Risk measuring under model uncertainty
aa r X i v : . [ q -f i n . R M ] D ec Risk Measuring under Model Uncertainty
Jocelyne BION-NADAL ∗ UMR 7641 CNRS CMAP Ecole Polytechnique, 91128 Palaiseau Cedex, France andMagali KERVAREC † Laboratoire analyse et probabilit´es, Universit´e d’ Evry, Bd F. Mitterrand, 91000 Evry France
Abstract
The framework of this paper is that of risk measuring under uncertainty, which is whenno reference probability measure is given. To every regular convex risk measure on C b (Ω), weassociate a unique equivalence class of probability measures on Borel sets, characterizing theriskless non positive elements of C b (Ω). We prove that the convex risk measure has a dualrepresentation with a countable set of probability measures absolutely continuous with respectto a certain probability measure in this class. To get these results we study the topologicalproperties of the dual of the Banach space L ( c ) associated to a capacity c .As application we obtain that every G -expectation IE has a representation with a countableset of probability measures absolutely continuous with respect to a probability measure P suchthat P ( | f | ) = 0 iff IE ( | f | ) = 0. We also apply our results to the case of uncertain volatility. classification: Key Words:
Risk Measure, duality theory, uncertainty, capacity
The purpose of this paper is to introduce a very general framework enabling the study of riskmeasures and dynamic risk measures in a context of model uncertainty, which is when no referenceprobability measure is given.In order to quantify the risk in finance, Artzner et al [1] have introduced the notion of coherent(i.e. sublinear) risk measure in the context of finite probability spaces. This notion has been ex-tended to general probability spaces [12] and then to the convex case ([21] and [22]). The notionof conditional risk measure has been considered in [17] and [6]. Dynamic risk measures have thenbeen studied in many papers, among them [13], [11] [24] [7] [8] [30]. For the particular case ofdynamic risk measures on a Brownian filtration one can cite [28] [3], [14]. Notice that in all thesepapers on dynamic risk measures, a reference probability space is fixed. This framework is richenough to study models with stochastic volatility or models with jumps, but not to deal with modeluncertainty.What means uncertainty? Usually in mathematical finance, in order to compute the risk or theprice associated to financial assets, one assumes that a reference family of liquid assets is given, andthat the dynamics of these reference assets is known. However in a context of model uncertainty ∗ tel: 33 1 69 33 46 25, fax: 33 1 69 33 30 11, e-mail : [email protected] † e-mail : [email protected] dX σt = b t X σt dt + σ t X σt dW t where σ t is allowed to vary inside an interval [ σ, σ ]. When σ describes the set of predictable pro-cesses varying inside this interval, the laws of the processes X tσ are not all absolutely continuouswith respect to some probability measure. Avellaneda et al [2], Denis and Martini [16] and Deniset al [15] have considered the problem of pricing for this family of models. Only few papers studyconvex risk measures in a context of uncertainty. F¨ollmer and Schied [21] have studied static riskmeasures defined on the vector space of all bounded measurable maps. This has been extendedby Bion-Nadal to the conditional case in [6]. Kervarec [25] has studied static risk measures whenmodel uncertainty is specified by a non dominated weakly compact set of probability measures.In this paper, motivated by the general context of model uncertainty, we study regular convex riskmeasures defined on C b (Ω), the set of continuous bounded functions on a Polish space Ω. Regularityis here equivalent to continuity with respect to a certain capacity c . Considering the completion L ( c ) of C b (Ω) with respect to the capacity c , this means that we study convex risk measures onthe Banach space L ( c ). Our main result is that for every regular convex risk measure on C b (Ω),there is a unique equivalence class of probability measures characterizing the riskless non positiveelements of C b (Ω), and that the convex risk measure has a dual representation with a countableset of probability measures all absolutely continuous with respect to a certain probability measurebelonging to this equivalence class. The tools of the proof are the capacities, topological propertiesof the dual of the Banach space L ( c ) associated to a capacity c , and convex duality for locallyconvex spaces.The paper is organized as follows. First, Section 2, we study the topological properties of thedual of L ( c ). We prove that the non negative part of the dual ball of L ( c ) is metric compact forthe weak* topology σ ( L ( c ) ∗ , L ( c )).Section 3 deals with convex risk measures on L ( c ). We prove that they satisfy the followingrepresentation formula: ρ ( X ) = sup Q ∈P ′ ( E Q [ − X ] − α ( Q )) (1.1)where P ′ is a set of probability measures belonging to the dual of L ( c ). There are two importantresults in this Section. The first one is the characterization of convex risk measures on L ( c )admitting a representation of the form (1.1) having a compact set P ′ of probability measures (forthe weak* topology σ ( L ( c ) ∗ , L ( c ))). In this case, the supremum in (1.1) is a maximum. Moreover,making use of the topological results of Section 2, we prove that every convex risk measure on L ( c )has a dual representation of the form (1.1) with a countable set of probability measures.In section 4 we assume that the capacity is defined on C b (Ω) by c p, P ( f ) = sup P ∈P E P ( | f | p ) p forsome weakly relatively compact set P of probability measures. We prove that the capacity c p, P isequal to the capacity c p, Q defined using a certain countable subset Q of P . We introduce a newequivalence relation on the set of non negative measures belonging to the dual of L ( c p, P ). When P is a singleton, it coincides with the usual equivalence relation on non negative measures. The mainresult of Section 4 is the existence of an equivalence class of probability measures characterizingthe null elements of L ( c p, P ) + , that is P belongs to this equivalence class if and only if for all f in L ( c p, P ), ( E P ( | f | ) = 0) ⇐⇒ ( c p, P ( | f | ) = 0).Section 5 deals with uniformly regular convex risk measures on C b (Ω). We prove that every such riskmeasure on C b (Ω) extends into a convex risk measure on L ( c ) for a certain capacity c associatedto a weakly compact set P of probability measures: c ( f ) = sup P ∈P E P ( f ). Therefore we can2ake use of the results obtained in Sections 4 and 3 in order to get the main result of the paper inTheorem 5.1: to every uniformly regular convex risk measure ρ on C b (Ω), one can associate a uniqueequivalence class of probability measures defined on the Borel sets, called c ρ -class, characterizing thenon positive elements of C b (Ω) with risk 0. The convex risk measure has then a dual representationwith a countable set of probability measures all absolutely continuous with respect to a certainprobability measure belonging to this c ρ -class.Section 6 deals with two examples. The first one is G -expectations introduced by Peng [26]. Thecapacity associated to a G -expectation IE is c ( f ) = IE ( | f | ). As application of our results we obtainthat there is a unique equivalence class of probability measures characterizing the non negativeelements f of C b (Ω) such that IE ( f ) = 0. The G-expectation IE has then a representation interms of a countable set of probability measures all absolutely continuous with respect to a certainprobability measure belonging to this class, IE ( X ) = sup n ∈ IN E Q n ( X ) (1.2)The second example, for which all our results apply, is the case where model uncertainty is char-acterized by a relatively weakly compact set of probability measures P . L ( c ) L ( c ) Let Ω be a metrizable and separable space. One classical example, furthermore a Polish space,is Ω = C ([0 , ∞ [ , IR d ) endowed with the topology of uniform convergence on compact subspaces. B (Ω) denotes the Borel σ -algebra on Ω. Denote M (Ω) the set of all bounded signed measures on(Ω , B (Ω)), and M + (Ω) the subset of non-negative finite measures.In the following L denotes a linear vector subspace of C b (Ω) containing the constants, generatingthe topology of Ω and which is a vector lattice. Recall the following definition of a capacity. Definition 2.1. a capacity on L is a semi norm c defined on L satisfying the following properties:1. monotonicity: ∀ , f, g ∈ L such that | f | ≤ | g | , c ( f ) ≤ c ( g )
2. regularity along sequences: for every sequence f n ∈ L decreasing to , inf c ( f n ) = 0The semi-norm c is extended as in [20] Section 2 to all real functions defined on Ω: ∀ f l.s.c. f ≥ , c ( f ) = sup { c ( φ ) | ≤ φ ≤ f, φ ∈ L} (2.1) ∀ g, c ( g ) = inf { c ( f ) | f ≥ | g | , f l.s.c. } (2.2)where l.s.c. means lower semi-continuous. L ( c ) denotes the closure of L in the set { g | c ( g ) < ∞} .From Proposition 10 of [20], L ( c ) contains C b (Ω). Let L ( c ) be the quotient of L ( c ) by the c nullelements. It is a Banach space. The following result shows that c (1 A ) can be expressed as the limitof a monotone sequence c ( f n ) for continuous functions f n with limit 1 A , as soon as A is either anopen subset or a closed subset of Ω. Proposition 2.1. • Let V be an open subset of Ω . There is an increasing sequence of nonnegative continuous functions h n on Ω such that V = lim n →∞ h n and c (1 V ) = lim n →∞ c ( h n ) . • Let F be a closed subset of Ω . There is a decreasing sequence of continuous functions g n ≤ on Ω such that F = lim n →∞ g n and c (1 F ) = lim n →∞ c ( g n ) . roof. • V is a non negative bounded l.s.c. function. Thus it is the limit of an increasingsequence of non negative continuous functions f n . On the other hand from definition of c (1 V )(equation (2.1)), there is a sequence of continuous functions g n ≤ V such that c (1 V ) =lim c ( g n ). Let h = g and for every n , h n +1 = sup( h n , f n , g n ). h n is an increasing sequenceof continuous functions with limit 1 V and such that c (1 V ) = lim c ( h n ). • Let F be a closed subset of Ω. By definition of the capacity, c (1 F ) = inf { ψ l.s.c., F ≤ ψ } c ( ψ ). Theinfimum of two l.s.c. functions is also l.s.c. , thus there is a decreasing sequence ψ n greateror equal to 1 F such that c (1 F ) = lim c ( ψ n ). Thus there is a strictly increasing sequence k ( n )such that for all n , c ( ψ k ( n ) ) ≤ c (1 F )+ n . Let ǫ n > − ǫ n )( c (1 F )+ n ) ≤ c (1 F )+ n .Let V n = { x | ψ k ( n ) ( x ) > − ǫ n } ∩{ x ∈ Ω; dist ( x, F ) < n } . As ψ k ( n ) is l.s.c., V n is an openset, furthermore F = ∩ n ∈ IN ∗ V n . For every n , there is a continuous function f n such that F ≺ f n ≺ V n . One can thus construct a decreasing sequence of continuous functions g n suchthat 1 F ≤ g n ≤ V n . Thus the sequence g n is decreasing to 1 F . As c (1 V n ) ≤ − ǫ n c ( ψ k ( n ) ) ≤ c (1 F ) + n , it follows that c (1 F ) ≤ c ( g n ) ≤ c (1 F ) + n .Further definitions and results on capacities are recalled in the Appendix (Section 7). We referalso to [20]. Partial order on L ( c ) Definition 2.2.
Let X ∈ L ( c ) . We say that X ≥ if there is a sequence ( f n ) n ∈ IN , f n ∈ L , f n ≥ such that for every g ∈ L ( c ) of class X , lim n →∞ c ( g − f n ) = 0 . Lemma 2.1. • let X, Y ∈ L ( c ) . If X ≥ and Y ≥ , then X + Y ≥ . • If there is in the class of X a non negative function f then X ≥ • Let X ∈ L ( c ) ; | X | ∈ L ( c ) . Furthermore X ≥ if and only if X = | X | in L ( c ) .Proof. • The first part of the lemma is trivial. • The second point follows from the inequality c ( | f | − | f n | ) ≤ c ( f − f n ) (2.3)Thus as f = | f | , c ( f − | f n | ) ≤ c ( f − f n ). • One can deduce from (2.3) that for all X ∈ L ( c ), | X | ∈ L ( c ). From point 2, | X | − X ≥ c ( | f | − f ) ≤ c ( | f | − f n ) + c ( f − f n ), it follows that X ≥ X = | X | in L ( c ). Proposition 2.2.
The relation X ≤ Y defined by Y − X ≥ defines a partial order on L ( c ) .Proof.
1. Reflexivity is trivial: take f n = 0 for all n
2. Antisymmetry. Let X ≥ Y and Y ≥ X . Let h in the class of X − Y . By definition thereare two sequences f n and g n of non negative functions in L such that lim n →∞ c ( f n − h ) = 0 andlim n →∞ c ( g n + h ) = 0. It follows that lim n →∞ c ( f n + g n ) = 0. As 0 ≤ | f n − g n | ≤ f n + g n , it followsthat lim n →∞ c ( | f n − g n | ) = 0. However lim n →∞ c ( f n − g n − h ) = 0. Thus X − Y , the class of h isequal to 0.3. Transitivity follows from the first part of Lemma 2.1.4 .2 Topological properties of the non negative part of the unit ball of L ( c ) ∗ For the definition of a Prokhorov capacity, see the Appendix (Section 7).
Proposition 2.3.
Let c be a Prokhorov capacity on a metrizable and separable space Ω . Everycontinuous linear form L on L ( c ) admits a representation: L ( f ) = Z f dµ ∀ f ∈ L ( c ) (2.4) where µ is a regular bounded signed measure defined on a σ -algebra containing the Borel σ -algebraof Ω .If L is a non negative linear form the regular measure µ is non negative finite. Following [5] a bounded signed measure µ is called regular if for all Borel set A , for all ǫ > F and an open set G such that F ⊂ A ⊂ G and | µ | ( G − F ) < ǫ .Notice that in [20], the existence of a bounded measure µ satisfying equation (2.4) is proved.However the statement of Proposition 11 of [20] does not give informations on the σ algebra onwhich the measure µ is defined. Therefore we have to go inside the proof. Proof. • A metrizable space is completely regular and c is a Prokhorov capacity so Proposition11 of [20] gives the existence of a measure µ satisfying equation (2.4). We want now provethat µ is defined on the Borel σ algebra. As in the proof of Proposition 11 of [20] let Z be acompactification of Ω, and c ′ the capacity defined on Z by c ′ ( g ) = c ( g | Ω ). As c is a Prokhorovcapacity, from Proposition 11 of [20], c ′ (1 Z − Ω ) = 0 and L ( c ) = L ( c ′ ). • As Z is a compact space, it follows from Theorem 3 of [19] that every non negative linear formon L ( c ′ ) can be represented by a non negative measure obtained from the Riesz representationtheorem applied to C ( Z ). Therefore this measure is defined on a σ -algebra containing the Borelsets of Z . From Theorem 6 of [19] every continuous linear form on L ( c ) is the differenceof two non negative linear forms, thus the bounded measure µ satisfying equation (2.4) isdefined on a Borel σ -algebra B containing the Borel σ -algebra of Z . • We want now prove that µ is defined on the Borel σ -algebra of Ω. µ is defined on the σ -algebra F obtained by completion of B with the µ -null sets. Notice that from Theorem 3 of[19], every c ′ -negligible set (i.e. c ′ (1 A ) = 0) is also µ -negligible. This is in particular the casefor Z − Ω which is therefore µ -measurable. Every open set V of Ω can be written V = U ∩ Ωfor some open set U of Z . Therefore V belongs to F . It follows that the measure µ definedon F is thus defined on the Borel σ -algebra of Ω. As Ω is a metric space and µ is defined onthe Borel σ -algebra of Ω, µ is regular from Theorem 1.1 of [5].Recall that the weak topology on M + (Ω) the set of non negative finite measures on (Ω , B (Ω))is the coarsest topology for which the mappings µ ∈ M + (Ω) → Z f dµ are continuous for every given f in C b (Ω). Proposition 2.4.
Let c be a Prokhorov capacity on a metrizable separable space. The set of nonnegative linear forms on the Banach space L ( c ) is a subset of M + (Ω) . The weak* topology (i.e.the σ ( L ( c ) ∗ , L ( c )) topology) on the non negative part K + of the unit ball of L ( c ) ∗ coincides withthe restriction to K + of the weak topology on M + (Ω) . roof. From Proposition 2.3, every non negative linear form on L ( c ) belongs to M + (Ω). Let µ ∈ K + . As C b (Ω) is dense in the Banach space L ( c ), the open sets V f ,f ,...f n ,ǫ ( µ ) = { ν ∈ K + |∀ i ∈ { , ...n } , | µ ( f i ) − ν ( f i ) | < ǫ } with f i ∈ C b (Ω) form a basis of neighborhoods of µ in K + for the weak* topology. Thus the weak*topology on K + coincides with the weak topology. Proposition 2.5.
Let c be a Prokhorov capacity on a metrizable separable space Ω . The set K + is compact metrizable for the weak* topology (i.e. the σ ( L ( c ) ∗ , L ( c )) topology), as well as for theweak topology.Proof. Prove first that K + is metrizable for the weak* topology. From Proposition 2.4, the weak*topology on K + coincides with the restriction to K + of the weak topology on M + (Ω). As Ω ismetrizable and separable, M + (Ω) is also metrizable and separable for the weak topology from [9]Section 5. Thus K + is metrizable for the weak* topology.From Banach Alaoglu Theorem, (theorem V 4 2 of [18]) the closed unit ball of the dual space of aBanach space is always compact for the weak* topology. As K + is a closed subset of this unit ballfor the weak* topology, it is also compact. This proves the result for the weak* topology. FromProposition 2.4, K + is also metrizable compact for the weak topology. Corollary 2.1.
Assume that Ω is a Polish space. For every capacity c on Ω , the set K + is compactmetrizable for the weak* topology.Proof. From [20], see also the Appendix (Section 7), every capacity on a Polish space is a Prokhorovcapacity, and thus the result follows from Proposition 2.5.In the particular case of a compact metrizable space, we obtain the following stronger result.
Proposition 2.6.
Let Ω be a metrizable compact space. Let c be a capacity on Ω . Then the Banachspace L ( c ) is separable and the unit ball of L ( c ) ∗ is metrizable compact for the weak* topology.Proof. As Ω is a metrizable compact space, C (Ω) is separable from Thm 1 Section 3 of [10]. Thusfor every capacity c on Ω, L ( c ) is also separable. Then from Theorem V 5 1 of [18], the unit ballof L ( c ) ∗ (and not only its non negative part) is metrizable compact for the weak* topology. L ( c ) In this section, c denotes a Prokhorov capacity on a metrizable separable space Ω. Recall that apartial order has been defined on L ( c ) in Section 2.1. We can define convex risk measures in theusual way as follows. Definition 3.1.
Let ρ : L ( c ) → IR . • ρ is monotonic if ρ ( X ) ≥ ρ ( Y ) for every X, Y ∈ L ( c ) , such that X ≤ Y . • ρ is convex if for every X, Y ∈ L ( c ) , for every ≤ λ ≤ , ρ ( λX + (1 − λ ) Y ≤ λρ ( X ) + (1 − λ ) ρ ( Y ) • ρ is translation invariant if ρ ( X + a ) = ρ ( X ) − a for every X ∈ L ( c ) and a ∈ IR . ρ is a convex risk measure if it satisfies all these conditions. .1 Representation for convex risk measures Duality results for risk measures are well known in other settings. A duality result was first provedin the case of risk measures on L ∞ spaces assuming furthermore continuity from below. Dualityresults are based on the Fenchel Legendre duality, generalized to the context of locally convextopological spaces by Rockafellar [29]. This is the generalized version that we need here. Noadditional hypothesis is needed in order to prove the dual representation result. The importantand new discussion will be developed in Subsection 3.2 using the topological results proved inSection 2.2. Theorem 3.1.
Let ρ be a convex risk measure on L ( c ) . Then, ρ is continuous and admits arepresentation of the form: ∀ X ∈ L ( c ) , ρ ( X ) = sup Q ∈P ′ ( E Q [ − X ] − α ( Q )) (3.1) where α ( Q ) = sup X ∈ L ( c ) ( E Q [ − X ] − ρ ( X )) (3.2) P ′ is the set of probability measures on (Ω , B (Ω)) belonging to L ( c ) ∗ .Proof. The continuity of ρ follows from Theorem 1 of [4].We call α the function on L ( c ) ⋆ defined by: ∀ µ ∈ L ( c ) ⋆ , α ( µ ) = sup X ∈ L ( c ) ( µ ( X ) − ρ ( X ))As the dual of L ( c ) ∗ (with the weak * topology) is L ( c ), the locally convex topological spaces L ( c ) and L ( c ) ∗ are paired in the sense of [29]. ρ is continuous, we can thus apply Theorem 5 inRockafellar [29]. We get the following equality: ∀ X ∈ L ( c ) , ρ ( X ) = sup µ ∈ L ( c ) ⋆ ( µ ( X ) − α ( µ ))In the supremum above, we can obviously restrict to the elements µ of L ( c ) ⋆ such that α ( µ ) < + ∞ .Let µ ∈ L ( c ) ⋆ such that α ( µ ) < + ∞ , we first prove that − µ is a positive linear form. Let X ∈ L ( c ) such that X ≥
0. For all λ >
0, using the monotonicity of ρ , ρ ( λX ) ≤ ρ (0), whichimplies that λµ ( X ) − α ( µ ) ≤ ρ (0) ρ (0) and α ( µ ) are finite and the above inequality is satisfied for all λ >
0, thus µ ( X ) ≤ − µ is represented by a finite non negative measure defined on (Ω , B (Ω)).Thanks to the translation invariance of ρ , for all λ ∈ IR , ρ ( λ ) = ρ (0) − λ , which means that: ρ (0) = λ + sup µ ∈ L ( c ) ⋆ ( λµ (1) − α ( µ )) ≥ λ (1 + µ (1)) − α ( µ )We conclude as above that 1 + µ (1) = 0. Thus, − µ is a probability measure on (Ω , B (Ω)) and − µ ∈ L ( c ) ∗ . 7 .2 Risk measures represented by a weakly relatively compact set of probabilitymeasures In this section we want to characterize risk measures ρ on L ( c ) admitting a dual representationwith a relatively compact set of probability measures for the weak* topology. Definition 3.2.
A convex risk measure ρ on L ( c ) is normalized if ρ (0) = 0 . Proposition 3.1.
Let ρ : L ( c ) → IR be a normalized convex risk measure. The following condi-tions are equivalent:1. ρ is majorized by a sublinear risk measure2. ∀ X ∈ L ( c ) , sup λ> ρ ( λX ) λ < ∞
3. there exits
K > such that ∀ X ∈ L ( c ) , | ρ ( X ) | ≤ Kc ( X ) ρ is represented by a set Q of probability measures in L ( c ) ∗ relatively compact for the weak*topology, i.e. ∀ X ∈ L ( c ) , ρ ( X ) = sup Q ∈Q ( E Q [ − X ] − α ( Q )) (3.3)Before giving the proof of the Proposition, we prove the following Lemma Lemma 3.1.
Let Q be a set of probability measures on (Ω , B (Ω)) such that Q ⊂ L ( c ) ∗ . Assumethat Q is relatively compact for the weak* topology σ (( L ( c ) ∗ , L ( c )) . Then Q is contained in someclosed ball of L ( c ) ∗ and the weak* closure of Q is also compact for the weak topology.Proof. Denote Q the closure of Q for the weak* topology. Q is compact. Let X ∈ L ( c ). Themap Q → E Q ( X ) is continuous for the weak* topology, thus sup Q ∈Q | E Q ( X ) | < ∞ . From BanachSteinhauss Theorem (cf [31]), it follows that Q is contained in some closed ball of L ( c ) ∗ , and thusin the non negative part of this closed ball. From Proposition 2.4, Q is weakly compact.We can now give the proof of Proposition 3.1. Proof.
Consider the dual representation of ρ given by equation (3.1). Denote Q = { Q ∈ P ′ | α ( Q ) < ∞} . Then ∀ X ∈ L ( c ) , ρ ( X ) = sup Q ∈Q ( E Q ( − X ) − α ( Q )) (3.4) . implies . Let ρ be a sublinear risk measure majorizing ρ . Then for every λ ∈ IR + ∗ , ρ ( λX ) ≤ λρ ( X ). Thus sup λ> ρ ( λX ) λ ≤ ρ ( X ), and is proved. . implies . For every X ∈ L ( c ), denote β X = sup λ> ρ ( λX ) λ . From the dual representation(3.4), applied with λX for every λ >
0, it follows that ∀ Q ∈ Q , E Q ( − X ) ≤ β X , and thussup Q ∈Q E Q ( − X ) ≤ β X < ∞ for every X ∈ L ( c ). With X = −| Y | , we get that ∀ Y ∈ L ( c ) , sup Q ∈Q | E Q ( Y ) | < ∞ (3.5) L ( c ) is a Banach space and from Theorem 3.1, every E Q is a continuous linear form on L ( c ).Denote || E Q || its norm. From Banach Steinhauss Theorem, equation (3.5) implies the existence of K > Q ∈Q || E Q || ≤ K . Notice that from the normalization condition ( ρ (0) = 0) it8ollows from equation (3.2) that for every Q , α ( Q ) ≥
0. Thus from the representation (3.4), forevery X ∈ L ( c ), ρ ( X ) ≤ Kc ( X ) (3.6)From the convexity, the monotonicity of ρ and ρ (0) = 0, it follows that − ρ ( X ) ≤ ρ ( − X ) ≤ ρ ( −| X | ) ≤ Kc ( −| X | ) = Kc ( X ) (3.7)Thus from equations (3.6) and (3.7), for every X ∈ L ( c ), | ρ ( X ) | ≤ Kc ( X )This proves . . implies . From the representation of ρ , equation (3.4) applied with − λ | X | for every λ >
0, itfollows from hypothesis . that for every Q ∈ Q || E Q || ≤ K . This means that Q is contained in aclosed ball of the dual of L ( c ). Every such closed ball is compact for the weak* topology (BanachAlaoglu Theorem). Thus Q is relatively compact for the weak* topology. . implies . ρ is represented by a set of probability measures Q ⊂ L ( c ) ∗ relatively compact forthe weak * topology. From Lemma 3.1, Q is contained in some closed ball of L ( c ) ∗ . Define ρ by ρ ( X ) = sup Q ∈Q E Q ( − X ). As Q is bounded, ρ ( X ) is finite for every X in L ( c ). It is easy toverify that ρ is a sublinear risk measure and that ρ is majorized by ρ . Theorem 3.2.
Let ρ be a convex risk measure on L ( c ) . Assume that ρ is represented by ρ ( X ) = sup Q ∈Q ( E Q ( − X ) − α ( Q )) where Q is a set of probability measures in L ( c ) ∗ relatively compact for the weak* topology. Let Q be the closure of Q for the weak* topology. Then • Q is metrizable compact both for the weak* topology and the weak topology. • For every X ∈ L ( c ) , there is a probability measure Q X ∈ Q such that ρ ( X ) = E Q X ( − X ) − α ( Q X ) (3.8) Proof. • From Lemma 3.1, Q is contained in a closed ball of L ( c ) ∗ and is compact both forthe weak and the weak* topology. From Proposition 2.5 it is metrizable compact. • Let X ∈ L ( c ). Let Q n be a sequence of elements in Q such that for every n , ρ ( X ) − n < E Q n ( − X ) − α ( Q n ) ≤ ρ ( X ) (3.9)As Q is metrizable compact for the weak* topology, there is a subsequence Q φ ( n ) convergingto ˜ Q ∈ Q , satisfying the inequality E ˜ Q ( − X ) − n < E Q φ ( n ) ( − X ) < E ˜ Q ( − X ) + 1 n (3.10)From inequality (3.9) applied with Q φ ( n ) , inequality (3.10) and the inequality φ ( n ) ≥ n , itfollows that 9 ˜ Q ( − X ) − ρ ( X ) − n < α ( Q φ ( n ) ) < E ˜ Q ( − X ) − ρ ( X ) + 2 n (3.11)Let Y ∈ L ( c ). Let ǫ >
0. There is N ( Y ) such that for every n > N ( Y ), E ˜ Q ( − Y )
0, it follows that α ( ˜ Q ) = sup Y ∈ L ( c ) ( E ˜ Q ( − Y ) − ρ ( Y )) ≤ E ˜ Q ( − X ) − ρ ( X )And thus ρ ( X ) = E ˜ Q ( − X ) − α ( ˜ Q ) Proposition 3.2.
Let ρ be a normalized convex risk measure on L ( c ) majorized by a sublinearrisk measure. There is a countable set { R n , n ∈ IN } of probability measures belonging to L ( c ) ∗ ,which is relatively compact for the weak* topology of L ( c ) ∗ and also for the weak topology and suchthat ∀ X ∈ L ( c ) , ρ ( X ) = sup n ∈ IN ( E R n [ − X ] − α ( R n )) (3.13) where α ( R ) = sup X ∈ L ( c ) ( E R [ − X ] − ρ ( X )) (3.14) Proof.
From Proposition 3.1, there is a set Q of probability measures in L ( c ) ∗ , relatively compactfor the weak* topology such that equation (3.3) is satisfied. From Lemma 3.1, Q is containedin mK + , the non negative part of a certain closed ball of L ( c ) ∗ . From Proposition 2.6, mK + ,is metrizable compact for the weak* topology. There is thus a countable dense set ( Q n ) n ∈ IN in mK + . Denote d a distance on mK + defining the weak* topology. For every Q ∈ mK + , let B ( Q, r ) = { R ∈ mK + | d ( Q, R ) ≤ r } . The set B ( Q, r ) is compact for the weak* topology. Thepenalty α defined on L ( c ) ∗ by equation 3.14 is l.s.c. thus for every n ∈ IN and k ∈ IN ∗ there is R n,k in B ( Q n , k ) such that α ( R n,k ) = min { α ( Q ) , Q ∈ B ( Q n , k ) } .Let X ∈ L ( c ). From Theorem 3.2, there is Q X ∈ Q such that ρ ( X ) = E Q X ( − X ) − α ( Q X ). Forall ǫ >
0, there is η > ∀ Q ∈ B ( Q X , η ), | E Q X ( − X ) − E Q ( − X ) | < ǫ . Let k such that k − < η . Let n such that Q X ∈ B ( Q n , k ) then E R n,k ( − X ) − α ( R n,k ) > ρ ( X ) − ǫ . It follows that { R n,k , n ∈ IN, k ∈ IN ∗ } is a countable set weakly relatively compact (as it is contained in mK + )satisfying the required condition. Theorem 3.3.
Every convex risk measure on L ( c ) can be represented by a countable set of prob-ability measures { R n , n ∈ IN } belonging to L ( c ) ∗ . ∀ X ∈ L ( c ) , ρ ( X ) = sup n ∈ IN ( E R n ( − X ) − α ( R n )) (3.15) where α ( R ) is given by equation (3.14). roof. From Theorem 3.1, ρ has a dual representation given by equation (3.1). Denote then ρ m ( X ) = sup Q ∈ mK + ( E Q ( − X ) − α ( Q )). Even if ρ m is not necessarily normalized, all the argumentsof the proof of Proposition 3.2 apply as mK + is metrizable compact for the weak* topology and α is l.s.c.. Thus ρ m has a representation with a countable set of probability measures. As ρ =sup m ∈ IN ρ m , this gives the result. Let Ω be a metrizable and separable space. In this section we study a capacity defined from aweakly relatively compact set of probability measures P possibly non dominated. Definition 4.1.
Let P be a weakly relatively compact set of probability measures on (Ω , B (Ω)) . Let ≤ p < ∞ . The capacity c p, P is defined on C b (Ω) by c p, P ( f ) = sup P ∈P E P ( | f | p ) p (4.1) and extended to every function on Ω as explained in Section 2.1, equations (2.1) and (2.2). Notice that as P is a weakly relatively compact set of probability measures, c p, P is a capacity(see Proposition I.3 of [25] or the Appendix, Section 7). The Banach space associated to thecapacity c p, P is denoted L ( c p, P ). When there is no ambiguity on the set P we simply write c p for c p, P .When P = { µ } , L ( c p, { µ } ) = L (Ω , B (Ω) , µ ). A non negative measure µ on (Ω , B (Ω)) belongsto the (usual) equivalence class of the probability measure µ if and only if ∀ A ∈ B (Ω) , µ ( A ) =0 ⇐⇒ µ ( A ) = 0Equivalently, for µ in the dual of L (Ω , B (Ω) , µ ), µ ∼ µ ⇐⇒ [ ∀ X ∈ L (Ω , B (Ω) , µ ) + , X = 0 ⇐⇒ Z Xdµ = 0]We address the following question: When P is weakly relatively compact can one associate aprobability measure P to L ( c p, P ) characterizing the null elements in the cone L ( c p, P ) + , i.e. suchthat ∀ X ∈ L ( c p, P ) + , X = 0 ⇐⇒ E P ( X ) = R XdP = 0 ? If yes, can one define a naturalequivalence relation so that one gets a unique equivalence class of such probability measures?Notice that when P is not finite, characteristic functions of Borelian sets are not all in L ( c p, P ). Lemma 4.1.
For all X in L ( c p, P ) , c p, P ( X ) = sup Q ∈P E Q ( | X | p ) p .Proof. Denote c p = c p, P . For all f, g in C b (Ω), for all Q ∈ P , | E Q ( | f | p ) p − E Q ( | g | p ) p | ≤ E Q ( | f − g | p ) p ≤ c p ( | f − g | )As C b (Ω) is dense in L ( c p ) for the c p norm it follows that for every X ∈ L ( c p ), g ∈ C b (Ω), and Q ∈ P , | E Q ( | X | p ) p − E Q ( | g | p ) p | ≤ c p ( | X − g | ) (4.2)11rom (4.2) it follows that E Q ( | X | p ) p ≤ c p ( X ) ∀ Q ∈ P (4.3)For every X ∈ L ( c p ), for every ǫ > g ∈ C b (Ω) such that c p ( X − g ) ≤ ǫ (4.4)From Definition 4.1, there is Q ∈ P such that c p ( g ) ≤ E Q ( | g | p ) p + ǫ (4.5)As c p ( X ) ≤ c p ( g )+ ǫ it follows from equations (4.2) (4.4) and(4.5) that c p ( X ) ≤ sup Q ∈P E Q ( | X | p ) p .The result follows from (4.3). Theorem 4.1.
Assume that Ω is a Polish space. There is a countable subset Q of P , Q = { P n , n ∈ IN } , such that for every X ∈ L ( c p, P ) , for every p ∈ [1 , ∞ [ , c p, P ( X ) = sup n ∈ IN ( E P n ( | X | p )) p (4.6) The capacities c p, P and c p, Q defined on C b (Ω) by equation (4.1) and extended to real functions usingformulas (2.1) and (2.2) are equal. The associated Banach spaces are equal: L ( c p, P ) = L ( c p, Q ) .Proof. From the previous Lemma, applied with p = 1, it follows that the set P is contained in K + ,the non negative part of the unit ball of the dual of L ( c , P ). Ω is a Polish space, so from Corollary2.1, K + is metrizable compact for the weak* topology. Thus P , the closure of P for the weak*topology, is metrizable compact. There is then in P a countable set ( P n ) n ∈ IN dense in P for theweak* topology. It follows that for every X ∈ L ( c , P ), sup Q ∈P E Q ( | X | ) = sup n ∈ IN E P n ( | X | ). Theequation (4.6) follows for every p ≥ X ∈ C b (Ω).The two capacities c p, P ( f ) = sup P ∈P E P ( | f | p ) p and c p, Q = sup Q ∈Q E Q ( | f | p ) p coincide on C b (Ω).By definition of the extension of a capacity to the set of all functions on Ω, these extensions arethe same. Therefore L ( c p, P ) = L ( c p, Q ).In the following proposition we study possible extensions of the equation (4.1). Proposition 4.1.
Let c p = c p, P . • For every non negative bounded lower semi-continuous map g , c p ( g ) = sup Q ∈P E Q ( g p ) p (4.7) • For every Borelian map f , sup Q ∈P E Q ( | f | p ) p ≤ c p ( f ) (4.8)12 roof. • The proof of the first part of Proposition 2.1 which was given for the characteristicfunction of an open set applies without any change to every non negative bounded l.s.c.function g . Thus there is an increasing sequence of continuous functions h n with limit g andsuch that c p ( g ) = lim c p ( h n ). As g is bounded, c p ( g ) is finite. Let ǫ >
0. There is n suchthat c p ( g ) − ǫ ≤ c p ( h n ) ≤ c p ( g ). By definition of c p on C b (Ω), there is Q n in P such that c p ( h n ) − ǫ ≤ E Q n ( h pn ) p ≤ c p ( h n ). Thus E Q n ( g p ) p ≥ c p ( g ) − ǫ (4.9)On the other hand for all Q in P , E Q ( h pn ) p ≤ c p ( h n ) ≤ c p ( g ). From the monotone convergencetheorem it follows that ∀ Q ∈ P , E Q ( g p ) p ≤ c p ( g ) (4.10)Thus from equations (4.9) and (4.10) we get that c p ( g ) = sup Q ∈P E Q ( g p ) p (4.11) • Let f be a Borelian map. If c p ( f ) = + ∞ , the result is trivial. Assume that c p ( f ) < ∞ . Let ǫ >
0. By definition of c p ( f ), (equation 2.2), there is g l.s.c., g ≥ | f | such that c p ( g ) < c p ( f )+ ǫ .As g is l.s.c., we already know that sup Q ∈P E Q ( | g | p ) p = c p ( g ). As f is Borel measurable,for all Q ∈ P , E Q ( | f | p ) p is defined. As g ≥ | f | it follows that E Q ( | f | p ) p ≤ c p ( f ) + ǫ . Thisinequality is true for every ǫ and every Q ∈ P . This proves the announced result for every f Borel measurable.
Remark 1. • For every open subset V of Ω , V is lower semi-continuous, so from Proposition4.1, c p (1 V ) = sup Q ∈P Q ( V ) p . • However there are Borelian subsets of Ω for which the equality c p (1 A ) = sup Q ∈P Q ( A ) p is notsatisfied.For example let Ω = [0 , . Let x n ∈ ]0 , be a sequence converging to . Let A = [0 , −{ x n , n ∈ IN } . Let Q n = δ x n . Let P = { Q n , n ∈ IN } . P is weakly relatively compact.Let f l.s.c. such that A ≤ f ≤ . For every η > , V = { x | f ( x ) > − η } is an openset containing A . As ∈ A , there is ǫ > such that [0 , ǫ [ ⊂ V . So there is N ∈ IN such that x n ∈ V ∀ n ≥ N . So E Q n ( f p ) = ( f ( x n )) p > (1 − η ) p . From equation (4.7), ≥ c p ( f ) = sup n ∈ IN ( E Q n ( f p )) p > − η for every η > . Thus c p ( f ) = 1 . It follows that c p (1 A ) = 1 . On the other hand Q n (1 A ) = 0 for all n ∈ IN . Therefore sup Q ∈P Q ( A ) p = 0 .This gives a counterexample. c p In all this section, we assume that Ω is a Polish space. We denote c p the capacity defined on C b (Ω)by c p ( f ) = sup Q ∈P E Q ( | f | p ) p . Definition 4.2. M + ( c p ) is the set of non negative finite measures on (Ω , B (Ω)) defining an elementof L ( c p ) ∗ . In the following we identify an element µ of M + ( c p ) with its associated linear form on L ( c p ).13 emark 2. A non negative finite measure µ on (Ω , B (Ω)) belongs to M + ( c p ) if and only if thereis a constant K > such that ∀ f ∈ C b (Ω) , | µ ( f ) | ≤ Kc p ( f ) . It follows easily that every element inthe weak closure of the convex hull of P defines an element of M + ( c p ) . Definition 4.3.
Define on M + ( c p ) the relation R c p by µ R c p ν ⇐⇒ (4.12) { X ∈ L ( c p ) , X ≥ | µ ( X ) = 0 } = { X ∈ L ( c p ) , X ≥ | ν ( X ) = 0 } The following lemma is trivial
Lemma 4.2. R c p defines an equivalence relation on M + ( c p ) . Definition 4.4.
Let µ ∈ M + ( c p ) . The c p -class of µ is the equivalence class of µ for the equivalencerelation R c p . Theorem 4.2.
To every weakly relatively compact set P of probability measures on (Ω , B (Ω)) , pos-sibly non dominated, can be associated canonically a c p -class of non negative measures on (Ω , B (Ω)) such that an element µ of M + ( c p ) belongs to this class if and only if ∀ X ∈ L ( c p ) , X ≥ , { µ ( X ) = 0 } ⇐⇒ { X = 0 in L ( c p ) } This class is referred to as the canonical c p -class.For every set { Q n , n ∈ IN } of probability measures on (Ω , B (Ω)) such that the equality (4.6) issatisfied for all X ∈ L ( c p ) , for α n > such that P n ∈ IN α n = 1 the probability measure P n ∈ IN α n Q n belongs to the canonical c p -class.Proof. Let p ∈ [1 , ∞ [. Let { Q n } be a countable set of probability measures such that the equality(4.6) is satisfied. Let Q = { Q n , n ∈ IN } . Let P = P n ∈ IN α n Q n . Let X ∈ L ( c p ) , X ≥
0, i.e. fromLemma 2.1, X = | X | . E P ( X ) = 0 if and only if E Q n ( | X | ) = 0 for all n ∈ IN .From equation (4.6), it follows that for X ≥ E P ( X ) = 0 if and only if c p ( X ) = 0 if and only if X = 0 in L ( c p ).This proves that the canonical c p -class is well defined (as it is not empty) and that P n ∈ IN α n Q n belongs to the canonical c p -class. Lemma 4.3.
Let P be a probability measure belonging to the canonical c p -class. Let X be anelement of L ( c p ) . Then X ≥ (for the order in L ( c p ) ) if and only X ≥ P a.s.
Proof.
For every X ∈ L ( c p ), | X | − X ≥
0. From Lemma 2.1 X ≥ | X | − X = 0in L ( c p ). By definition of the canonical c p -class this is equivalent to | X | − X = 0 P a.s., i.e. X ≥ P a.s. Remark 3.
When P = { P } the canonical c p -class is the restriction to M + ( c p ) of the usualequivalence class of the probability measure P .When P is a finite set, P = { P , ...P n } the canonical c p -class is the restriction to M + ( c p ) of theequivalence class (in the usual sense) of the probability measure P = P ≤ i ≤ n P i n . Our next goal is to give a description of L ( c p ) ∗ .14 heorem 4.3. There is a regular probability measure P belonging to the canonical c p -class, and acountable subset D = { L n , n ∈ IN } of the set L ( c p ) ∗ + of non negative continuous linear forms on L ( c p ) such that • { L n , n ∈ IN } is dense in L ( c p ) ∗ + = M + ( c p ) for the weak* topology. • Every L n is represented by a non negative measure on (Ω , B (Ω)) absolutely continuous withrespect to P .Every continuous linear form Φ on L ( c p ) is the weak* limit of a sequence Φ n where every Φ n isthe difference of two elements of D .Furthermore for every X ≥ in L ( c p ) , X = 0 iff P ( X ) = 0 , iff L n ( X ) = 0 for all n ∈ IN .Proof. Denote nK + = { L ∈ L ( c p ) ∗ , L ≥ || L || ≤ n } . From Corollary 2.1, every nK + ismetrizable compact for the weak* topology. There is then in nK + a dense countable set D n .Thus D = ∪ n ∈ IN D n is countable and dense in L ( c p ) ∗ + for the weak* topology. Enumerate theelements of D , D = { L n , n ∈ IN } . From Proposition 2.3, every L n is represented by a nonnegative finite measure µ n on (Ω , B (Ω)). Let α n > P n ∈ IN α n || L n || < ∞ . Then˜ L = P n ∈ IN α n L n ∈ L ( c p ) ∗ + . From Proposition 2.3, ˜ L is represented by a non negative finitemeasure µ . Denote P the probability measure P = µµ (Ω) . P is a probability measure on (Ω , B (Ω)), P ∈ M + ( c p ). Furthermore every µ n is absolutely continuous with respect to P , and P is regularfrom Theorem 1.1 of [5].We prove now that P belongs to the canonical c p -class. Every L n belongs to L ( c p ) ∗ . Thus forevery X in L ( c p ) such that X = 0 in L ( c p ), L n ( X ) = 0 and thus ˜ L ( X ) = 0. It follows that P ( X ) = 0. Conversely let X ≥ L ( c p ) such that P ( X ) = 0. It follows that ˜ L ( X ) = 0. Every L n belongs to L ( c p ) ∗ + , and X ≥
0, thus L n ( X ) ≥ n . From the equality ˜ L ( X ) = 0, it followsthat L n ( X ) = 0 ∀ n ∈ IN . { L n , N ∈ IN } is dense in L ( c p ) ∗ + for the weak* topology, therefore L ( X ) = 0 for all L ∈ L ( c p ) ∗ + . From the representation result of continuous linear forms on L ( c p )(Proposition 2.3) and the Jordan decomposition of bounded signed measures on (Ω , B (Ω)), it followsthat every Φ ∈ L ( c p ) ∗ is represented by a bounded measure µ = µ + − µ − . There is a Borelian set A such that R f dµ + = R f A dµ for every f ∈ C b (Ω). | µ | = µ + + µ − is defined on (Ω , B (Ω)) and isthus regular from Theorem 1.1 of [5]. ∀ ǫ > , ∃ V open, A ⊂ V such that | µ | (1 V − A ) ≤ ǫ V is lower semi-continuous so it is the increasing limit of a sequence of continuous functions h n .From the monotone convergence theorem, and equation (4.13), it follows that ∀ ǫ > , ∃ h ∈ C b (Ω) , ≤ h ≤ V , such that Z | A − h | d | µ | < ǫ (4.14)Thus | Z f A dµ − Z f hdµ | < || f || ∞ ǫ (4.15)By definition of µ , ∀ f ∈ C b (Ω) , | Z f hdµ | < || Φ || c p ( f h ) ≤ || Φ || c p ( f ) (4.16)From (4.15) and (4.16), we get | R f dµ + | = | R f A dµ | ≤ || Φ || c p ( f ). It follows that µ + defines anelement of L ( c p ) ∗ + . It is the same for µ − . Thus for every Φ ∈ L ( c p ) ∗ , Φ( X ) = 0. From HahnBanach Theorem, it follows that X = 0 in L ( c p ). This proves that P belongs to the canonical15 p -class.We have proved that every Φ ∈ L ( c p ) ∗ can be written Φ = Φ + − Φ − , Φ + , Φ − ∈ L ( c p ) ∗ + . Theresult follows then from the density of D in L ( c p ) ∗ + .The results of the previous section on convex risk measures on L ( c ) can be specified when thecapacity is c p = c p, P . Proposition 4.2.
Let ρ be a convex risk measure on L ( c p ) . There is a probability measure Q in the canonical c p -class and a countable set { Q n , n ∈ IN } of probability measures all absolutelycontinuous with respect to Q such that ρ ( X ) = sup n ∈ IN [ E Q n ( − X ) − α ( Q n )] ∀ X ∈ L ( c p ) (4.17) Proof.
From Theorem 3.3, there is a countable set { Q n , n ∈ IN } of probability measures such thatequation (4.17) is satisfied. From Theorem 4.2 there is a probability measure P in the canonical c p -class. Let Q = P + P n ∈ IN Q n n +2 . It is easy to verify that Q satisfies the required conditions. Remark 4.
Even if the capacity c p is defined from a weakly relatively compact set of probabilitymeasures, the set of probability measures { Q n , n ∈ IN } in the above dual representation (4.17) ofa convex risk measure ρ on L ( c p ) is not always relatively compact for the weak* topology. FromProposition 3.1, { Q n , n ∈ IN } is relatively compact iff ρ is majorized by a sublinear risk measure. C b (Ω) Notice that in a context of uncertainty, which is when no reference probability measure is given,it is natural to consider risk measures defined on the space C b (Ω) or more generally on a latticevector subspace of C b (Ω). As in Section 2.1, L denotes a linear vector subspace of C b (Ω) containingthe constants, generating the topology of Ω and which is a vector lattice. Definition 5.1. ρ : L → IR is a convex risk measure on L if it satisfies the axioms of Definition3.1, replacing everywhere L ( c ) by L . It is normalized if ρ (0) = 0 . • A sublinear risk measure ρ on L is regular if for every decreasing sequence X n of elements of L with limit , ρ ( − X n ) tends to . • A normalized convex risk measure is uniformly regular if for all X sup λ> ρ ( λX ) λ < ∞ , and forevery decreasing sequence X n of elements of L with limit , ρ ( − λX n ) λ converges to uniformlyin λ . Remark 5.
For sublinear risk measures, the two notions of regularity and uniform regularity areequivalent.
From now on in this section ρ is a normalized convex risk measure on L . Lemma 5.1.
Assume that ρ is uniformly regular. ρ min ( X ) = sup λ> ρ ( λX ) λ defines a regularsublinear risk measure on L . It is the minimal sublinear risk measure on L majorizing ρ . roof. The convexity, monotonicity and translation invariance of ρ min follow easily from the sameproperties of ρ . The homogeneity of ρ min follows from its definition. Thus ρ min is a sublinear riskmeasure on L majorizing ρ . The regularity of ρ min follows from the uniform regularity of ρ . Forevery sublinear risk measure ρ majorizing ρ , for every X ∈ L , ρ min ( X ) ≤ ρ ( X ). Thus ρ min isminimal. Lemma 5.2.
For every Y in L , for every sequence λ n of real numbers decreasing to , the sequence ρ ( λ n Y ) converges to the limit ρ ( Y ) .Proof. As λ n is a decreasing sequence with limit 1, one can assume that 2 > λ n ≥
1. Write λ n = 1 + ǫ n , 0 ≤ ǫ n <
1. From the convexity of ρ and ρ (0) = 0, it follows that ρ ((1 + ǫ n ) Y ) ≥ (1 + ǫ n ) ρ ( Y ) (5.1)(1 + ǫ n ) Y = (1 − ǫ n ) Y + ǫ n (2 Y ). Using the convexity of ρ , it follows that ρ ((1 + ǫ n ) Y ) ≤ (1 − ǫ n ) ρ ( Y ) + ǫ n ρ (2 Y ) (5.2)From inequations (5.1) and (5.2),(1 + ǫ n ) ρ ( Y ) ≤ ρ ((1 + ǫ n ) Y ) ≤ (1 − ǫ n ) ρ ( Y ) + ǫ n ρ (2 Y ) (5.3)Passing now to the limit in inequality (5.3), it follows that the sequence ρ ((1 + ǫ n ) Y ) has a limitequal to ρ ( Y ).Using the preceding Lemma, we prove now that every normalized uniformly regular convex riskmeasure can be extended into a convex risk measure on L ( c ) for some capacity c . Therefore wewill be able to apply the representation results of Section 3. Lemma 5.3.
Assume that ρ is uniformly regular. Denote ρ a regular sublinear risk measure on L such that ρ ≤ ρ . • c ( X ) = ρ ( −| X | ) defines a capacity on L . • ρ has a unique continuous extension into a sublinear risk measure ρ on L ( c ) . • ρ has a unique continuous extension into a normalized convex risk measure ρ on L ( c ) ma-jorized by ρ .Proof. • The sublinearity, monotonicity and regularity of ρ imply that c is a capacity on L .As usual, this leads to the Banach space L ( c ). • As ρ is sublinear, for every X, Y ∈ L , ρ ( X ) ≤ ρ ( Y ) + ρ ( X − Y ).Exchanging X and Y and using the monotonicity of ρ and the definition of c , it follows that | ρ ( X ) − ρ ( Y ) | ≤ c ( X − Y ). Thus ρ is uniformly continuous on L for the c semi-norm. Itextends uniquely into a continuous function ρ on L ( c ). ρ is a sublinear risk measure. • let ǫ n > X = 11 + ǫ n [(1 + ǫ n ) Y ] + ǫ n ǫ n [ 1 + ǫ n ǫ n ( X − Y )]17rom the convexity of ρ , the majoration of ρ by ρ and the homogeneity of ρ (cf ρ issublinear), it follows that ρ ( X ) ≤
11 + ǫ n ρ ((1 + ǫ n ) Y ) + ρ ( X − Y ) (5.4)From inequation (5.4) and Lemma 5.2 applied with (1 + ǫ n ) Y , passing to the limit, it followsthen that ρ ( X ) − ρ ( Y ) ≤ ρ ( X − Y ) ≤ c ( X − Y ). Exchanging X and Y , this proves the uniformcontinuity of ρ for the c semi-norm. ρ extends then uniquely into a continuous function ρ on L ( c ). ρ is a convex risk measure on L ( c ) majorized by ρ . Definition 5.2.
Let ρ be a normalized uniformly regular convex risk measure on L . The capacity c ρ defined as c ρ ( X ) = ρ min ( −| X | ) is called the capacity canonically associated with ρ . In this section, we assume that Ω is a Polish space. Taking into account the liquidity risk in afinancial market, we introduce the following definition for a riskless asset, which means that allinvestment in this asset is risk-free.
Definition 5.3.
A non positive element X of C b (Ω) is riskless if for all λ > , ρ ( λX ) = 0 (orequivalently for all λ > , ρ ( λX ) ≤ ). Theorem 5.1.
Let ρ be a normalized uniformly regular convex risk measure on L .Then ρ extends uniquely to C b (Ω) and admits the following representation ∀ X ∈ C b (Ω) ρ ( X ) = sup n ∈ IN ( E Q n ( − X ) − α ( Q n )) (5.5) for a certain weakly relatively compact set { Q n , n ∈ IN } of probability measures. Furthermore for α n > such that P n ∈ IN α n = 1 the probability measure P = P n ∈ IN α n Q n characterizes the risklessnon negative elements of C b (Ω) , that is X ≤ is riskless iff X = 0 P a.s.For every X ∈ C b (Ω) , there is a probability measure Q X in the weak closure of { Q n , n ∈ IN } , suchthat ρ ( X ) = E Q X ( − X ) − α ( Q X ) (5.6) Proof.
Let c ρ ( X ) = ρ min ( −| X | ) be the capacity canonically associated with ρ (definition 5.2). AsΩ is a Polish space, every capacity is a Prokhorov capacity. Denote ρ (resp ρ min ) the extensions of ρ (resp ρ min ) to L ( c ρ ) given by Lemma 5.3.As ρ is majorized by ρ min , the representation result with a countable weakly relatively compactset Q = { Q n } follows from Proposition 3.2. We can of course restrict to Q n such that α ( Q n ) < ∞ .Then c ρ ( X ) = sup n ∈ IN E Q n ( | X | ) i.e. c ρ = c , Q . From Theorem 4.2 the probability measure P = P n ∈ IN α n Q n belongs to the canonical c ρ -class. Let X ≤ C b (Ω), X is riskless iff ρ ( λX ) =0 ∀ λ >
0, iff c ρ ( − X ) = 0, iff X = 0 P a.s. The existence of Q X follows from Theorem 3.2. In all this section, Ω = C ([0 , ∞ [ , IR d ), the set of continuous functions f defined on [0 , ∞ [ withvalues in IR d such that f (0) = 0. C ([0 , ∞ [ , IR d ) endowed with the topology of uniform convergence18n compact spaces is a Polish space.Peng introduced the notion of sublinear expectation and of G-expectations [26] [27] defined ona vector lattice H of real functions containing 1 and included in C b (Ω). For the definition of asublinear expectation IE on H we refer to [15] section 3. G-expectations are defined from solutionsof P.D.E. in [26] and [27]. A G-expectation is up to a minus sign a sublinear risk measure.It is proved in [15] and [23] that every G -expectation IE has a representation with respect to aweakly relatively compact set of probability measures P : IE ( f ) = sup P ∈P E P ( f ) for all f in H . IE extends naturally to C b (Ω): IE ( f ) = sup P ∈P E P ( f ) ∀ f ∈ C b (Ω) (6.1)As P is weakly relatively compact, ρ ( f ) = IE ( − f ) is a sublinear regular risk measure on C b (Ω).Denote c IE = c ρ the corresponding capacity c IE ( X ) = IE ( | X | ) ∀ X ∈ C b (Ω).Notice that alternatively, regularity could be proved directly for G-expectations. Theorem 5.1would thus give the representation result (equation 6.1). Proposition 6.1.
There is a countable weakly relatively compact set { Q n , n ∈ IN } of probabilitymeasures, Q n ∈ P such that ∀ X ∈ C b (Ω) IE ( X ) = sup n ∈ IN E Q n ( X ) (6.2) Let P = P n ∈ IN ∗ Q n n +1 . For all f ≥ in C b (Ω) , IE ( f ) = 0 iff f = 0 P a.s.For every X ∈ C b (Ω) , there is a probability measure Q X in the weak closure of { Q n , n ∈ IN ∗ } , suchthat IE ( X ) = E Q X ( X ) .Proof. The result follows from Theorem 5.1.
We consider a framework introduced in [16]. Let Ω = C ([0 , T ] , IR d ) the space of continuousfunctions on [0 , T ] null in zero. For every t ≤ T , let Ω t = C ([0 , t ] , IR d ). Ω t is identified with thesubset of Ω of elements which are constant on [ t, T ]. Let B t be the σ -algebra on Ω generated bythe open sets of Ω t . Denote B t the coordinate process. A probability measure Q on (Ω , B (Ω)) iscalled an orthogonal martingale measure if the coordinate process ( B t ) is a martingale with respectto B t under Q and if the martingales (( B i ) t ) ≤ i ≤ d are orthogonal in the sense that for all i = j , < B i , B j > Qt = 0 Q a.s. . < B i , B j > Q denotes the quadratic covariational process corresponding to B i and B j , under Q and < B > Q the quadratic variation of B under Q . Fix for all i ∈ { , . . . , d } two finite deterministic H¨older-continuous measures µ i and µ i on [0 , T ] and consider the set P oforthogonal martingale measures such that ∀ i ∈ { , . . . , d } , dµ i,t ≤ d < B i > Qt ≤ dµ i,t . M. Kervarec has proved in [25], Lemma 1.3 that the set P is weakly relatively compact. Thus c ( f ) = sup Q ∈P E Q ( | f | ) defines a capacity on C b (Ω) (see Appendix, Section 7). As in Section4, L ( c ) denotes the corresponding Banach space, containing C b (Ω) as a dense subset. FromTheorem 4.1, and Theorem 4.2, there is a countable set ( P n ) n ∈ IN , P n ∈ P such that ∀ X ∈ L ( c ), c ( X ) = sup n ∈ IN E P n ( | X | ) and such that P = P n ∈ IN P n n belongs to the canonical c -class. Lemma 6.1.
For every probability measure R defining an element of L ( c ) ∗ , ∀ i ∈ { , . . . , d } , dµ i,t ≤ d < B i > Rt ≤ dµ i,t . R in L ( c ) ∗ does not necessarily belongs to P and thereforethe result is not trivial. Proof.
From [16], ( B i ) s ∈ L ( c ) for every s , thus R t ( B i ) s d ( B i ) s can be defined as an element of L ( c ). We thus define the quadratic variation of B in L ( c ) by < B i > c t = ( B i ) t − Z t ( B i ) s d ( B i ) s (6.3)This equation is satisfied in L ( c ) thus it is satisfied R a.s. for every probability measure R definingan element of L ( c ) ∗ . Let s ≤ t . Let A = { ω | < B i > c t − < B i > c s > µ i [ s, t ] } ∪ { ω | < B i > c t − < B i > c s < µ i [ s, t ] } . By hypothesis P n ( A ) = 0. Thus P ( A ) = 0. The inequality µ i [ s, t ] ≥ < B i > c t − < B i > c s ≥ µ i [ s, t ] (6.4)is thus satisfied P a.s. From Lemma 4.3, inequality (6.4) is then satisfied in L ( c ) and then also R a.s. for every probability measure defining an element of L ( c ) ∗ . Proposition 6.2.
The set P is convex metrizable compact for the weak* topology σ ( L ( c ) ∗ , L ( c )) and also for the weak topology.Proof. The convexity of P is obvious. Denote as in Section 2, K + the non negative part of the unitball of L ( c ) ∗ . From the definition of c it follows that P ⊂ K + . Thus the weak*closure P of P isa subset of K + . From Lemma 6.1 it follows that every element Q ∈ P satisfies ∀ i ∈ { , . . . , d } , dµ i,t ≤ d < B i > Qt ≤ dµ i,t From Corollary 2.1, K + is metrizable compact for the weak* topology thus for every Q ∈ P , thereis a sequence Q n , Q n ∈ P converging to Q for the weak* topology.From [16], | ( B i ) t | k ∈ L ( c ) for k = 1 or 2, so ( E Q n − E Q )( | ( B i ) t | k ) →
0. Passing to the limit, E Q | ( | ( B i ) t | ) ≤ c ( | ( B i ) t | ) and E Q | ( | ( B i ) t | ) ≤ c ( | ( B i ) t | ) (6.5)Let g in C b (Ω s ). g can be identified with the element ˜ g of C b (Ω) defined by ˜ g ( x ) = g ( x | [0 ,s ] ). Itfollows from the inequality c ( Xg ) ≤ || g || ∞ c ( | X | ) that ∀ u ≥ s , ( B i ) u g ∈ L ( c ), so ∀ g ∈ C b (Ω s ) ∀ λ ∈ IR , ( E Q n − E Q )(( B i ) u ( g + λ )) → B i ) t is a martingale for Q n , thus passing to the limit in (6.6), with u = t and u = s , we obtain ∀ g ∈ C b (Ω s ) ∀ λ ∈ IR , E Q (( B i ) t ( g + λ )) = E Q (( B i ) s ( g + λ )) (6.7)From (6.5), ( B i ) u ∈ L (Ω , B u , Q ) for u = t, s , and { g + λ, g ∈ C b (Ω s ) , λ ∈ IR } is dense in L (Ω , B s , Q ), thus the equality (6.7) is satisfied for every g ∈ L (Ω , B s , Q ). This proves that ( B i ) t is a martingale for Q . A very similar proof leads to the fact that the martingales ( B i ) t and ( B j ) t are mutually orthogonal for i = j . Thus P is closed for the weak* topology. As P ⊂ K + , P ismetrizable compact for the weak* topology. The result follows from Proposition 2.4 for the weaktopology. 20or every P ∈ P let β ( P ) ≥
0. Let ρ be defined by ∀ X ∈ C b (Ω) ρ ( X ) = sup P ∈P ( E P ( − X ) − β ( P )) (6.8)As P is metrizable compact for the weak topology, ρ − ρ (0) is a uniformly regular convex riskmeasure. Thus Theorem 5.1 applies.The link between the two previous examples is studied in [15]. The convex weakly compact setcharacterizing the G-expectation IE is in fact contained in the set P of orthogonal martingalemeasures introduced in [16] and considered in Section 6.2. Let Ω be a metrizable separable space and L as in Section 2 a lattice of continuous boundedfunctions, containing constants and generating the topology of Ω. We now recall some definitionsand propositions proved in Section 2 of [20]. A capacity is defined as in definition 2.1, Section 2. Definition 7.1.
A capacity c defined on L is regular if it satisfies:For all decreasing net f α ∈ L converging to , inf c ( f α ) = 0 . Definition 7.2.
A capacity c defined on L is a Prokhorov capacity if:For all ǫ > , there exists a compact set K such that c ( f ) ǫ for all f ∈ L such that | f | ≤ Ω \ K . Proposition 7.1. If Ω is a Lindel¨of space then every capacity is a regular capacity. Proposition 7.2. If Ω is locally compact or a Polish space then every regular capacity is aProkhorov capacity. Remark 6. If Ω is a Polish space, then it is a Lindel¨of space and thus every capacity is a Prokhorovcapacity. Proposition 7.3. If P is weakly relatively compact c defined on C b (Ω) by c ( f ) = sup P ∈P ( E P [ | f | p ]) p is a capacity. The proof follows from Dini Theorem (see Proposition I.3 in [25] for more details).
Acknowledgements
We thank an anonymous referee for very helpful comments.