Lower semicontinuity of monotone functionals in the mixed topology on Cb
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Nendel, Max Article — Published Version Lower semicontinuity of monotone functionals in the mixed topology on Cb Finance and Stochastics Provided in Cooperation with: Springer Nature Suggested Citation: Nendel, Max (2024) : Lower semicontinuity of monotone functionals in the mixed topology on Cb, Finance and Stochastics, ISSN 1432-1122, Springer, Berlin, Heidelberg, Vol. 29, Iss. 1, pp. 261-287, https://doi.org/10.1007/s00780-024-00552-2 This Version is available at: https://hdl.handle.net/10419/315065 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/
Finance and Stochastics (2024) 29:261–287 https://doi.org/10.1007/s00780-024-00552-2 Lower semicontinuity of monotone functionals in the mixed topology on Cb Max Nendel1 Received: 18 October 2022 / Accepted: 23 July 2024 / Published online: 28 November 2024 © The Author(s) 2024 Abstract The main result of this paper characterises the continuity from below of monotone functionals on the space Cbof bounded continuous functions on an arbitrary Polish space as lower semicontinuity in the mixed topology. In this particular situation, the mixed topology coincides with the Mackey topology for the dual pair (Cb,ca), where ca denotes the space of all countably additive signed Borel measures of finite variation. Hence lower semicontinuity in the mixed topology is for convex monotone maps Cb→Requivalent to a dual representation in terms of countably additive measures. Such representations are of fundamental importance in finance, e.g. in the context of risk measures and superhedging problems. Based on the main result, regularity properties of capacities and dual representations of Choquet integrals in terms of countably additive measures for 2-alternating capacities are studied. Moreover, a well-known characterisation of star-shaped risk measures on L∞is transferred to risk measures on Cb. In a second step, the paper provides a characterisation of equicontinuity in the mixed topology for families of convex monotone maps. As a consequence, for every convex monotone map on Cbtaking values in a locally convex vector lattice, continuity in the mixed topology is equivalent to continuity on normbounded sets. Keywords Risk measure ·Monotone functional ·Choquet integral ·Continuity from below ·Lower semicontinuity ·Mixed topology ·Mackey topology ·Star-shaped Mathematics Subject Classification 91G70 ·46A20 ·28A12 JEL Classification C02 ·C65 1Introduction In this paper, we study continuity properties for monotone maps Cb→R, where Cb=Cb() denotes the space of all bounded continuous functions on a Polish M. Nendel [email protected] 1Center for Mathematical Economics, Bielefeld University, Universitätsstraße 25, 33615 Bielefeld, Germany
262 M. Nendel space with values in R. Monotone functionals Cb→Rappear in many applications. Special instances of such maps, in the context of finance and actuarial science, are – risk measures or nonlinear expectations; cf. Denis et al. [11], Föllmer and Schied [15, Chap. 4] and Peng [22, Chap. 1]. – superhedging functionals; cf. Cheridito et al. [7] and Cheridito et al. [8]. – robust expected utilities or loss functions; cf. Delbaen [9,10]. – Choquet integrals, e.g. in the context of insurance premia; cf. Wang et al. [25]. In order to obtain dual representations of convex monotone functionals in terms of countably additive measures, additional continuity properties are usually required. The two most prominent continuity properties in this context are continuity from above and continuity from below; cf. Föllmer and Schied [15, Lemma 4.21 and Theorem 4.22]. For convex monotone functionals, continuity from below is usually a weaker requirement than continuity from above; see for instance Cheridito et al. [8]. In the field of mathematical finance, these continuity properties have been studied in many contexts in the past decades. For risk measures, continuity from above is (up to a different sign convention) related to the Lebesgue property, whereas continuity from below is closely tied to the Fatou property. The Lebesgue and Fatou properties refer to sequential continuity and lower semicontinuity, respectively, for uniformly bounded pointwise convergent sequences of measurable functions, and Fatou closedness is a fundamental ingredient in no-arbitrage theory; see e.g. Burzoni and Maggis [5] and Herdegen and Khan [18]. Fixing a reference measure and working on L∞, it is well known that continuity from below of convex risk measures is equivalent to a dual representation in terms of countably additive measures; cf. Föllmer and Schied [15, Theorem 4.33]. If the risk measure is also law-invariant, continuity from below is automatically satisfied if the underlying probability space is assumed to be atomless; cf. Jouini et al. [19]. On the other hand, monetary risk measures are closely linked to nonlinear expectations and the topic of model uncertainty in finance. In this context, risk measures which are not dominated by a single probability measure that deems events to be negligible or not play a crucial role. An example for such a risk measure is the G-expectation; cf. Denis et al. [11] and Peng [22, Chap. 2]. However, on the space Bbof bounded measurable functions without a reference probability, continuity from below alone is not sufficient to guarantee a dual representation of convex monotone functionals in terms of countably additive measures, despite the fact that it implies sequential lower semicontinuity of such functionals in the weak topology σ(Bb,ca) of the dual pair (Bb,ca). In Denk et al. [12, Example 3.6], an example is given for a coherent risk measure which is continuous from below on Bb, but does not have a single countably additive minorant. On the other hand, continuity from above of a risk measure on the space of bounded measurable functions already implies the existence of a dominating reference measure; see e.g. Denk et al. [12, Remark 3.3]. One way out of this dilemma is to restrict attention to continuous claims. For the space Cbof bounded continuous functions as an underlying function space, it is well known that continuity from above, for a convex monotone functional Cb→R,is sufficient but not necessary for a dual representation in terms of countably additive
Lower semicontinuity of monotone functionals on Cb263 measures; see e.g. Cheridito et al. [8]. However, the question whether such a representation is equivalent to the weaker notion of continuity from below on general Polish state spaces has remained unanswered for almost a decade, as discussed in the introduction of Delbaen [10]. In a series of papers, this question has been answered positively by Delbaen [9,10], and as a consequence, convex monotone functionals on Cbwhich are continuous from below are lower semicontinuous in the Mackey topology μ(Cb,ca)of the dual pair (Cb,ca), where ca denotes the space of countably additive signed Borel measures with finite variation. From a mathematical perspective, this is a remarkable result, since the Mackey topology μ(Cb,ca)is not metrisable and continuity from below is a requirement for sequences, so that nonmetrisability poses a major problem. Therefore in [9,10], another path is chosen and the proofs there rely on compactification methods; more precisely, they use the fact that every Polish space can be embedded as a Gδinto a compact metric space. Theorem 2.2 in the present paper generalises the main result of Delbaen [9]by showing that for any monotone functional Cb→R, continuity from below is equivalent to lower semicontinuity in the mixed topology. The latter is a classical general concept in analysis, cf. Wiweger [27], and coincides with the Mackey topology μ(Cb,ca)in this particular setting. Moreover, Theorem 2.2 shows that for monotone functionals Cb→R, upper or lower semicontinuity in the mixed topology are equivalent to sequential upper or lower semicontinuity in the mixed topology, respectively, despite the fact that the mixed topology is not metrisable. Since the Mackey topology μ(Cb,ca)is the finest topology leading to the dual space ca of countably additive signed Borel measures with finite variation, it is a natural choice for duality theory on Cb. In particular, Theorem 2.2 implies that every convex monotone functional on Cbadmits a dual representation in terms of countably additive measures; cf. Corollary 2.5. However, using the explicit representation of a local base at the origin for the mixed topology allows us to further characterise continuity from below in terms of proximity on compact sets also for nonconvex monotone functionals; see Theorem 2.2. In Corollary 2.6, we turn our focus to capacities and Choquet integrals. In a first step, we characterise the regularity of general capacities, defined on open sets, in terms of continuity from below of the Choquet integral on the set Lbof all bounded lower semicontinuous functions →Rand of the capacity along sequences of open sets. In a second step, we characterise 2-alternating capacities, for which the related Choquet integral admits a dual representation in terms of countably additive measures on Lb, in terms of continuity from below along sequences of open sets and regularity of the capacity, partially extending the results in Adamski [1]. Corollary 2.7 extends a well-known characterisation of star-shaped risk measures on L∞, discussed in Castagnoli et al. [6], to general risk measures on the space Cb. While the proof follows closely that in [6], Corollary 2.5 allows the transition from L∞to Cb. Another question we address in this paper is a characterisation of continuity of convex monotone maps in the mixed topology. Continuity in the mixed topology is of fundamental importance in many situations in robust finance. In the context of superhedging, it has been studied in Cheridito et al. [7]. In the context of dynamic risk measures and semigroups related to stochastic processes under model uncertainty, it
264 M. Nendel appears in Blessing et al. [3], Goldys et al. [17] and with a different language, it is also used in the analysis of large deviation principles based on max-stable risk measures; cf. Kupper and Zapata [21]. Building on Theorem 2.2, we discuss the equicontinuity of families of convex monotone maps in the mixed topology in Theorem 2.8. There, a characterisation of equicontinuity in terms of continuity from above and uniform equicontinuity on supremum-norm-bounded sets is given. From a mathematical perspective, the mixed topology has two striking features. On the one hand, unless is compact, it has no neighborhood of zero which is bounded with respect to the supremum norm. On the other hand, it has the intrinsic property that for linear operators taking values in an arbitrary locally convex space, continuity is equivalent to continuity on supremum-norm-bounded sets. Corollary 2.11, which is a consequence of Theorem 2.8, extends this intrinsic property by showing that for convex monotone maps taking values in a locally convex vector lattice, μ(Cb,ca)-continuity is equivalent to μ(Cb,ca)-continuity on supremum-normbounded sets. A particular instance of a locally convex vector lattice is Cbitself, endowed with the mixed topology, which in a financial context corresponds for example to the case of conditional risk measures or conditional nonlinear expectations. The rest of the paper is organised as follows. In Sect. 2, we state the main results and their corollaries. Section 3contains all proofs. In Appendix A, we prove an auxiliary result for capacities and Choquet integrals, and in Appendix B, we prove an auxiliary result on locally convex vector lattices. 2 Main results Throughout,letbe a Polish space and Cb=Cb() the space of all bounded continuous functions →R. We consider the local base V2:= {{g∈Cb:g∞<r}:r>0} at 0 ∈Cbfor the topology induced by the supremum norm ·∞, and the local base V1:=g∈Cb:sup x∈C|g(x)|<r :r>0,C⊆compact at 0 ∈Cbfor the vector topology of uniform convergence on compacts. Let Vdenote the system consisting of all sets n∈N n ∑ k=1 (V 1 k∩kV 2)with (V 1 k)k∈N⊆V1and V2∈V2,(2.1) where kV2:= {kg :g∈V2}for all k∈Nand n ∑ k=1 Vk:= n ∑ k=1 gk:g1∈V1,...,g n∈Vn
Lower semicontinuity of monotone functionals on Cb265 for n∈Nand nonempty subsets V1,...,V nof Cb. Then Vis a local base at 0 ∈Cb for a Hausdorff locally convex topology β, which is known as the mixed topology.We refer to Wiweger [27] for a detailed discussion on the mixed topology in a more general setting. Clearly, the mixed topology βis finer than the weak topology σ(Cb,ca) of the dual pair (Cb,ca), where ca denotes the space of all countably additive signed Borel measures of finite variation. A well-known fact, which we do not make use of, is that the mixed topology βcoincides with the Mackey topology of the dual pair (Cb,ca). Moreover, βbelongs to the class of strict topologies; cf. Wheeler [26]. We also refer to Fremlin et al. [16] and Sentilles [24] for additional fine properties of mixed or strict topologies. We say that a functional U:Cb→Ris monotone if U(f) ≤U(g) for all f, g ∈Cbwith f≤g, where for functions →R, the relation ≤and all other order-related objects refer to the pointwise order. For a sequence (fn)n∈N⊆Cband a function f:→R, we write fnf as n→∞if fn≤fn+1for all n∈Nand f(x) =limn→∞ fn(x) for all x∈. Analogously, we write fnfas n→∞if fn≥fn+1for all n∈Nand f(x)=limn→∞ fn(x) for all x∈. Definition 2.1 a) We say that a monotone functional U:Cb→Ris continuous from below if U(f) =limn→∞ U(fn)for all sequences (fn)n∈N⊆Cband f∈Cbwith fnfas n→∞. b) We say that a monotone functional U:Cb→Ris continuous from above if U(f) =limn→∞ U(fn)for all sequences (fn)n∈N⊆Cband f∈Cbwith fnf as n→∞. Theorem 2.2 Let U:Cb→Rbe monotone.Then the following are equivalent: (i) Uis continuous from below. (ii) Uis lower semicontinuous in the mixed topology β. (iii) Uis sequentially lower semicontinuous in the mixed topology β. (iv) For all f∈Cb,ε>0and r≥0, there exist δ>0and a compact C⊆ such that U(f) ≤U(f +e) +ε for all e∈Cbwith e∞≤rand supx∈C|e(x)|≤δ. For an arbitrary function U:Cb→R, its conjugate function U:Cb→Ris given by U(f) := −U(−f)for all f∈Cb. Then we obtain the following corollary. Corollary 2.3 Let U:Cb→Rbe monotone.Then the following are equivalent: (i) Uis continuous from above. (ii) Uis upper semicontinuous in the mixed topology β. (iii) Uis sequentially upper semicontinuous in the mixed topology β. (iv) For all f∈Cb,ε>0and r≥0, there exist δ>0and a compact C⊆ such that U(f +e) ≤U(f)+ε for all e∈Cbwith e∞≤rand supx∈C|e(x)|≤δ.
266 M. Nendel A combination of Theorem 2.2 and Corollary 2.3 leads to the following characterisation of continuity in the mixed topology for monotone functionals. Corollary 2.4 Let U:Cb→Rbe monotone.Then the following are equivalent: (i) Uis continuous from above and below. (ii) Uis continuous in the mixed topology β. (iii) Uis sequentially continuous in the mixed topology β. (iv) For all f∈Cb,ε>0and r≥0, there exist δ>0and a compact C⊆ such that |U(f +e) −U(f)|≤ε for all e∈Cbwith e∞≤rand supx∈C|e(x)|≤δ. As a direct consequence of Theorem 2.2, we obtain the main result in Delbaen [9]. We denote by ca+the set of all positive elements of ca, i.e., the set of all finite Borel measures. Corollary 2.5 Let U:Cb→Rbe convex and monotone.Then the following are equivalent: (i) Uis continuous from below. (ii) Uis lower semicontinuous in the mixed topology β. (iii) Uis lower semicontinuous in the weak topology σ(Cb,ca). (iv) There exist a nonempty set M⊆ca+and a function α:M→Rsuch that U(f) =sup μ∈M( fdμ−α(μ))for all f∈Cb.(2.2) We apply Corollary 2.5 to the case of Choquet integrals. In the sequel, let Odenote the family of all open subsets of , i.e., the topology on , and Lbthe set of all bounded lower semicontinuous functions →R.Acapacity (on O)is a map c:O→[0,∞)with c(∅)=0 and c(B1)≤c(B2)for all B1,B 2∈Owith B1⊆B2. For a capacity c:O→[0,∞), we define the Choquet integral with respect to cas fdc:=∞ 0c({f>s})ds+0 −∞ (c({f>s})−c())dsfor all f∈Lb. By definition, the Choquet integral is positively homogeneous, i.e., (λf ) dc=λ fdcfor all f∈Lband λ>0, and constant additive, i.e., (f +m) dc= fdc+mc() for all f∈Lband m∈R.
Lower semicontinuity of monotone functionals on Cb267 A well-known fact is that the Choquet integral is subadditive, i.e., (f1+f2)dc≤ f1dc+ f2dcfor all f1,f 2∈Lb, if and only if the capacity cis 2-alternating, i.e., c(B1∪B2)+c(B1∩B2)≤c(B1)+c(B2)for all B1,B 2∈O. For the reader’s convenience, we provide a proof of this statement in Appendix A. Another consequence of Theorem 2.2 and Corollary 2.5 is the following result concerning the regularity of general capacities and dual representations of Choquet integrals in terms of countably additive measures for 2-alternating capacities. We point out that the equivalence between (i) and (ii) is a standard result from Choquet theory and can be found for example in the textbook by König [20, Exercise 11.18]. The equivalence between (iii) and (iv) has been discussed in a more general setting in Adamski [1]. The main novelty is the implication (ii) ⇒(iii), which is a consequence of Theorem 2.2 and together with Corollary 2.5 facilitates the proof of the remaining equivalences. For the reader’s convenience, we provide a self-contained proof of all equivalences. Corollary 2.6 Let c:O→[0,∞)be a capacity.Then the following are equivalent: (i) For every sequence (Bn)n∈N⊆Owith Bn⊆Bn+1for all n∈N, c( n∈N Bn)=lim n→∞c(Bn). (ii) The Choquet integral is continuous from below on Lb,i.e., fdc=lim n→∞ fndc for all f∈Lband any sequence (fn)n∈N⊆Lbwith fnfas n→∞. (iii) The capacity cis regular,i.e., for all B∈O, c(B) =sup CB inf A∈O A⊇C c(A), (2.3) wherewewriteCBfor C⊆B⊆with Ccompact. If cis 2-alternating,the statements (i)–(iii) are equivalent to the following statement: (iv) There exists a nonempty set M⊆ca+with μ() =c() for all μ∈Mand fdc=sup μ∈M fdμfor all f∈Lb. We now present an application of Corollary 2.5 for star-shaped risk measures on Cb. In the following, we say that a monotone functional R:Cb→Ris a risk
268 M. Nendel measure if R(0)=0 and R(f +m) =R(f) +mfor all f∈Cband all constants m∈R. We say that a risk measure R:Cb→Ris star-shaped if R(λf) ≤λR(f ) for all f∈Cband λ∈[0,1]. For a detailed discussion on risk measures, we refer to Föllmer and Schied [15, Chap. 4], and for a survey on star-shaped risk measures to Castagnoli et al. [6]. The following corollary is a variant of [6, Proposition 5] in our setting. The proof heavily uses the insights obtained in the proof of [6, Theorem 2]. We point out that [6, Proposition 5] covers only the case of dominated risk measures, i.e., risk measures on L∞, which is discussed in detail in [15, Sect. 4.3], whereas we consider general risk measures restricted to the space of bounded continuous functions. Corollary 2.7 Let R:Cb→R.Then the following are equivalent: (i) The map Ris a star-shaped risk measure. (ii) There exist a nonempty set Iand a family (αi)i∈Iof functions ca1 +→[0,∞] with infμ∈ca1 +αi(μ) =0for all i∈Iand R(f) =min i∈Isup μ∈ca1 +( fdμ−αi(μ))for all f∈Cb. We conclude this section with various characterisations of continuity in the mixed topology for convex monotone maps. We start with the following theorem which is our second main result. Recall Vfrom (2.1). Theorem 2.8 Let Ibe a nonempty index set and (Ui)i∈Ia family of convex and monotone maps Cb→Rwith sup i∈I(Ui(r) −Ui(0))<∞for all constants r≥0.(2.4) Then the following are equivalent: (i) For every sequence (fn)n∈N⊆Cbwith fn0as n→∞, lim n→∞sup i∈I(Ui(fn)−Ui(0))=0. (ii) For every r≥0and every ε>0, there exists some V∈Vwith sup f∞≤r sup i∈I|Ui(f +e) −Ui(f )|<ε for all e∈V. (iii) For every r≥0and every ε>0, there exist a compact C⊆and a constant M≥0such that sup i∈I|Ui(f1)−Ui(f2)|≤Msup x∈C|f1(x) −f2(x)|+ε for all f1,f 2∈Cbwith max{f1∞,f2∞}≤r.
Lower semicontinuity of monotone functionals on Cb275 Temam [14, Proposition 3.1], (ii) is equivalent to the fact that Uadmits a dual representation of the form U(f) =sup μ∈M(μf −α(μ))for all f∈Cb withasetMof β-continuous linear functionals on Cband a function α:M→R. Let μ∈Mand f∈Cbwith f≥0. Since Uis monotone, it follows that 1 λ(μ(−λf ) −α(μ))≤U(−λf ) λ≤U(0) λfor all λ>0. Hence 0=lim λ→∞−U(0)+α(μ) λ≤μf, which shows that every linear functional in Mis positive. The remaining equivalences and in particular the dual representation (2.2) now follow from the fact that by Theorem 2.2 and the Daniell–Stone theorem, cf. Bogachev [4, Theorem 7.8.1], a positive linear functional μ:Cb→Ris continuous in the mixed topology βif and only if it belongs to ca+. Proof of Corollary 2.6 1) We first prove the implication (i) ⇒(ii). To that end, let (fn)n∈N⊆Lband f∈Lbwith fnfas n→∞. Then for all s∈R, lim n→∞c({fn>s})=c({f>s}). Using the monotone convergence theorem, it follows that lim n→∞ fndc=lim n→∞(∞ 0c({fn>s})ds+0 −∞ (c({fn>s})−c())ds) =∞ 0c({f>s})ds+0 −∞ (c({f>s})−c())ds = fdc. (3.7) 2) For the implication (ii) ⇒(iii), first observe that c(B) ≥sup CB inf A∈O A⊇C c(A) for all B∈O. In order to show the converse inequality, let B∈Oand ε>0. In a first step, we consider the case B=. Then by Theorem 2.2, there exist δ>0 and a compact set C⊆such that 1dc≤ gdc+ε
276 M. Nendel for all g∈Cbwith g∞≤1 and supx∈C|g(x) −1|≤δ.NowletA∈Owith A⊇C. Since Ais open, there exists some m∈Nsuch that g:→Rgiven by g(x) := sup y∈C(1−md(x, y))∨0 for all x∈ satisfies g(x) =0forx∈\A. Since 0 ≤g≤1 and g(x) =1 for all x∈C,it follows that c() = 1dc≤ gdc+ε≤ 1Adc+ε=c(A) +ε. We have therefore shown that c() ≤inf A∈O A⊇C c(A) +ε Taking the supremum over all Cand letting ε→0 yields (2.3). For general B∈O, the statement now follows from the fact that B, endowed with the subspace topology OB:={A∩B:A∈O}={A∈O:A⊆B}⊆O, is again a Polish space, together with the observation that a subset of Bis compact in the subspace topology OBif and only if it is compact in the original topology O. 3) To prove that (iii) implies (i), let (Bn)n∈N⊆Owith Bn⊆Bn+1for all n∈N and ε>0. Then there exists some compact C⊆n∈NBn=:Bwith c(B) ≤inf A∈O A⊇C c(A) +ε. Since Cis compact, C⊆B=n∈NBnand Bnis open with Bn⊆Bn+1for all n∈N, there exists some n0∈Nsuch that C⊆Bn0. Hence inf A∈O A⊇C c(A) ≤c(Bn0), and it follows that sup n∈Nc(Bn)≤c(B) ≤sup n∈Nc(Bn)+ε=lim n→∞c(Bn)+ε. Letting ε→0, it follows that c(B) =limn→∞ c(Bn). 4) Now we assume that the capacity cis 2-alternating. In order to prove that (iv) implies (i), let (Bn)n∈N⊆Owith Bn⊆Bn+1for all n∈N. Then c( n∈N Bn)=sup μ∈M μ( n∈N Bn) =sup μ∈M sup n∈Nμ(Bn)=sup n∈Nsup μ∈M μ(Bn)=lim n→∞c(Bn).
Lower semicontinuity of monotone functionals on Cb277 5) In the last step, we prove that (ii) implies (iv). By Corollary 2.5, there exist a set M⊆ca+and a function α:M→Rsuch that fdc=sup μ∈M( fdμ−α(μ))for all f∈Cb. Since the Choquet integral is positively homogeneous, it follows that fdμ−α(μ) λ=1 λ( λf dμ−α(μ))≤1 λ λf dc= fdc for all f∈Cb,μ∈Mand λ>0. Letting λ→∞, it follows that sup μ∈M fdμ≤ fdcfor all f∈Cb. On the other hand, −α(μ) = 0dμ−α(μ) ≤ 0dc=0 for all μ∈M. Hence for all f∈Cb, sup μ∈M fdμ≤ fdc=sup μ∈M( fdμ−α(μ))≤sup μ∈M fdμ. In particular, μ() = 1dμ≤ 1dc=c() for all μ∈M. Since the Choquet integral is constant additive, it follows that 0=c() + (−1)dc≥c() + (−1)dμ=c() −μ() for all μ∈M. We have therefore shown that μ() =c() for all μ∈M. Defining for f∈Lb,x∈and n∈Nthe quantity fn(x) := infy∈(f (y) +nd(x, y)) with ametricdconsistent with the topology on , there exists for all f∈Lba sequence (fn)n∈N⊆Cbwith fnfas n→∞, and so fdc=sup n∈N fndc =sup n∈Nsup μ∈M fndμ=sup μ∈M sup n∈N fndμ=sup μ∈M fdμ, where the first equality uses (3.7) and the last the monotone convergence theorem.
278 M. Nendel Proof of Corollary 2.7 First assume that (ii) is satisfied, i.e., there exist a set I=∅and a family (αi)i∈Iof functions ca1 +→[0,∞] with infμ∈ca1 +αi(μ) =0 for all i∈I and R(f) =min i∈Isup μ∈ca1 +(fdμ−αi(μ))for all f∈Cb. Then one readily verifies that Ris monotone with R(f +m) =R(f ) +mfor all f∈Cband m∈R. Since infμ∈ca1 +αi(μ) =0 for all i∈I, it follows that R(0)=min i∈Isup μ∈ca1 +(−αi(μ))=min i∈I(−inf μ∈ca1 + αi(μ))=0. We have therefore shown that Ris a risk measure. To prove that Ris star-shaped, let f∈Cband λ∈[0,1]. Then R(λf) =min i∈Isup μ∈ca1 +( λf dμ−αi(μ)) ≤min i∈I(λsup μ∈ca1 + fdμ−αi(μ))=λR(f ). To prove the converse implication (i) ⇒(ii), assume that Ris a star-shaped risk measure. Following the proof of Castagnoli et al. [6, Theorem 2], we define Aϕ:= {f∈Cb:∃λ∈[0,1]with f≤λ(ϕ−R(ϕ))} for all ϕ∈Cb. Then for all ϕ∈Cb,thesetAϕis convex with g∈Aϕfor all g∈Cbwith g≤f for some f∈Aϕ, and 0 ∈Aϕ,m/∈Aϕfor all m∈(0,∞). Indeed, for the latter, assume towards a contradiction that there exist m∈(0,∞)and λ∈(0,1] with m≤λ(ϕ −R(ϕ)). Then ϕ≥m λ+R(ϕ), which contradicts the fact that Ris a risk measure. Hence by Föllmer and Schied [15, Proposition 4.7], the functional Rϕ:Cb→Rgiven by Rϕ(f ) := inf{m∈R:f−m∈Aϕ}for all f∈Cb defines a convex risk measure on Cb.Letf∈Cband m∈Rwith m>infϕ∈CbRϕ(f ). Then there exists some ϕ∈Cbwith m>R ϕ(f ), and so f−m≤λ(ϕ−R(ϕ))for some λ∈[0,1]. Since Ris a star-shaped risk measure, it follows that R(f) −m=R(f −m) ≤Rλ(ϕ−R(ϕ))≤λR(ϕ−R(ϕ))=0. Hence R(f) ≤infϕ∈CbRϕ(f ) for all f∈Cb. Moreover, for all ϕ∈Cband m∈R with m<R(ϕ), it follows that ϕ−m>ϕ−R(ϕ) so that Rϕ(ϕ) =R(ϕ) by the
Lower semicontinuity of monotone functionals on Cb279 definition of Aϕ. Indeed, if ϕ−R(ϕ) ≤0, it follows that R(ϕ) =supx∈ϕ(x).In this case, the inequality m<R(ϕ)implies that there exists some x∈with ϕ(x) −m>0≥λ(ϕ(x) −R(ϕ))for all λ∈[0,1]. We have therefore shown that R(f) =min ϕ∈Cb Rϕ(f ) for all f∈Cb. In view of Corollary 2.5 and [15, Theorem 4.16], it remains to prove that Rϕ:Cb→Ris continuous from below for all ϕ∈Cb. To that end, let ϕ∈Cb, (fn)n∈N⊆Cbwith fnf∈Cbas n→∞, and (mn)n∈N⊆Rwith mn>supk∈NRϕ(fk)for all n∈Nand limn→∞ mn=supk∈NRϕ(fk). Then there exists a sequence (λn)n∈N⊆[0,1]such that fn−mn≤λn(ϕ−R(ϕ))for all n∈N. Since [0,1]is compact, by passing to a subsequence, we may without loss of generality assume that λn→λ∈[0,1]as n→∞. Then f−sup k∈NRϕ(fk)=lim n→∞(fn−mn)≤lim n→∞λn(ϕ−R(ϕ))=λ(ϕ −R(ϕ)), which proves that Rϕ(f ) ≤supk∈NRϕ(fk). Since Rϕ:Cb→Ris monotone, it follows that Rϕ(f ) =limn→∞ Rϕ(fn). Proof of Theorem 2.8 First observe that by convexity of Uifor all i∈Iand (2.4), sup i∈I(Ui(f1)−Ui(f2))≤sup i∈I(Ui(f1)+Ui(−f2)−2Ui(0)) ≤2sup i∈I(Ui(r) −Ui(0))<∞(3.8) for all r≥0 and f1,f 2∈Cbwith max{f1∞,f2∞}≤r. 1) The implication (iii) ⇒(i) follows from Remark 3.1 a). To prove that (i) implies (ii), we first show that for every f∈Cband every ε>0, there exists some V∈V with sup i∈I|Ui(f +e) −Ui(f )|<ε for all e∈V. To that end, let f∈Cband consider the monotone maps Uf,Uf:Cb→Rgiven by Uf(g) := sup i∈I(Ui(f +g) −Ui(f )), Uf(g) := sup i∈I(Ui(f ) −Ui(f −g))for all g∈Cb. Observe that by (3.8), Ufand Ufare well defined and Uf(−g) =sup i∈I(Ui(f ) −Ui(f +g))for all g∈Cb.
280 M. Nendel Moreover, for any V∈V,e∈Vif and only if −e∈V. Hence the auxiliary statement follows from Corollary 2.3 once we have shown that both Ufand Ufare continuous from above. To that end, let g∈Cband (gn)n∈N⊆Cbwith gngas n→∞. Moreover, let ε>0, n∈Nand λ∈(0,1). Then using the fact that Uiis convex for all i∈I, Uf(gn)−Uf(g) ≤sup i∈I(Ui(f +gn)−Ui(f +g)) ≤λsup i∈I(Uign−g λ−Ui(0))+λsup i∈I(Ui(0)−Ui(f +g)) +(1−λ) sup i∈I(Uif+g 1−λ−Ui(f +g)) and Uf(gn)−Uf(g) ≤sup i∈I(Ui(f −g) −Ui(f −gn)) ≤sup i∈I(Ui(f −2g+gn)−Ui(f −g)) ≤λsup i∈I(Uign−g λ−Ui(0))+λsup i∈I(Ui(0)−Ui(f −g)) +(1−λ) sup i∈I(Uif−g 1−λ−Ui(f −g)). Since the maps R→R,γ→ sup i∈IUi(γ(f ±g))−Ui(f ±g) are convex and therefore continuous, we obtain λsup i∈I(Ui(0)−Ui(f ±g))+(1−λ) sup i∈I(Uif±g 1−λ−Ui(f ±g))<ε 2 for λ∈(0,1)sufficiently small. Moreover, by assumption, λsup i∈I(Uign−g λ−Ui(0))<ε 2 for n∈Nsufficiently large since gn−g λ0asn→∞. We have therefore shown that 0≤Uf(gn)−Uf(g) < ε and 0 ≤Uf(gn)−Uf(g) < ε for n∈Nsufficiently large, and so both Ufand Ufare continuous from above. We have thus proved the auxiliary statement and are now ready to prove the implication (i) ⇒(ii). For i∈Iand f∈Cb,letUi,f :Cb→Rbe given by Ui,f (g) := Ui(f +g) −Ui(f ) for all g∈Cb.
Lower semicontinuity of monotone functionals on Cb281 Then Ui,f is convex and monotone with Ui,f (0)=0 for all i∈Iand f∈Cb. Moreover, for all λ∈(0,1),i∈Iand f, g ∈Cb, Ui,f (g) ≤λ(Uig λ−Ui(f ))+(1−λ)(Uif 1−λ−Ui(f )) ≤λ(Uig λ−Ui(0))+λ(Ui(−f)−Ui(0)) +(1−λ)(Uif 1−λ−Ui(f )), where the second inequality uses the fact that Ui(0)−Ui(f ) ≤Ui(−f)−Ui(0)for all i∈I.Letr≥0, (gn)n∈N⊆Cbwith gn0asn→∞and ε>0. Then by (3.8), the map R→R,γ→ sup f∞≤r sup i∈I(Ui(γf ) −Ui(f )) is convex and well defined. Therefore it is continuous and it follows that λ(Ui(−f)−Ui(0))+(1−λ)(Uif 1−λ−Ui(f ))<ε 2 for λ∈(0,1)sufficiently small. Hence we get sup f∞≤r sup i∈I Ui,f (gn)≤λsup i∈I(Uign λ−Ui(0))+ε 2<ε for n∈Nsufficiently large, and we have shown that lim n→∞ sup f∞≤r sup i∈I Ui,f (gn)=0. Using the auxiliary statement for the convex monotone functions Ui,f with i∈Iand f∈Cbwith f∞≤r, there exists some V∈Vsuch that sup f∞≤r sup i∈I|Ui(f +e) −Ui(f )|= sup f∞≤r sup i∈I|Ui,f (e)|≤εfor all e∈V. 2) It remains to prove the implication (ii) ⇒(iii). Let r≥0 and ε>0. Then there exists some V∈Vsuch that sup f∞≤r sup i∈I|Ui(f +e) −Ui(f )|≤εfor all e∈V. By the definition (2.1) of the local base V, there exist some compact C⊆and some δ>0 such that e∈Cb:sup x∈C|e(x)|<δ ∩{e∈Cb:e∞≤2r}⊆V.
282 M. Nendel Now let f1,f 2∈Cbwith max{f1∞,f2∞}≤r. Then by the triangle inequality, f1−f2∞≤2r.Ifsup x∈C|f1(x) −f2(x)|<δ, then sup i∈I|Ui(f1)−Ui(f2)|≤ε. On the other hand, if supx∈C|f1(x) −f2(x)|≥δ, then by (3.8), it follows that sup i∈I|Ui(f1)−Ui(f2)|≤Msup x∈C|f1(x) −f2(x)| with M:= 2 δsupi∈I(Ui(r) −Ui(0)). The proof is complete. Proof of Corollary 2.9 Since (L, τ) is a locally convex vector lattice, (iv) implies (i). The implication (i) ⇒(iii) is trivial, and by Corollary 2.3, (ii) and (iii) are equivalent. By Theorem 2.8 and Lemma B.1, (ii) ⇒(iv) and (ii) ⇒(v), since (ii) together with Dini’s lemma implies that for every convex and weak∗compact set Kof nonnegative τ-continuous linear functionals, the map UK:Cb→R,f→ sup μ∈K μ(U(f)−U(0)) is convex with limn→∞ UK(fn)=0 for every sequence (fn)n∈N⊆Cbwith fn0 as n→∞. Since for every nonnegative τ-continuous linear functional λ:L→R, there exists a τ-continuous lattice seminorm p:L→[0,∞)with |λu|≤p(u) for all u∈L, (v) implies (ii) by Dini’s lemma. Appendix A: Capacities and Choquet integrals The setup and notation in this section follow that of the main part. The following lemma is a sort of folklore result; cf. König [20, Property 11.8 and Theorem 11.11]. For the reader’s convenience, we nevertheless provide a short proof. Lemma A.1 Let c:O→[0,∞)be a capacity.Then the following are equivalent: (i) For all B1,B 2∈O, c(B1∪B2)+c(B1∩B2)≤c(B1)+c(B2). (ii) For all f1,f 2∈Lb, (f1+f2)dc≤ f1dc+ f2dc. Proof We first prove the implication (ii) ⇒(i). To that end, let B1,B 2∈O. Then c(B1∪B2)+c(B1∩B2)= (1B1+1B2)dc ≤ 1B1dc+ 1B2dc=c(B1)+c(B2).
Lower semicontinuity of monotone functionals on Cb283 We proceed with the proof of the implication (i) ⇒(ii). In the first step, we prove by induction over n∈Nthat n ∑ i=1 1Bidc≤ n ∑ i=1 c(Bi)for all B1,...,B n∈O.(A.1) For n=1, the statement is trivial. Assume that (A.1)isprovedforsomen∈Nand let B1,...B n+1∈O. Then n+1 ∑ k=1 1Bi=1n+1 i=1Bi+ n ∑ k=1 1(k i=1Bi)∩Bk+1. Using (A.1) and (i), we obtain that n+1 ∑ i=1 1Bidc=c(n+1 i=1 Bi)+ n ∑ k=1 1(k i=1Bi)∩Bk+1dc ≤c(n+1 i=1 Bi)+ n ∑ k=1 c(k i=1 Bi∩Bk+1)≤ n+1 ∑ i=1 c(Bi). Now let f1,f 2∈Lb. Since the Choquet integral is constant additive, we may without loss of generality assume that f1≥0 and f2≥0. Let r:= max{f1∞,f2∞}.For i=1,2, n∈Nand k=1,...,2n, define Bk i,n := {fk>k2−nr}. Then for i=1,2, fi− 2n ∑ k=1 2−nr1Bk i,n ∞≤2−nr−→ 0asn→∞. But positive homogeneity of the Choquet integral and (A.1)give 2n ∑ k=1 2−nr(1Bk 1,n +1Bk 2,n )dc=2−nr 2n ∑ k=1 (1Bk 1,n +1Bk 2,n )dc ≤2−nr 2n ∑ k=1(c(Bk 1,n)+c(Bk 2,n)) = 2n ∑ k=1 2−nr1Bk 1,n dc+ 2n ∑ k=1 2−nr1Bk 2,n dc, and so it follows that (f1+f2)dc≤ f1dc+ f2dc.
284 M. Nendel Appendix B: Locally convex vector lattices Thoughout this section,let(L, τ) be a locally convex vector lattice, i.e., a vector lattice Ltogether with a locally convex topology τon Lwhich is generated by a family of lattice seminorms. Let L+:= {u∈L:u≥0}and Lbe the topological dual space of L, i.e., the space of all continuous linear functionals L→R. Moreover, let L+:= {λ∈L:λu ≥0,∀u∈L+} be the set of all positive continuous linear functionals on L.Foru∈L,weusethe standard notation u+:= u∨0 and u−:= −(u ∧0). Then u=u+−u−and |u|:=u++u−for all u∈L. The following lemma can be deduced from Aliprantis and Border [2, Theorem 8.24 and Corollary 8.25] together with the fact that every linear functional that is bounded by a lattice seminorm is order bounded. For the sake of a self-contained exposition, we provide a short proof. Lemma B.1 Let p:L→[0,∞)be a continuous lattice seminorm. a) For every λ∈Lwith |λu|≤p(u) for all u∈L,there exist λ+,λ −∈L+with λu =λ+u−λ−uand max{λ+|u|,λ −|u|} ≤ p(u) for all u∈L. (B.1) b) There exists a convex and weak∗compact set K⊆L+with max μ∈K|μu|≤max μ∈Kμ|u|=p(u) ≤2max μ∈K|μu|for all u∈L. Proof a) Since pis a lattice seminorm, it follows that p(u) =p(v) for all u, v ∈L with |u|=|v|. In particular, p(u) =p(|u|)for all u∈L. (B.2) Let λ∈Lwith |λu|≤p(u) for all u∈L, and define λ+u:= sup{λv :v∈L+,v ≤u}for all u∈L+. Since pis a lattice seminorm, 0 ≤λ+u≤p(u) and λ+(αu) =αλ+ufor all u∈L+and α≥0. In order to prove that λ+is additive, let u1,u 2∈L+. Then for v1,v 2∈L+with v1≤u1and v2≤u2, λv1+λv2≤λ(v1+v2)≤λ+(u1+u2). Hence λ+u1+λ+u2≤λ+(u1+u2). On the other hand, for v∈L+with v≤u1+u2, let v1:= (v −u2)+≥0,v 2:= v−v1=v+(u2−v) ∧0=v∧u2≤u2. Moreover, v1≤u1since u1≥0, and v2=v∧u2≥0 since v≥0 and u2≥0. Hence λv =λv1+λv2≤λ+u1+λ+u2.