We study continuity properties of law-invariant (quasi-)convex functions $${f:L^\infty(\Omega, \mathcal{F}, \mathbb{P}) \to (-\infty,\infty]}$$ over a non-atomic probability space $${(\Omega, \mathcal{F}, \mathbb{P})}$$. This is a supplementary note to Jouini etal. (Adv Math Econ 9:49-71, 2006

Svindland, Gregor

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Math Finan Econ (2010) 3:39–43DOI 10.1007/s11579-010-0026-xContinuity properties of law-invariant (quasi-)convexrisk functions on L∞Gregor SvindlandReceived: 12 February 2010 / Accepted: 15 March 2010 / Published online: 27 March 2010© Springer-Verlag 2010Abstract We study continuity properties of law-invariant (quasi-)convex functions f :L∞(,F, P) → (−∞,∞] over a non-atomic probability space (,F, P). This is a sup-plementary note to Jouini et al. (Adv Math Econ 9:49–71, 2006).Keywords Law-invariant (quasi-)convex risk measures · Duality · Fatou propertyJEL Classification C02 · C601 Introduction and statement of the resultThroughout this note we assume that the reader is familiar with basic duality theory andrelated terminology as outlined in e.g. [2,6,14]. Moreover, since this is meant to be a shortsupplementary text, we also presume knowledge of the concept of convex risk measures andrelated continuity problems as outlined in [9,8], and in particular [11], the latter being the mainreference of this note. In their seminal paper [11] Jouini, Schachermayer and Touzi prove thatgiven any standard non-atomic probability space (,F, P) every law-invariant convex riskmeasure ρ : L∞(,F, P) =: L∞ → R satisfies the Fatou property. Law-invariant meansthat ρ(X) = ρ(Y ) for any X, Y ∈ L∞ being identically distributed (X ∼ Y ). The Fatouproperty equals lower semi-continuity (l.s.c.) of ρ with respect to the σ(L∞, L1)-topologyand is desirable since it yields a dual representation of ρ as a supremum over weightedprobabilities, i.e.ρ(X) = supQPEQ[−X ] − ρ∗(Q), X ∈ L∞, (1.1)whereG. Svindland (B)EPFL CDM SFI CSF EXTRA 218, 1015 Lausanne, Switzerlande-mail: gregor.svindland@epfl.ch12340 Math Finan Econ (2010) 3:39–43ρ∗(Q) := supX∈L∞EQ[−X ] − ρ(X)for any probability measure Q  P on (,F). More generally, the results of [11] imply thatgiven any standard non-atomic probability space (,F, P), any l.s.c. (with respect to the‖·‖∞-topology) law-invariant convex function f : L∞ → (−∞,∞] (not necessarily a con-vex risk measure) is automatically l.s.c. with respect to the σ(L∞, L1)-topology. In this notewe show that the assumption of standardness of the probability space can be dropped (seeAppendix for a short survey on standardness, and Example 1.4 for a non-standard probabilityspace). This may be no surprise, it seems obvious that the standardness of the underlyingprobability space should not play a prominent role, however we feel the necessity of clar-ifying this fact since the main result of [11] is often cited (see [1,3,7,8,10,12,13,17], andmany more). Thus many of the results building on [11] in fact hold on general non-atomicprobability spaces, which meets the common urge not to impose other requirements on theunderlying probability space than that it be rich enough to support continuously distributedrandom variables, i.e. that it be non-atomic. Moreover, we also generalize to the quasi-convex case which is becoming increasingly popular (see e.g. [3,5,4] for a duality theory forquasi-convex functions). Our aim is to proveProposition 1.1 Let (,F, P) be a non-atomic probability space and let f : L∞ →(−∞,∞] be a law-invariant l.s.c. (with respect to ‖ · ‖∞-topology) quasi-convex function,then f is l.s.c. with respect to any σ(L∞, Lq)-topology for every q ∈ [1,∞].So in case of a law-invariant convex risk measure ρ we may even obtain a dual rep-resentation (1.1) over probabilities with bounded densities dQ/dP. A version of Proposi-tion 1.1 for the standard case and a convex function has already been shown in [8]. Proposi-tion 1.1 immediately follows from the following Proposition 1.2 because l.s.c. of the functionf : L∞ → (−∞,∞] with respect to the σ(L∞, Lq)-topology is equivalent to the levels setsEk := {X ∈ L∞ | f (X) ≤ k}, k ∈ R, being closed in the σ(L∞, Lq)-topology, whereasquasi-convexity of f (i.e. f (λX + (1 − λ)Y ) ≤ max{ f (X), f (Y )} for all X, Y ∈ L∞ andλ ∈ [0, 1]) is equivalent to convexity of the level sets.Proposition 1.2 Let (,F, P) be a non-atomic probability space and let C ⊂ L∞ be alaw-invariant (i.e. X ∈ C and Y ∼ X implies Y ∈ C) convex set which is closed in the‖ · ‖∞-topology (‖ · ‖∞-closed). Then, C is closed with respect to any σ(L∞, Lq)-topologyfor every q ∈ [1,∞].The core of the proof of Proposition 1.2 is the following Lemma 1.3 which is the analogin our setting of Lemma 4.2 of [11]. The idea of its proof is similar to the original proof forstandard probability spaces as presented in [11], the difference being that we do not deal withmeasure preserving maps which do not necessarily exist if the underlying probability spaceis not assumed to be standard, but argue by means of the quantile function.Lemma 1.3 Let (,F, P) be a non-atomic probability space and let C ⊂ L∞ be a law-invariant convex ‖ · ‖∞-closed set, then for all sub-σ -algebras A ⊂ F and all X ∈ C wehave that E[X | A] ∈ C.Proof The proof follows the lines of the corresponding proof in [11].Step 1: First of all we prove the assertion for the expectation, i.e. A = {∅,}. Let X ∈ Cand  > 0. Without loss of generality we may assume that X ≥ 0 (otherwise weshift C by a constant). Denote by123Math Finan Econ (2010) 3:39–43 41qX : [0, 1] → R, qX (s) :=⎧⎪⎨⎪⎩essinf X s = 0,inf{x | P(X ≤ x) ≥ s} s ∈ (0, 1),esssup X s = 1the (left-continuous) quantile function of X . Note that 0 ≤ qX (s) ≤ qX (1) =‖X‖∞ < ∞, so the quantile function is bounded. Let n ∈ N be such that qX (1)n ≤ ,and let Ai := ( i−1n , in ] for i = 1, . . . , n − 1 and An := (1 − 1n , 1). Moreover, letB1, . . . , Bn be a partition of  such that P(Bk) = 1/n (this exists according to Theo-rem 9.51 of [2]). Since (,F, P(· | Bk)) is a non-atomic probability space for everyk ∈ {1, . . . , n}, for any j ∈ {1, . . . , n} there exists a random variable U kj which is uni-formly distributed on A j under P(· | Bk), i.e. P(U kj ≤ s | Bk) = ns − j +1, s ∈ A j(Proposition A.27 of [9]). Then, for any permutation π : {1, . . . .n} → {1, . . . .n},the random variable Uπ := ∑nk=1 U kπ(k)1Bk is uniform on (0, 1) under P. HenceXπ := qX (Uπ ) = ∑nk=1 qX (U kπ(k))1Bk has the same distribution as X and is thusan element of C . As C is convex we infer thatXn := 1n!∑π∈SnXπ ∈ Cwhere Sn denotes the set of all permutations π : {1, . . . .n} → {1, . . . .n}. Moreover,we have (by monotonicity of the quantile function) that‖Xn − E[X ]‖∞ = ‖Xn −∫ 10qX (s)ds‖∞ ≤ 1nn∑i=1(qX(in)− qX(i − 1n))≤ qX (1)n≤ .Since  > 0 was arbitrary and C is closed, we conclude that E[X ] ∈ C .Step 2: Now we continue as in [11] and argue for the case A = σ(D1, . . . , Dr ) for somefinite partition D1, . . . , Dr ∈ F of  with P(Di ) > 0 for all i = 1, . . . , r, r ∈ N.Let X ∈ C . We consider the non-atomic probability spaces (Di ,Fi , Pi ) whereFi := {A ∩ Di | A ∈ F} and Pi : Fi → [0, 1], A → P(A)/P(Di ), and the law-invariant convex sets Ci := {Y|Di | Y ∈ C} ⊂ L∞(Di ,Fi , Pi ), i = 1, . . . , r . Givenany  > 0, by the same arguments as in step 1 applied to Ci over (Di ,Fi , Pi ), weobtain some Xin = 1n!∑π∈Sn Xi,π ∈ Ci such that Xi,π and X |Di are identically dis-tributed under Pi for every π ∈ Sn and ‖Xin − EPi [X |Di ]‖∞ ≤ . (We may assumethat the n is independent of i because when applying step 1 to Ci we may alwayschoose the maximal “n over all i = 1, . . . , r”) LetXn :=r∑i=1Xin1Di =1n!∑π∈Snr∑i=1Xi,π1Di .Then Xn ∈ C , since ∑ri=1 Xi,π1Di and X are identically distributed under P. More-over, we have that‖Xn − E[X |A]‖∞ = ‖Xn −r∑i=1EPi [X |Di ]1Di ‖∞ ≤ .As  > 0 was arbitrary and as C is closed, we conclude that E[X |A] ∈ C .12342 Math Finan Econ (2010) 3:39–43Step 3: For the final step we note that for any sub-σ -algebra A ⊂ F there is a sequence offinite sub-σ -algebras An ⊂ F, n ∈ N, such that ‖E[X |An] − E[X |A]‖∞ → 0 forn → ∞, so the assertion follows from step 2 and closedness of C . unionsqNow the proof of Proposition 1.2 is literally the same as the proof ofLemma 2.4 (i) in [8]. For the sake of completeness we repeat it here.Proof of Proposition 1.2. Step 1: Let G ⊂ F be a sub-σ -algebra. As in [11] we define theconditional expectation on L∞∗ (the dual space of (L∞, ‖ · ‖∞)) as a functionE[· | G] : L∞∗ → L∞∗ where E[μ | G] is given by〈E[μ | G], X〉 := 〈μ, E[X | G]〉 ∀ X ∈ L∞.If G is finite, i.e. G = σ(A1, . . . , An) for a finite partition A1, . . . , An ∈ F of with P(Ai ) > 0 for all i = 1, . . . , n, then E[μ | G] ∈ L∞ since for all X ∈ L∞ wehave〈E[μ | G], X〉 = 〈μ, E[X | G]〉 =n∑i=1E[X1Ai ]μ(Ai )P(Ai ).Hence, E[μ | G] = ∑ni=1 μ(Ai )P(Ai ) 1Ai ∈ L∞.Step 2: If C = ∅, the assertion of Proposition 1.2 is obvious. For the remainder of this proof,we assume that C = ∅. Now let (Xi )i∈I be a net in C converging to some X ∈ L∞in the σ(L∞, Lq)-topological sense, i.e. E[Z Xi ] → E[Z X ] for all Z ∈ Lq . Then,in view of step 1, if G is finite, we have E[E[μ | G]Xi ] → E[E[μ | G]X ] for allμ ∈ L∞∗. But this equals 〈μ, E[Xi | G]〉 → 〈μ, E[X | G]〉 for all μ ∈ L∞∗. Inother words, the net (E[Xi | G])i∈I converges to E[X | G] in the σ(L∞, L∞∗)-topology. Since, according to Lemma 1.3, E[Xi | G] ∈ C for all i ∈ I , we concludethat E[X | G] ∈ C , because C is closed and convex and thus σ(L∞, L∞∗)-closed.Hence, E[X | G] ∈ C for all finite sub-σ -algebras G ⊂ F . Recalling that wecan approximate X in (L∞, ‖ · ‖∞) by a sequence of conditional expectations(E[X | Gn])n∈N in which the Gn are all finite, we conclude by means of the ‖ · ‖∞-closedness of C that X ∈ C . Thus C is σ(L∞, Lq)-closed. unionsqExample 1.4 We give a simple example of a non-atomic probability space which is notstandard. To this end, consider the non-atomic standard probability space per se which is([0, 1],B([0, 1]), λ) where B([0, 1]) denotes the Borel-σ -algebra on [0, 1] and λ is the Le-besgue measure. Now let  := [0, 1]×{0, 1} and denote by π the projection of  onto [0, 1].We construct a probability space by equipping  with the σ -algebra F := {π−1(B)|B ∈B([0, 1])}, and with the probability measure P(A) := λ(π(A)), A ∈ F . Then, (,F, P) isnon-atomic, but not standard since there exists no measurable bijection f :  → [0, 1].Acknowledgements Support from Swissquote is gratefully acknowledged. We wish to thank an anonymousreferee for helpful comments.AppendixStandard probability spaceTwo probability spaces (,F, P) and (′,B, Q) are isomorphic mod 0 if there exists null-setsA ∈ F and B ∈ B and a bijection f : \A → ′\B such that both f and f −1 are measur-123Math Finan Econ (2010) 3:39–43 43able and measure-preserving (i.e. P(C ∩ Ac) = Q( f (C ∩ Ac)) for all C ∈ F) on the restrictedprobability spaces. The map f is called isomorphism mod 0. An non-atomic probability space(,F, P) is standard if it is isomorphic mod 0 to the probability space ([0, 1],B([0, 1]), λ)where B([0, 1]) denotes the Borel-σ -algebra over [0, 1] and λ is the Lebesgue-measure onB([0, 1]) (see [15], Sect. 2). A mapping τ :  →  is a measure preserving transformationif it is an isomorphism mod 0. Given an non-atomic standard probability space (,F, P) andtwo sets A, B ∈ F such that P(A) = P(B), there exists a measure preserving transformationτ :  →  such that τ(A) = B P-a.s. and τ(B) = A P-a.s. and τ = Id on Ac ∩ Bc P-a.s.This is a direct consequence of the definition of standardness and the fact that for everysubset A ∈ F such that P(A) > 0 the restricted probability space with conditional measureis again standard (see [15], Sect. 2, in particular 2.3 and 2.4). For instance, if  is a completeseparable metric space, F the corresponding σ -algebra of Borel-sets, and P a probabilitymeasure on (,F), then (,F, P) is standard (see e.g. [16], Theorem 9, p. 327).References1. Acciaio, B.: Optimal risk sharing with non-monotone monetary functionals. Finan. Stoch. 11, 267–289 (2007)2. Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis. Springer, NY (1999)3. Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M., Montrucchio, L.: Risk Measures: Rationality andDiversification. 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Continuity properties of law-invariant (quasi-)convex risk functions on L ∞

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Continuity properties of law-invariant (quasi-)convex risk functions on L ∞

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