ⊆ K + , it follows that π ψ ∈ K + M . Moreover, note that ψ ∈ ker( π ψ ) ⊥ . As a result, we have ψ ∈ bar( A ) ∩ ker( π ψ ) ⊥ = bar( A + ker( π ψ )) = bar(cl( A + ker( π ψ ))) . Since x ∈ cl( A + ker( π ψ )) by our assumption, we can use (2.1) again to get ψ ( x ) ≥ σ cl( A +ker( π ψ )) ( ψ ) = σ A +ker( π ψ ) ( ψ ) = σ A ( ψ ) , where the last equality holds because ψ ∈ ker( π ψ ) ⊥ . This concludes the proof.7he next lemma records a representation of the map R that will immediately yield our desired multi-utilityrepresentation with (upper) semicontinuous functionals. Lemma 4.10 ( Dual representation of R ). For every x ∈ L the set R ( x ) can be represented as R ( x ) = \ π ∈ K + M \{ } { m ∈ M : π ( m ) ≥ ρ ∗ π ( x ) } = \ π ∈ K + M \{ } \ ψ ∈ ext( π ) { m ∈ M : π ( m ) ≥ σ A ( ψ ) − ψ ( x ) } . (If A is a cone, then σ A = 0 on bar( A ) and the representation simplifies accordingly).Proof. Fix x ∈ L . It follows from the representation in (2.1) and Lemma 4.9 that R ( x ) = \ π ∈ K + M \{ } \ ψ ∈ ker( π ) ⊥ { m ∈ M : ψ ( m ) ≥ σ A ( ψ ) − ψ ( x ) } . (4.2)To establish the desired representation of R ( x ) it then suffices to show that the set ker( π ) ⊥ in the right-hand side of (4.2) can be replaced by ext( π ). To this effect, let m ∈ M satisfy π ( m ) ≥ σ A ( ψ ) − ψ ( x )for all nonzero π ∈ K + M and ψ ∈ ext( π ). Moreover, take an arbitrary nonzero π ∈ K + M and an arbitrary ψ ∈ ker( π ) ⊥ . To conclude the proof, we have to show that ψ ( m ) ≥ σ A ( ψ ) − ψ ( x ) . (4.3)This is clear if ψ / ∈ bar( A ) or ψ ∈ ext( π ). Hence, assume that ψ ∈ bar( A ) \ ext( π ). Note that, since π isnonzero and K − K = L , we find n ∈ K M such that π ( n ) >
0. Since bar( A ) ⊆ K + , two situations arepossible. On the one hand, if ψ ( n ) >
0, then ψ belongs to ext( π ) up to a strictly-positive multiple andtherefore (4.3) holds. On the other hand, if ψ ( n ) = 0, then we must have ψ ∈ M ⊥ . To deal with this case,note first that we always find a nonzero π ∗ ∈ K + M satisfying ext( π ∗ ) ∩ bar( A ) = ∅ , for otherwise everyfunctional in bar( A ) ∩ ker( π ∗ ) ⊥ would annihilate the entire M and it would follow from (2.1) and (4.2)that R ( y ) = M for every y ∈ A , which is against Proposition 3.2. Now, take ϕ ∈ ext( π ∗ ) ∩ bar( A ) andset ϕ k = ϕ + kψ ∈ ext( π ∗ ) for each k ∈ N . It follows that π ∗ ( m ) = sup k ∈ N ϕ k ( m ) ≥ sup k ∈ N { σ A ( ϕ k ) − ϕ k ( x ) }≥ σ A ( ϕ ) − ϕ ( x ) + sup k ∈ N { k ( σ A ( ψ ) − ψ ( x )) } . This implies that ψ ( m ) = 0 ≥ σ A ( ψ ) − ψ ( x ) must hold, establishing (4.3). Theorem 4.11.
The preference (cid:23) R can be represented by the multi-utility family U ∗ = { u ∗ π : π ∈ K + M \ { } , ext( π ) ∩ bar( A ) = ∅} . Proof.
Note that ρ ∗ π ( x ) = −∞ for every x ∈ L whenever ext( π ) ∩ bar( A ) = ∅ for some π ∈ K + M . Hence,the desired assertion follows immediately from Lemma 4.10; see also the proof of Theorem 4.5. Remark 4.12.
The statement of Lemma 4.10 provides a unifying formulation for the dual representationsof set-valued risk measures in the literature. This is further illustrated in Section 5. The strategy used inthe proof is different from the ones adopted in the literature, which are often based on results from set-valued duality, thereby offering a complementary perspective on the existing proofs; see also Remark 4.15.8he next proposition shows the link between the two multi-utility representations we have established. In asense made precise below, the representation U ∗ can be seen as the regularization of U by means of (upper)semicontinuous hulls. Before we show this, it is useful to single out the following dual representation ofthe augmented acceptance set, which should be compared with Theorem 1 in Farkas et al. [20]. Lemma 4.13.
For every π ∈ K + M \ { } such that ext( π ) ∩ bar( A ) = ∅ we have cl( A + ker( π )) = \ ψ ∈ ext( π ) { x ∈ L : ψ ( x ) ≥ σ A ( ψ ) } . Proof.
In view of (2.1) and Lemma 4.9, the assertion is equivalent to \ ψ ∈ ker( π ) ⊥ { x ∈ L : ψ ( x ) ≥ σ A ( ψ ) } = \ ψ ∈ ext( π ) { x ∈ L : ψ ( x ) ≥ σ A ( ψ ) } . We only need to show the inclusion “ ⊇ ”. To this end, we mimic the argument in the proof of Lemma 4.10.Let x ∈ L belong to the right-hand side above and take ψ ∈ ker( π ) ⊥ . We have to show that ψ ( x ) ≥ σ A ( ψ ) . (4.4)This is clear if ψ / ∈ bar( A ) or ψ ∈ ext( π ). Hence, assume that ψ ∈ bar( A ) \ ext( π ). Note that, since π isnonzero and K − K = L , we find n ∈ K M such that π ( n ) >
0. Since bar( A ) ⊆ K + , two situations arepossible. On the one hand, if ψ ( n ) >
0, then ψ belongs to ext( π ) up to a strictly-positive multiple andtherefore (4.4) holds. On the other hand, if ψ ( n ) = 0, then we must have ψ ∈ M ⊥ . In this case, take anyfunctional ϕ ∈ ext( π ) ∩ bar( A ) and set ϕ k = kψ + ϕ ∈ ext( π ) for every k ∈ N . Then, ψ ( x ) + 1 k ϕ ( x ) = 1 k ϕ k ( x ) ≥ k σ A ( ϕ k ) ≥ σ A ( ψ ) + 1 k σ A ( ϕ )for every k ∈ N . Letting k → ∞ yields (4.4) and concludes the proof.For a given map f : L → [ −∞ , ∞ ] we denote by lsc( f ) the largest lower semicontinuous map dominatedby f and, similarly, by usc( f ) the smallest upper semicontinuous map dominating f . Proposition 4.14.
For every π ∈ K + M \ { } such that ext( π ) ∩ bar( A ) = ∅ the following statements hold:(i) ρ ∗ π = lsc( ρ π ) .(ii) u ∗ π = usc( u π ) .Proof. Fix a nonzero π ∈ K + M such that ext( π ) ∩ bar( A ) = ∅ . Clearly, we only need to show (i). To thiseffect, recall that ρ ∗ π is lower semicontinuous and note that it is dominated by ρ π . Indeed, for every x ∈ L and for every m ∈ M such that x + m ∈ A sup ψ ∈ ext( π ) { σ A ( ψ ) − ψ ( x ) } ≤ sup ψ ∈ ext( π ) { ψ ( x + m ) − ψ ( x ) } = π ( m ) , showing that ρ ∗ π ( x ) ≤ ρ π ( x ). Now, take a lower semicontinuous map f : L → [ −∞ , ∞ ] such that f ≤ ρ π .We claim that f ≤ ρ ∗ π as well. To show this, suppose to the contrary that f ( x ) > ρ ∗ π ( x ) for some x ∈ L .Note that ρ ∗ π ( x ) = inf { λ ∈ R : x + λm ∈ cl( A + ker( π )) } by Lemma 4.13, where m ∈ M is any element satisfying π ( m ) = 1 (which exists because π is nonzero).As a result, we must have f ( x ) > λ for some λ ∈ R such that x + λm ∈ cl( A + ker( π )). Hence, there existtwo nets ( x α ) ⊆ A and ( m α ) ⊆ ker( π ) such that x α + m α → x + λm . Since { f > λ } is open by lowersemicontinuity, it eventually follows from the translativity of ρ π that λ < f ( x α + m α − λm ) ≤ ρ π ( x α + m α − λm ) = ρ π ( x α ) + λ ≤ λ. Since this is impossible, we infer that f ≤ ρ ∗ π must hold, concluding the proof.9 emark 4.15. (i) The preceding proposition shows that the dual representation in Lemma 4.10 and,hence, the multi-utility representation in Theorem 4.11 can be equivalently stated in terms of the semicon-tinuous hulls of the functionals ρ π and u π , respectively. This should be compared with the representationin Lemma 5.1 in Hamel and Heyde [28].(ii) The preceding proposition also suggests the following alternative path to establishing Lemma 4.10:(1) Start with the representation in Lemma 4.4. (2) Show that the functionals ρ π there can be replacedby their lower semicontinuous hulls lsc( ρ π ). (3) Show that we can discard from the representation all thefunctionals π ∈ K + M \ { } such that lsc( ρ π ) is not proper or, equivalently, ext( π ) ∩ bar( A ) = ∅ . (4) UseProposition 4.14 to replace the functionals lsc( ρ π ) with the more explicit functionals ρ ∗ π . The advantageof the strategy pursued in the proof of Lemma 4.10 is that it avoids passing through semicontinuous hullsand the analysis of their properness.The representing functionals belonging to the multi-utility representation in Theorem 4.11 are, by def-inition, upper semicontinuous. As a final step, we want to find conditions ensuring a multi-utility rep-resentation consisting of continuous functionals only. To achieve this, we exploit the link between thefunctionals ρ π and their regularizations ρ ∗ π established in Proposition 4.14. Lemma 4.16.
Let π ∈ K + M \ { } be such that ext( π ) ∩ bar( A ) = ∅ . If int( A ) = ∅ and ρ π ( x ) < ∞ forevery x ∈ L , then ρ π is finite valued and continuous. In particular, ρ π = ρ ∗ π .Proof. First of all, we claim that ρ π ( x ) > −∞ for every x ∈ L . To see this, take any functional ψ ∈ ext( π ) ∩ bar( A ) and note that for every x ∈ Lρ π ( x ) ≥ ρ ∗ π ( x ) ≥ σ A ( ψ ) − ψ ( x ) > −∞ . As a result, ρ π is finite valued. Note that, by definition, ρ π is bounded above on A by 0. Since A hasnonempty interior and ρ π is convex, we infer from Theorem 8 in Rockafellar [40] that ρ π is continuous.The last statement is a direct consequence of Proposition 4.14.The following multi-utility representation with continuous utility functionals is a direct consequence ofTheorem 4.11 and Lemma 4.16. Theorem 4.17.
Assume that int( A ) = ∅ and that ρ π ( x ) < ∞ for all π ∈ K + M \{ } with ext( π ) ∩ bar( A ) = ∅ and x ∈ L . Then, the preference (cid:23) R can be represented by the multi-utility family U ∗∗ = { u π : π ∈ K + M \ { } , ext( π ) ∩ bar( A ) = ∅} . In addition, every element of U ∗∗ is finite valued and continuous. We conclude by showing a number of sufficient conditions for the finiteness assumption in Lemma 4.16to hold. This should be compared with the results in Section 3 in Farkas et al. [20]. The recession coneof A is denoted by rec( A ) := { x ∈ L : x + y ∈ A, ∀ y ∈ A } . Note that rec( A ) is the largest convex cone such that A + rec( A ) ⊆ A . In particular, if A is a cone, thenrec( A ) = A . Moreover, for any convex cone C ⊆ L we denote byqint( C ) := { x ∈ C : ψ ( x ) > , ∀ ψ ∈ C + \ { }} the quasi interior of C . Note that we always have int( C ) ⊆ qint( C ). Proposition 4.18.
Let π ∈ K + M \ { } satisfy ext( π ) ∩ bar( A ) = ∅ . Then, ρ π ( x ) < ∞ for every x ∈ L ifany of the following conditions holds: i) L = A + M .(ii) M ∩ qint( K ) = ∅ .(iii) M ∩ qint(rec( A )) = ∅ .Proof. The desired assertion clearly holds under (i). Since K ⊆ rec( A ) by assumption (A1), we see thatqint( K ) ⊆ qint(rec( A )). Hence, it suffices to establish that (iii) implies the desired assertion. So, assumethat (iii) holds and take m ∈ M ∩ qint(rec( A )). If ρ π ( x ) = ∞ for some x ∈ L , then we must have( x + M ) ∩ A = ∅ . It follows from a standard separation result, see e.g. Theorem 1.1.3 in Z˘alinescu [43],that we find a nonzero functional ψ ∈ L ′ satisfying ψ ( x + λm ) ≤ σ A ( ψ ) for every λ ∈ R . This is onlypossible if ψ ( m ) = 0, which cannot hold because ψ ∈ bar( A ) ⊆ (rec( A )) + . As a result, we must have ρ π ( x ) < ∞ for every x ∈ L . In this final section we specify the general dual representation of R to a number of concrete situations.The explicit formulation of the corresponding multi-utility representation can be easily derived as inTheorem 4.11 and Theorem 4.17. Throughout the section we consider a probability space (Ω , F , P ) andfix an index d ∈ N . For every p ∈ [0 , ∞ ] and every Borel measurable set S ⊆ R d we denote by L p ( S )the set of all equivalence classes with respect to almost-sure equality of d -dimensional random vectors X = ( X , . . . , X d ) : Ω → R d with p -integrable components such that P [ X ∈ S ] = 1. As usual, we neverexplicitly distinguish between an equivalence class in L p ( S ) and any of its representative elements. Wetreat R d as a linear subspace of L p ( R d ). For all vectors a, b ∈ R d we set h a, b i := d X i =1 a i b i . The expectation with respect to P is simply denoted by E . For every p ∈ [1 , ∞ ] the space L p ( R d ) can benaturally paired with L q ( R d ) for q = pp − via the bilinear form( X, Z ) E [ h X, Z i ] . Here, we adopt the usual conventions := ∞ and ∞∞ := 1. Finally, for every random vector X ∈ L ( R d )we use the compact notation E [ X ] := ( E [ X ] , . . . , E [ X d ]) . We consider a financial market where d different currencies are traded. Every element of L ( R d ) isinterpreted as a vector of capital positions expressed in our different currencies at some future point intime. For a pre-specified acceptance set A ⊆ L ( R d ) we look for the currency portfolios that have to beset up at the initial time to ensure acceptability. As a first step, we consider a one-period market with dates 0 and 1. In this setting, we focus on thecurrency portfolios that we have to build at time 0 in order to ensure acceptability of currency positionsat time 1. This naturally leads to defining the set-valued map R : L ( R d ) ⇒ R d by R ( X ) := { m ∈ R d : X + m ∈ A} . ssumption 5.1. In this subsection we work under the following assumptions:(1) A is norm closed, convex, and satisfies A + L ( R d + ) ⊆ A .(2) R ( X ) / ∈ {∅ , R d } for some X ∈ L ( R d ).We derive the following representation by applying our general results to( L, L ′ , K, A, M ) = (cid:16) L ( R d ) , L ∞ ( R d ) , L ( R d + ) , A , R d (cid:17) . This result should be compared with the dual representation established in Jouini et al. [32], Kulikov [35],and Hamel and Heyde [28].
Proposition 5.2.
For every X ∈ L ( R d ) the set R ( X ) can be represented as R ( X ) = \ w ∈ R d + \{ } \ Z ∈ L ∞ ( R d + ) , E [ Z ]= w { m ∈ R d : h m, w i ≥ σ A ( Z ) − E [ h X, Z i ] } . In addition, if A is a cone, then we can simplify the above representation using that σ A ( Z ) = ( Z ∈ A + , −∞ otherwise . Proof.
Note that K + M can be identified with R d + and that bar( A ) is contained in L ∞ ( R d + ) by assumption(1). Since, for all w ∈ R d and Z ∈ L ∞ ( R d ), the random vector Z (viewed as a functional on L ( R d )) isan extension of w (viewed as a functional on R d ) precisely when E [ Z ] = w , the desired representationfollows immediately from Lemma 4.10. Remark 5.3.
Note that, in the above framework, the set qint( K ) consists of all the random vectors in L ( R d ) with components that are strictly positive almost surely and, hence, M ∩ qint( K ) = ∅ . This can beused to ensure multi-utility representations with continuous representing functionals; see Proposition 4.18. Example 5.4 ( Multidimensional Expected Shortfall).
For every X ∈ L ( R ) and every α ∈ (0 , we denote by ES α ( X ) the Expected Shortfall of X at level α , i.e. ES α ( X ) := − α Z α q X ( β ) dβ, where q X is any quantile function of X . The multi-dimensional acceptance set based on Expected Shortfallintroduced in Hamel et al. [31] is given by A = { X ∈ L ( R d ) : ES α i ( X i ) ≤ , ∀ i ∈ { , . . . , d }} for a fixed α = ( α , . . . , α d ) ∈ (0 , d . Note that assumptions (1) and (2) hold. In particular, we have R (0) = R d + . In addition, A is a cone. Note that Z ∞ w ( α ) := { Z ∈ L ∞ ( R d ) : E [ Z ] = w } ∩ A + = n Z ∈ L ∞ ( R d + ) : E [ Z ] = w, Z ≤ wα o for every w ∈ R d + (where wα is understood component by component). This follows from the standard dualrepresentation of Expected Shortfall; see Theorem 4.52 in F¨ollmer and Schied [24]. As a result, the dualrepresentation in Proposition 5.2 reads R ( X ) = \ w ∈ R d + \{ } \ Z ∈Z ∞ w ( α ) { m ∈ R d : h m, w i ≥ − E [ h X, Z i ] } for every random vector X ∈ L ( R d ) . .1.2 The dynamic case As a next step, we consider a multi-period financial market with dates t = 0 , . . . , T and informationstructure represented by a filtration ( F t ) satisfying F = {∅ , Ω } and F T = F . In this setting, currencyportfolios can be rebalanced through time. A (random) portfolio at time t ∈ { , . . . , T } is represented byan F t -measurable random vector in L ( R d ). We denote by C t the set of F t -measurable portfolios that canbe converted into portfolios with nonnegative components by trading at time t . This means that, for all F t -measurable portfolios m t and n t , we can exchange m t for n t at time t provided that m t − n t ∈ C t . Thesets C t are meant to capture potential transaction costs. A flow of portfolios is represented by an adaptedprocess ( m t ). More precisely, for every date t ∈ { , . . . , T − } , the portfolio m t is set up at time t andheld until time t + 1. The portfolio flows belonging to the set C := { ( m t ) : m t − m t +1 ∈ C t +1 , ∀ t ∈ { , . . . , T − }} are said to be admissible. The admissibility condition is a direct extension of the standard self-financingproperty in frictionless markets.We look for all the initial portfolios that can be rebalanced in an admissible way until the terminal datein order to ensure acceptability. This leads to the set-valued map R : L ( R d ) ⇒ R d defined by R ( X ) := { m ∈ R d : ∃ ( m t ) ∈ C , n T ∈ L ( R d ) : m − m ∈ C , m T − n T ∈ C T , X + n T ∈ A} . In words, the above set consists of all the initial portfolios that give rise, after a convenient exchange atdate 0, to an admissible rebalancing process making the outstanding currency position acceptable aftera final portfolio adjustment at time T . This setting can be embedded in our framework because we canequivalently write R ( X ) = (cid:26) m ∈ R d : X + m ∈ A + T X t =0 C t (cid:27) . Assumption 5.5.
In this subsection we work under the following assumptions:(1) A is norm closed, convex, and satisfies A + L ( R d + ) ⊆ A .(2) C t is convex and contains L t ( R d + ) for every t ∈ { , . . . , T } .(3) ( A + P Tt =0 C t ) ∩ L ( R d ) is norm closed.(4) R ( X ) / ∈ {∅ , R d } for some X ∈ L ( R d ).We derive the following representation by applying our general results to( L, L ′ , K, A, M ) = L ( R d ) , L ∞ ( R d ) , L ( R d + ) , (cid:18) A + T X t =0 C t (cid:19) ∩ L ( R d ) , R d ! . For convenience, we also set C T := (cid:18) T X t =1 C t (cid:19) ∩ L ( R d ) . For later use note thatbar( C T ) ⊆ bar T X t =1 (cid:16) C t ∩ L ( R d ) (cid:17)! = T \ t =1 bar (cid:16) C t ∩ L ( R d ) (cid:17) . The next result should be compared with the dual representation established in Hamel et al. [29] in thespecial setting of Example 5.8. 13 roposition 5.6.
For every X ∈ L ( R d ) the set R ( X ) can be represented as R ( X ) = \ w ∈ R d + \{ } \ Z ∈ L ∞ ( R d + ) , E [ Z ]= w { m ∈ R d : h m, w i ≥ σ A , C ( Z ) − E [ h X, Z i ] } . where we have set for every Z ∈ L ∞ ( R d ) σ A , C ( Z ) := σ A ( Z ) + σ C ( Z ) + σ C T ( Z ) In addition, if A is a cone, the above representation can be simplified by using that σ A ( Z ) = ( Z ∈ A + , −∞ otherwise . Moreover, if C is a cone, then σ C ( Z ) = ( Z ∈ L ∞ ( R d + ) , E [ Z ] ∈ C +0 , −∞ otherwise . Similarly, if C t is a cone for every t ∈ { , . . . , T } , then σ C T ( Z ) = ( Z ∈ ( C T ) + ⊆ T Tt =1 ( C t ∩ L ( R d )) + , −∞ otherwise . Proof.
The assertion follows from Proposition 5.2 because C ⊂ R d implies (cid:18) A + T X t =0 C t (cid:19) ∩ L ( R d ) = A + C + C T and observe that σ A + C + C T = σ A + σ C + σ C T . Remark 5.7.
Note that, as in the static case, we have M ∩ qint( K ) = ∅ . This can be used to ensuremulti-utility representations with continuous representing functionals; see Proposition 4.18. Example 5.8 ( Superreplication under proportional transaction costs).
We adopt the discreteversion of the model by Kabanov [33]. For every t ∈ { , . . . , T } we say that a set-valued map S : Ω ⇒ R d is F t -measurable provided that { ω ∈ Ω : S ( ω ) ∩ U 6 = ∅} ∈ F t for every open set U ⊂ R d . In this case, we denote by L ( S ) the set of all random vectors X ∈ L ( R d ) suchthat P [ X ∈ S ] = 1 . This set is always nonempty if S has closed values; see Corollary 14.6 in Rockafellarand Wets [41]. Now, let K t : Ω ⇒ R d be an F t -measurable set-valued map such that K t ( ω ) is a polyhedralconvex cone (hence K t ( ω ) is closed) containing R d + for every ω ∈ Ω and set C t = L ( K t ) . Moreover, we consider the worst-case acceptance set A = L ( R d + ) . Assumptions (1) and (2) are easily seen to be satisfied. Moreover, A as well as each of the sets C t is acone. As proved in Theorem 2.1 in Schachermayer [42], assumption (3) always holds under the so-called robust no-arbitrage” condition. Finally, as ∈ R (0) , assumption (4) holds if and only if R d is notentirely contained in P Tt =0 C t . Note also that A + = L ∞ ( R d + ) . As a result, Proposition 5.6 yields R ( X ) = \ w ∈ K +0 \{ } \ Z ∈ L ∞ ( R d + ) , Z ∈ ( C T ) + , E [ Z ]= w { m ∈ R d : h m, w i ≥ − E [ h X, Z i ] } for every X ∈ L ( R d ) . The dual elements Z in the above representation can be linked to consistent pricingsystems, see e.g. Schachermayer [42]. To see this, note that, for every t ∈ { , . . . , T } , the set-valued map K + t : Ω ⇒ R d defined by K + t ( ω ) = (cid:0) K t ( ω ) (cid:1) + is F t -measurable, see e.g. Exercise 14.12 in Rockafellar and Wets [41], and such that (cid:16) C t ∩ L ( R d ) (cid:17) + = L ( K + t ) ∩ L ∞ ( R d ) by measurable selection, see the argument in the proof of Theorem 1.7 in Schachermayer [42]. As a result,every dual element Z in the above dual representation satisfies E [ Z ] ∈ K +0 , Z ∈ ( C T ) + ⊆ T \ t =1 (cid:16) C t ∩ L ( R d ) (cid:17) + ⊆ T \ t =1 L ( K + t ) . This shows that the d -dimensional adapted process ( E [ Z |F t ]) , where the conditional expectations are takencomponentwise, satisfies E [ Z |F T ] = Z and E [ Z |F t ] ∈ L ( K + t ) for every t ∈ { , . . . , T } and thus qualifiesas a consistent pricing system. In other words, the above dual elements Z can be viewed as the terminalvalues of consistent pricing systems. Remark 5.9. (i) It is worth noting that our approach provides a different path, compared to the strategypursued in Schachermayer [42], to establish the existence of consistent pricing systems under the robustno-arbitrage assumption (admitting the closedness of the reference target set). Moreover, by rewriting theabove dual representation in terms of consistent pricing systems, we recover the (localization to L ( R d )of the) superreplication theorem by Schachermayer [42].(ii) The above dual representation was also obtained in Hamel et al. [29]. Differently from that paper, wehave not derived it from the superreplication theorem in Schachermayer [42] but from a direct applicationof our general results. We consider a single-period economy with dates 0 and 1 and a financial system consisting of d entities.Every element of L ∞ ( R d ) is interpreted as a vector of capital positions of the various financial entities attime 1. The individual positions can be aggregated through a function Λ : R d → R . For a pre-specifiedacceptance set A ⊆ L ∞ ( R ) we look for the cash injections at time 0 that ensure the acceptability of theaggregated system. This leads to the set-valued map R : L ∞ ( R d ) ⇒ R d defined by R ( X ) := { m ∈ R d : Λ( X + m ) ∈ A} . This setting can be easily embedded in our framework because we can write R ( X ) = { m ∈ R d : X + m ∈ Λ − ( A ) } . Assumption 5.10.
In this subsection we work under the following assumptions:151) A is convex and satisfies A + L ∞ ( R + ) ⊆ A .(2) Λ is nondecreasing and concave.(3) Λ − ( A ) is σ ( L ∞ ( R d ) , L ( R d ))-closed.(4) R ( X ) / ∈ {∅ , R d } for some X ∈ L ∞ ( R d ).We derive the following representation by applying our general results to( L, L ′ , K, A, M ) = (cid:16) L ∞ ( R d ) , L ( R d ) , L ∞ ( R d + ) , Λ − ( A ) , R d (cid:17) . The result should be compared with the dual representations established in Ararat and Rudloff [1].
Proposition 5.11.
For every X ∈ L ∞ ( R d ) the set R ( X ) can be represented as R ( X ) = \ w ∈ R d + \{ } \ Z ∈ L ( R d + ) , E [ Z ]= w { m ∈ R d : h m, w i ≥ σ Λ − ( A ) ( Z ) − E [ h X, Z i ] } . If Λ − ( A ) is a cone, then we can simplify the above representation using that σ Λ − ( A ) ( Z ) = ( Z ∈ (Λ − ( A )) + , −∞ otherwise . Proof.
Note that Λ − ( A ) is convex and satisfies Λ − ( A ) + L ∞ ( R d + ) ⊆ Λ − ( A ) by assumptions (1) and(2). Note also that K + M can be identified with R d + . In addition, we have bar(Λ − ( A )) ⊆ L ( R d + ). Since,for all w ∈ R d and Z ∈ L ( R d ), the random vector Z (viewed as a functional on L ∞ ( R d )) is an extensionof w (viewed as a functional on R d ) precisely when E [ Z ] = w , the desired representation follows fromLemma 4.10. Example 5.12 ( Weighted aggregated losses).
Let α = ( α , . . . , α d ) ∈ (1 , ∞ ) d and consider theaggregation function Λ : R d → R defined by Λ( x ) = d X i =1 max( x i ,
0) + d X i =1 α i min( x i , . Moreover, define the acceptance set