Some formulas for the conjugate of convex risk measures

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2 Some formuls for the conjugte of convex risk mesures Rdu Ion Boț Nicole Lorenz Gert Wnk Abstrct. The im of this pper is to give formuls for the conjugte functions of different convex risk mesures. To this end we use, on the one hnd, some clssicl results from convex nlysis nd, on the other hnd, some tools from the conjugte dulity theory. The chrcteriztions of the so-clled devition mesures recently given in the literture (see [8]) follow immeditely from our results s nturl conseuences. Keywords. conjugte functions, conjugte dulity, convex risk mesures, convex devition mesures. Introduction In mny prcticl pplictions like those which pper in the portfolio optimiztion, the notion of risk plys n importnt role. It reflects the uncertinty of some processes nd one chllenge consists in untifying it by n pproprite mesure. Until now in the literture different formultions for such so-clled risk mesure hve been mde. The clssicl ppliction in finncil mthemtics is the portfolio optimiztion problem treted by Mrkowitz (cf. [7]), where the risk of portfolio ws mesured by mens of the stndrd devition nd vrince, respectively. In 998 Artzner et l. (cf. []) first gve n xiomtic definition of wht they clled coherent risk mesure. The properties in the definition of this clss of mesures seem to be common in mny prcticl problems. In 2002 Rockfellr et l. (cf. [0]) introduced new clss of mesures closely relted to the coherent risk mesures, clled devition mesures. An importnt representtive of this clss of mesures is the vrince. It is remrkble tht the coherent risk mesures s well Fculty of Mthemtics, Chemnitz University of Technology, D-0907 Chemnitz, Germny, e-mil: rdu.bot@mthemtik.tu-chemnitz.de. Fculty of Mthemtics, Chemnitz University of Technology, D-0907 Chemnitz, Germny, e-mil: nicole.lorenz@mthemtik.tu-chemnitz.de. Fculty of Mthemtics, Chemnitz University of Technology, D-0907 Chemnitz, Germny, e-mil: gert.wnk@mthemtik.tu-chemnitz.de

3 s the devition mesures fulfill some positive homogeneity nd subdditivity properties. As mny risk mesures used in prctice do not fulfill these properties, the clss of coherent risk mesures hs been extended to the clss of convex risk mesures (see for exmple the pper of Föllmer nd Schied, [5]), in which definition the sublinerity ws replced by convexity. Recent ppers, where some theoreticl results concerning convex risk mesures hve been given, re the works of Pflug (cf. [8]) nd Ruszczynski nd Shpiro (cf. [2]). In [2] some necessry nd sufficient conditions for the optiml solutions of optimiztion problems with convex risk mesures s objective functions re given, wheres in [8] the uthor gives some dul representtions for number of convex risk nd devition mesures with prcticl relevnce. In this pper we consider different convex risk nd devition mesures, defined in nlogy to Pflug s pper, nd clculte their conjugte functions. To this end we use the powerful theory of conjugte functions from convex nlysis s well s some dulity results for convex optimiztion problems in loclly convex spces. By using the Fenchel Moreu theorem we succeed to give dul representtion for ll mesures we del with. In this wy we extend nd improve the results obtined by Pflug ([8]). The pper is orgnized s follows. In the next section we introduce some nottions nd preliminry results coming from convex nlysis s well s from the theory of stochstics. Further, in Section 3, we introduce the notion of convex risk mesure nd, closely connected with it, tht of convex devition mesures. Then we give some exmples for these two clsses of mesures. Section 4 is devoted to the clcultion of the conjugtes of some clssicl convex risk nd devition mesures. In Section 5 we del with some elborted convex risk nd devition mesures nd we clculte their conjugtes by using the generl formul of the conjugte of composed convex function. Finlly, in the lst section, we derive some dul representtions for the convex risk nd devition mesures considered in the previous two sections nd compre our results with the ones given by Pflug in [8]. 2 Nottions nd preliminry results In this section we introduce some nottions nd preliminry results used lter in the pper. Let Z be nontrivil loclly convex spce nd Z its topologicl dul spce endowed with the wek topology. We denote by x, x : x (x) the vlue of the liner continuous functionl x Z t x Z. For set D Z we denote by cl(d) the closure of D, by int(d) its interior nd by core(d) d D : x Z ε > 0 : λ [ ε, ε] d + λx D its lgebric interior. The indictor function δ D : Z R R ± of the set D is defined 2

4 by 0, x D, δ D (x) By tking function f : Z R we consider the (Fenchel-Moreu) conjugte function of f, f : Z R defined by f (x ) sup x, x f(x). x Z Similrly, the biconjugte function of f, f : Z R is defined by f (x) sup x Z x, x f (x ). Further, for the function f : Z R we consider lso its epigrph epi(f) (x, r) : x Z, r R : f(x) r nd its effective domin dom(f) x Z : f(x) < +. We sy tht f is proper if dom(f) nd f(x) >, x Z. We cn stte now very importnt result coming from convex nlysis: Theorem 2.. (Fenchel-Moreu) Let f : Z R be proper, convex nd lower semicontinuous function. Then it holds f f. The following theorem gives sufficient condition for the formul of the conjugte of the precomposition of convex function with liner continuous mpping (see [9]). In wht follows U is nother nontrivil loclly convex spce. Let us mention tht ll round this pper we write min (mx) insted of inf (sup) when the infimum (supremum) is ttined. Theorem 2.2. Let f : Z R be proper nd convex function nd A : U Z liner continuous mpping. Assume tht there exists x A (dom(f)) such tht f is continuous t Ax. Then (f A) (u ) minf (z ) : A z u, u U. The next result we recll in this section dels with Lgrnge dulity for the optimiztion problem with cone constrints (P ) inf f(x), x X, g(x) K where X Z is non-empty convex set, K U is closed, convex cone, f : Z R is convex, continuous function nd g : Z U is K-convex (g(z) + K is convex), continuous function. For hving strong dulity between (P ) nd the Lgrnge dul problem (D L ) sup inf f(x) + λ, g(x), λ K x X 3

5 some wek closedness type constrint ulifictions hve been recently introduced in the literture (cf. [3], [4], [6]). In the definition of (D L ), by K λ U : λ, u 0, u K we denote the dul cone of K. Let further λ, g : Z R be the function defined by λ, g (x) : λ, g(x), x X. In order to formulte the strong dulity between (P ) nd (D L ) let us introduce first the following so-clled closed cone constrint ulifiction (see [4], [6]): (CCCQ) epi(( λ, g + δ X ) ) is wek closed set. λ K For the optimiztion problem (P ) we denote by v(p ) its optiml objective vlue. Theorem 2.3. ([4], [6]) Under the ssumptions mde bove for (P ), if (CCCQ) is fulfilled, then v(p ) v(d L ) nd the Lgrnge dul hs n optiml solution. Consider now the probbility spce (Ω, F, P), where Ω is bsic spce, F σ- lgebr on Ω nd P probbility mesure on the mesureble spce (Ω, F). For mesurble rndom vrible x : Ω R the expecttion vlue is defined with respect to P by E(x) x(ω)dp(ω). The essentil supremum of x is Ω essup x inf R : P(ω : x(ω) > ) 0. Furthermore for p [, + ) let L p be the following liner spce of rndom vribles: L p : L p (Ω, F, P, R) x : Ω R, x mesureble, x(ω) p dp(ω) < +. The spce L p euipped with the norm x p (E( x p )) p for x Lp is Bnch spce. It is well-known tht the dul spce of L p is L : L (Ω, F, P, R), where (, + ] fulfills +. The spce p L : L (Ω, F, P, R) x : Ω R, x mesureble, essup x < + is considered to be euipped with the essentil supremum norm. In order to mke these spces pired we consider on L p the norm topology nd on L the wek topology. If p (, + ) then L p nd L re reflexive Bnch spces nd they re pired spces euipped with the norm topologies. The closed unit bll in L is denoted by B (0, ). Ω 4

6 For x L p, x L nd x x : Ω R, defined by (x x)(ω) : x (ω) x(ω), one cn define now x, x : E(x x) x (ω)x(ω)dp(ω). Eulities nd ineulities between rndom vribles re to be viewed in the sense of holding lmost surely (.s.). Thus for x, y : Ω R when we write x y or x y we men x y.s. or x y.s., respectively. For p [, + ), the cone (L p ) + x L p : x 0.s. is inducing the prtil ordering denoted by. The dul cone of (L p ) + is (L ) +, where (, + ] fulfills +. The p prtil ordering induced by (L ) + is lso denoted by. As these orderings re given in different liner spces, no confusion is possible. Hving rndom vrible x : Ω R which tkes the constnt vlue c R, i.e. x c.s., we identify it with the rel number c R. For n rbitrry rndom vrible x : Ω R we lso define x, x + : Ω R in the following wy: x (ω) : mx( x(ω), 0) Ω ω Ω nd x + (ω) : mx(x(ω), 0) ω Ω. One cn esily see tht x x + x, x + ( x) nd x ( x) +. 3 Risk mesures nd devition mesures In this section we give some forml definitions of convex risk nd devition mesures. In 2002 Föllmer nd Schied (cf. [5]) first introduced the convex risk mesures s n extension of the well-known coherent risk mesures. The ltter hve been introduced in [], where for the first time n xiomtic wy for defining risk mesures hs been given. Rockfellr nd his couthors (see [0]) introduced long the coherent risk mesures the so-clled devition mesures nd studied the reltion between these concepts. In this pper we del with the brod clss of convex risk mesures s done by Ruszczynski nd Shpiro (cf. [2]) nd Pflug (see [8]), respectively. We wnt to notice tht lrge number of risk functions mentioned in the litertur do not hve the sublinerity properties sked by the xioms of coherent risk mesure, however they fulfill the properties in the definition of convex risk mesure. In the following definition we introduce the notion of convex risk mesure s done in [8]. Definition 3.. The function ρ : L p R is clled convex risk mesure if the following properties re fulfilled: (R) Trnsltion euivrince: ρ(x + b) ρ(x) b, x L p, b R; (R2) Strictness: ρ(x) E(x), x L p ; (R3) Convexity: ρ(λx+( λ)y) λρ(x)+( λ)ρ(y), λ [0, ], x, y L p. 5

7 For certin pplictions it cn be usefull to postulte some monotonicity properties of the risk mesure, like, for exmple, the monotonicity with respect to the pointwise ordering: x y ρ(x) ρ(y), x, y L p. Closely relted to the risk mesure we cn define the so-clled convex devition mesure. Definition 3.2. The function d : L p R is clled convex devition mesure if the following properties re fulfilled: (R) Trnsltion invrince: d(x + b) d(x), x L p, b R; (R2) Strictness: d(x) 0, x L p ; (R3) Convexity: d(λx+( λ)y) λd(x)+( λ)d(y), λ [0, ], x, y L p. The following theorem sttes the connection between convex risk nd convex devition mesures (see [8],[0], []). Theorem 3.. The function ρ : L p R is convex risk mesure if nd only if d : L p R, d(x) ρ(x) + E(x), x L p, is convex devition mesure. Next we give some exmples for convex risk mesures nd for the corresponding devition mesures. Exmple 3.. First we consider, for p 2, ρ : L 2 R defined by ρ(x) x E(x) 2 2 E(x), x L 2. This is convex risk mesure nd it is closely relted to the clssicl vrince σ 2 (x) which is the corresponding devition mesure: d(x) σ 2 (x) x E(x) 2 2, x L 2. Exmple 3.2. Let be gin p 2 nd ρ : L 2 R defined by ρ(x) x E(x) 2 E(x), x L 2. The relted convex devition mesure is the stndrd devition σ(x) d(x) σ(x) x E(x) 2, x L 2. The convex risk nd devition mesures in the previous exmples re specil cses of some generl clsses of risk nd devition mesures, respectively, which re described in the following. 6

8 Exmple 3.3. For p [, + ) nd we define the convex risk mesure ρ : L p R, ρ(x) x E(x) p E(x), x L p. The corresponding convex devition mesure is d : L p R, d(x) x E(x) p, x L p. In cse p, d is clled men bsolute devition. Exmple 3.4. Similr to Exmple 3.3, for p [, + ) nd we consider the following pirs of convex risk nd devition mesures, ρ : L p R nd d : L p R defined by ρ(x) (x E(x)) p E(x), d(x) (x E(x)) p nd ρ(x) (x E(x)) + p E(x), d(x) (x E(x)) + p, respectively. The devition mesures we get by tking p re the so-clled lower nd upper semidevition, respectively. For p 2 nd we get the stndrd lower nd upper semidevition, respectively. 4 Conjugtes of convex devition mesures: the cse In this section we del with the formuls of the conjugte functions of some convex devition mesures, including those in Exmple 3.3 nd 3.4, for p [, + ) nd. Hving these formuls we cn esily clculte the formuls of the conjugte functions of the corresponding risk mesures. The following reltion proves this, s for x L it holds ρ (x ) sup x L p x, x ρ(x) sup x L p x, x d(x) + E(x) sup x L p x, x d(x) +, x sup x L p x +, x d(x) d (x + ). In order to derive the formuls for the conjugtes of the convex devition mesures we need the following preliminry results. Exmple ()

9 Let be f : L p R, f (x) x p. It is well-known tht the conjugte function of f is f : L R, 0, x f (x ), Exmple 4.2. We consider now f 2 : L p R, f 2 (x) x p. For x L one obtins the following formul for the conjugte function of f 2, f 2 : L R: f 2 (x ) inf x L p x p x, x inf x L p mx( x, 0) p x, x. Hving for n rbitrry z L p with the property z mx( x, 0) 0 tht z p mx( x, 0) p, one gets further f 2 (x ) inf z p x, x x L p,z L p, z mx( x,0) inf (x,z) L p L p, x z 0, z 0 Let (P ) be the following convex optimiztion problem: (P ) inf (x,z) L p L p, x z 0, z 0 The Lgrnge dul problem of (P ) looks like z p x, x. z p x, x. (D L ) sup inf z p x, x λ, z λ 2, x + z λ,λ 2 (L ) + (x,z) L p L p sup inf x, x λ 2, x + inf z p λ, z λ 2, z λ,λ 2 (L ) + x L p z L p sup inf x + λ 2, x + inf z p λ + λ 2, z λ,λ 2 (L ) + x L p z L p sup δl p (x + λ 2 ) ( p ) (λ + λ 2 ). λ,λ 2 (L ) + Since nd 0, x δl p (x + λ 2 ) λ 2, 0, ( p ) λ + λ (λ + λ 2 ) 2, 8

10 the optiml objective vlue of the Lgrnge dul (D L ) cn be written s 0, λ (L v(d L ) ) + : λ x nd x (L ) +,, otherwise, ) 0, x (B (0, ) + (L ) + (L ) +,, otherwise, 0, x B (0, ) (L ) +,, otherwise. The euivlence of ) the lst two formuls comes from the eulity of the sets (B (0, ) + (L ) + (L ) + nd B (0, ) (L ) +. As the inclusion B (0, ) ) (L ) + (B (0, ) + (L ) + (L ) + is trivil, we prove the opposite one. ) Let be x (B (0, ) + (L ) + (L ) +. Then x t + z 0, where t B (0, ) nd z (L ) +. Since x 0 nd z 0 we hve 0 x t z t nd so x x t t. Thus x B (0, ) (L ) +. In order to identify f 2 (x ) with the optiml objective vlue of (D L ) we hve to prove tht between (P ) nd (D L ) strong dulity holds. As (P ) is convex optimiztion problem one needs constrint ulifiction for closing the gp between these duls. Let us notice tht, since int((l p ) + ) core((l p ) + ), one cnnot use the generlized interior-point constrint ulifictions given in the literture for convex optimiztion problems with cone ineulity constrints. But, wht we prove now is tht (CCCQ) is fulfilled. Let be g : L p L p Z, g(x, z) ( x z, z) nd λ (λ, λ 2 ) (L ) + (L ) +. One hs (x, z, r) epi(( λ, g + δ Lp L p ) ) epi( λ, g ) λ, g (x, z ) r sup x L p,z L p x, x + z, z λ, x z λ 2, z r sup x, x λ, x + sup z, z λ + λ 2, z r x L p z L p sup x L p x + λ, x + sup z L p z + λ + λ 2, z r x λ, z λ λ 2, r [0, + ). In order to prove tht (CCCQ) is fulfilled we hve to show tht M : λ λ λ 2 [0, + ) λ,λ 2 (L ) + 9

11 is closed in L L R with respect to the product topology induced by the norm topology on L nd the Eucliden topology on R. One cn notice tht since M is convex this is the sme with M is wek closed. Tke (x, z, r) cl(m) nd (x n, zn, r n ) M such tht (x n, zn, r n ) (x, z, r) (n + ). Then there exists n N, (λ n ), (λ n 2) (L ) + nd r n 0 such tht x n λ n nd zn λ n λ n 2. It follows x (L ) +, x z lim ( λ n z n n) lim (λ n n 2) (L ) + nd r 0. Thus (x, z, r) ( ( x ), ( x ) (x z ), r) M nd the set M turns out to be closed. By Theorem 2.3 it follows tht 0, x f2 (x ) v(p ) v(d L ), x 0, (2) Finlly we give n euivlent formultion since the domin of the conjugte f2 cn be restricted to the set of those x L which fulfill x 0 nd E(x ) 0. (Note tht x 0 implies E(x ) 0 nd tht x implies E(x ).) This leds to the following formul for the conjugte of f 2 (see lso [8]): 0, x f2 (x ) 0, x, E(x ) 0, (3) In the next exmple we del with the conjugte function of the devition mesure d (x) x E(x) p, which will be derived from more generl formul. Exmple 4.3. Consider d : L p R, d (x) x E(x) p nd A : L p L p, Ax x E(x). Here we hve to interpret E(x) R s (constnt) element of L p. Denoting for C F by C : Ω R the rndom indictor function, ω C, C (ω) 0, otherwise, the liner continuous mpping A cn be written s Ax x E(x) Ω. Hving now d (x) Ax p, x L p, in order to clculte d, we cn use Theorem 2.2. Since p is continuous on L p, the regulrity condition is fulfilled nd we hve x L : d (x ) min( p ) (y ) : A y x 0, y L : A y x nd y, From the clcultion bove one cn see tht we need the djoint opertor of A. In the following we show tht A is self-djoint, i.e. A A. For x L p nd x L it holds: x, Ax x, x E(x) x, x x, E(x) Ω. 0

12 The second term cn be written s follows (we pply here the Theorem of Fubini): x,e(x) Ω x (ω)e(x)dp(ω) x (ω) x(τ)dp(τ) dp(ω) Ω x(τ) x (ω)dp(ω) dp(τ) Ω Ω x(τ)e(x )dp(τ) E(x ) Ω, x. Ω Ω Ω So we hve x, Ax x E(x ) Ω, x, x L, x L p nd, in conclusion, A x x E(x ) Ω x E(x ). Thus the conjugte function of d becomes x L, 0, y d (x ) L : y E(y ) x nd y, We prove now tht y L such tht y E(y ) x nd y if nd only if E(x ) 0 nd c R such tht x c. Let be n y L fulfilling y E(y ) x nd y. Then E(x ) E(y E(y )) 0 nd for c : E(y ) one hs x c. On the other hnd, ssume tht E(x ) 0 nd tht c R with the property x c. Defining y : x c L one hs y nd y E(y ) x. So the conjugte of d turns out to be x L, 0, E(x d (x ) ) 0 nd c R : x c, 0, E(x ) 0 nd min c R x c, Considering the convex risk mesure ρ : L p R, ρ (x) d (x) E(x) x E(x) p E(x), by (), one cn esily deduce the formul for the conjugte of ρ : L p R. This looks like 0, E(x ) nd min ρ (x ) d (x + ) c R x c, (4) In the lst exmple we consider in this section, we clculte the conjugte function of the convex devition mesure know lso s lower semidevition. After tht, we derive the formul for the conjugte of the corresponding convex risk mesure. Exmple 4.4. Let be d 2 : L p R, d 2 (x) (x E(x)) p. Denoting gin by A : L p L p the liner continuous mpping defined by Ax x E(x), we hve tht d 2 f 2 A.

13 Since f 2 is convex continuous function with rel vlues, by Theorem 2.2, one hs for ll x L d 2(x ) minf 2 (y ) : A y x, which cn be further written s (see (3)) d 2(x ) 0, y L : A y x, y 0, y, E(y ) 0, Since A y y E(y ) (see Exmple 4.3) we obtin x L d 2(x ) 0, y L : y E(y ) x, y 0, y, E(y ) 0, Like in Exmple 4.3 one cn show tht there exists y L such tht y E(y ) x, y 0, y nd E(y ) 0 if nd only if E(x ) 0 nd there exists c R fulfilling 0 c, x c nd x c. Thus 0, E(x d 2(x ) ) 0 nd c R : 0 c, x c, x c, Let us prove now tht for x L the reltions nd E(x ) 0 nd c R : 0 c, x c, x c (5) E(x ) 0, x, essup x x (6) re euivlent. Assuming tht (5) holds, one hs x c. Further we hve essup x c nd this mens tht c x essup x x 0, implying c x essup x x. Reltion (6) is so proved. On the other hnd, if (6) holds, one cn tke c essup x. Tht c nd x c is obvious. Assuming now tht c < 0, this would men tht E(x ) < 0. In conclusion, reltion (5) must lso hold. This led us to the following formul for d 2 x L, 0, E(x d 2(x ) ) 0, x, essup x x, (7) As in the previous exmple, the formul of the conjugte function of the corresponding convex risk mesure ρ 2 : L p R, ρ 2 (x) (x E(x)) p E(x) cn be lso clculted. By () we hve x L, 0, E(x ρ 2(x ) d 2(x + ) ), x 0, essup x x, 2

14 Remrk 4.. One cn notice tht the formuls for the conjugtes of f 2 nd d 2 llow us to clculte the formuls for the conjugtes of the functions x x + p nd x (x E(x)) + p, s these re nothing but f 2 ( x) nd d 2 ( x), respectively. In generl, hving h : Z R, h(x) f( x) it holds x Z h (x ) sup x, x h(x) sup x, x f( x) x Z x Z sup x, x f(x) f ( x ). x Z 5 Conjugtes of convex devition mesures: the cse > In this section we extend our investigtions to the conjugte functions of convex devition mesures given in Exmple 3.3 nd Exmple 3.4, but in the cse > (like before, p [, + )). We use reltion () in order to clculte the conjugte functions of the corresponding convex risk mesures. In our pproch we use the very well-developed clculus existing in the theory of conjugte functions. The functions considered in this section will be viewed s compositions of convex incresing function with convex function. The conjugtes will be obtined by using formul existing in the literture for the conjugte of composed convex function (see [2] for more on this subject). Let us stte now the min theorem used in this section: Theorem 5.. Let Z be nontrivil loclly convex spce nd f : Z R, g : R R be convex functions such tht g is incresing on f(z) + [0, + ). We ssume tht there exists x Z such tht f(x ) dom(g) nd g is continuous t f(x ). Then for ll x Z one hs (g f) (x ) min β R + g (β) + (βf) (x ). (8) We pply Theorem 5., by tking for f the convex devition mesures considered in the previous section nd for g : R R the function defined for > by x g(x), x 0, The set f(l p ) + [0, + ) is eul [0, + ) nd one cn see tht both f nd g re convex functions nd tht g is incresing on [0, + ). The regulrity condition is lso fulfilled, so formul (8) will hold. The following lemm provides the formul for the conjugte of the function g. 3

15 Lemm 5.. The conjugte function of g, g : R R is g (β) ( ) ( ) β, β 0, 0, otherwise. Proof. By definition it holds g (β) sup x R (xβ g(x)) sup (xβ x ). x 0 In the cse β 0, one gets g (β) 0. In cse β > 0, we consider h : [0, + ) R, h(x) xβ x. One hs h (x) 0 x ( β ttins its mximum t x ( β g (β) β ) ( ) ( β (β ) ) β > 0 nd so ) > 0. Since h is concve it ( ) [ β ] ( ) ( ) β. In conclusion, we get g (β) ( ) ( ) β, β 0, 0, otherwise. In order to clculte the formuls for the conjugte functions of the convex devition mesures in Exmple 3.3 nd Exmple 3.4 we need the following intermedite formuls. Exmple 5.. Let be f 3 (x) x p. For x L p we hve f 3 (x) (g f )(x). Let us for β R + first clculte the formul for (βf ) (x ). Since for β > 0 nd x L it holds (see Exmple 4.) ( ) ( ) β (βf ) (x ) βf β x β( p ) x 0, x β, 0, x β, nd for β 0 nd x L one hs 0, x (βf ) (x ) 0, 4

16 we finlly get β 0, x L, 0, x (βf ) (x ) β, Thus with Theorem 5. the conjugte of f 3 becomes x L, f 3 (x ) min g (β) β 0, x β min ( ) β 0, x β ( ) β ( ) x. Exmple 5.2. Let be f 4 : L p R, f 4 (x) x p, x L p. One cn see tht in this cse f 4 g f 2. In order to use the reltion in (8), we hve to clculte (βf 2 ) for β 0. If β 0 one hs gin 0, x (βf 2 ) (x ) 0, while if β > 0, by (2), it holds ( ) (βf 2 ) (x ) βf2 β x 0, β x, β x 0, 0, x β, x 0, By (8) we obtin tht for ll x L such tht x (L ) +, f 4 (x ) min β 0 g (β) + (βf 2 ) (x ) ( ) x. min β 0, x β ( ) If x (L ) +, f4 (x ) +, so one hs x L, f4 (x ( ) ) x, x (L ) +, +, otherwise. ( ) β In the next exmple, we come to the convex devition mesure considered in Exmple 3.3, d 3 (x) x E(x) p, x L p. 5

17 Exmple 5.3. The convex devition mesure d 3 : L p R, d 3 (x) x E(x) p cn be written s d 3 g d. Let be x L. For β 0 one hs 0, x (βd ) (x ) 0, while if β > 0 it holds, (βd ) (x ) βd ( ) β x ( ) 0, E β x 0 nd min c R β x c, 0, E(x ) 0 nd min c R x c β, Then by (8), we hve x L such tht E(x ) 0, d 3(x ) min g (β) min ( ) β 0, c R (x c) β min c R x c nd d 3(x ) +, if E(x ) 0. We conclude tht d 3(x min ( ) ) c R (x c) +,, E(x ) 0, otherwise. The conjugte function of the corresponding convex risk mesure ρ 3 : L p R, ρ 3 (x) d 3 (x) E(x) x E(x) p E(x), turns out to be x L p (cf. ()), ρ 3(x ) d 3(x + ) min ( ) c R (x c) +,, E(x ), otherwise. The lst conjugte function we derive now is tht of the convex devition mesure given in Exmple 3.4. Exmple 5.4. Considering d 4 : L p R, d 4 (x) (x E(x)) p, x L p, one cn see tht d 4 g d 2. Let us clculte now for ll β 0, (βd 2 ). We fix n x L. If β 0, then 0, x (βd 2 ) (x ) 0, 6

18 while for β > 0 one hs (see (7)) ( ) (βd 2 ) (x ) βd 2 β x ( ) ( ) 0, E β x 0, β x, essup β x β x, 0, E(x ) 0, x β, essup x x β, Let us notice tht for E(x ) 0 if β essup x x, then β E(essup x x ) essup x, which implies tht (βd 2 ) (x ) is nothing else thn 0, E(x (βd 2 ) (x ) ) 0, essup x x β, By (8) we get x L such tht E(x ) 0, ( ) β d 4(x ) inf ( ) ( ) β 0, (essup x x ) β essup x x while if E(x ) 0, d 4(x ) +. Thus d 4(x ( ) ) (essup x x ), E(x ) 0, +, otherwise. The conjugte function of the corresponding convex risk mesure ρ 4 : L p R, ρ 4 (x) d 4 (x) E(x) (x E(x)) p E(x) follows x L (cf. ()), ρ 4(x ) d 4(x ( ) + ) (essup x x ), E(x ), +, otherwise. 6 Dul representtion of convex risk mesures In this section we give for the convex risk nd devition mesures considered in this pper some dul representtions which will follow by pplying the Fenchel- Moreu theorem (Theorem 2.). For p [, + ) nd f : L p R proper, convex nd lower-semicontinuous function we hve x L p, f(x) f (x) sup x L x, x f (x ) sup x L E(x x) f (x). (9) As ll convex risk nd devition mesures fulfill the hypotheses of Theorem 2., by using the formuls of the conjugtes derived in the previous sections, we obtin 7,

19 in very nturl wy the desired dul representtions. Our representtions turn out to be generliztions of the recently published results by Pflug (cf. [8]). More thn tht, we show the usefulness of the powerful theory of conjugte functions from the convex nlysis in this field s well. Exmple 6.. The first convex devition mesure we tret is d : L p R, d (x) x E(x) p. We proved tht x L, 0, E(x ) 0 nd min d (x ) c R x c, nd so, by (9), x L p, d (x) supe(x x) : x L, E(x ) 0 nd min c R x c. Anlogously, by (4), we obtin x L p, ρ (x) sup E(x x) : x L, E(x ) nd min c R x c. Pflug (cf. [8, Proposition 3]) lso gives for p (, + ) representtions for these convex risk nd devition mesures, which re ctully generliztions of the stndrd devition. The formuls given by Pflug re not uite ccurte, s he considers inf insted of min. But, s we hve seen in the previous chpters, the existence of c R, such tht x c, is indispensble. Let us lso notice tht Pflug uses insted of convex risk mesures so-clled cceptbility functionls (we denote them like in [8] by A). They re linked to the convex risk mesures in our pper by the reltion A(x) ρ(x), x L p. Exmple 6.2. Tke now d 2 : L p R, d 2 (x) (x E(x)) p, s in Exmple 4.4. For the conjugte function d 2 it holds x L (see (7)), 0, E(x d 2(x ) ) 0, x, essup x x, nd so one gets the following dul representtion x L p, d 2 (x) supe(x x) : x L, E(x ) 0, x, essup x x. Similry, we get x L p, ρ 2 (x) supe(x x) : x L, E(x ), x 0, essup x x. The lst two eulities re ctully the formuls proved in Proposition 5 in [8]. 8

20 Exmple 6.3. Let be d 3 (x) x E(x) p, x L p, the convex risk mesure considered in Exmple 5.3 for >. The conjugte funtion d 3 : L R is x L, d 3(x ) min ( ) c R (x c) +,, E(x ) 0, otherwise. By (9) we get the following dul representtion x L p, d 3 (x) sup E(x x) min ( ) c R (x c) : x L, E(x ) 0. Similrly, we get the representtion of the convex risk mesure x L p, ρ 3 (x) sup E(x x) min ( ) c R (x c) : x L, E(x ). Pflug gives in Proposition 2 in [8] the formul just for the specil cse when p nd p (, + ). More thn tht, his formul my be improved by mentioning tht the inner infimum is ttined. Exmple 6.4. Finlly let be d 4 : L p R, d 4 (x) (x E(x)) p, x L p, where >. From Exmple 5.4, one hs x L, d 4(x ( ) ) (essup x x ), E(x ) 0, +, otherwise, nd so, x L p, d 4 cn be represented s d 4 (x) sup E(x x) ( ) (essup x x ) : x L, E(x ) 0. Agin, for the convex risk mesure ρ 4 (x) (x E(x)) p E(x) we get x L p, ρ 4 (x) sup E(x x) ( ) (essup x x ) : x L, E(x ) These ssertions generlize Proposition 4 in [8] where the formuls hve been given just in the cse p nd p (, + ).. 9

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