Convex Risk Measures Beyond Bounded Risks, or The Canonical Model Space for Law-Invariant Convex Risk Measures is L^1

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1 Working Paer Series Working Paer No. 2 Convex Risk Measures Beyond Bounded Risks, or The Canonical Model Sace for Law-Invariant Convex Risk Measures is L^1 Damir Filiović and Gregor Svindland First version: July 27 Current version: March 28

2 Convex Risk Measures Beyond Bounded Risks, or The Canonical Model Sace for Law-invariant Convex Risk Measures is L 1 Damir Filiović Gregor Svindland 16 March 28 Abstract In this aer we rovide a rigorous toolkit for extending convex risk measures from L to L, for 1. Our main result is a one-to-one corresondence between law-invariant convex risk measures on L and L 1. This roves that the canonical model sace for the redominant class of law-invariant convex risk measures is L 1. Some significant counterexamles illustrate the many itfalls with convex risk measures and their extensions to L. Key words: convex risk measures, extensions to L, law-invariant convex functions 1 Introduction Convex risk measures are best known on L. Indeed, Artzner et al. [2] introduced the seminal axioms of coherence, which then were further generalised to the convex case by Föllmer and Schied [12] and Frittelli and Rosazza-Gianin [14], on L. However, there is a growing mathematical finance literature dealing with convex risk measures beyond L, see e.g. [3, 4, 7, 16, 17, 19]. This extended aroach is vital since imortant risk models, such as normal distributed random variables, are not contained in L. In some of the above mentioned articles, the model sace is chosen such that some reselected risk measure remains finite valued. In contrast, we believe that A former version of this aer circulated under the title Risk Measures on L. We thank Patrick Cheridito, Volker Krätschmer and Michael Kuer for helful discussions. Filiović is suorted by WWTF (Vienna Science and Technology Fund). Address: Vienna Institute of Finance, University of Vienna, and Vienna University of Economics and Business Administration, Heiligenstädter Strasse 46-48, A-119 Wien, Austria. damir.filiovic@vif.ac.at Svindland gratefully acknowledges financial suort from Munich Re Grant for doctoral students. Address: Mathematics Institute, University of Munich, Theresienstrasse 39, D München, Germany. svindla@mathematik.uni-muenchen.de 1

3 the model sace should be maximal ossible to cature the universe of financial risks from the outset. On the other hand, from a comutational oint of view, the model sace should be standard and endowed with a toological structure suorting convex duality. In sum, we roose the model sace L for 1. The aim of this aer is threefold. First, the last ten years of risk measure theory has been mainly focused on L. This requires and we rovide a rigorous toolkit for extending convex risk measures from L to L. Second, we establish a one-to-one corresondence between law-invariant convex risk measures on L and L 1. This roves that the canonical model sace for the redominant class of law-invariant convex risk measures is L 1. Third, we rovide some significant counter-examles which illustrate the itfalls with convex risk measures and their extensions to L. Notwithstanding the above mentioned literature, we believe that such elaboration is needed. In fact, our aer can be seen as a comletion of the interlay of convex risk measures on and beyond L. The structure of the aer is as follows. In section 2 we recall some fundamental roerties of convex risk measures on L and conclude with a remarkable dichotomy: a convex risk measure ρ on L is either continuous on L or its domain has emty interior. Section 3 contains our main results. Point of dearture is an arbitrary given function f on L. We then discuss existence and uniqueness of closed convex extensions of f to L. It turns out that such extensions do not always exist (section 5.3), and if they exist are not unique in general (section 5.2). We define and show that the L -closure f of f is the greatest closed convex function on L majorised by f on L (theorem 3.2). Moreover, if a closed convex extension of f to L exists, then f is the greatest of such extensions (theorem 3.7). As an advice against using value at risk (VaR), we show that there exists no non-trivial closed convex function majorised by VaR (section 5.5). Our main result (theorem 3.8) states that there is a one-to-one corresondence between law-invariant closed convex functions on L and L 1. In articular, for any law-invariant closed convex function f on L, the L 1 -closure f 1 is the unique law-invariant closed convex extension to L 1. We conclude that the canonical model sace for law-invariant convex risk measures is L 1. The roof of this theorem is given in sections 6 7. In section 4, we discuss an alternative su inf -extension of convex risk measures from L to L, which was roosed by Krätschmer [17]. However, it turns out that this aroach has itfalls and is actually the source for some of the above mentioned (counter-)examles. For the sake of readability of the aer, we collected all examles, which illustrate the itfalls with convex risk measures and their extensions, in section 5. Finally, section 8 contains the roof of theorem Convex Risk Measures on L Throughout, we fix a standard robability sace (Ω, F, P), i.e. (Ω, F, P) has no atoms and L 2 (Ω, F, P) is searable (in fact, this assumtion is only needed in 2

4 theorem 3.8 and sections 5 7 below). All equalities and inequalities between random variables are understood in the P-almost sure (a.s.) sense. The toological dual sace of L = L (Ω, F, P), for [1, ], is denoted by L, the ositive order cone by L + and its olar cone by L. It is well known that L L 1 can be identified with ba, the sace of all bounded finitely additive signed measures on (Ω, F) which vanish on P-null sets. With some facilitating abuse of notation, we shall write (X, Z) E[XZ] for the dual airing on (L, L ) also for the case =. We suose the reader is familiar with standard terminology and basic duality theory for convex functions as outlined in [18] or [8]. We recall that a function f : L [, ] is (i) convex if its eigrah ei f := {(X, y) f(x) y} is a convex subset of L R, (ii) roer if f > and its domain dom f := {f < }, (iii) closed if either f or f or f is roer and lower semi-continuous (l.s.c.), (iv) law-invariant if f(x) = f(y ) for all identically distributed X Y. The conjugate f (Z) = su X L (E[XZ] f(x)) of f is a closed convex function on L. The Fenchel Moreau theorem (roosition 4.1 in [8]) states that f = f if and only if f is closed convex. The indicator function δ( C) of a set C is defined as zero on C and elsewhere. We denote by C the closure of a set C in L. Definition 2.1. A convex function ρ : L (, ] is called convex risk measure if it is (i) cash-invariant: ρ() R and ρ(x + m) = ρ(x) m for all m R, (ii) monotone: ρ(x) ρ(y ) for X Y. We denote by A ρ := {ρ } the accetance set of ρ. A ositively homogeneous convex risk measure ρ is called coherent. Remark 2.2. Examle 5.1 below shows that convex risk measures are not closed in general. Convex risk measures on L and more general model saces have been studied, among others, in [3, 4, 16, 19]. The next theorem summarises some fundamental roerties that will be used in the sequel. For the sake of comleteness, we rovide a roof in section 8. Theorem 2.3. Let ρ be a convex risk measure on L. (i) int dom ρ if and only if ρ is continuous on L. (ii) A ρ is convex with A ρ + L + A ρ, R A ρ, and A ρ L. 3

5 (iii) For any A L satisfying roerties (ii) instead of A ρ, ρ A (X) := inf{m R X + m A} defines a convex risk measure on L. Moreover, if A is closed, then A ρa = A and ρ A is closed. (iv) dom ρ P := {Z L ρ (Z) = su X Aρ E[ZX]. E[1Z] = 1} and for all Z P we have Part (i) yields a remarkable dichotomy for convex risk measures ρ on L : either ρ is continuous on L or int dom ρ =. 3 L -closures This section contains our main results, which are valid not only for convex risk measures. We therefore fix [1, ] and some function f : L [, ]. Its conjugate f (Z) = su (E[ZX] f(x)) X L is a closed convex function on L, and hence on L. The following is thus well defined. Definition 3.1. The L -closure of f is defined as f (X) := su (E[ZX] f (Z)), X L. (3.1) Z L We next rove some fundamental roerties of f. Theorem 3.2. (i) f is the greatest closed convex function on L majorised by f on L. (ii) (f ) = f L. (iii) f is roer if and only if f is roer and dom f L. (iv) If either f is finite or f is roer then ei f = co ei f, where the right hand side denotes the L R-closure of the convex hull of ei f. Proof. By construction, f is a closed convex function on L with f f on L. (3.2) Now let g be any closed convex function on L with g f on L. Then, for all Z L, g (Z) = su E[XZ] g(x) su E[XZ] f(x) = f (Z). (3.3) X L X L 4

6 Hence, g(x) = g (X) and (i) is roved. Now let Z L. By definition we obtain su E[XZ] f (Z) = f (X), Z L (f ) (Z) = su X L (E[XZ] su Y L (E[XY ] f (Y ))) f (Z). On the other hand, from (3.3) we infer (f ) (Z) f (Z). This roves (ii). Proerty (iii) is obvious. As for (iv), inequality (3.2) imlies ei f ei f on L R. By convexity and closedness of ei f we thus have co ei f ei f. To show the converse inclusion, we note that g(x) = inf{a (X, a) co ei f } defines a l.s.c. convex function on L with f g on L and g f. Thus, if either f is finite or f is roer then g is closed. In view of the first art of the lemma, we conclude that g = f and thus co ei f = ei f. Thus the theorem is roved. Remark 3.3. Note that, for r, even if f is roer, we do not necessarily have that f L r = f r, see examle 5.6 below. Moreover, theorem 3.2(iv) does not hold without requiring that f be finite or f be roer. Indeed, examle 5.7 below shows that co ei f ei f is ossible. The next corollary shows that monotonicity and cash-invariance of convex risk measures is reserved under L -closure. Corollary 3.4. Let ρ be a convex risk measure on L. Then ρ is a closed convex risk measure on L if and only if A ρ L. In either case, ρ = ρ Aρ and A ρ = A ρ. Proof. Since any convex risk measure ρ on L is finite (due to monotonicity and cash-invariance), we infer from theorem 3.2(iv) that ei ρ = ei ρ, (3.4) A ρ = {X L (X, ) ei ρ } = {X L (X, ) ei ρ } = A ρ. (3.5) Hence, if ρ is a closed convex risk measure on L, then A ρ must have the roerties stated in theorem 2.3(ii), in articular A ρ L. Conversely, suose that A ρ L. We claim that A ρ satisfies the conditions of theorem 2.3(ii). Firstly, convexity and A ρ R are obvious, and secondly we have A ρ L by assumtion. In order to verify the yet missing condition, let X A ρ and Y L +. Choose (X n ) n N A ρ converging to X. As Y n L + for all n N, we have Y n := Y n + X n A ρ for all 5

7 n N. Since Y n converges to X + Y w.r.t. the -norm, we conclude that (X + Y ) A ρ. Consequently, according to theorem 2.3(iii), ρaρ is a closed convex risk measure on L. As ei ρ Aρ = ei ρ, we infer from (3.4) and (3.5) that ρ = ρ Aρ and A ρ = A ρ. Definition 3.5. A function g : L (, ] is called an extension of f to L if g = f on L. In the following we elaborate on the existence and uniqueness of closed convex extensions. Let us first discuss uniqueness. For illustration, consider a closed convex function g on L. Obviously, g L is a closed convex extension of g L to L. That is, g = g L on L. However, examle 5.3 below illustrates that this equality may fail on L. Hence uniqueness does not hold in general. In fact, an immediate consequence of theorem 3.2(ii) is the following: Corollary 3.6. Let g be a closed convex function on L. Then g = g L if and only if g = (g L ) on L. Examles below show convex risk measures on L which admit no closed convex extension to L. We now give necessary and sufficient conditions for the existence of closed convex extensions and illustrate the articular role of the L -closure. Theorem 3.7. The following roerties are equivalent: (i) There exists a closed convex extension of f to L. (ii) f is an extension of f to L. (iii) f is convex and σ(l, L )-closed. In either case, f is the greatest closed convex extension of f to L. Proof. (i) (ii): let g be a closed convex extension of f to L. Theorem 3.2(i) imlies g f and f = g f f on L. This roves (ii) and also the last statement of the theorem. (ii) (iii): this is obvious since f L is convex and σ(l, L )-l.s.c. (iii) (ii): the Fenchel Moreau theorem yields f(x) = su Z L (E[XZ] f (Z)) = f (X). If we restrict to law-invariant closed convex functions, then existence and uniqueness of closed convex extensions always holds. This is our main result, the roof of which is given in sections 6 and 7 below. Theorem 3.8. For any law-invariant closed convex function f on L and [1, ), the L -closure f equals f 1 L and is the unique law-invariant closed convex extension of f to L. In articular, f is σ(l, L )-closed. Hence there exists a one-to-one corresondence between law-invariant closed convex risk measures on L and L 1. In this sense, we may conclude that the canonical model sace for law-invariant convex risk measures is L 1. 6

8 4 An Alternative Extension? Let ρ be a convex risk measure on L. alternative extension of ρ to L : The following ma seems to be an ˆρ(X) := su inf n N ρ(x+ n X m), X L. (4.6) m N This formula has been suggested by Krätschmer [17]. Clearly, ˆρ = ρ on L. However, we demonstrate in examles that ˆρ is no closed convex risk measure on L in general, even if ρ is a closed convex risk measure (see examle 5.6). We also remark that in case [1, ) there are closed convex risk measures on L which do not admit a reresentation of tye (4.6). This means that lugging a closed convex risk measure ρ on L into the formula (4.6), we may have that ˆρ ρ. This is illustrated in examle 5.3. Trivially, if ρ is continuous, then ρ = ˆρ. Corollary 4.1. Suose that ˆρ is closed and convex on L. Then ˆρ = ρ. Proof. Since ˆρ is a closed convex extension of ρ, theorem 3.7 states that ˆρ ρ and equality holds on L. Monotonicity and l.s.c. of ρ (corollary 3.4) yield the converse inequality, for X L, ˆρ(X) = su lim inf m N n ρ (X + n X m) su ρ (X + X m) m N lim inf m ρ (X + X m) ρ (X). Remark 4.2. If ρ is law-invariant and continuous from below (i.e. for every bounded sequence (X n ) n N L such that X n X we have ρ(x n ) ρ(x)) then we know that ˆρ = ρ. This is roved in a forthcoming aer [11]. However, it is still an oen roblem whether ˆρ is closed and convex on L if ρ is closed convex on L in general. 5 Examles For the sake of readability of the aer, we collect all examles, which illustrate the itfalls with convex risk measures and their extensions, in this section. 5.1 Non-closed convex risk measures The first examle of this section shows that there are non-closed convex risk measures on L, [1, ). Examle 5.1. For [1, ), consider ρ : L (, ], ρ(x) = E[X] + δ(x L ). 7

9 This law-invariant coherent risk measure assigns to an endowment X the value in case the ossible losses are not bounded, and E[ X] else. Clearly, the accetance set A ρ is not closed, so ρ is not closed. For Z P we have that ρ (Z) = su E[XZ] ρ(x) = su E[X(Z + 1)] X L X {Y L Y L } su E[k(Z + 1)1 {Z> 1} ]. k R Hence ρ = δ( { 1}) = ( E), and thus ρ = E. 5.2 Non-uniqueness of closed convex extensions The following lemma will be used to show that closed convex extensions of convex risk measures, if they exist, are not unique in general. Lemma 5.2. Let T L 1 satisfy essinf T < and esssu T =. Denote by A the closed convex cone generated by T and L 1 +. Then ρ A is a closed coherent risk measure on L 1 such that ρ A essinf and ρ A L = essinf L. Proof. Theorem 2.3 imlies that ρ A is a closed coherent risk measure on L 1 such that A ρa = A. Moreover, ρ A essinf on L 1 since ρ A (T ) < essinf T by construction. We claim that A L = L +, (5.7) imlying that ρ A L = essinf L. As for the roof of (5.7), note that A = A 1 where A = {tx X B, t } and B = conv(t, L 1 +) + L 1 +. The inclusion L + A L follows from construction. To show the converse, L \ L + L \ (A L ), we choose any S L such that P(S < ) >. Since S L 1 + is bounded whereas T is unbounded from above, S cannot be an element of the convex hull conv(t, L 1 +), and neither of its monotone hull B, because any convex combination in conv(t, L 1 +) is either P-a.s. ositive or unbounded from above, so it cannot be dominated by S. But then, clearly S A too. Now suose S A \ A. Then there would be a sequence (S n ) n N A converging to S in L 1. By monotonicity of A we may assume that S n S for all n N (otherwise Ŝn := S n S A will do), and shifting to a subsequence if necessary, we may assume that S n S P-a.s. Clearly, there is some N N such that P(S n < ) > for all n N. By construction of A there are t n, α n (, 1] and X n, Z n L 1 + such that S n = t n (α n T + (1 α n )X n + Z n ). Thus {S n < } {T < }, and consequently we have P-a.s. that {S < } = {S n < } {T < }. k N m k 8

10 Let ɛ := E[ S1 {S<} ] > and δ := E[ T 1 {s<} ] >. Choose N 1 N such that for all n N 1 : S S n 1 = E[S n S] ɛ 2. Then, for n N 1 we have: ɛ 2 E[S n S] E[(S n S)1 {S<} ] = E[(t n α n T + t n (1 α n )X n + t n Z n S)1 {S<} ] t n α n E[T 1 {S<} ] + E[ S1 {S<} ] = t n α n δ + ɛ. Consequently, t n α n ɛ 2δ =: r >, and thus S n rt, for all n N 1. But this contradicts the boundedness of S. Hence S / A and (5.7) is roved. Examle 5.3. Let T and ρ A be as in lemma 5.2, and let ˆρ A be the function defined by (4.6). It is easily verified that ˆρ A = essinf = ρ A L 1 on L 1. Hence both ˆρ A and ρ A are closed convex risk measures that extend essinf to L 1. This shows that such extensions are not unique in general. 5.3 Non-extendable convex risk measures The next two examles show convex risk measure on L extended to L. which cannot be Examle 5.4. Let [1, ) and Z P \ P. Define ρ(x) = E[XZ]. Then ρ is a convex risk measure on L with L -closure ρ. Hence, we know by theorem 3.7 that there exists no closed convex extension of ρ to L. In articular ˆρ as defined in (4.6) cannot be closed convex on L. Examle 5.5. Let (Ω, F, P) = ((, 1], B(, 1], λ) where λ denotes the Lebesgue measure restricted to the Borel-σ-algebra B(, 1], and let A n := (, 1 2 ], n N. n Moreover, let P n ( ) := P( A n ), and we denote by essinf Pn (X) the essential infimum of a random variable X under the measure P n. Define ρ(x) := lim n essinf P n (X), X L. In fact, the function ρ := ρ L is a coherent risk measure on L. However, it is easily verified that for [1, ) there are X L such that ρ(x) =, so ρ fails to be roer on L. Moreover, A ρ = L. The domain of ρ is a subset of (L ) \ L 1, because for any Z P L 1 we have ρ (Z) = su E[XZ] su E[ k1 A c n Z] = su k(1 + E[Z1 An ]) =. X A ρ k,n N k,n N That is condition (iii) of theorem 3.2 is not satisfied. Hence, ρ =, and, according to theorem 3.7, ρ is a coherent risk measure on L which admits no closed convex extension to L, [1, ). Next we illustrate by a simle examle that we cannot exect the L -closure to be an extension in general, even if it is a closed convex function on L. 9

11 Examle 5.6. Let ρ(x) := max{e[ X], E[ZX], E[ ZX]}, X L, for some Z P 2 \ L and Z P \ L 1. It is easily verified that ρ is a coherent risk measure on L and that dom ρ is the convex hull co{ 1, Z, Z}. We have ρ (X) = max{e[ X], E[ZX]} for all [2, ), but ρ 1 = E[ X]. Clearly, ρ 1 ρ 2 on L 2 and ρ ρ on L for all [1, ). Hence, ρ is no extension of ρ although ρ is a continuous coherent risk measure on L. Moreover, we observe that L -closures of ρ for different may differ. As regards (4.6), theorem 3.7 imlies that ˆρ (which is roer due to ˆρ(X) E[ X], and an extension of ρ by definition) cannot be closed convex, because otherwise ρ would be an extension. In articular, we have that ρ ˆρ for every [1, ). 5.4 A counter-examle related to theorem 3.2 The next examle shows that the requirements in theorem 3.2(iv) cannot be droed in general. Examle 5.7. Recall examle 5.5 and define f : L (, ], X ρ(x) + δ(x C) where C := {X L 1 X a.s. on [1/2, 1]}. Clearly, C L is convex and closed for every [1, ]. Hence, f is a closed convex function on L. Next we rove by similar arguments as in examle 5.5 that dom f L 1 = imlying that f = for all [1, ). To this end, note that ρ(k1 [,1/2] ) = k for all k R. Consequently, for any Z L we have f (Z) su k(e[z1 [,1/2] ] + 1), k R so either E[Z1 [,1/2] ] = 1 or Z dom f. Now let Z L 1 such that E[Z1 [,1/2] ] = 1. Then P({Z < } [, 1/2]) >, and since X k,n := k1 A c n 1 [,1/2] 1 {Z<} C satisfy ρ(x k,n ) = for all k, n N, we obtain f (Z) su E[ZX k,n ] su ke[z1 {Z<} 1 [,1/2] ] =. k,n N k N Hence, f = for all [1, ) which is equivalent to ei f = L R. However, it is easily verified that co ei f = (C L ) R L R. 5.5 Value at risk cannot be fixed The value at risk at some level α (, 1) of a random variable X is defined as VaR(X) = q α (X) 1

12 where q α denotes the uer α-quantile of X. Desite its non-convexity, VaR is still the industry standard risk measure. In view of its redominance in real alications, it would thus be desirable to relace VaR by the greatest closed convex risk measure majorised by VaR, i.e. its L -closure. Unfortunately, it turns out that, for any [1, ], VaR =. (5.8) Indeed, VaR is a continuous cash-invariant and monotone function on L. Hence dom (VaR ) P (see e.g. Lemma 3.2 in [9]). Now let Z P. Since (Ω, F, P) is atomless, there exists some A F with E[Z1 A ] < and < P[A] < α (consider any finite artition Ω = A i with < P[A i ] < α). Then VaR( k1 A ) = for all k N, and hence VaR (Z) su(e[ k1 A Z] VaR( k1 A )) = su ke[z1 A ] =. k k We conclude that VaR =, which roves (5.8). 6 Law-Invariant Convex Functions on L In the following we collect and rove results on law-invariant convex functions which will lay a fundamental role in the roof of theorem 3.8 which is stated in section 7. Throughout this section we let [1, ] and q := 1 where 1 := and 1 := 1. One of the main ingredients for roving theorem 3.8 is lemma 6.2 below. In fact, this lemma is an extension of results by Jouini, Schachermayer, and Touzi in [15]. Let G F be a sub-σ-algebra. As in [15] we define the conditional exectation on L as a function E[ G] : L L where E[µ G] is given by E[E[µ G]X] := E[µE[X G]] X L. Clearly, this definition is consistent with the ordinary conditional exectation in case µ L 1 L. Remark 6.1. If G = σ(a 1,..., A n ) is finite, then E[µ G] L. In order to verify this, note that for all X L we have E[E[µ G]X] = E[µE[X G]] = Hence, E[µ G] = n µ(a i) i=1 P(A 1 i) A i L. n i=1 Note that (L, L r ) is a dual air for every r [q, ]. E[X1 Ai ] µ(a i) P(A i ). Lemma 6.2. A law-invariant convex function f : L [, ] is closed (w.r.t. σ(l, L )) if and only if it closed w.r.t. any σ(l, L r )-toology for every r [q, ]. 11

13 For the roof of lemma 6.2 we will need the following result. Lemma 6.3. Let D L be a. -closed convex law-invariant set. Then D is σ(l, L r )-closed for every r [q, ]. Proof.. If D =, the assertion is obvious. For the remainder of this roof, we assume that D. 1. According to lemma 4.2 in [15], for all Y D and all sub-σ-algebras G F we have E[Y G] D. 2. Let (X i ) i I be a net in D converging to some X L in the σ(l, L r )- toological sense, i.e. E[ZX i ] E[ZX] for all Z L r. Then, in view of remark 6.1, if G is finite, we have E[E[µ G]X i ] E[E[µ G]X] for all µ L. But by definition this equals E[µE[X i G]] E[µE[X G]] for all µ L. In other words, the net (E[X i G]) i I converges to E[X G] in the σ(l, L )- toology. Since, according to 1., E[X i G] D for all i I, we conclude that E[X G] D, because D is closed and convex and thus σ(l, L )-closed. Hence, E[X G] D for all finite sub-σ-algebras G F. Recalling that we can aroximate X in (L, ) by a sequence of conditional exectations (E[X G n ]) n N in which the G n s are all finite, we conclude by means of the norm-closedness of D that X D. Thus D is σ(l, L r )-closed. Proof of lemma 6.2. Let f be closed. Then, for every k R the level sets {X L f(x) k} are. -closed, convex, and law-invariant. Hence, lemma 6.3 yields the σ(l, L r )-closedness of the level sets, i.e. f is closed with resect to the σ(l, L r )-toology. The converse imlication is trivial. Besides lemma 6.2, the roof of theorem 3.8 draws heavily on lemmas 6.4 and 6.6 below. It will require some working with quantiles and some further analysis of law-invariant convex functions in order to deriving these lemmas. For the roof of lemma 6.4 we will need the following fact. (F1) For X L and Z L q we have that q X (s)q Z (s) ds = su E[ ˆXZ] = su E[XẐ]. ˆX X Ẑ Z The relation (F1) is stated and roved in Föllmer/Schied [13] lemma 4.55 for the case X L 1 and Z L. This result in turn can be easily extended to X L and Z L q by suitable aroximation, so we omit a roof here. Lemma 6.4. Let f : L [, ] be a closed convex function. Then the following conditions are equivalent: (i) f is law-invariant. (ii) f is σ(l, L q )-closed and f (res. f L 1 if = ) is law-invariant. 12

14 Moreover, if either holds, then: and f (Z) = su X L f(x) = su Z L q q X (s)q Z (s) ds f(x), Z L q, q X (s)q Z (s) ds f (Z), X L. Proof. (i) (ii): the first statement is trivial if [1, ) and roved in lemma 6.2 if =. Moreover, for any Z L q we gather from (F1) that ( ) f (Z) = su E[XZ] f(x) = su X L = su X L X L q X (s)q Z (s) ds f(x) su E[ ˆXZ] ˆX X in which the latter exression deends on the law of Z only. (ii) (i): again by (F1) ( ) f(x) = f (X) = su Z L q = su Z L q for all X L. Hence, f is law-invariant. su E[XZ] Ẑ Z f (Z) q X (s)q Z (s) ds f (Z) f(x) Next we introduce two orders on L 1 which are well-known from utility theory. To this end, recall that a utility function is a strictly concave and strictly increasing function u : R R. Definition 6.5. For any two X, Y L 1 we define (i) the concave order: X c Y E[u(X)] E[u(Y )] for all concave functions u : R R, (ii) the second order stochastic order: X Y E[u(X)] E[u(Y )] for all utility functions u. A function f : L [, ] is said to be c ( )-monotone if f(x) f(y ) whenever X c Y (X Y ). Clearly, both X Y and X c Y imly X Y. However, and c are not related in general. The subsequent lemma 6.6 is a main result in Dana [6] where she roves it for {1, }. In the roof of theorem 3.8 we need this result for general [1, ], so we rovide a self-contained roof below. However, this roof builds on the following two facts which in turn are roved in Dana [6]. Let X, Y L 1, then 13

15 (F2) X Y if and only if q X (s)g(s) ds q Y (s)g(s) ds for all increasing g : (, 1) (, ] such that both integrals exist. (F3) X c Y if and only if q X (s)g(s) ds q Y (s)g(s) ds for all increasing g : (, 1) R such that both integrals exist. Lemma 6.6. Let f : L [, ] be a closed convex function. Equivalent are: (i) f is law-invariant. (ii) f is c -monotone. Moreover, if in addition f is monotone, then (i) and (ii) are equivalent to (iii) f is -monotone. In articular, if either of the conditions (i), (ii) or (iii) holds, then (iv) f(e[x G]) f(x) for all X L and all sub-σ-algebras G F. Proof. (i) (ii): let X c Y. Then by lemma 6.4 and (F3): f(x) = su Z L q su Z L q q X (s)q Z (s) ds f (Z) q Y (s)q Z (s) ds f (Z) = f(y ). (ii) (i): conversely, suose that f is c -monotone and let X Y. Trivially, X c Y and Y c X, so f(x) = f(y ). (i) (iii): let X Y. Since f is monotone, we have that dom f L q L q (see [9] lemma 3.2). Hence, by lemma 6.4 and (F2), f(x) = su Z L q su Z L q q X (s)q Z (s) ds f (Z) q Y (s)q Z (s) ds f (Z) = f(y ). (iii) (i): if X Y, then X Y and Y X. Thus, f(x) = f(y ). (iv): for any concave function u : R R Jensen s inequality yields E[u(E[X G])] E[E[u(X) G]] = E[u(X)]. Hence, E[X G] c X. Now aly (ii). 14

16 Remark 6.7. Note that the roof of lemma 6.6 relies on lemma 6.2, and thus on lemma 4.2 in [15], only in case =. We recall that lemma 4.2 in [15] states that if D L is a convex law-invariant and -closed set, then E[X G] D for all X D and all sub-σ-algebras G F. Indeed, for every such set D, the indicator function δ( D) is a law-invariant closed convex function. Therefore, according to lemma 6.6(iv), δ(e[x G] D) δ(x D). This imlies E[X G] D whenever X D. Hence, we have derived an alternative roof of lemma 4.2 in [15] for the cases [1, ) (but clearly not for = ). 7 Proof of Theorem 3.8 According to lemma 6.2 any law-invariant closed convex function f : L [, ] is σ(l, L )-closed. Hence, by theorem 3.7 f is a closed convex extension of f to L. Theorem 3.2(ii) and lemma 6.4 yield the law-invariance of f. In order to rove that f is the unique law-invariant closed convex extension of f to L, let g be any such extension. For every X L and all m N there exists a finite artition A m 1,..., A m n of Ω such that the L -distance between X and the simle random variable X m := E[X σ(a m 1,..., A m n )] L is less than 1/m. On the one hand, lemma 6.6(iv) imlies that g(x m ) g(x) for all m N. On the other hand, by l.s.c. of g, we know that g(x) lim inf m g(x m ). Hence, g(x) = lim m g(x m ). Since the latter observation in articular holds for f, we obtain g(x) = lim m g(x m) = lim m f(x m) = lim m f (X m ) = f (X) and uniqueness is roved. Finally, it immediately follows that f 1 L = f. 8 Proof of Theorem 2.3 (i): clearly, if ρ is continuous, then e.g. int dom ρ. In order to rove the converse imlication note that according to [19] roosition 3.1 ρ is continuous on int dom ρ. Hence, it suffices to rove that int dom ρ = dom ρ = L. We show this by means of contradiction. Suose for the moment that there is a X L such that ρ( X) =. Since the interior of the convex set dom ρ is non-emty by assumtion and X dom ρ, an aroriate version of the Hahn-Banach searating hyerlane theorem ensures the existence of a nontrivial Z L such that su E[ZY ] E[Z X]. Y dom ρ Since L dom ρ we obtain te[zx] E[Z X] for all X L and t R, and thus E[XZ] = for all X L. As L is dense in L, we infer that Z must be trivial. But this is a contradiction. Therefore, dom ρ = L. 15

17 (ii): we only rove A ρ L, the other three roerties are obvious. Suose we had A ρ = L. Then for every n N there is a X n A ρ such that X n ( n) = X n + n 1 2 n. By monotonicity we may assume that X n + n. Then, the sequence Y n := n k=1 X k + k, n N, converges to ( ) Y := X k + k L, Y 1, k=1 and Y Y n X n + n for all n N. Thus, by monotonicity, cash-invariance, and X n A ρ we infer ρ(y ) ρ(x n + n) ρ(x n ) n n for all n N. Consequently ρ(y ) =. But this is a contradiction to the roerness of ρ. Hence, A ρ L. (iii): it is easily verified that ρ A is a convex, cash-invariant, and monotone function such that ρ A () <. It remains to rove that ρ A is roer. In doing so, it suffices to show that ρ A is roer because ρ A ρ A. Observe that A satisfies roerties (ii) because A does. Hence, ρ A is convex, cash-invariant, and monotone too, and ρ A () <. If we had ρ A () =, then it follows that R A, and thus L A, so actually A = L which is a contradiction to the assumtion A L. Consequently, ρ A () >. Clearly, A = A ρa, so ρ A is l.s.c. Since any l.s.c. convex function which assumes the value cannot take any finite value (see [8] roosition 2.4.), and since ρ A () R, we conclude that ρ A >, i.e. ρ A is roer. (iv): for a roof of dom ρ P, lease consult [9] lemma 3.2. The stated reresentation of ρ (Z) for any Z P is easily verified in view of E[1Z] = 1 and cash-invariance of ρ. References [1] Alirantis, C.D. and Border K.C. (1999), Infinite Dimensional Analysis, Sringer. [2] Artzner, P., Delbaen, F., Eber, J.M. and Heath, D. (1999), Coherent measures of risk, Mathematical Finance, 9, [3] Biagini, S. and Frittelli, M. (26), On Continuity Proerties and Dual Reresentation of Convex and Monotone Functionals on Frechet Lattices, Working Paer. [4] Cheridito, P. and Li, T. (27) Risk Measures on Orlicz Hearts, rerint. 16

18 [5] Cherny, A.S. and Madan, D.B. (26), CAPM, Rewards, and Emirical Asset Pricing with Coherent Risk, SSRN Working Paer. [6] Dana, R-A. (25), A Reresentation Result for Concave Schur Concave Functions, Mathematical Finance, 15, [7] Delbaen, F. (2), Coherent Risk Measures, Cattedra Galileiana. Scuola Normale Sueriore di Pisa. [8] Ekeland, I. and Témam, R. (1999), Convex Analysis and Variational Problems, Chater I, SIAM. [9] Filiović, D. and Kuer, M. (27), Monotone and Cash-Invariant Convex Functions and Hulls, Insurance: Mathematics and Economics, 41, [1] Filiović, D. and Svindland, G. (27), Otimal Caital and Risk Allocations for Law-Invariant Convex Cash-Invariant Functions, rerint. [11] Filiović, D. and Svindland, G. (28), Subgradients of law-invariant convex risk measures, work in rogress. [12] Föllmer, H. and Schied, A. (22), Convex measures of risk and trading constraints, Finance and Stochastics, 6, [13] Föllmer, H. and Schied, A. (22), Stochastic Finance, An Introduction in Discrete Time, de Gruyter Studies in Mathematics 27. [14] Frittelli, M. and Rosazza Gianin, E. (22), Putting order in risk measures, Journal of Banking and Finance, 26, [15] Jouini, E., Schachermayer, W. and Touzi, N. (26), Law invariant risk measures have the Fatou roerty, Advances in Mathematical Economics, Vol. 9, [16] Kaina, M. and Rüschendorf, L. (27), On Convex Risk Measures on L, Working Paer. [17] Krätschmer, V. (27), On σ-additive robust reresentation of convex risk measures for unbounded financial ositions in the resence of uncertainty about the market model, SFB 649 Discussion Paer 27-1, Humboldt University Berlin. [18] Rockafellar, R.T. (1997), Convex Analysis, Princeton University Press. [19] Ruszczyński, A. and Shairo, A. (26), Otimization of Convex Risk Functions, Math. Oers. Res., 31,