Key words. quasiconvex functions, dual representation, quasiconvex optimization, dynamic risk measures, conditional certainty equivalent.

Transcription

1 DUAL REPRESENTATION OF QUASICONVEX CONDITIONAL MAPS MARCO FRITTELLI AND MARCO MAGGIS Abstract. We provide a dual representation of quasiconvex maps π : L F L G, between two locally convex lattices of random variables, in terms of conditional expectations. This generalizes the dual representation of quasiconvex real valued functions π : L F R and the dual representation of conditional convex maps π : L F L G. These results were inspired by the theory of dynamic measurements of risk and are applied in this context. Key words. quasiconvex functions, dual representation, quasiconvex optimization, dynamic risk measures, conditional certainty equivalent. MSC (2010): primary 46N10, 91G99, 60H99; secondary 46A20, 46E Introduction 1. Quasiconvex analysis has important applications in several optimization problems in science, economics and in finance, where convexity may be lost due to absence of global risk aversion, as for example in Prospect Theory [KT92]. The first relevant mathematical findings on quasiconvex functions were provided by De Finetti [DF49] and since then many authors, as [Fe49], [Cr80], [PP84] and [PV90] - to mention just a few, contributed significantly to the subject. More recently, a Decision Theory complete duality involving quasiconvex real valued functions has been proposed by [CMM09b]. For a review of quasiconvex analysis and its applications and for an exhaustive list of references on this topic we refer to Penot [Pe07]. A function f : L R := R { } { } defined on a vector space L is quasiconvex if for all c R the lower level sets {X L f(x) c} are convex. In a general setting, the dual representation of such functions was shown by Penot and Volle [PV90]. The following theorem, reformulated in order to be compared to our results, was proved by Volle [Vo98], Th As shown in the Appendix 5.2, its proof relies on a straightforward application of Hahn Banach Theorem. Theorem 1.1 ([Vo98]). Let L be a locally convex topological vector space, L be its dual space and f : L R := R { } { } be quasiconvex and lower semicontinuous. Then where R : R L R is defined by f(x) = sup X L R(X (X), X ) (1.1) R(t, X ) := inf ξ L {f(ξ) X (ξ) t}. The generality of this theorem rests on the very weak assumptions made on the domain of the function f, i.e., on the space L. On the other hand, the fact that only real valued maps are admitted considerably limits its potential applications, especially in a dynamic framework. To the best of our knowledge, a conditional version of this representation is lacking in the literature. When (Ω, F, (F t ) t 0, P) is a filtered probability space, many Forthcoming on SIAM J. Financial Mathematics Dipartimento di Matematica, Università degli Studi di Milano. Dipartimento di Matematica Università degli Studi di Milano. 1 Acknowledgements We wish to thank Dott. G. Aletti for helpful discussion on this subject. 1

2 2 M. FRITTELLI AND M. MAGGIS problems having dynamic features lead to the analysis of maps π : L t L s between the subspaces L t L 0 (Ω, F t, P) and L s L 0 (Ω, F s, P), 0 s < t (see Section 1.1 for some examples). In this paper we consider quasiconvex maps of this form and analyze their dual representation. We provide (see Theorem 2.9 and 2.10 for the exact statements) a conditional version of (1.1): where π(x) = ess sup R(E Q [X F s ], Q), (1.2) Q L t P R(Y, Q) := ess inf ξ L t {π(ξ) E Q [ξ F s ] Q Y }, L t L 1 t is the order continuous dual space of L t and { } dq P =: Q << P and Q probability = { ξ L 1 + E P [ξ ] = 1 }. dp With a slight abuse of notation, we write Q L t P instead of dq dp L t P. Furthermore, we prove in Proposition 2.13 that π is quasiconvex, monotone, continuous from below and regular if and only if (1.2) holds with R belonging to the class R of maps S : L 0 s L t L 0 s such that S(, ξ ) is quasiconvex, monotone, continuous from below and regular. If the map π is quasiconvex, monotone and cash additive then we derive from (1.2) the well known representation of a convex conditional risk measure, as in [DS05]. Of course, this is of no surprise since cash additivity and quasiconvexity imply convexity, but it supports the correctness of our dual representation. The formula (1.2) is obtained under quite weak assumptions on the space L t which allow us to consider maps π defined on the typical spaces used in the literature: L (Ω, F t, P), L p (Ω, F t, P), the Orlicz spaces L Ψ (Ω, F t, P). In Theorem 2.9 we assume that π is lower semicontinuous, with respect to the weak topology σ(l t, L t ). As shown in Proposition 2.5 this condition is equivalent to continuity from below, a natural requirement in this context. In Theorem 2.10 instead we provide the dual representation under a strong upper semicontinuity assumption. The proof of our main Theorems 2.9 and 2.10 is not based on techniques similar to those applied in the quasiconvex real valued case [Vo98], nor to those used for convex conditional maps [DS05]. Indeed, the so called scalarization of π via the real valued map X E P [π(x)] does not work, since this scalarization preserves convexity but not quasiconvexity. The idea of our proof is to apply (1.1) to the real valued quasiconvex map π A : L t R defined by π A (X) := ess sup ω A π(x)(ω), A F s, and to approximate π(x) with π Γ (X) := A Γ π A(X)1 A, where Γ is a finite partition of Ω of F s - measurable sets A Γ. As explained in Section 4.1, some delicate issues arise when one tries to apply this simple and natural idea to prove that: ess = ess inf Γ sup Q L t P ess inf ξ L t {π(ξ) E Q [ξ F s ] Q E Q [X F s ]} ess sup Q L t P ess inf ξ L t { π Γ (ξ) E Q [ξ F s ] Q E Q [X F s ] }. (1.3)

3 DUAL REPRESENTATION OF QUASICONVEX CONDITIONAL MAPS 3 The uniform approximation result here needed is stated in the key Lemma 4.3 and the Appendix 5.1 is devoted to prove it. It has recently been shown 2 in [FM10] that results of the same nature, but technically quite different, of those of Theorems 2.9 and 2.10 hold for maps defined on Modules of the L p -type (see [FKV09] for details on this setting). The module approach used in [FM10] permits to prove that the maximum in (2.5) is attained, under the same assumptions of Theorem 2.10, when the maps are defined on Modules of the L p -type. In the present paper we limit ourselves to consider conditional maps π : L t L s and we defer to a forthcoming paper the study of the temporal consistency of the family of maps (π s ) s [0,t], π s : L t L s. The paper is organized as follows. In Section 2 we introduce the key definitions in order to have all the ingredients to state, in Section 2.1, our main results. Section 3 is a collection of a priori properties about the maps we use to obtain the dual representation. Theorems 2.9 and 2.10, are proved in Section 4 and a brief outline of the proof is there reported to facilitate its understanding. The technical important Lemmas are left to the Appendix, where we also report the proof of Theorem Applications to Finance. As a further motivation for our findings, we give some examples of quasiconvex (quasiconcave) conditional maps arising in economics and finance, which will also be analyzed in details in a future paper. Certainty Equivalent in dynamic settings. Consider a stochastic dynamic utility u : R [0, ) Ω R where: the function x u(x, t, ω) is strictly increasing and concave on R, for almost any ω Ω and for t [0, ); the function u(x, t, ) is F t - measurable for all (x, t) R [0, ). These functions have been recently considered in [MZ06] to develop the theory of forward utility. In [FM11] we defined the Conditional Certainty Equivalent (CCE) of a random variable X L t, as the random variable π(x) L s that is the solution of the equation: u(π(x), s) = E P [u(x, t) F s ]. Thus the CCE defines the valuation operator π : L t L s, π(x) = u 1 (E P [u(x, t) F s ]), s). We showed in [FM11] that the CCE, as a map π : L t L s is monotone, quasi concave, regular and that it admits the (concave version) representation as in (1.2). Static Risk Measures. Our interest in quasiconvex analysis was triggered by the recent paper [CMM09a] on quasiconvex risk measures, where the authors show that it is reasonable to weaken the convexity axiom in the theory of convex risk measures, introduced in [FS02] and [FR02]. In fact when one replaces cash additivity with cash subadditivity (as explained in [ER09]), quasiconvexity and convexity are not anymore equivalent. But the quasiconvexity property is the literal translation of the principle diversification should not increase the risk. The recent interest in quasiconvex static risk measures is also testified by a second paper [DK10] on this subject, that was inspired by [CMM09a] and disclosed after the first version of the present paper. 2 Added in the revision

4 4 M. FRITTELLI AND M. MAGGIS Dynamic Risk Measures. As already mentioned the dual representation of a conditional convex risk measure can be found in [DS05]. The findings of the present paper show the dual representation of conditional quasiconvex risk measures when cash additivity does not hold true. For a better understanding we give a concrete example: consider a non empty convex set C L (Ω, F t, P) such that C + L + C. The set C represents the future positions considered acceptable by the supervising agency. Let s [0, t]. For all m R denote by v s (m, ω) the price at time s of m euros at time t. The function v s (m, ) will be in general F s measurable as in the case of stochastic discount factor where v s (m, ω) = D s (ω)m. By adapting the definitions in the static framework of [ADEH99] and [CMM09a] we set: ρ s,vs (X)(ω) = ess inf Y L 0 Fs {v s (Y, ω) X + Y C}. When v s is linear, then ρ s,vs is a convex monetary dynamic risk measure, but the linearity of v s may fail when zero coupon bonds with maturity t are illiquid. It seems reasonable to assume that v s (, ω) is increasing and upper semicontinuous and v s (0, ω) = 0, for P almost every ω Ω. In this case ρ s,vs (X)(ω) = v s (ess inf Y L 0 Fs {Y X + Y C}, ω) = v s (ρ s (X), ω), where ρ s (X) is the convex monetary dynamic risk measure induced by the set C. Thus in general ρ s,vs is neither convex nor cash additive, but it is quasiconvex and eventually cash subadditive (under further assumptions on v s ). Acceptability Indices. As studied in [CM09] the index of acceptability is a map α from a space of random variables L(Ω, F, P) to [0, + ) which measures the performance or quality of the random X which may be the terminal cash flow from a trading strategy. Associated with each level x of the index there is a collection of terminal cash flows A x = {X L α(x) x} that are acceptable at this level. The authors in [CM09] suggest four axioms as the stronghold for an acceptability index in the static case: quasiconcavity (i.e., the set A x is convex for every x [0, + )), monotonicity, scale invariance and the Fatou property. It appears natural (see also the recent paper [BCZ10]) 3 to generalize these kind of indices to the conditional case and to this aim we propose a couple of basic examples: i) Conditional Gain Loss Ratio: let F s F t and: CGLR(X F s ) = E P[X F s ] E P [X F s ] 1 {E P [X F s]>0}. This measure is clearly monotone, scale invariant and well defined on L 1 (Ω, F t, P). It can be proved that it is continuous from below and quasiconcave. ii) Conditional Coherent Risk-Adjusted Return on Capital: let F s F t and suppose a coherent conditional risk measure ρ : L(Ω, F t, P) L 0 (Ω, F s, P) is given, where L(Ω, F t, P) L 1 (Ω, F t, P) is any vector space. We define CRARoC(X F s ) = E P[X F s ] 1 {E[X Fs]>0}. ρ(x) 3 Added in the revision

5 DUAL REPRESENTATION OF QUASICONVEX CONDITIONAL MAPS 5 We use the convention that CRARoC(X F s ) = + on the F s -measurable set where ρ(x) 0. Again CRARoC( F s ) is well defined on the space L(Ω, F t, P) and takes values in the space of extended random variables; moreover is monotone, quasiconcave, scale invariant and continuous from below whenever ρ is continuous from above. 2. The dual representation. The probability space (Ω, F, P) is fixed throughout the paper and G F is any sigma algebra contained in F. As usual we denote with L 0 (Ω, F, P) the space of F measurable random variables that are P a.s. finite and by L 0 (Ω, F, P) the space of extended real-valued random variables. The L p (Ω, F, P) spaces, p [0, ], will simply be denoted by L p, unless it is necessary to specify the sigma algebra, in which case we write L p F. In presence of an arbitrary probability measure Q, if confusion may arise, we will explicitly write = Q (resp. Q ), meaning Q almost surely. Otherwise, all equalities/inequalities among random variables are meant to hold P-a.s. Moreover the essential (P almost surely) supremum ess sup λ (X λ ) of an arbitrary family of random variables X λ L 0 (Ω, F, P) will be simply denoted by sup λ (X λ ), and similarly for the essential infimum (see [FS04] Section A.5 for reference). Here we only notice that 1 A sup λ (X λ ) = sup λ (1 A X λ ) for any F measurable set A. Hereafter the symbol denotes inclusion and lattice embedding between two lattices; (resp. ) denotes the essential (P almost surely) maximum (resp. the essential minimum) between two random variables, which are the usual lattice operations. We consider a lattice L F := L(Ω, F, P) L 0 (Ω, F, P) and a lattice L G := L(Ω, G, P) L 0 (Ω, G, P) of F (resp. G) measurable random variables. Therefore, the range of a map π : L F L G includes extended real-valued random variables. Definition 2.1. A map π : L F L G is said to be (MON) monotone increasing if for every X, Y L F X Y π(x) π(y ) ; (QCO) quasiconvex if for every X, Y L F, Λ L 0 G and 0 Λ 1 π(λx + (1 Λ)Y ) π(x) π(y ) ; (LSC) τ lower semicontinuous if the set {X L F π(x) Y } is closed for every Y L G with respect to a topology τ on L F. (CFB) continuous from below if X n X P a.s. π(x n ) π(x) P a.s. (USC) τ strong upper semicontinuous if the set {X L F π(x) < Y } is open for every Y L G with respect to a topology τ on L F and there exists θ L F such that π(θ) < +. Remark 2.2. (On quasiconvexity) As it happens for real valued maps, the definition of (QCO) is equivalent to the fact that Y L G the lower level sets A(Y ) = {X L F π(x) Y } are conditionally convex, i.e., for all X 1, X 2 A(Y ) and for all G-measurable random variables Λ, 0 Λ 1, one has that ΛX 1 +(1 Λ)X 2 A(Y ). Remark 2.3. (On upper semicontinuity) When the map π is real valued then (USC) is equivalent to {X L F π(x) Y } is closed Y R. But when the range of π is L G (a space of random variables) this equivalence does not hold true. Our strong definition (USC) implies that if a net {X α } α L F satisfies

6 6 M. FRITTELLI AND M. MAGGIS τ X α X then lim supα π(x α ) π(x). Furthermore, this last condition implies that the set {X L F π(x) Y } is closed, i.e., the usual upper semicontinuity (USC) condition, so that (USC) (USC). We are assuming that there exists at least one θ L F such π(θ) < + ; otherwise the set {X L F π(x) < Y } is always empty (and then open) and the condition (USC) looses any meaning. In [BF09] it is proved the equivalence between (CFB) and σ(l F, L F )-(LSC) for monotone convex real valued functions. In the next proposition, we state that this equivalence remains true for monotone quasiconvex conditional maps, under the same assumption on the topology adopted in [BF09]. Therefore, in Theorem 2.9 the σ(l F, L F )-(LSC) condition can be replaced by (CFB), which is often easy to check. Definition 2.4 ([BF09]). A linear topology τ on a Riesz space has the C-property τ if X α X implies the existence of of a sequence {Xαn } n and a convex combination o Z n conv(x αn,...) such that Z n X. As explained in [BF09], the assumption that σ(l F, L F ) has the C-property is very weak and is satisfied in all cases of interest. Proposition 2.5. Suppose that σ(l F, L F ) satisfies the C-property and L F is order complete. If π : L F L G is (MON) and (QCO) then: π is σ(l F, L F )-(LSC) if and only if π is (CFB). We omit the proof since it is a simple extension of the one written in the cited reference. Definition 2.6. A vector space L F L 0 F satisfies the property 1 F if X L F and A F = (X1 A ) L F. (1 F ) Suppose L F (resp. L G ) satisfies property (1 F ) (resp 1 G ). A map π : L F L G is: (REG) regular if for every X, Y L F and A G or equivalently if π(x1 A )1 A = π(x)1 A. π(x1 A + Y 1 A C ) = π(x)1 A + π(y )1 A C 2.1. The representation theorems and their consequences. Standing assumptions In the sequel of the paper it is assumed that: (a) G F and the lattice L F (resp. L G ) satisfies the property (1 F ) (resp 1 G ). Both L G and L F contains the constants as a vector subspace. (b) The order continuous dual of (L F, ), denoted by L F = (L F, ), is a lattice ([AB05], Th. 8.28) that satisfies L F L1 F and property (1 F). (c) The space L F endowed with the weak topology σ(l F, L F ) is a locally convex Riesz space. The condition (c) requires that the order continuous dual L F is rich enough to separate the points of L F, so that (L F,σ(L F, L F )) becomes a locally convex TVS and Proposition 5.5 can be applied. Remark 2.7. Many important classes of spaces satisfy these conditions, such as - The L p -spaces, p [1, ]: L F = L p F, L F = Lq F L1 F. - The Orlicz spaces L Ψ for any Young function Ψ: L F = L Ψ F, L F = LΨ F L1 F, where Ψ denotes the conjugate function of Ψ; - The Morse subspace M Ψ of the Orlicz space L Ψ, for any continuous Young function Ψ: L F = MF Ψ, L F = LΨ F L1 F.

7 DUAL REPRESENTATION OF QUASICONVEX CONDITIONAL MAPS 7 Define K : L F (L F P) L 0 G and R : L0 G L F as K(X, Q) := inf {π(ξ) E Q [ξ G] Q E Q [X G]}, (2.1) R(Y, ξ ) := inf {π(ξ) E P [ξ ξ G] Y }. (2.2) The function K is well defined on L F (L F P), while the actual domain of R is: Σ := {(Y, ξ ) L 0 G L F ξ L F s.t. E P [ξ ξ G] Y }. (2.3) Obviously (E P [ξ X G], ξ ) Σ for every X L F, ξ L F. Notice that K(X, Q) depends on X only through E Q [X G]. Moreover, for every λ > 0, R(E P [ξ X G], ξ ) = R(E P [λξ X G], λξ ). Thus we can consider R(E P [ξ X G], ξ ), ξ 0, ξ 0, always defined on the normalized elements Q L F P. It is easy to check that [ ] dq E P dp ξ G and for Q L F P we deduce E P [ dq dp X G K(X, Q) = R ] E Q [ξ G] Q E Q [X G], ( [ ] ) dq E P dp X G, Q. Remark 2.8. Since the order continuous functional on L F are contained in L 1, then Q(ξ) := E Q [ξ] is well defined and finite for every ξ L F and Q L F P. In particular this and property (1 F ) imply that E Q [ξ G] is well defined. Moreover, (1 F ) guarantees that dq dp 1 A L F whenever Q L F and A F. Theorem 2.9. Suppose that σ(l F, L F ) has the C-property and L F is order complete. If π : L F L G is (MON), (QCO), (REG) and σ(l F, L F )-(LSC) then π(x) = sup K(X, Q). (2.4) Theorem If π : L F L G is (MON), (QCO), (REG) and τ-(usc) then π(x) = sup K(X, Q). (2.5) Notice that in (2.4) and (2.5) the supremum is taken over the set L F P. In the next corollary, proved in Section 4.2, we show that we match the conditional convex dual representation, by restricting our optimization problem on the set { } dq P G =: Q P and Q = P on G. dp Clearly, when Q P G then L 0 (Ω, G, P) = L 0 (Ω, G, Q) and comparison of G measurable random variables is understood to hold indifferently for P or Q. Corollary Under the same hypothesis of Theorem 2.9 or Theorem 2.10, suppose that for X L F there exists η L F and δ > 0 such that P(π(η) + δ < π(x)) = 1. Then π(x) = sup K(X, Q). (2.6) G

8 8 M. FRITTELLI AND M. MAGGIS If S : Σ L 0 G then: S(, ξ ) is (REG) if S(Y 1 A, Q)1 A = S(Y, Q)1 A A G; S(, ξ ) is (CFB) if Y n Y, P a.s., and (Y n, ξ ) Σ, (Y, ξ ) Σ imply S(Y n, ξ ) S(Y, ξ ). As shown in the next Proposition, the dual characterization of maps π : L F L G that are (MON), (REG), (QCO) and (CFB) is obtained throught the class: R := { S : Σ L 0 G such that S(, ξ ) is (MON), (REG) and (CFB) }. Remark Any map S : Σ L 0 G such that S(, ξ ) is (MON) and (REG) is automatically (QCO) in the first component. Indeed, let Y 1, Y 2, Λ L 0 G, 0 Λ 1 and define B = {Y 1 Y 2 }, S(, Q) = S( ). Then S(Y 1 1 B ) S(Y 2 1 B ) and S(Y 2 1 B C ) S(Y 1 1 B C ) so that from (MON) and (REG) S(ΛY 1 + (1 Λ)Y 2 ) S(Y 2 1 B + Y 1 1 B C ) = S(Y 2 )1 B + S(Y 1 )1 B C S(Y 1 ) S(Y 2 ). Proposition Suppose that σ(l F, L F ) satisfies the C-property and L F is order complete. The map π : L F L G is (MON), (QCO), (REG) and (CFB) if and only if there exists S R such that π(x) = sup S ( [ ] ) dq E dp X G, Q. (2.7) The proof is based on Theorem 2.9 and is postponed to Section 4.2. In [CMM09b] the authors provide a complete duality for real valued quasiconvex (either USC or LSC) functionals when the space L F is an M-space (such as L ): the idea is to prove a one to one relationship between quasiconvex monotone functionals π and the function R in the dual representation. Obviously R will be unique only in an opportune class of maps satisfying certain properties. A similar result has been recently obtained in [DK10] for general TVS in the LSC real valued case. In the conditional case, uniqueness is a very delicate issue which has only recently been addressed (see [FM10]) 4, but only in the module framework developed in [FKV09]. In the setting of the present paper, a partial result can be easily derived, from the static case, when G is countably generated by a partition {A n } n N and thus the map π is constant on each atom A n. In the following Corollary (proved in Section 4.2) we show that the (MON) property implies that the constraint E Q [ξ G] Q E Q [X G], in the definition of K(X, Q), may be restricted to E Q [ξ G] = Q E Q [X G]. We also show that we may recover, in agreement with [DS05], the dual representation of a dynamic risk measure when π also satisfies the following property: A map π : L F L G is said to be (CAS) cash additive if for all X L F and Λ L G L F π(x + Λ) = π(x) + Λ. Corollary Suppose that E Q [ξ G] L F Q L F P G and ξ L F. (i) If Q L F P G and if π : L F L G is (MON) and (REG), then 4 Added in the revision K(X, Q) = inf {π(ξ) E Q [ξ G] = E Q [X G]} ; (2.8)

9 DUAL REPRESENTATION OF QUASICONVEX CONDITIONAL MAPS 9 if in addition π is (CAS), then: K(X, Q) = E Q [X G] π (Q), (2.9) where π (Q) := sup ξ LF {E Q [ξ G] π(ξ)}. (ii) Under the same hypotheses of Theorem 2.9 or Theorem 2.10 and if π is also (CAS), then: π(x) = sup {E Q [X G] π (Q)}. (2.10) G 3. Preliminary results. In the sequel of the paper π :L F L G is assumed(reg) Properties of R(Y, ξ ). We remind that Σ is the actual domain of R as given in (2.3). For any (Y, ξ ) Σ, we have R(Y, ξ ) = inf A(Y, ξ ) where A(Y, ξ ) := {π(ξ) ξ L F, E P [ξ ξ G] Y }. By convention R(Y, ξ ) = + for every (Y, ξ ) (L 0 G L F ) \ Σ Lemma 3.1. For every (Y, ξ ) Σ the set A(Y, ξ ) is downward directed and therefore there exists a sequence {η m } m=1 L F such that E P [ξ η m G] Y and as m, π(η m ) R(Y, ξ ). Proof. We have to prove that for every π(ξ 1 ), π(ξ 2 ) A(Y, ξ ) there exists π(ξ ) A(Y, ξ ) such that π(ξ ) min{π(ξ 1 ), π(ξ 2 )}. Consider the G-measurable set G = {π(ξ 1 ) π(ξ 2 )} then min{π(ξ 1 ), π(ξ 2 )} = π(ξ 1 )1 G + π(ξ 2 )1 G C = π(ξ 1 1 G + ξ 2 1 G C ) = π(ξ ), where ξ = ξ 1 1 G + ξ 2 1 G C. Hence E P [ξ ξ G] = E P [ξ ξ 1 G]1 G + E P [ξ ξ 2 G]1 G C that we can deduce π(ξ ) A(Y, ξ ). Lemma 3.2. Properties of R(Y, ξ ). i) R(, ξ ) is monotone, for every ξ L F. ii) R(λY, λξ ) = R(Y, ξ ) for any λ > 0, Y L 0 G and ξ L F. iii) For every A G, (Y, ξ ) Σ Y so R(Y, ξ )1 A = inf {π(ξ)1 A E P [ξ ξ G] Y } (3.1) = inf {π(ξ)1 A E P [ξ ξ1 A G] Y 1 A } = R(Y 1 A, ξ )1 A. (3.2) iv) For every Y 1, Y 2 L 0 G (a) R(Y 1, ξ ) R(Y 2, ξ ) = R(Y 1 Y 2, ξ ) (b) R(Y 1, ξ ) R(Y 2, ξ ) = R(Y 1 Y 2, ξ ) v) The map R(, ξ ) is quasi-affine, i.e., for every Y 1, Y 2, Λ L G and 0 Λ 1, R(ΛY 1 + (1 Λ)Y 2, ξ ) R(Y 1, ξ ) R(Y 2, ξ ) (quasiconcavity) R(ΛY 1 + (1 Λ)Y 2, ξ ) R(Y 1, ξ ) R(Y 2, ξ ) (quasiconvexity). vi) inf Y L 0 G R(Y, ξ 1) = inf Y L 0 G R(Y, ξ 2) for every ξ 1, ξ 2 L F. Proof. (i) and (ii) are trivial. (iii) By definition of the essential infimum one easily deduce (3.1). To prove (3.2), for every ξ L F such that E P [ξ ξ1 A G] Y 1 A we define the random variable η = ξ1 A + ζ1 A C where E P [ξ ζ G] Y. Then E P [ξ η G] Y and we can conclude {η1 A η L F, E P [ξ η G] Y } = {ξ1 A ξ L F, E P [ξ ξ1 A G] Y 1 A }.

10 10 M. FRITTELLI AND M. MAGGIS Hence from (3.1) and (REG): 1 A R(Y, ξ ) = inf η L F {π(η1 A )1 A E P [ξ η G] Y } = inf {π(ξ1 A )1 A E P [ξ ξ1 A G] Y 1 A } = inf {π(ξ)1 A E P [ξ ξ1 A G] Y 1 A }. The second equality in (3.2) follows in a similar way. (iv) a): Since R(, ξ ) is monotone, the inequality R(Y 1, ξ ) R(Y 2, ξ ) R(Y 1 Y 2, ξ ) holds true. To show the opposite inequality, define the G-measurable sets: B := {R(Y 1, ξ ) R(Y 2, ξ )} and A := {Y 1 Y 2 } so that R(Y 1, ξ ) R(Y 2, ξ ) = R(Y 1, ξ )1 B +R(Y 2, ξ )1 B C R(Y 1, ξ )1 A +R(Y 2, ξ )1 A C (3.3) Set: D(A, Y ) = {ξ1 A ξ L F, E P [ξ ξ1 A G] Y 1 A } and check that D(A, Y 1 ) + D(A C, Y 2 ) = {ξ L F E P [ξ ξ G] Y 1 1 A + Y 2 1 A C } := D From (3.3) and using (3.2) we get: R(Y 1, ξ ) R(Y 2, ξ ) R(Y 1, ξ )1 A + R(Y 2, ξ )1 A C = inf {π(ξ1 A)1 A } + ξ1 A D(A,Y 1) inf η1 A C D(A C,Y 2) = inf ξ1 A D(A,Y 1) {π(ξ1 A )1 A + π(η1 A C )1 A C } η1 A C D(A C,Y 2) = inf {π(ξ1 A + η1 A C )} (ξ1 A +η1 A C ) D(A,Y 1)+D(A C,Y 2) {π(η1 A C )1 A C } = inf ξ D {π(ξ)} = R(Y 11 A + Y 2 1 A C, ξ ) = R(Y 1 Y 2, ξ ). Simile modo: iv) b). (v) Follows from item (iv) and Remark (vi) Trivial generalization of Theorem 2 (H2) in [CMM09b]. Consider the map R + : Σ L 0 G defined by R + (Y, ξ ) = ess sup R(Y, ξ ). (3.4) Y <Y Lemma 3.3. If π : L F L G is (REG) and (MON) then R + R. Proof. Clearly R + (, Q) inherits from R(, Q) the properties (REG) and (MON). From Remark 2.12 we then know that R + (, Q) is (QCO). We show that it is also (CFB). Let Y n Y. It is easy to check that (MON) of R(, ξ ) implies that the set {R(η, ξ ) η < Y } is upward directed. Then for any ε, δ > 0 we can find η ε < Y s.t. P(R + (Y, ξ ) R(η ε, ξ ) < ε) > 1 δ (3.5) There exists an n ε such that P(Y n > η ε ) > 1 δ for every n > n ε. Denote by A n = {Y n > η ε } so that from (REG) we have R + (Y n, ξ )1 An R(η ε, ξ )1 An. This last inequality together with equation (3.5) implies P(R + (Y, ξ ) R + (Y n, ξ ) < ε) > 1 2δ n > n ε, i.e., R + (Y n, Q) P R + (Y, Q). From (MON), R + (Y n, Q) R + (Y, Q) P-a.s.

11 DUAL REPRESENTATION OF QUASICONVEX CONDITIONAL MAPS 11 Proposition 3.4 is in the spirit of [CMM09b]: as a consequence of the dual representation the map π induces on R (resp. R + ) its characteristic properties and so does R (resp. R + ) on π. Recall that (E P [ξ X G], ξ ) Σ for every X L F, ξ L F. Proposition 3.4. Consider a map S : Σ L G. (a) Let χ L F, X L F and π(x) = sup S(E P [Xξ G], ξ ). ξ χ Then for every (Y, ξ ) Σ, A G, λ R +, Λ L G L F i) S(Y 1 A, ξ )1 A = S(Y, ξ )1 A = π (REG); ii) Y S(Y, ξ ) (MON) = π (MON); iii) Y S(Y, ξ ) (QCO) = π (QCO); iv) Y S(Y, ξ ) (CFB) = π (CFB); v) S(λY, ξ ) = S(Y, ξ ) = π(λx) = π(x); vi) S(λY, ξ ) = λs(y, ξ ) = π(λx) = λπ(x); vii) S(E P [(X + Λ)ξ G], ξ ) = S(E P [Xξ G], ξ ) + Λ = π(x + Λ) = π(x) + Λ; viii) S(E P [(X + Λ)ξ G], ξ ) S(E P [Xξ G], ξ ) + Λ = π(x + Λ) π(x) + Λ. (b) When the map S is replaced by R defined in (2.2), all the above items - except (iv) - hold true replacing = by. (c) When the map S is replaced by R + defined in (3.4), all the above items (i)-(viii) hold true replacing = by. Proof. The proofs of all items in (a) are trivial. (b): The implication = in (i) and (ii) follow from Lemma 3.2. We omit the remaining elementary proofs. (c) the implications = in (i) (ii) (iii) and (iv) are proved in Lemma 3.3. The remaining items are easily proved from the corresponding properties of R Properties of K(X, Q) and H(X).. Lemma 3.5. Properties of K(X, Q). Let Q L F P and X L F. i) K(, Q) is monotone and quasi affine. ii) K(X, ) is scaling invariant: K(X, λq) = K(X, Q) for every λ > 0. iii) K(X, Q)1 A = inf ξ LF {π(ξ)1 A E Q [ξ1 A G] Q E Q [X1 A G]} for all A G. iv) There exists a sequence { ξ Q m} m=1 L F such that E Q [ξ Q m G] Q E Q [X G] m 1, π(ξ Q m) K(X, Q) as m. v) The set K = {K(X, Q) Q L F P} is upward directed, i.e., for every K(X, Q 1), K(X, Q 2 ) K there exists K(X, Q) K such that K(X, Q) K(X, Q 1 ) K(X, Q 2 ). vi) Let Q 1 and Q 2 be elements of L dq1 F P and B G. If dp 1 B = dq2 dp 1 B then K(X, Q 1 )1 B = K(X, Q 2 )1 B. Proof. The monotonicity property in (i), (ii) and (iii) are trivial; from Lemma 3.2 v) it follows that K(, Q) is quasi affine; (iv) is a consequence of Lemma 3.1. (v) Define F = {K(X, Q 1 ) K(X, Q 2 )} and let Q given by d Q dp := 1 F dq1 dp + 1 F C dq2 dp ; up to a normalization factor (from property (ii)) we may suppose Q L F P. We need to show that K(X, Q) = K(X, Q 1 ) K(X, Q 2 ) = K(X, Q 1 )1 F + K(X, Q 2 )1 F C. From E Q[ξ G] = Q E Q1 [ξ G]1 F + E Q2 [ξ G]1 F C we get E Q[ξ G]1 F = Q1 E Q1 [ξ G]1 F and E Q[ξ G]1 F C = Q2 E Q2 [ξ G]1 F C. For i = 1, 2, consider the sets  = {ξ L F E Q[ξ G] Q E Q[X G]} A i = {ξ L F E Qi [ξ G] Qi E Qi [X G]}.

12 12 M. FRITTELLI AND M. MAGGIS For every ξ A 1 define η = ξ1 F + X1 F C Q 1 << P η1 F = Q1 ξ1 F E Q[η G]1 F Q E Q[X G]1 F Q 2 << P η1 F C = Q2 X1 F C E Q[η G]1 F C = Q E Q[X G]1 F C Then η Â and π(ξ)1 F = π(ξ1 F )1 F = π(η1 F )1 F = π(η)1 F. Viceversa, for every η Â define ξ = η1 F + X1 F C. Then ξ A 1 and again π(ξ)1 F = π(η)1 F. Hence inf π(ξ)1 F = inf π(η)1 F. ξ A 1 η Â In a similar way: inf ξ A2 π(ξ)1 F C = inf η Â π(η)1 F C and we can finally deduce K(X, Q 1 ) K(X, Q 2 ) = K(X, Q). (vi). By the same argument used in (v), it can be shown that inf ξ A1 π(ξ)1 B = inf ξ A2 π(ξ)1 B and the thesis. For X L F we define H(X) := sup K(X, Q) = and notice that for all A G H(X)1 A = sup sup inf {π(ξ) E Q [ξ G] Q E Q [X G]} inf {π(ξ)1 A E Q [ξ G] Q E Q [X G]}. Applying Proposition 3.4 we already know the one to one correspondence between the properties of π and those of H. In particular, H is (MON) and the regularity of π implies that H is (REG) (i.e., H(X1 A )1 A = H(X)1 A for any A G). As an immediate consequence of Lemma 3.5 iv) and v) we deduce the next result. Lemma 3.6. Let X L F. There exist a sequence { Q k} k 1 L F P and, { k 1, a sequence ξ Qk m } m 1 L F satisfying E Q k[ξ Qk m G] Q k E Q k[x G] and π(ξ Qk m ) K(X, Q k ) as m, K(X, Q k ) H(X) as k, (3.6) H(X) = lim k lim m π(ξqk m ). (3.7) 3.3. On the map π A. Given π : L F L G we define for every A G, the map π A : L F R by π A (X) := ess sup π(x)(ω). ω A Proposition 3.7. Under the assumptions of Theorem 2.9 or of Theorem 2.10 and for any A G π A (X) = sup inf {π A (ξ) E Q [ξ G] Q E Q [X G]}. (3.8) Proof. Notice that the map π A inherits from π the properties (MON) and (QCO). Under the assumptions of Th. 2.9 we get, from Proposition 2.5, that π is (CFB) and this obviously implies that π A is (CFB). Applying to π A Proposition 2.5, which holds also for real valued maps, we deduce that π A is σ(l F, L F )-(LSC).

13 DUAL REPRESENTATION OF QUASICONVEX CONDITIONAL MAPS 13 Under the assumptions of Th we prove that π A is τ-(usc) by showing that c R the set B c := {ξ L F π A (ξ) < c} is τ open. W.l.o.g. B c. If we fix an arbitrary η B c, we may find δ > 0 s.t. π A (η) < c δ. Define B := {ξ L F π(ξ) < (c δ)1 A + (π(η) + δ)1 A C }. Since (c δ)1 A + (π(η) + δ)1 A C L G and π is (USC) we deduce that B is τ open. Moreover π A (ξ) c δ for every ξ B, i.e., B B c, and η B since π(η) < c δ on A and π(η) < π(η) + δ on A C. Applying Theorem 1.1 in the (LSC) case and Theorem 5.4 in the (USC) one then deduce the representation of π A : π A (X) = sup sup inf {π A (ξ) E Q [ξ] E Q [X]} inf {π A (ξ) E Q [ξ G] Q E Q [X G]} π A (X). Notice that in case π A is (USC) the sup can be replaced by a max. 4. Proofs of the main results. Notations: In the following, we will only consider finite partitions Γ = { A Γ} of G measurable sets A Γ Γ and we set π Γ (X) : = A Γ Γ π A Γ(X)1 A Γ, K Γ (X, Q) : = inf { π Γ (ξ) E Q [ξ G] Q E Q [X G] } H Γ (X) : = sup K Γ (X, Q) 4.1. Outline of the proof. We anticipate an heuristic sketch of the proof of Theorem 2.9 and Theorem 2.10, pointing out the essential arguments involved in it and we defer to the following section the details and the rigorous statements. Unfortunately, we cannot prove directly that ε > 0, Q ε L F P s.t. {ξ L F E Qε [ξ G] Qε E Qε [X G]} {ξ L F π(ξ) > π(x) ε} (4.1) relying on Hahn-Banach Theorem, as it happened in the real case (see the proof of Theorem 1.1, equation (5.26), in Appendix). Indeed, the complement of the set in the right hand side of (4.1) is not any more a convex set - unless π is real valued - regardless of the continuity assumption made on π. Also the idea applied in the conditional convex case [DS05] can not be used here, since the map X E P [π(x)] there adopted preserves convexity but not quasiconvexity. Then our method is to apply an approximation argument and the choice of approximating π( ) by π Γ ( ), is forced by the need to preserve quasiconvexity. I The first step is to prove (see Proposition 4.4) that: H Γ (X) = π Γ (X). This is based on the representation of the real valued quasiconvex map π A in Proposition 3.7. Therefore, the assumptions (MON), (REG), (QCO) and (LSC) or (USC) on π are here all needed. II Then it is a simple matter to deduce π(x) = inf Γ π Γ (X) = inf Γ H Γ (X), where the inf is taken with respect to all finite partitions. III As anticipated in (1.3), the last step, i.e., proving that inf Γ H Γ (X) = H(X), is more delicate. It can be shown easily that is possible to approximate H(X)

14 14 M. FRITTELLI AND M. MAGGIS with K(X, Q ε ) on a set A ε of probability arbitrarily close to 1. However, we need the following uniform approximation: For any ε > 0 there exists Q ε L F P such that for any finite partition Γ we have HΓ (X) K Γ (X, Q ε ) < ε on the same set A ε. This key approximation result, based on Lemma 4.3, shows that the element Q ε does not depend on the partition and allows us (see equation (4.7)) to conclude the proof Details. The following two lemmas are straightforward applications of measure theory; the third is the already mentioned key result and it is proved in the Appendix, for it needs a pretty long argument. Lemma 4.1. For every Y L 0 G there exists a sequence Γ(n) of finite partitions such that Γ(n) (sup A Y ) 1 Γ(n) A converges in probability, and P-a.s., to Y, so that Γ(n) for any ε, δ > 0 we may find N such that for Γ = Γ(N) we have: P { ω Ω A Γ Γ ( sup Y A Γ ) 1 A Γ(ω) Y (ω) < δ Lemma 4.2. For each X L F and Q L F P inf Γ KΓ (X, Q) = K(X, Q) } > 1 ε. (4.2) where the infimum is taken with respect to all finite partitions Γ. Lemma 4.3. Let X L F and let P and Q be arbitrary elements of L F P. Suppose there exists ε 0 and B G s.t.: K(X, P )1 B >, π B (X) < + and K(X, Q)1 B K(X, P )1 B + ε1 B. Then for every partition Γ = {B C, Γ}, where Γ is a partition of B, we have K Γ (X, Q)1 B K Γ (X, P )1 B + ε1 B. Since π Γ assumes only a finite number of values, we may apply Proposition 3.7 and deduce the dual representation of π Γ. Proposition 4.4. Suppose that the assumptions of Theorem 2.9 or of Theorem 2.10 hold true and Γ is a finite partition. If for every X L F, π(x) < c with c [0, + ), then: H Γ (X) = π Γ (X) π(x) (4.3) and therefore inf Γ H Γ (X) = π(x). Proof. First notice that K Γ (X, Q) H Γ (X) π Γ (X) < + for all Q L F P. Consider the sigma algebra G Γ := σ(γ) G, generated by the finite partition Γ. From Proposition 3.7 we then have for every A Γ Γ π A Γ(X) = sup inf {π A Γ(ξ) E Q [ξ G] Q E Q [X G]}. (4.4) Moreover H Γ (X) is constant on A Γ since it is G Γ -measurable as well. Using the fact that π Γ ( ) is constant on each A Γ, for every A Γ Γ we then have: H Γ { (X)1 A Γ = sup inf π Γ (ξ)1 A Γ E Q [ξ G] Q E Q [X G] } = sup inf {π A Γ(ξ)1 A Γ E Q [ξ G] Q E Q [X G]} = π A Γ(X)1 A Γ = π Γ (X)1 A Γ (4.5)

15 DUAL REPRESENTATION OF QUASICONVEX CONDITIONAL MAPS 15 where the first equality in (4.5) follows from (4.4). consequence of (4.3) and Lemma 4.1 The remaining statement is a Proof. [of Theorems 2.9 and 2.10] Obviously π(x) H(X), since X satisfies the constraints in the definition of H(X). Step 1. First we assume that π is uniformly bounded, i.e., there exists c > 0 such that for all X L F π(x) c. Then H(X) >. From Lemma 3.6, eq. (3.6), there exists a sequence Q k L F P s.t. K(X, Q k ) H(X), as k. Therefore, for any ε > 0 we may find Q ε L F P and A ε G, P(A ε ) > 1 ε s.t. Since H(X) K(X, Q) Q L F P, H(X)1 Aε K(X, Q ε )1 Aε ε1 Aε. (K(X, Q ε ) + ε)1 Aε K(X, Q)1 Aε Q L F P. This is the basic inequality that enable us to apply Lemma 4.3, replacing there P with Q ε and B with A ε. Only notice that sup Ω π(x) c and K(X, Q) > for every Q L F P. This Lemma assures that for every partition Γ of Ω (K Γ (X, Q ε ) + ε)1 Aε K Γ (X, Q)1 Aε Q L F P. (4.6) From the definition of essential supremum of a class of random variables equation (4.6) implies that for every Γ (K Γ (X, Q ε ) + ε)1 Aε sup K Γ (X, Q)1 Aε = H Γ (X)1 Aε. (4.7) Since π Γ c, applying Proposition 4.4, equation (4.3), we get (K Γ (X, Q ε ) + ε)1 Aε π(x)1 Aε. Taking the infimum over all possible partitions, as in Lemma 4.2, we deduce: Hence, for any ε > 0 (K(X, Q ε ) + ε)1 Aε π(x)1 Aε. (4.8) (K(X, Q ε ) + ε)1 Aε π(x)1 Aε H(X)1 Aε K(X, Q ε )1 Aε which implies π(x) = H(X), since P(A ε ) 1 as ε 0. Step 2. Now we consider the case when π is not necessarily bounded. We define the new map ψ( ) := arctan(π( )) and notice that ψ(x) is a G-measurable r.v. satisfying ψ(x) Π 2 for every X L F. Moreover ψ is (MON), (QCO) and ψ(x1 G ) = ψ(x)1 G for every G G. In addition, ψ inherits the (LSC) (resp. the (USC) ) property from π. The first is a simple consequence of (CFB) of π. For the second we may apply Lemma 4.5 below. The map ψ is uniformly bounded and by step 1 we may conclude ψ(x) = H ψ (X) := sup K ψ (X, Q), where

16 16 M. FRITTELLI AND M. MAGGIS K ψ (X, Q) := inf {ψ(ξ) E Q [ξ G] Q E Q [X G]}. Applying again Lemma 3.6, equation (3.6), there exists Q k L F such that H ψ (X) = lim k K ψ (X, Q k ). We will show below that Admitting this, we have for P-almost every ω Ω K ψ (X, Q k ) = arctan K(X, Q k ). (4.9) arctan(π(x)(ω)) = ψ(x)(ω) = H ψ (X)(ω) = lim k K ψ (X, Q k )(ω) = lim k arctan K(X, Q k )(ω)) = arctan(lim k K(X, Q k )(ω)), where we used the continuity of the function arctan. This implies π(x) = lim k K(X, Q k ) and we conclude: π(x) = lim k K(X, Q k ) H(X) π(x). It only remains to show (4.9). We prove that for every fixed Q L F P K ψ (X, Q) = arctan (K(X, Q)). Since π and ψ are regular, from Lemma 3.5 iv), there exist ξ Q h L F and η Q h L F such that E Q [ξ Q h G] Q E Q [X G], E Q [η Q h G] Q E Q [X G], h 1, (4.10) ψ(ξ Q h ) K ψ(x, Q) and π(η Q h ) K(X, Q), as h. From (4.10) and the definitions of K(X, Q), K ψ (X, Q) and by the continuity and monotonicity of arctan we get: K ψ (X, Q) lim h ψ(η Q h ) = lim h arctan π(ηq h ) = arctan lim h π(ηq h ) = arctan K(X, Q) arctan lim h π(ξ Q h ) = lim h ψ(ξq h ) = K ψ(x, Q). Let Y be G-measurable and define: A := {ξ L F π(ξ) < tan(y )} B := {ξ L F arctan(π(ξ)) < Y }, where tan(y ) = + on { } { } Y π 2 and tan(y ) = on Y π 2. Notice that A = { ξ B π(ξ) < on { }} Y > π 2. Lemma 4.5. Suppose that π is regular and there exists θ L F such that π(θ) < +. For any G-measurable random variable Y, if A is open then also B is open. Proof. We may assume Y π 2, otherwise B =. Let ξ B, θ L F such that π(θ) < +. Define ξ 0 := ξ1 {Y π 2 } + θ1 {Y > π 2 }. Then ξ 0 A. Since A is open, we may find a neighborhood U of 0 such that: ξ 0 + U A. Define: V := (ξ 0 + U)1 {Y π 2 } + (ξ + U)1 {Y > π 2 } = ξ + U1 {Y π 2 } + U1 {Y > π 2 }.

17 DUAL REPRESENTATION OF QUASICONVEX CONDITIONAL MAPS 17 It is easy to show that ξ V and V B. Finally V is a neighborhood of ξ, since the set U1 {Y π 2 } + U1 {Y > π 2 } contains U and therefore is a neighborhood of 0. Proof. [of Proposition 2.13] The if part is trivial, as the various properties are easy to check. For the only if part we already know from Theorem 2.9 that ( [ ] ) dq π(x) = sup R E dp X G, Q, where R is defined in [ (2.2). For every Q L F P we consider R+ (, Q) R(, Q) and denote X Q = E dq dp ]. X G We observe that π(x) δ>0 sup R + (X Q, Q) = sup sup sup R(X Q δ, Q) X Q δ<x Q sup R(Y, Q) Y <X Q = sup sup R(E P [(X δ) dq/dp G], Q) = sup π(x δ) δ>0 and so we have the representation π(x) = δ>0 sup R + (E Q [X G], Q) (CF B) = π(x) and we already know from Lemma 3.3 that R + R. Remark 4.6. Take Q P such that Q P on G and define the probability [ dp Q(F ) := E Q dq G 1 F ] where G dp dq [ dp =: E Q G dq ], F F. Then Q(G) = P(G) for all G G, and so Q P G. Moreover, it is easy to check that for all X L F and Q L F P such that Q P on G we have: E Q[X G] = E Q [X G] which implies K(X, Q) = K(X, Q). Proof. [of Corollary 2.11] Consider the probability Q ε L F P built up in Theorem 2.9, satisfying equation (4.8). We claim that Q ε is equivalent to P on A ε. By contradiction there exists B G, B A ε, such that P(B) > 0 but Q ε (B) = 0. Consider η L F, δ > 0 such that P(π(η)+δ < π(x)) = 1 and define ξ = X1 B C +η1 B so that E Qε [ξ G] Qε E Qε [X G]. By regularity π(ξ) = π(x)1 B C + π(η)1 B which implies for P-a.e. ω B π(ξ)(ω) + δ = π(η)(ω) + δ < π(x)(ω) K(X, Q ε )(ω) + ε π(ξ)(ω) + ε which is impossible for ε δ. So Q ε P on A ε for all small ε δ. Consider Q ε such that d Q ε dp = dqε dp 1 A ε + dp dp 1 (A ε) C. Up to a normalization factor Q ε L F P and is equivalent to P. Moreover from Lemma 3.5 (vi), K(X, Q ε )1 Aε = K(X, Q ε )1 Aε and from Remark 4.6 we may define Q ε P G such that K(X, Q ε )1 Aε =

18 18 M. FRITTELLI AND M. MAGGIS K(X, Q ε )1 Aε = K(X, Q ε )1 Aε. From (4.8) we finally deduce: K(X, Q ε )1 Aε + ε1 Aε π(x)1 Aε, and the thesis then follows from Q ε P G. Proof. [of Corollary 2.14] First we prove equation (2.8): let us denote with k(x, Q) the right hand side of equation (2.8) and notice that K(X, Q) k(x, Q). By contradiction, suppose that P(A) > 0 where A =: {K(X, Q) < k(x, Q)}. As shown in Lemma 3.5 iv), there exists a random variable ξ L F s.t.: E Q [ξ G] Q E Q [X G] and Q(E Q [ξ G] > E Q [X G]) > 0. K(X, Q)(ω) π(ξ)(ω) < k(x, Q)(ω) for P-a.e. ω B A and P(B) > 0. Set Z = Q E Q [ξ X G]. By assumption, Z L F and it satisfies Z Q 0 and, since Q P G, Z 0. Then, thanks to (MON), π(ξ) π(ξ Z). From E Q [ξ Z G] = Q E Q [X G] we deduce the contradiction: K(X, Q)(ω) π(ξ)(ω) < k(x, Q)(ω) π(ξ Z)(ω) for P-a.e. ω B. Second we show (2.9). From (2.8) we deduce: K(X, Q) = inf {π(ξ) E Q [ξ G] = Q E Q [X G]} = E Q [X G] + inf {π(ξ) E Q [X G] E Q [ξ G] = Q E Q [X G]} = E Q [X G] + inf {π(ξ) E Q [ξ G] E Q [ξ G] = Q E Q [X G]} = E Q [X G] sup {E Q [ξ G] π(ξ) E Q [ξ G] = Q E Q [X G]} = E Q [X G] π (Q), where the last equality follows from Q P G and π (Q) = sup {E Q [ξ + E Q [X ξ G] G] π(ξ + E Q [X ξ G])} = sup η L F {E Q [η G] π(η) η = ξ + E Q [X ξ G]} sup {E Q [ξ G] π(ξ) E Q [ξ G] = Q E Q [X G]} π (Q). (ii) The (CAS) property implies that for every X L F and δ > 0, P(π(X 2δ) + δ < π(x)) = 1. So the hypothesis of Corollary 2.11 holds true and (2.10) is a consequence of (2.9) and (2.6). As noticed in [DS05] and in [CDK06] when L F = L F, the assumption that π is (REG) is not necessary, as (CAS) and (MON) or (CAS) and convexity already imply regularity. 5. Appendix Proof of the key approximation Lemma 4.3. We will adopt the following notations: If Γ 1 and Γ 2 are two finite partitions of G-measurable sets then Γ 1 Γ 2 := {A 1 A 2 A i Γ i, i = 1, 2} is a finite partition finer than each Γ i. Lemma 5.1 is the generalization of Lemma 3.1 to the approximated problem. Lemma 5.1. For every partition Γ, X L F and Q L F P, the set A Γ Q(X) {π Γ (ξ) ξ L F and E Q [ξ G] Q E Q [X G]} is downward directed. This implies that there exists exists a sequence { η Q m} m=1 L F, depending also on Γ, such that E Q [η Q m G] Q E Q [X G] m 1, π Γ (η Q m) K Γ (X, Q) as m.

19 DUAL REPRESENTATION OF QUASICONVEX CONDITIONAL MAPS 19 Proof. To show that the set A Γ Q (X) is downward directed we use the notations and the results in the proof of Lemma 3.1 and check that π Γ (ξ ) = π Γ (ξ 1 1 G + ξ 2 1 G C ) min { π Γ (ξ 1 ), π Γ (ξ 2 ) }. For any given sequence of partition there exists one sequence that works for all: Lemma 5.2. For any fixed, at most countable, family of partitions {Γ(h)} h 1 and Q L F P, there exists a sequence { ξ Q m} m=1 L F such that E Q [ξ Q m G] Q E Q [X G] for all m 1 π(ξ Q m) K(X, Q) as m and for all h π Γ(h) (ξ Q m) K Γ(h) (X, Q) as m. Proof. Apply Lemma 3.1 and Lemma 5.1 and find {ϕ 0 m} m, {ϕ 1 m} m,..., {ϕ h m} m,... such that for every i and m we have E Q [ϕ i m G] Q E Q [X G] and π(ϕ 0 m) K(X, Q) as m and for all h π Γ(h) (ϕ h m) K Γ(h) (X, Q) as m. For each m 1 consider m i=0 π(ϕi m): then there will exists a (non unique) finite partition of Ω, {Fm} i m i=1 such that m m π(ϕ i m) = π(ϕ i m)1 F i m. i=0 i=0 Denote ξm Q m =: i=0 ϕi m1 F i and notice that m m i=0 π(ϕi (REG) m)1 F i m = π ( ξm) Q and E Q [ξm G] Q Q E Q [X G] for every m. Moreover π(ξm) Q is decreasing and π(ξm) Q π(ϕ 0 m) implies π(ξm) Q K(X, Q). For any fixed h, π(ξm) Q π(ϕ h m) for all h m and: π Γ(h) (ξ Q m) π Γ(h) (ϕ h m) implies π Γ(h) (ξ Q m) K Γ(h) (X, Q) as m. Finally, we state the basic step used in the proof of Lemma 4.3. Lemma 5.3. Let X L F and let P and Q be arbitrary elements of L F P. Suppose there exists ε 0 and B G s.t.: K(X, P )1 B >, π B (X) < + and K(X, Q)1 B K(X, P )1 B + ε1 B. Then for any δ > 0 and any partition Γ 0 there exists Γ Γ 0 for which K Γ (X, Q)1 B K Γ (X, P )1 B + ε1 B + δ1 B Proof. By our assumptions we have: < K(X, P )1 B π B (X) < + and K(X, Q)1 B π B (X) < +. Fix δ > 0 and the partition Γ 0. Suppose by contradiction that for any Γ Γ 0 we have P(C) > 0 where C = {ω B K Γ (X, Q)(ω) > K Γ (X, P )(ω) + ε + δ}. (5.1)

20 20 M. FRITTELLI AND M. MAGGIS Notice that C is the union of a finite number { of} elements in the partition Γ. Lemma 3.5 guarantees the existence of ξ Q h L F satisfying: π(ξ Q h ) K(X, Q), as h,, E Q[ξ Q h G] Q E Q [X G] h 1. (5.2) Moreover, for each partition Γ and h 1 define: { Dh Γ := ω Ω π Γ (ξ Q h )(ω) π(ξq h )(ω) < δ } G, 4 h=1 and observe that π Γ (ξ Q h ) decreases if we pass to finer partitions. From Lemma 4.1 equation (4.2), we deduce that for each h 1 there exists a partition( Γ(h) such ) that ( ) h P D Γ(h) h 1 1. For every h 1 define the new partition Γ(h) = Γ(h) Γ 2 h 0 j=1 ( ) so that for all h 1 we have: Γ(h + 1) Γ(h) Γ 0, P D Γ(h) h 1 1 and 2 h ( π(ξ Q h ) + δ ) ) 1 Γ(h) 4 D (π Γ(h) (ξ Qh ) 1 Γ(h) h D, h 1. (5.3) h Lemma 5.2 guarantees that for the fixed sequence of partitions {Γ(h)} h 1, there exists a sequence { ξ P m} m=1 L F, which does not depend on h, satisfying For each m 1 and Γ(h) define: C Γ(h) m := E P [ξ P m G] P E P [X G] m 1, (5.4) π Γ(h) (ξ P m) K Γ(h) (X, P ), as m, h 1. (5.5) { ω C π Γ(h) (ξ P m)(ω) K Γ(h) (X, P )(ω) δ 4 } G. Since the expressions in the definition of Cm Γ(h) assume only a finite number of values, from (5.5) and from our assumptions, which imply that K Γ(h) (X, P ) K(X, P ) > ( on B, we deduce ) that for each Γ(h) there exists an index m(γ(h)) such that: P C \ C Γ(h) m(γ(h)) = 0 and Set E h = D Γ(h) h ( K Γ(h) (X, P )1 Γ(h) C π Γ(h) (ξm(γ(h)) P ) δ ) 1 Γ(h) m(γ(h)) 4 C, h 1. (5.6) m(γ(h)) C Γ(h) m(γ(h)) G and observe that From (5.3) and (5.6) we then deduce: ( π(ξ Q h ) + δ 4 K Γ(h) (X, P )1 Eh 1 Eh 1 C P a.s. (5.7) ) ) 1 Eh (π Γ(h) (ξ Qh ) ( π Γ(h) (ξ P m(γ(h)) ) δ 4 1 Eh, h 1, (5.8) ) 1 Eh, h 1. (5.9)

21 DUAL REPRESENTATION OF QUASICONVEX CONDITIONAL MAPS 21 We then have for any h 1 π(ξ Q h )1 E h + δ 4 1 E h (π Γ(h) (ξ Qh ) ) 1 Eh (5.10) K Γ(h) (X, Q)1 Eh (5.11) ( ) K Γ(h) (X, P ) + ε + δ 1 Eh (5.12) (π Γ(h) (ξ Pm(Γ(h)) ) δ4 ) + ε + δ 1 Eh (5.13) (π(ξ Pm(Γ(h)) ) + ε + 34 ) δ 1 Eh. (5.14) (in the above chain of inequalities, (5.10) follows from (5.8); (5.11) follows from (5.2) and the definition of K Γ(h) (X, Q); (5.12) follows from (5.1); (5.13) follows from (5.9); (5.14) follows from the definition of the maps π A Γ(h)). Recalling (5.4) we then get, for each h 1, π(ξ Q h )1 E h ( π(ξm(γ(h)) P ) + ε + δ ) ( 1 Eh K(X, P ) + ε + δ ) 1 Eh >. (5.15) 2 2 From equation (5.2) and (5.7) we have π(ξ Q h )1 E h K(X, Q)1 C P-a.s. as h and so from (5.15) ( 1 C K(X, Q) = lim π(ξ Q h )1 E h lim 1 Eh K(X, P ) + ε + δ ) ( = 1 C K(X, P ) + ε + δ ) h h 2 2 which contradicts the assumption of the Lemma, since C B and P(C) > 0. Proof. [of Lemma 4.3] First notice that the assumptions of this Lemma are those of Lemma 5.3. Assume by contradiction that there exists Γ 0 = {B C, Γ 0 }, where Γ 0 is a partition of B, such that P(ω B K Γ0 (X, Q)(ω) > K Γ0 (X, P )(ω) + ε) > 0. (5.16) By our assumptions we have K Γ0 (X, P )1 B K(X, P )1 B > and K Γ0 (X, Q)1 B π B (X)1 B < +. Since K Γ0 is constant on every element A Γ0 Γ 0, we denote with K AΓ 0 (X, Q) the value that the random variable K Γ0 (X, Q) assumes on A Γ0. From (5.16) we deduce that there exists ÂΓ0 B, Â Γ0 Γ 0, such that Let then d > 0 be defined by + > KÂΓ 0 (X, Q) > KÂΓ 0 (X, P ) + ε >. d =: KÂΓ 0 (X, Q) KÂΓ 0 (X, P ) ε. (5.17) Apply Lemma 5.3 with δ = d 3 : then there exists Γ Γ 0 (w.l.o.g. Γ = {B C, Γ} where Γ Γ 0 ) such that K Γ (X, Q)1 B ( K Γ (X, P ) + ε + δ ) 1 B. (5.18)

22 22 M. FRITTELLI AND M. MAGGIS Considering only the two partitions Γ and Γ 0, we may apply Lemma 5.2 and conclude that there exist two sequences {ξ P h } h=1 L F and {ξ Q h } h=1 L F satisfying as h : E P [ξ P h G] P E P [X G], π Γ0 (ξ P h ) K Γ0 (X, P ), π Γ (ξ P h ) K Γ (X, P ) (5.19) E Q [ξ Q h G] Q E Q [X G], π Γ0 (ξ Q h ) KΓ0 (X, Q), π Γ (ξ Q h ) KΓ (X, Q) (5.20) Since K Γ0 (X, P ) is constant and finite on ÂΓ0, from (5.19) we find h 1 1 s.t. πâγ 0 (ξ P h ) KÂΓ 0 (X, P ) < d 2, h h 1. (5.21) From equation (5.17) and (5.21) we deduce that πâγ 0 (ξh P ) < KÂΓ 0 (X, P ) + d 2 = KÂΓ 0 (X, Q) ε d + d 2, h h 1, and therefore, knowing from (5.20) that KÂΓ 0 (X, Q) πâγ 0 (ξ Q h ), πâγ 0 (ξ P h ) + d 2 < π Â Γ 0 (ξq h ) ε h h 1. (5.22) We now take into account all the sets A Γ ÂΓ0 B. For the convergence of π A Γ(ξ Q h ) we distinguish two cases. On those sets A Γ for which K AΓ (X, Q) > we may find, from (5.20), h 1 such that Then using (5.18) and (5.19) we have π A Γ(ξ Q h ) KAΓ (X, Q) < δ 2 h h. so that π A Γ(ξ Q h ) < KAΓ (X, Q) + δ 2 KAΓ (X, P ) + ε + δ + δ 2 π A Γ(ξP h ) + ε + δ + δ 2 π A Γ(ξ Q h ) < π A Γ(ξP h ) + ε + 3δ 2 h h. On the other hand, on those sets A Γ for which K AΓ (X, Q) = the convergence (5.20) guarantees the existence of ĥ 1 for which we obtain again: π A Γ(ξ Q h ) < π A Γ(ξP h ) + ε + 3δ 2 h ĥ (5.23) (notice that K Γ (X, P ) K(X, P )1 B > and (5.19) imply that π A Γ(ξh P ) converges to a finite value, for A Γ B). Since the partition Γ is finite there exists h 2 1 such that equation (5.23) stands for every A Γ ÂΓ0 and for every h h 2 and for our choice of δ = d 3 (5.23) becomes π A Γ(ξ Q h ) < π A Γ(ξP h ) + ε + d 2 h h 2 A Γ ÂΓ0. (5.24) Fix h > max{h 1, h 2 } and consider the value πâγ 0 (ξ Q h ). Then among all A Γ ÂΓ0 we may find B Γ ÂΓ0 such that π B Γ(ξ Q h ) = πâγ 0 (ξ Q h ). Thus: πâγ 0 (ξ Q h ) = π B Γ(ξ Q h ) (5.24) < π B Γ(ξh P ) + ε + d 2 π Â Γ 0 (ξp h ) + ε + d (5.22) < πâγ 2 0 (ξ Q h ). which is a contradiction.