On the extension of the Namioka-Klee theorem and on the Fatou property for Risk Measures

Transcription

1 On the extensin f the Namika-Klee therem and n the Fatu prperty fr Risk Measures Sara Biagini, Marc Frittelli In: Optimality and risk: mdern trends in mathematical nance. The Kabanv Festschrift (2009) Editrs: F. Delbaen, M. Rasnyi, Ch. Stricker Abstract This paper has been mtivated by general cnsideratins n the tpic f Risk Measures, which essentially are cnvex mntne maps de ned n spaces f randm variables, pssibly with the s-called Fatu prperty. We shw rst that the celebrated Namika-Klee therem fr linear, psitive functinals hlds als fr cnvex mntne maps n Frechet lattices. It is well-knwn amng the specialists that the Fatu prperty fr risk measures n L 1 enables a simpli ed dual representatin, via prbability measures nly. The Fatu prperty in a general framewrk f lattices is nthing but the lwer rder semicntinuity prperty fr. Our secnd gal is thus t prve that a similar simpli ed dual representatin hlds als fr rder lwer semicntinuus, cnvex and mntne functinals de ned n mre general spaces X (lcally cnvex Frechet lattices). T this end, we identify a link between the tplgy and the rder structure in X - the C-prperty - that enables the simpli ed representatin. One main applicatin f these results leads t the study f cnvex risk measures de ned n Orlicz spaces and f their dual representatin. Acknwledgements The rst authr wuld like t thank B. Rudl, P. Cheridit and A. Hamel fr sme discussins while she was visiting the ORFE Department at Princetn University. The secnd authr wuld like t thank Marc Maggis, PhD student at Milan University fr helpful discussin n this subject. 1 Intrductin The analysis in this paper was triggered by recent develpments in the thery f Risk Measures in Mathematical Finance. Cnvex risk measures were 1

2 independently intrduced by [FS02] and [FR02] as generalizatin f the cncept f a cherent risk measure develped in [ADEH99]. Cnsider a space f nancial psitins X (real-valued, measurable functins n a xed measurable space (; F)) cntaining the cnstants. A cnvex risk measure n X is a map : X! ( 1; +1] with the fllwing prperties: 1. (0) = 0 (s is prper, i.e. it des nt cincide with +1) 2. mntncity: if X; Y 2 X ; X Y, then (X) (Y ) 3. cnvexity: if 2 [0; 1], then (X + (1 )Y ) (X) + (1 )(Y ) fr any X; Y 2 X 4. cash additivity: if m 2 R then (X + m) = (X) m fr any X 2 X When is als psitive hmgeneus, i.e. 5. (X) = (X) fr all 0 the risk measure is called cherent. If X is als tplgical space (as it is always the case in the applicatins), it is f curse useful t have a result n the degree f smthness f the risk measure. Strangely enugh, when this paper was rst written t ur knwledge there was yet n general result. This is exactly the message f the extended Namika- Klee Therem, stated belw in Therem 1. The (tplgical vectr) space f psitins X hwever must have sme ther prperties, i.e. it must be a Frechet lattice. Recall that a tplgical vectr space (X ; ) is a Frechet lattice if: its tplgy is induced by a cmplete distance d X is a lattice, that is it has an rder structure (X ; ) and each pair X 1 ; X 2 2 X has a supremum X 1 _ X 2 in X X is lcally slid, that is the rigin 0 has a fundamental system f slid neighbrhds (a neighbrhd U f 0 is slid if fr any X 2 U, Y 2 X ; jy j jxj ) Y 2 U where jxj = X _ ( X)). Nte that a Frechet lattice is nt necessarily lcally cnvex. Examples f cmmn Frechet lattices are the spaces L p n a prbability space (; F; P ), fr p 2 [0; 1) (with the natural, a.s. pintwise rder). When p 1, L p is als Frechet lattice, but with an extra prperty. The tplgy is induced by the L p -nrm (and thus the space becmes lcally cnvex). Mrever, the nrm has a mntnicity prperty: jxj jy j ) kxk p ky k p. S L p ; p 2 [1; +1] is in 2

3 fact a Banach lattice. Other Banach lattices imprtant fr ur applicatins belng t the family f Orlicz spaces - dented with L fr a Yung functin - which are described in details in Sectin 5. Finally, we present the abstract statement f the extended Namika-Klee Therem, prved in Sectin 2 (where there is als an extensive cmparisn with the existing literature, as we discvered that there are a cuple f recent, very similar results). The Therem is stated fr cnvex, mntne increasing maps, nt necessarily cash additive. But a similar result clearly hlds fr mntne decreasing maps. Dm() indicates here and in the rest f the paper the subset f X where is nite. The tplgical dual space is dented by X 0 and the set X+ 0 indicates the cnvex cne f thse functinals Y in X 0 that are psitive, i.e. hy; Xi 0 fr all X 2 X ; X 0. The symbl h; i indicates the bilinear frm fr the duality (X ; X 0 ). The map : X 0! ( 1; +1] is the cnvex cnjugate f, als knwn as Fenchel transfrm, and it is de ned as (Y ) = sup fhy; Xi X2X (X)g: Therem 1 (Extended Namika-Klee) Any prper cnvex and mntne increasing functinal : X! ( 1; +1] n a Frechet lattice (X ; ) is cntinuus and subdi erentiable n int(dm()) (the interir f Dm()). Mrever, it admits a dual representatin as (X) = max fhy; Xi (Y )g 8X 2 int(dm()): (1) Y 2X+ 0 T give an idea abut the genesis f the secnd and mst innvative part f the paper, let us g back t the nancial setup and let us fcus rst n the case L 1. A risk measure n L 1 has the pleasant prperty f being always nitevalued, thanks t the bundedness f its elements and t the mntnicity and cash additivity prperty. The therem abve ensures that is cntinuus and subdi erentiable n the entire L 1. This implies the existence f a well-knwn dual representatin fr ver L 1, namely (X) = max fe Q[ X] (Q)g (2) Q2M 1;f (P ) where: a) M 1;f (P ) indicates the set f psitive, nitely additive measures Q n (; F) that are abslutely cntinuus w.r. t P and are nrmalized (Q(I ) = 1); b) is the cnvex cnjugate f and shuld be interpreted as a penalty functinal. These results are knwn and prved e.g. [FS04, Therem 4.12]), where subdi erentiability is prved by hand fr the speci c case study L 1. Nw, let us recall the de nitin f the Fatu prperty fr risk measures (see e.g. [De00] r Sectin 4, [FS04], in the case L 1 ): 3

4 De nitin 2 (Fatu prperty) A risk measure : L p! R[ f1g, p 2 [1; 1]; has the Fatu prperty (F.P.) if given any sequence fx n g n dminated in L p and cnverging P a.s. t X we have: (X) lim inf n!1 (X n): This prperty enables a simpli ed dual representatin f. Instead f the nitely additive measures M 1;f (P ), if a cnvex risk measure : L 1! R has the F.P. ne can write (X) = sup fe Q [ X] (Q)g; (3) Q prbability; QP s the supremum can be taken nly ver prbabilities, the additive elements f M 1;f, see [FS04, Therem 4.26]. There is a price t pay: the abve supremum may nt be attained ver prbability measures, but nly n M 1;f. What can be said abut the representatin prblem f a cnvex risk measures de ned n subspaces f L 0? The spaces L p are typical examples f spaces f nancial psitins in the applicatins (see fr example, [FR02], [FS07], [RK08]). Mrever, in [BF08] it is shwn that Orlicz spaces that can be assciated with a utility functin are the right framewrk fr the utility maximizatin prblems which cmmnly arises in nancial prblems. Mtivated by this idea, in the rst versin f this paper 1, we initiated the study f risk measures de ned n Orlicz spaces L and mre generally n Frechet lattices. Independently, in [CL07] cnvex risk measures were de ned n the Mrse subspace M f the Orlicz space L. As we shall see in Sectin 5 sme f the ndings in [CL07] are special case ur results, while ther prperties d nt hld in the general case, essentially because the tplgy n the (whle) Orlicz space L is nt rder cntinuus. Our generalizatin f the representatin in (3) and its varius implicatins will be stated fr maps, de ned f general Frechet lattice, that are cnvex, mntne and increasing, nt necessarily translatin invariant. This latter implies that the set f dual variables ver which the supremum - r the maximum - is taken will nt be nrmalized in general. T begin with, let us recall a few ntins abut Riesz spaces, i.e. linear spaces that are lattices (see als Sectin 3). The rst is that f rder cnvergence. A generalized sequence, r net, (X ) in a Riesz space R is rder cnvergent t sme X 2 R, ntatin X! X; if there is a net (Z ) in R satisfying Z # 0 and jx Xj Z fr each (4) 1 Presented by the secnd authr at the Wrkshp n Risk Measures, University f Evry, France, July and at the Cnference n Risk Measures and Rbust Cntrl in Finance, The Bendheim Center, Princetn University, Octber 6-7,

5 (Z # 0 means that (Z ) is mntne decreasing and its in mum is 0). A functinal f : R! R de ned n R is rder cntinuus if X! X ) f(x )! f(x); and a tplgy n R is rder cntinuus if X! 0 ) X! 0: These de nitins readily imply that if the tplgy is rder cntinuus then f is (tplgically) cntinuus ) f is rder cntinuus. We dente with (Xn ) the cne f rder cntinuus linear functinals n X. By the classic Namika Therem, see Sectin 2., (Xn ) + cincides with X+, 0 the psitive elements f tplgical dual space X 0. We recall that in a Banach lattice X the nrm tplgy is rder cntinuus if and nly if Xn = X 0 and that the fllwing three classes f spaces all have rder cntinuus nrm: (a) L p when p 2 [1; +1); (b) M when is a nite valued Yung functin; (c) L when is a Yung functin satisfying the 2 cnditin (in this case L = M ). The abve implicatin, tgether with Therem 1, readily imply the fllwing Crllary 3 If Frechet lattice (X ; ) has an rder cntinuus tplgy and : X! R is cnvex and mntne (increasing), is already rder cntinuus n X. Thus it admits a dual representatin as (X) = where (X n ) + = X 0 +. max Y 2(X n )+ fhy; Xi (Y )g ; X 2 X ; (5) a) In the speci c case X = L p ; p 2 [1; +1) the representatin abve becmes (X) = max Y 2(L q ) + fe[y X] (Y )g ; X 2 L p ; b) In the speci c case X = M and is a nite valued Yung functin, the representatin abve becmes (X) = max Y 2(L ) + fe[y X] (Y )g ; X 2 M ; (6) where is the cnjugate functin f. Hwever, the rder cntinuity f a tplgy is a strng assumptin, which is nt satis ed by e.g. L 1, r by L fr general Yung functins r by ther 5

6 Frechet lattices, as shwn in Sectin 5. Mrever, in general (X n ) + is nly a subspace f X 0 +. This is exactly what happens with L 1 : ((L 1 ) n ) + = L 1 + and (L 1 ) 0 + = L 1 + S + ; where S are the purely nitely additive measures. It is then natural t investigate whether admits a representatin n (X n ) + under cnditins, linking tplgy and rder structure, less restrictive than the rder cntinuity f the tplgy. As we shall see in Remark 17, when X = L p, p 2 [0; 1], the Fatu prperty cincides with rder lwer semicntinuity, which is the apprpriate cncept in the present general setting. De nitin 4 A functinal : R! ( 1; +1] de ned n a Riesz space R is rder lwer semicntinuus if X! X implies (X) lim inf (X ). Frm nw n, lcal cnvexity is needed and in what fllws the Frechet lattice X is als suppsed lcally cnvex. As a cnsequence f the Hahn-Banach therem in any lcally cnvex Frechet lattice if the prper, increasing cnvex map : X! ( 1; +1] is als (X ; X n ) lwer semicntinuus then (X) = sup fhy; Xi (Y )g ; X 2 X ; (7) Y 2(Xn )+ where is the cnvex cnjugate f. Therefre, as the pintwise supremum f a family f rder cntinuus functinals, is als rder lwer semicntinuus One culd then cnjecture that the cnverse always hlds true i.e. any rder lwer semicntinuus n any lcally cnvex Frechet lattice admits a representatin in terms f (Xn ) + ; as in (7) r, in lucky cases, (5). The cnjecture is nt true in general, see Example 6 at the end f this Intrductin. Prpsitin 24 cntains the main result f the paper, that is there exists an additinal assumptin "à la Kmlś", linking the tplgy and the rder structure, that enables the representatin ver (Xn ) +. De nitin 5 (C-prperty) A linear tplgy n a Riesz space has the C- prperty 2 if X! X implies the existence f a subsequence (Xn ) n and cnvex cmbinatins Z n 2 cnv(x n ; ) such that Z n! X. This prperty is quite reasnable, all the details are in Sectin 4. In particular, whenever a lcally cnvex Frechet lattice (X ; ) can be embedded in L 1 with a linear lattice embedding, then all the tplgies:, (X ; X 0 ) and (X ; X n ) 2 The "C" stands fr "cnvex cmbinatins"... 6

7 have the C-prperty. A relevant example f spaces with an assciated cllectin f tplgies (nrm, weak and (X ; X n )) satisfying the C-prperty is the family f Orlicz spaces (Sectin 5.1). In the case X = L 1, it is well knwn that the sup in (7) in general is nt a max, but the sup is attained under sme strnger cntinuity cnditin. In the general case, in Lemma 27 we shw that fr a nite valued cnvex increasing map which is rder upper semicntinuus the sup in (7) is indeed a max. Finally, in Sectin 5.2 we analyze cnvex risk measures de ned n Orlicz spaces and with values in R [ f+1g. This new setup allws fr an extensin f the knwn dual representatin n L 1 : We further prvide sme new results n the cnvex risk measures assciated t utility functins, as in the case f the entrpic risk measure. Example 6 (When the C-prperty fails) When the (X ; X n )- tplgy des nt satisfy the C-prperty, there may be rder l.s.c. cnvex functinals (even - cntinuus!) that are nt (X ; X n )- l.s.c. Take X = C([0; 1]), the Banach lattice f the cntinuus functins n [0; 1] with the supremum nrm and the pintwise rder. The dual X 0 cnsists f the Brel signed measures n [0; 1] and it is knwn (see e.g. [Za83, Example 87.5]) that there is n nn zer rder-cntinuus functinal in X 0. The tplgy (X ; X n ) = (X ; f0g) is therefre the indiscrete ne and clearly it desn t have the C-prperty. Cnsider then the cnvex, increasing best case functinal (X) = max t2[0;1] X(t) which is nite valued, s that the extended Namika-Klee Therem implies that is nrm cntinuus and subdi erentiable and it admits a representatin n X+ 0 as (X) = max fhy; Xi (Y )g Y 2X+ 0 But it evidently des nt admit a representatin n (Xn ) + = f0g because it is nt cnstant. T shw that is rder-l.s.c. let X! X and suppse by cntradictin that there exists a subnet (X ) (as a subnet, still rder cnvergent t X) such that (X) > lim (X ). Let t 2 argmax(x). Then and evidently (X ) X (t ) X(t ) = (X) > lim (X ) lim sup X (t ); 7

8 which cntradicts pintwise cnvergence. 2 The Extended Namika Therem The fllwing is the statement f the well-knwn Namika-Klee Therem in the case f linear functinals '. Therem 7 (Namika-Klee) Any linear and psitive functinal ' : X! R n a Frechet lattice X is cntinuus (see [Na57]). In rder t prvide a technically straightfrward, but quite relevant, extensin f Namika-Klee Therem t cnvex functinals ; the psitivity assumptin 0 Y ) 0 '(Y ) has t be replaced with the requirement that is mntne increasing X Y ) (X) (Y ) Mntnicity and psitivity are equivalent fr linear functinals, but it is straightfrward t see that fr cnvex mntnicity implies psitivity (assuming (0) = 0). And it is easy t prduce a psitive but nn-mntne cnvex map by taking (X) = jxj n X = R. S, while n ne hand ne relaxes the linearity assumptin, n the ther hand a strnger link with the rder structure is required. The prperties in the next Lemma are straightfrward cnsequences f the de nitins. Lemma 8 Let R be a Riesz space and let : R! ( increasing and (0) = 0: Then: 1; +1] be cnvex, i- (X) (X) ; 8 2 [0; 1]; 8X 2 R; ii- (X) (X); 8 2 ( 1; 0] [ [1 + 1); 8X 2 R; iii- j(x)j (jxj) 8X 2 R: Prf f Therem 1. Step 1: Cntinuity. The prf f Namika-Klee Therem (see e.g. [AB05, Therem 9.6]) can be adapted, in a straightfrward manner, t deal with the current weaker assumptins. We repeat the argument s that the paper is self- cntained. W.l..g. it can be assumed that the interir f Dm(), int(dm()), is nt empty that 0 2 int(dm()) and (0) = 0: Let B r be the centered pen ball f radius r > 0 in a metric that generates. Take any sequence fx n g n such that X n! 0: Fix r small enugh, s that B2r int(dm()). Then pick a 8

9 cuntable base fv n g n f slid neighbrhds f zer satisfying V 1 +V 1 B r and V n+1 + V n+1 V n fr each n: Then V n+1 V n B r fr each n: By passing t a subsequence f X n, ne can suppse X n 2 1 n V P n fr each n. Set Y n = n ijx i j and nte that Y n Y n+1 and njx n j Y n. In additin Y n+p Y n = n+p X i=n+1 ijx i j 2 V n+1 + V n+2 + ::: + V n+p V n : Therefre Y n 2 B r fr each n and fy n g n is a Cauchy sequence, s Y n! Y fr sme Y in X. Since Y n 2 B r, Y 2 B r B 2r int(dm()), (Y ) is nite. This Y is an upper bund fr the sequence (actually, sup n Y n = Y ). In fact, x any n: Since Y m Y n 2 X + fr each m n; the sequence fy m Y n : m ng in X + satis es Y m Y n! Y Yn, as m! 1: Since X + is -clsed ([AB05, Therem ]), Y Y n 2 X + fr each n: Hence Y n Y fr each n: Frm Lemma 8, j(x n )j (jx n j) 1 n (njx nj): By mntnicity f we derive j(x n )j 1 n (njx nj) 1 n (Y n) 1 n (Y )! 0 which shws that is cntinuus at zer and therefre is cntinuus n the whle int(dm()) ([AB05, Therem 5.43]). Step 2: Subdi erentiability. Fr all X 2 int(dm()) we must exhibit a subgradient Y 2 X 0, i.e. a Y such that (X) (X ) hy ; X X i fr all X 2 X : (8) T this end, again w.l..g. we can suppse X = 0 2 int(dm()) and (0) = 0. Then, the directinal derivative functinal D in 0 D(X) := lim t#0 (tx) t satis es D thanks t Lemma 8. It is nite valued and cnvex and thus the rst part f this prf implies that it is cntinuus. By the Hahn-Banach Therem (see e.g. [AB05, Therem 5.53]) there exists a linear functinal Y which satis es hy ; Xi D(X) n X whence Y is a cntinuus subgradient fr at 0. Step 3: Representatin. Fix any X 2 int(dm()). It is an exercise t shw that increasing implies is nite at mst ver X 0 +. Fix any subgradient Y (which is then psitive) f at X. Reshu ing eq. (8), this means i=1 hy ; X i (X ) = max X2X fhy ; Xi (X)g = (Y ) 9

10 where the last equality fllws frm the de nitin f. This chain f equalities in turn implies that (X ) = hy ; X i (Y ) = max Y 2X 0fhY; X i (Y )g as the inequality (X ) hy; X i (Y ) autmatically hlds fr any Y 2 X 0. Remark 9 In [CL07] there is a frmula identical t (1) fr de ned n Banach lattices. Crllary 10 Every nite-valued cnvex and mntne functinal n a Banach lattice is nrm-cntinuus and subdi erentiable. Crllary 11 If a Frechet lattice X supprts a nn-cnstant cnvex mntne map, then necessarily X 0 6= f0g. As a generic Frechet lattice X is nt necessarily lcally cnvex, it may happen that the tplgical dual X 0 is very pr r even f0g : This is the case, fr example, f the spaces L p (; F; ), p 2 (0; 1); when is a nnatmic measure (see [AB05, Therem 13.31]) and f the space L 0 (; F; ), when is a nnatmic nite measure ( [AB05, Therem 13.41] ). Therefre, the nly cnvex mntne s n these spaces are the cnstants. 2.1 The current literature Surprisingly enugh given their imprtance in the applicatins, it seems that results n cntinuity and subdi erentiability fr cnvex mntne maps have appeared nly very recently in the literature. After nishing the rst versin f the paper, which did nt cntain the subdi erentiability additinal result, we came t knw that in the recent articles [MMR] and [RS06] there are statements very clse t thse f Therem 1. T start, in [MMR] it is shwn that: if L is an rdered Banach space, with L + clsed and such that L = L + L + then any cnvex mntne : U! R de ned n an pen set U f L is cntinuus. These hyptheses are strnger than urs n the tplgical part as L must be a Banach space, but milder n the rder part. In fact, their cnditins: the psitive cne L + is clsed and generating L = L + L + are always satis ed in a Frechet lattice. Nte that nthing is said abut subdi erentiability. On the cntrary, in [RS06] the authrs were the rst t prve subdi erentiability f cnvex mntne maps, but with the strnger assumptin that is de ned n a Banach lattice L: 10

11 If L is a Banach lattice, : L! R is prper, cnvex and mntne, then it is cntinuus and subdi erentiable n the interir f the prper dmain. The line f their prf is the fllwing. Fr any xed X 2 int(dm()), rst ne exhibits a psitive subgradient, which is then cntinuus by classic Namika-Klee therem. This implies lwer semicntinuity f at X, which in turn implies cntinuity. Inspired by this wrk we als prve subdi erentiability f, in the case f Frechet lattices. Hwever, we reverse the rder, since rst ne prves cntinuity f n int(dm()) and then subdi erentiability (this latter in the same way as dne by [RS06]). This is nly a matter f taste and it wuld nt be di cult t extend the results in [RS06] with the same line f reasning frm Banach t Frechet lattices. The nly interesting aspect in prving rst cntinuity is that ne realizes that the same prf f "classic" Namika fr linear psitive functinals still hlds, basically unchanged, fr cnvex mntne maps. The interested reader is als referred t [CL07] fr further develpments. 3 On rder lwer semicntinuity in Riesz spaces Let us recall sme basic facts abut Riesz spaces. The same ntatin is used fr the rder relatins in R, in ( 1; +1] and fr the directin f index sets f nets, as the meaning will be clear frm the cntext. A subset A f a Riesz space R is rder bunded if there exists X 1 2 R and X 2 2 R such that X 2 X X 1 fr all X 2 A: A net (X ) in R is increasing; written X "; if implies X X. A net (X ) in R is increasing t sme X 2 R; written X " X; if X " and sup X = X: A subset A f R is rder clsed if X 2 A and X! X implies X 2 A: The space R is rder cmplete when each rder bunded subset A has a supremum (least upper bund) and an in mum (largest lwer bund). Recall ([AB05, Th. 8.15]) that the lattice peratins are rder cntinuus. In additin ([AB05, Th 8.16]), if a net (X ) is rder bunded and R is rder cmplete, then lim inf X, sup inf X and lim sup X, inf sup X are well de ned, and X! X i X = lim inf X = lim sup X : The next Lemma is an immediate cnsequence f the facts and de nitins abve and f (4). Lemma 12 Let R be a Riesz space. (i) Let X! X. Then there exists such that (X ) is rder bunded, i.e. the net is de nitely rder bunded. In case the index set f the net has a minimum then (X ) is rder bunded. 11

12 (ii) Let R be rder-cmplete and let X! X. If Y, (inf X ) ^ X, then Y " X: Example 13 (Order cnvergence in L p ) In L p spaces, p 2 [0; 1]; the ntin f rder is the very familiar pintwise ne, i.e. Y X i Y (!) X(!) P -a.e. As L p is rder separable, see the next sectin, sequences can be used instead f nets t characterize rder cnvergence. A sequence (X n ) n in L p is rder bunded i it is dminated in L p (i.e. there exists a Y 2 L p + such that jx n j Y ). The rder cnvergence in the L p case is just dminated pintwise cnvergence: X n! X, Xn P a:e:! X and (X n ) n is dminated in L p (9) Therefre, the L p -nrm tplgies are rder cntinuus fr all p < +1, as the abve equivalence implies that Lebesgue dminated cnvergence therem can be L applied t cnclude X n! X ) p Xn! X. 3.1 Equivalent frmulatins f rder l.s.c. De nitin 14 A functinal : R! ( 1; +1] de ned n a Riesz space R (a) is cntinuus frm belw if X "X ) (X ) " (X) (a ) is cntinuus frm belw if X n " X ) (X n ) " (X) (b) is rder lwer semicntinuus if X! X ) (X) lim inf (X ) (b ) is rder lwer semicntinuus if X n! X ) (X) lim inf (Xn ): (10) Nte that the pintwise supremum f a family f rder l.s.c. functinals is rder l.s.c. As shwn in the next Lemma, if is increasing and R pssesses mre structure then the cnditins (a); (a ); (b); (b ) are all equivalent. The rder separability f R (any subset A which admits a supremum in R cntains a cuntable subset with the same supremum) allws t frmulate the rder-l.s.c. prperty with sequences instead f nets (i.e. (b), (b )). Lemma 15 Let R be an rder cmplete Riesz space and : R! ( 1; +1] be increasing. Then: (a), (b); (a ), (b ); (a) ) (a ); (b) ) (b ): If in additin R is rder separable then (a); (a ); (b); (b ) are all equivalent. 12

13 Prf. (a) ) (b): Let X! X and set Y = (inf X ) ^ X. By Lemma 12 (ii), Y " X and s (X) (a) = lim (Y ) (mn) lim inf (X ): (b) (b) ) (a). Since X " X implies X! X; we get: (X) lim (X ) (mn) (X): (a ), (b ) fllws in the same way as (a), (b); while (a) ) (a ) and (b) ) (b ) are bvius. T shw the last sentence it is su cient t prve e.g. (a ) ) (a). Fr any net X " X we can nd a cuntable subnet X n such that X n " X. Hence (X) (a) = lim (X n ) lim (X ) (mn) (X): n Remark 16 (On rder separability) A su cient cnditin fr R t be rder separable is that, fr every principal ideal R X, there exists a psitive linear functinal n R which is strictly psitive n R X (see [Za83, Therem 84.4]). All Banach lattices with rder cntinuus nrm verify this cnditin, as shwn in [ABu85, Therem 12.14]. Anther su cient cnditin fr rder separability is the existence f a linear functinal n R which is strictly psitive n the entire R. This implies that all the Orlicz Banach lattices L = L (; F; P ) (and hencefrth all the L p spaces, p 2 [1; 1]) are rder separable (and rder cmplete as well). See Sectin 5.1. Remark 17 (On the Fatu Prperty) Frm (9), De nitin (2) and De - nitin (14), we immediate see that when R = L p, p 2 [0; 1], rder lwer semicntinuity cincides with the Fatu Prperty. Remark 18 (On decreasing functinal) Analgus cnsideratins hld fr decreasing functinals: if R is an rder cmplete and rder separable Riesz space and if is decreasing, then the cnditins: (b); (b ); cntinuity frm abve [i.e.: X # X ) (X ) " (X)] and cntinuity frm abve [i.e.: X n # X ) (X n ) " (X)] are all equivalent. These equivalent frmulatins will be used t study sme prperties f cnvex risk measures in Sectin The rder cntinuus dual X n Given a Frechet lattice X, the space f rder bunded linear functinals X (thse which carry rder bunded subset f X t rder bunded sets f R) cincides with the tplgical dual X 0. This is a cnsequence f Namika-Klee Therem 7. Frm the general thery (see [Za83]) n the decmpsitin f X X 0 = X = X n X s 13

14 where Xn is the rder clsed ideal (band) f X f all the rder cntinuus linear functinals n X and it is called the rder cntinuus dual f X. The space f singular functinals Xs is de ned as the band disjint cmplement f Xn in X : Examples f this decmpsitin are given in Sectin 5. The main gal f the next Sectin is t prvide sme criteria that guarantee the C-prperty f the tplgy (X ; Xn ). 4 On the C-prperty The C-prperty is veri ed by the strng tplgy f all Frechet lattices withut passing t cnvex cmbinatins, as shwn belw. Lemma 19 Let (X ; ) be a Frechet lattice. If (X n ) n -cnverges t X, then there exists a subsequence which is rder cnvergent. Prf. Call d a cmplete distance that induces, which is als abslute, i.e. d(x; 0) = d(jxj; 0). Suppse d(x n ; X)! 0 and select a subsequence such that P k0 d(x n k ; X) = P k0 d(x n k X; 0) < +1. Set Y = P k0 jx n k Xj. By cmpleteness f d, Y 2 X. Nw, if Y k := P hk jx n h Xj then clearly Y k # and Y k! 0 s by [AB05, Therem 8.43] Y k # 0. As jx nk Xj Y k ne deduces that X nk rder cnverges t X. We will be mainly cncerned with the C-prperty f weak tplgies in lcally cnvex Frechet lattice. This is the reasn why the mst t hpe fr is t extract an rder cnvergent subsequence f cnvex cmbinatins frm a tplgically cnvergent net, e.g. exactly the C-prperty. Lemma 20 Let (X ; ) be a lcally cnvex Frechet lattice. Then the (X ; X 0 ) tplgy veri es the C-prperty. Prf. Let W! W in the weak tplgy. By Hahn-Banach Therem, W belngs t the -clsure f cnv(w ; ) fr all and as the tplgy is rst cuntable there exists a subsequence ( n ) n and a sequence Y n 2 cnv(w n ; :::) which cnverges t W in the tplgy. Lemma 19 ensures that we can extract a subsequence (Y nk ) k that rder cnverges t W. Remark 21 The lcal cnvexity assumptin cannt be drpped in the statement f the previus Lemma. An immediate cunterexample is given by the Frechet lattice L 0, since when P has n atms (L 0 ) 0 = f0g. S, the weak tplgy (L 0 ; (L 0 ) 0 ) is the indiscrete ne and desn t satisfy the C-prperty. 14

15 Hwever, even under the lcal cnvexity assumptin, the C-prperty is nt preserved if ne keeps weakening the tplgy, frm (X ; X 0 ) t (X ; X n ). An extreme situatin is the ne already encuntered in Example 6 where the Banach lattice X = C([0; 1]) has dual X 0 cnsisting f the signed measures f nite variatin n [0; 1], but n is rder cntinuus apart frm the null measure. S the fllwing Lemma may be helpful. Lemma 22 Let (L; L ); (X ; ) be lcally cnvex Frechet lattices and suppse there exists a linear, injective lattice mrphism such that Then (X ; X n ) veri es the C-prperty. (X ; ) i! (L; L ) fy i j Y 2 L 0 g X n : (11) Prf. We specify that by "linear, injective lattice mrphism" i we mean that i is linear, injective, tplgically cntinuus and preserves the lattice structure. Nte that X needs nt t be hmemrphic t i(x ). (X ;X Let (X ) be a net such that X n )! X. The cnditin (11) implies that W := i(x ) cnverges t W := i(x) in the (L; L 0 )-tplgy. Applying the same argument and using the same ntatins f the prf f Lemma 20, there exists (Y nk ) k cnverging in rder t W in L and s the inverse image Z k = i 1 (Y nk ) veri es Z k 2 cnv(x nk ; ) and Z k! X. Cnditin (11) is evidently satis ed in case L 0 = L n, which is equivalent t the assumptin that L is rder cntinuus. If this hlds, essentially the abve Lemma applies t any lcally cnvex Frechet lattice X that can be identi ed with a sublattice f L, s the rder structure is identical f that f L, but with pssibly ner tplgy than the ne inherited frm L. This is the cntent f the next Crllary, that will be applied fr the Orlicz Banach lattice L. Crllary 23 Any lcally cnvex Frechet lattice X f randm variables that can be injected int L 1 by a linear lattice mrphism has, (X ; X 0 ) and (X ; X n ) tplgies with the C-prperty. 4.1 The C-prperty in the representatin f cnvex and mntne functinals We present the result n the equivalence between the (X ; Xn ) l.s.c. prperty fr cnvex functinals n lcally cnvex Frechet lattices and the rder-l.s.c. prperty, under the assumptin that the tplgy (X ; Xn ) has the C-prperty. 15

16 Prpsitin 24 Let (X ; ) be a lcally cnvex Frechet lattice and cnsider the fllwing cnditins fr a prper, cnvex functinal : X! ( 1; +1]: 1. is (X ; X n ) l.s.c. 2. admits the representatin 3. is rder l.s.c. (X) = sup fhy; Xi (Y )g ; X 2 X ; (12) Y 2Xn Then 1) ) 2) ) 3). If (X ; X n ) has the C-prperty, the three cnditins are equivalent. If is in additin mntne increasing, the cnclusins are identical and in the representatin (12) X n can be replaced by (X n ) +. Prf. 1)) 2) fllws frm (X ; (X ; Xn )) 0 = Xn and frm Fenchel-Mreau Therem (see e.g. [BR83, Chap I]); 2)) 3) Since is the pintwise supremum f a family f rder cntinuus functinals, it is als rder l.s.c. Suppse nw that (X ; Xn ) has the C-prperty and that 3) hlds. T prve 1) we shw that fr any real k the sublevel A k = fx 2 X j (X) kg is (X ; Xn (X ;X ) clsed. Suppse that X 2 A k and X n )! X: By the C- prperty, there exists Y n 2 cnv(x n ; ) such that Y n! X: The cnvexity f implies that (Y n ) k fr each n. Frm rder l.s.c. f s that X 2 A k. (X) lim inf (Y n ) k Remark 25 Nte that the C-prperty culd have been stated with rder cnverging subnets f cnvex cmbinatins instead f subsequences (as in fact it was in the rst versin f the present paper). Hwever the current presentatin is given with subsequences as the applicatins rely nly n Crllary 23, which in turn is based n Lemma 20. A natural questin is whether the sup in frmula (12) is attained when is nite valued. In general, the answer is n, as shwn in the example belw, where the max is attained ver X 0 + thanks t (1) but nt ver (X n ) +. Example 26 Cnsider the classic cunterexample [FS04, Example 4.36] translated in the language f mntne increasing maps, that is take : L 1! R, (X) = ess sup X. This map is cnvex, increasing, psitively hmgeneus and 16

17 rder l.s.c. n L 1. Fr later use, bserve als that it is nt rder u.s.c. Frm (12), taking int accunt the "cash additivity" prperty (X +c) = (X)+c; c 2 R, (X) = sup fq prbab. ;QP g E Q [X]; (13) Similarly, frm (1) and the gd prperties f, (X) = max E Q[X] Q2M 1;f (P ) which is exactly the representatin in (2) with zer penalty functin. If X is selected s that its ess-sup is nt attained, the sup in (13) cannt be a maximum. We will cnsider a similar case in the Example 5.3. If is nite valued and rder u.s.c., then interestingly enugh admits a representatin as in (12) with the supremum replaced by a maximum, withut the C-prperty requirement. Lemma 27 Let (X ; ) be a lcally cnvex Frechet lattice and : X! R be a cnvex increasing map. If is rder u.s.c. then (X) = and thus a frtiri is rder cntinuus. Prf. Frm (1), max fhy; Xi (Y )g ; X 2 X ; Y 2(Xn )+ (X) = max fhy; Xi (Y )g sup fhy; Xi (Y )g Y 2X+ 0 Y 2(Xn )+ We nw prve that any Y attaining the max n X+ 0 is rder cntinuus. In fact, suppse by cntradictin that the max is attained n a psitive, nn rder cntinuus Y 0. Then, there exists Z! 0 such that lim sup hy 0 ; Z i > 0 and (X) = fhy 0 ; Xi (Y 0 )g < lim supfhy 0 ; X+Z i which is a cntradictin with rder u.s.c. f. (Y 0 )g lim sup (X+Z ) 5 Orlicz spaces and applicatins t Risk Measures 5.1 Orlicz spaces have the C-prperty The fllwing Orlicz spaces and the L p spaces; p 2 [0; +1]; are de ned n the same prbability space (; F; P ). A Yung functin is an even, cnvex functin : R! R [ f+1g with the prperties: 17

18 1. (0) = 0; 2. (1) = +1; 3. < +1 in a neighbrhd f 0. Nte that may jump t +1 utside f a bunded neighbrhd f 0. In case is nite valued hwever, it is als cntinuus by cnvexity. The Orlicz space L is then de ned as L = fx 2 L 0 j 9 > 0 E[ (X)] < +1g: It is a Banach space with the Luxemburg (r gauge) nrm X N (X) = inf c > 0 j E 1 : c With the usual pintwise lattice peratins, L nrm satis es the mntnicity cnditin is als a Banach lattice, as the jy j jxj ) N (Y ) N (X): Since is bunded in a neighbrhd f 0 and it is cnvex and ges t +1 when jxj! 1, it is rather easy t prve that L 1 i!l i!l 1 (14) with linear, injective lattice mrphisms (the inclusins i). The dual (L ) 0 admits the general decmpsitin in rder cntinuus band and singular band (L ) 0 = (L ) n (L ) s (15) and (L ) n can be identi ed with the Orlicz space L where (y) = supfyx (x)g x2r is the Yung functin cnjugate f. The examples belw illustrate di erent cases and shw that the L p are in fact particular Orlicz spaces. 1. Suppse p 2 (1; +1) and = p where p(x) = jxjp p then L p = L p. Since this space has an rder cntinuus tplgy, the dual cnsists nly f rder cntinuus functinals. As ( p) = q with q =, ne recvers the classic p p 1 (L p ) 0 = L q : 18

19 2. = 1, where 1(x) = 0 if jxj 1 +1 therwise Then the assciated Orlicz space L 1 is exactly L 1 and as ( 1) (y) = jyj, L ( 1) = L 1. The decmpsitin f the dual prvided in (15) is nthing but the Ysida- Hewitt decmpsitin (L 1 ) 0 = L 1 (L 1 ) s and the singular band (L 1 ) s cnsists f the purely nitely additive measures. 3. e(x) = e jxj 1 is a genuine example f Yung functin which induces an Orlicz space di erent frm the L p. L e is the space f randm variables with sme nite expnential mment, i.e. L e = fx 2 L 0 j 9 > 0 s.t. E[e jxj ] < +1g: Analgusly t what happens fr L 1, this space has a tplgy which is nt rder cntinuus. Thus the dual has the full general decmpsitin (15), with nn-null singular band, as (L e ) 0 = L ( e) (L e ) s where the cnjugate ( e) is given by the functin jyj(ln jyj 1) + 1 if jyj > 1 0 therwise which will be indicated with. b As better explained belw, since b desn t grw t fast the Orlicz L b displays a behavir similar t that f the L p ; 1 p < +1, in the sense that its tplgy is rder cntinuus. Then, its dual (L b ) 0 cincides with L b = L e. The cnsequence is that the tplgy induced n L e by the rder cntinuus functinals, (L e ; L b ), is nthing but the weak* tplgy n L e. As anticipated in the examples abve, when veri es a slw- grwth cnditin, knwn in the literature as 2 cnditin (see e.g. [RR91]): 9t > 0; 9K > 0 s.t. (2t) K (t) fr all t > t then (L ) 0 = (L ) n = L, that is the nrm-tplgy n L is rder cntinuus. S by Lemmata 19 and 20 the nrm tplgy and the weak tplgy (L ; (L ) 0 ) = (L ; L ) have the C-prperty. In general, by (14) and Crllary 23, the fllwing tplgies n L all have the C-prperty: (a) the nrm tplgy, (b) the weak tplgy, (c) the (L ; L )-tplgy. 19

20 We remark that when it is that satis es the 2 cnditin, as in example 3 abve, then the dual space f L cincides with L. Therefre in this case the tplgy (L ; L ) is nthing but the weak* tplgy n L and it has the C-prperty. One may als cnsider the Mrse subspace M f the Orlicz space L : M = X 2 L j E[ (kx)] < +1 8k > 0 : (16) When is nite-valued, M is a nrm clsed band f L and its dual (M ) 0 = L, s (M ; L ) has the C-prperty t. In the cntext f expected utility maximizatin, the spaces M were rst used in [B07]. They are the bject f study in [CL07] and applied t risk measures. In [CL07] it has als been shwn that a risk measure de ned n M has nn empty tplgical interir if and nly if it is nite valued. As the dual f the Mrse space M can be identi ed with a space f functins, the Orlicz L, these spaces are easier t handle than the whle L. In particular, since M has rder cntinuus nrm, the dual representatin (6) fllws immediately frm the Extended Namika Klee Therem 1. In [BF08] and in [BFG07] it has been shwn that the full duality (L bu ; (L bu ) 0 ) can als be successfully emplyed t cver new cases in the applicatins t expected utility maximizatin and indi erence pricing. In fact the Orlicz space L bu, de ned by the Yung functin bu(x) = u( jxj) + u(0) assciated t the utility functin u; is the natural envirnment fr such investigatin. And the results n the indi erence price fr claims in the general L bu btained in [BFG07] shw that in the general setup the result by [CL07] fails: a cnvex risk measure n L bu can have nn empty tplgical interir withut being nite valued everywhere. Fr ther examples f this situatin self-cntained in the present paper, see the next Sectin 5.2 where there are sme ther interesting applicatins f the full duality t risk measures. 5.2 New insights n the dwnside risk and risk measures assciated t a utility functin u We assume that the investment pssibilities at a certain date in the future are mdelled by elements X f L 0. As straightfrward cnsequences f Prpsitin 24 we have the fllwing representatins f decreasing functinals de ned n subspaces f L 0 : Crllary 28 Let (X ; ) be a lcally cnvex Frechet lattice cntained in L 0 : If : X! ( 1; +1] is a prper cnvex rder l.s.c. decreasing functinal and (X ; Xn ) has the C-prperty, then admits the representatin (X) = sup Y 2(X n )+ fhy; Xi ( Y )g ; X 2 X : (17) 20

21 If in additin satis es the cash additivity prperty then (X) = (X + c) = (X) c; 8c 2 R 8X 2 X ; (18) sup fhy; Xi ( Y )g ; X 2 X : (19) Y 2(Xn )+ ; hy;1i=1 If in additin is psitively hmgeneus, then there exists a cnvex subset C fy 2 (X n ) + j hy; 1i = 1g such that (X) = suphy; Y 2C Xi: Let us cnsider an agent, whse preferences n the investments X can be represented via expected utility. We assume that the utility functin u : R! R is increasing and cncave (thugh nt necessarily strictly cncave) and satis es lim x! 1 u(x) = 1. Withut lss f generality, suppse u(0) = 0: The gal is that f describing a natural framewrk assciated t the expected utility f the agent, i.e. t the functinal t the related dwnside risk E[u(X)]; (X) := E[ u(x)] and t sme assciated cnvex risk measures. As it is nt required that u is strictly cncave, u can be identically 0 n R + and in this case is nthing but the s-called shrtfall risk ([FS04]). It turns ut that a gd setup is that f an Orlicz spaces duality induced by the functinal itself. As shwn in [B07] and [BF08] the functin bu(x) = u( jxj) is a Yung functin and de nes the Orlicz space L bu assciated t u: Call (y) = supfu(x) x2r xyg the cnvex cnjugate f u. Since bu is nite n R, then, as bserved right after the de nitin (16), M bu is a nrm-clsed band f L bu and its dual is L b. 21

22 It is clear that there must be a link between and, b the Yung functin cnjugate t bu. In fact 0 if jyj b(y) = (jyj) if jyj > where 0 is the right derivative f bu at 0, namely = D + bu(0) = D u(0). If u is di erentiable, nte that = u 0 (0) and it is the unique slutin f the equatin 0 (y) = 0. T x the ideas, cnsider the fllwing examples. 1. Fix > 0 and take u (x) = e x + 1 (20) whence cu (x) = e jxj 1 and (y) = y ln y y + 1 and (y) b = (j y j ln j y j j y j + 1)I fj y j1g. It is nt di cult t see that the assciated Orlicz spaces d nt depend n (in the sense that they are physically the same and changing amunts t a dilatin f the Luxemburg nrm) and therefre, as pinted ut in Sectin 4.1 Example 3, n L cu = X 2 L 0 j 9 > 0 s.t. E[e jxj ] < +1 ; M cu = n h X 2 L 0 j 8 > 0 E e jxji < +1 ; L c = Y 2 L 0 j E (jy j ln jy j )I fjy j>1g < +1 and (L bu ) 0 = L b (L bu ) s : 2. Let u be the quadratic- at utility, i.e. ( x 2 u(x) = 2 if x 0 0 if x 0 (21) In this case, bu(x) = x2 2 = b (x), and all the spaces L bu ; M bu ; L b are equal and cincide (mdul an ismrphism) with L 2. Let us recall that the Orlicz class f L bu is de ned as L bu = fx 2 L 0 j E[bu(X)] < +1g and it is a cnvex subset (nt necessarily clsed) f L bu. The fllwing Lemma is a nice cnsequence f the right chice f the spaces. 22

23 Lemma 29 The dwnside risk : L bu! ( 1; +1], (X) = E[ u(x)], is a well-de ned, prper, cnvex and mntne decreasing functinal which is rder l.s.c. In additin, and Dm() = fx 2 L bu j X 2 L bu g int(dm()) = fx 2 L bu j 9 > 0 (1 + )X 2 L bu g M bu : (22) Mrever, admits the representatin: (X) = sup Y 2L b + fe[ XY ] E[(Y )]g: (23) Prf. If X 2 L bu, then by Jensen s inequality E[ u(x)] u(e[x]) > 1 (24) since E[X] 2 R frm (14) and u < +1 n R. S the de nitin is well-psed and is clearly cnvex and mntne decreasing. T prve the characterizatin f Dm(), simply nte that X 2 Dm() i E[u(X)] > 1 i E[u( X )] > 1 i E[bu(X )] < +1 where the secnd equivalence abve is due t the fact that E[u(X + )] is always nite as u(0) u(x + ) ax + + b fr sme a; b 2 R, s that u(0) E[u(X + )] ae[x + ] + b < +1. T prve (22), if X 2 int(dm()) then clearly fr sme > 0 E[ u(x X )] is nite, that is E[bu((1 + )X )] is nite. Cnversely, suppse (1 + )X 2 L bu. Then, (1 + )X 2 Dm() and fr any Z with Luxemburg nrm N bu (Z) <, X + Z 2 Dm(). In fact: h E[ u(x + Z)] = E 1+ i 1 u 1+ ((1 + )X) + Z 1 1+E[ u((1 + )X)] + 1+ E u 1+ Z < +1 since 1+ Z has Luxemburg nrm less than 1 and thus E 1 + u Z E u 1 + Z 1 + = E bu Z 1 Thanks t Remark 18, in rder t shw that is rder l.s.c. ne just needs t check whether is -cntinuus frm abve. But this is an immediate cnsequence f the mntne cnvergence therem and (24). Finally, the (L bu ; L b ) 23

24 tplgy has the C-prperty s the representatin (17) n the rder cntinuus dual L b applies (X) = sup Y 2L b + fe[ Y X] ( Y )g: By Kzek s results [Kz79] (r directly by hand),, the cnvex cnjugate f, (Y ) = sup fe[y X] (X)g X2L bu veri es ( Y ) = E[(Y )]; if Y 2 L b : (25) Clearly satis es all the requirements f a cnvex risk measure but cash additivity. As shwn in [BK06] in the L 1 case, the greatest cnvex risk measure smaller than a cnvex functinal : L 1! R can be cnstructed by taking the infcnvlutin wrst f with wrst = L 1 +, which is the risk measure assciated t the acceptance set L 1 +. Then the penalty functin f wrst is equal t 0 n M 1;f (P ), and is equal t 1 utside M 1;f (P ). Since the penalty functin f wrst is the sum f the penalty functin f and f wrst ; the representatin f wrst will have the same penalty functin f ; but the supremum in such representatin is restricted t the set M 1;f (P ); i.e. t thse psitive elements in the dual space that are als nrmalized. The same cnclusin hlds in ur setting, as shwn in the fllwing result. Prpsitin 30 The map u : L bu! ( u (X) = sup QP; dq dp 2L b + 1; +1] de ned by dq dp E Q [ X] E[ ] (26) is a well-de ned rder l.s.c. cnvex risk measure and it is the greatest rder l.s.c. cnvex risk measure smaller than and hence u = L bu + : Mrever, the sup n in (26) can equivalently be cmputed n the set Q prbab.; Q P j E[ dq dp Prf. It is clear that u is an rder l.s.c cnvex risk measure. Frm (23) we als have: u : We need nly t prve that if e : L bu! ( 1; +1] is an rder l.s.c. cnvex risk measure such that e ; then e u. Let e(y ) = e ( Y ) be the penalty functin assciated with e in the representatin (19) e(x) = sup fe[ XY ] e(y )g: Y 2L b + ;E[Y ]=1 ] < +1 : 24

25 By cash additivity, e(x + (X)) = e(x) (X) 0, fr all X 2 L bu, s that e(x + (X)) = This implies that, if Y 2 L b + ; E[Y ] = 1; and, by (25), Therefre, e(x) = sup fe[ Y X] (X) e(y )g 0: Y 2L b + ;E[Y ]=1 e(y ) E[ Y X] (X) fr all X 2 L bu e(y ) sup X2L bu fe[ Y X] (X)]g = ( Y ) = E[(Y )]: sup fe[ Y X] e(y )g sup fe[ Y X] E[(Y )] = u (X): Y 2L b + ;E[Y ]=1 Y 2L b + ;E[Y ]=1 Since the integrability cnditin E[(Y )] < +1 n Y 0 is mre severe than the requirement Y 2 L b +, the last sentence is bvius. T any utility functin satisfying ur assumptins, ne can als assciate the map u : L bu! ( 1; +1] de ned by: u (X) = inf fc 2 R j X + c 2 A u g ; (27) where the set A u is de ned as n A u := X 2 L bu j E[u(X)] u(0) = 0 = Lemma 31 A u has the prperties: 1. it is cnvex; 2. if X 2 A u and Z 2 L bu ; Z X, then Z 2 A u ; 3. inffc 2 R j c 2 A u g > 1; n X 2 L bu j (X) 0 4. fr any X 2 A u and Z 2 L bu, the set ft 2 [0; 1] j (1 t)x + tz 2 A u g is clsed in [0; 1]. Prf. We nly prve item 4, as the thers are simple cnsequences f the prperties f u. Fix any X 2 A u and call = ft 2 [0; 1] j (1 t)x + tz 2 A u g. Fr any cluster pint t f, there exists a sequence (t n ) n 2 ; t n! t. But then, (1 t n )X + t n Z rder cnverges t (1 t )X + t Z. Frm Lemma 29, is rder l.s.c., s ((1 t )X + t Z) lim inf n ((1 t n )X + t n Z) 0 which means t 2. 25

26 Prpsitin 32 u : L bu! ( 1; +1] is a well-de ned rder l.s.c. cnvex risk measure that admits the representatin: where u (X) = sup QP; dq dp 2L b + fe Q [ X] (Q)g; (28) (Q) = sup X2A u fe Q [ X]g: (29) Mrever, A u := X 2 L bu j u (X) 0, the acceptance set f u, satis es A u = A u ; and, as a cnsequence, if e : L bu! R [ f+1g is an rder l.s.c. cnvex risk measure such that e ; then e u. Prf. The facts that u is a cnvex risk measure and that its acceptance set A u cincides with A u are cnsequences f the abve Lemma and Prpsitins 2,4 in [FS02]. Nw, since is rder l.s.c., A u is rder clsed, s that the acceptance set f u is rder-clsed. And since (L bu ; L b ) has the C-prperty the acceptance set A u = A u is (L bu ; L b )-clsed. Hence, by a classic result, as its sublevels are (L bu ; L b )-clsed, u is (L bu ; L b )-l.s.c. But this implies that it is als u rder l.s.c. by the rst part f the statement in Prpsitin 24. Then, the representatin (19) n the rder cntinuus dual L b applies: u (X) = sup QP; dq dp 2L b + fe Q [ X] (Q)g; X 2 L bu ; (Q), u( Q) = sup X2L bu fe Q [ X] u (X)g It is straightfrward t see that the penalty functinal admits the representatin (Q) = sup E Q [ X] X2A u If e : L bu! ( 1; +1] is an rder l.s.c. cnvex risk measure such that e then A e = X 2 L bu j e(x) 0 A u = A u, and this implies e u : Remark 33 Obviusly, u u ; but u = u if and nly if u which in general is nt true. Nte that the inequality u wuld imply: (X + (X)) (X + u (X)) 0 (this latter inequality fllws frm X + u (X) 2 A u = A u )), but in general (X + (X)) 0 des nt hld. S, u and u may be di erent (see the Subsectin 5.3 belw). In Subsectin 5.4 there is a case where u = u. 26

27 5.3 Quadratic- at utility If u is the quadratic- at utility functin (21), then u and u are di erent. Indeed, L bu = L 2 and (X) = 1 2 E[(X 1 )2 ] = sup E[ Y X] Y 2L 2 2 E[Y 2 ] ; X 2 L 2 ; + ( " dq #) 2 1 u (X) = sup E Q [ X] dq 2 E ; X 2 L 2 : dp dp 2L2 + Since A u = X 2 L 2 j (X) 0 = L 2 +, we have: u (X) = inf fc 2 R j X + c 0g = wrst (X) := ess inf(x): The dual representatin in (28) becmes u (X) = sup E Q [ X] QP; dq dp 2L2 + since, frm (29), the penalty term is given by (Q) = sup E Q [ X2L 2 ;(X)0 X] = 0; if dq dp 2 L2 +: Nte als that u (X) = sup QP; dq dp 2L2 + E Q [ X] = sup E Q [X ] E[(X )2 ] QP; E[X ] dq dp 2L2 + and if 0 < E[X ] < 2, u (X) > (X). Mrever, u is nt even nitevalued. Therefre, while and u are nite valued and thus cntinuus and subdi erentiable n L 2, Dm( u ) has empty interir thanks t the cited result f [CL07] fr risk measures n Mrse subspaces (here, L bu = M bu = L 2 ). 5.4 Expnential utility Let u(x) = e x + 1 be the expnential utility functin cnsidered in (20). W.l..g. we set = 1: Then, (X) = E[e X ] 1, X 2 L bu ; and (y) = y ln y y + 1: Frm the de nitin (27) we have u (X) = inf c 2 R j E[e X c 1] 0] = ln E[e X ]; with the cnventin ln E[e X ] = +1 if E[e X ] = +1. Clearly u (X) = ln E[e X ] E[e X ] 1 = (X) and therefre, in this case, u = u : S, frm 27

28 (26) we recver the entrpic risk measure tgether with its dual variatinal identity dq ln E[e X ] = sup E Q [ X] E QP; dq dp 2L dp b + dq = sup E Q [ X] E Q ln ; X 2 L bu : QP; dq dp 2L dp b + The nvelty here is that the space where this representatin hlds is L bu, naturally induced by u and nt an arbitrarily selected subspace f L 0 (traditinally, the entrpic risk measure is de ned n L 1 and the frmula abve is prvided fr X 2 L 1 L bu, see [FS04] and the remarks belw). And u is a genuine example f a risk measure n the general Orlicz space L bu which is nt nite valued everywhere and still has dmain with n empty interir: as Dm( u ) = Dm(), the interir f the dmain has been cmputed in (22). T cnclude, let us fcus n the restrictin f u = u t the subspace M bu. Crllary 34 The restrictin u f u t the subspace M bu is a well-de ned nrm cntinuus (hence rder cntinuus) cnvex risk measure u : M bu! R that admits the representatin u (X) = max QP; dq dp 2L b + E Q [ X] E dq ; X 2 M bu : dp We thus recver the representatin frmulae prvided by [CL07] n Mrse subspaces and the frmula with the max fr the entrpic risk measure n L 1 M bu. References [AB05] [ABu85] C. D. Aliprantis and K. C. Brder. In nite dimensinal analysis. Springer, 2005 (third editin). C. D. Aliprantis and O. Burkinshaw. Psitive Operatrs. Academic Press INC, [ADEH99] Artzner P., F. Delbaen, J.M. Eber and D. Heath (1999): Cherent measures f risk, Mathematical Finance, 4, [BK06] P. Barrieu and N. El Karui, Pricing, hedging and ptimally designing derivatives via minimizatin f risk measures, T appear in Vlume n Indi erence Pricing (ed: Rene Carmna), Princetn University Press. 28

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