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Open accessJournal ArticleDOI: 10.1137/20M1341258

Law-Invariant Functionals on General Spaces of Random Variables

04 Mar 2021-Siam Journal on Financial Mathematics (Society for Industrial and Applied Mathematics)-Vol. 12, Iss: 1, pp 318-341
Abstract: We establish general versions of a variety of results for quasiconvex, lower-semicontinuous, and law-invariant functionals. Our results extend well-known results from the literature to a large clas...

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Open accessJournal ArticleDOI: 10.1016/J.INSMATHECO.2021.03.002
Abstract: We discuss when law-invariant convex functionals "collapse to the mean". More precisely, we show that, in a large class of spaces of random variables and under mild semicontinuity assumptions, the expectation functional is, up to an affine transformation, the only law-invariant convex functional that is linear along the direction of a nonconstant random variable with nonzero expectation. This extends results obtained in the literature in a bounded setting and under additional assumptions on the functionals. We illustrate the implications of our general results for pricing rules and risk measures.

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Topics: Invariant (mathematics) (61%), Affine transformation (55%), Bounded function (55%) ... read more

8 Citations


Open accessPosted Content
Abstract: We establish general "collapse to the mean" principles that provide conditions under which a law-invariant functional reduces to an expectation. In the convex setting, we retrieve and sharpen known results from the literature. However, our results also apply beyond the convex setting. We illustrate this by providing a complete account of the "collapse to the mean" for quasiconvex functionals. In the special cases of consistent risk measures and Choquet integrals, we can even dispense with quasiconvexity. In addition, we relate the "collapse to the mean" to the study of solutions of a broad class of optimisation problems with law-invariant objectives that appear in mathematical finance, insurance, and economics. We show that the corresponding quantile formulations studied in the literature are sometimes illegitimate and require further analysis.

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Topics: Quasiconvex function (59%), Invariant (mathematics) (58%), Convexity (56%) ... read more

3 Citations


Open accessPosted Content
Abstract: In this note, we show that, on a wide range of rearrangement-invariant spaces, a law-invariant bounded linear functional is a scalar multiple of the expectation. We also construct a rearrangement-invariant space on which this property fails.

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Topics: Linear form (59%), Bounded function (58%), Invariant (mathematics) (57%) ... read more

1 Citations


Open accessPosted Content
Abstract: In this paper, we show that, on classical model spaces including Orlicz spaces, every real-valued, law-invariant, coherent risk measure automatically has the Fatou property at every point whose negative part has a thin tail.

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1 Citations


Open accessPosted Content
Felix-Benedikt Liebrich1Institutions (1)
Abstract: We consider the problem of finding Pareto-optimal allocations of risk among finitely many agents. The associated individual risk measures are law invariant, but with respect to agent-dependent and potentially heterogeneous reference probability measures. Moreover, we assume that the individual risk assessments are consistent with the respective second-order stochastic dominance relations. We do not assume their convexity though. A simple sufficient condition for the existence of Pareto optima is provided. Its proof combines local comonotone improvement with a Dieudonne-type argument, which also establishes a link of the optimal allocation problem to the realm of "collapse to the mean" results. Finally, we extend the results to capital requirements with multidimensional security markets.

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Topics: Stochastic dominance (54%), Convexity (54%), Pareto principle (53%)

1 Citations


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Open accessJournal ArticleDOI: 10.1007/S007800200072
Hans Föllmer1, Alexander Schied1Institutions (1)
Abstract: We introduce the notion of a convex measure of risk, an extension of the concept of a coherent risk measure defined in Artzner et al. (1999), and we prove a corresponding extension of the representation theorem in terms of probability measures on the underlying space of scenarios. As a case study, we consider convex measures of risk defined in terms of a robust notion of bounded shortfall risk. In the context of a financial market model, it turns out that the representation theorem is closely related to the superhedging duality under convex constraints.

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Topics: Dynamic risk measure (68%), Entropic risk measure (66%), Coherent risk measure (64%) ... read more

1,174 Citations


Book ChapterDOI: 10.1007/978-4-431-67891-5_4
Shigeo Kusuoka1Institutions (1)
01 Jan 2001-
Abstract: The idea of coherent risk measures has been introduced by Artzner, Delbaen, Eber and Heath [1] We think of a special class of coherent risk measures and give a characterization of it Let (Ω, ℱ, P) be a probability space We denote L ∞(Ω, ℱ, P) by L ∞ Following [1], we give the following definition

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677 Citations


Open accessBook ChapterDOI: 10.1007/4-431-34342-3_4
01 Jan 2006-
Abstract: S. Kusuoka [K01, Theorem 4] gave an interesting dual characterization of law invariant coherent risk measures, satisfying the Fatou property. The latter property was introduced by F. Delbaen [D 02]. In the present note we extend Kusuoka’s characterization in two directions, the first one being rather standard, while the second one is somewhat surprising. Firstly we generalize — similarly as M. Fritelli and E. Rossaza Gianin [FG 05] — from the notion of coherent risk measures to the more general notion of convex risk measures as introduced by H. Follmer and A. Schied [FS 04]. Secondly — and more importantly — we show that the hypothesis of Fatou property may actually be dropped as it is automatically implied by the hypothesis of law invariance.

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243 Citations


Open accessJournal ArticleDOI: 10.1111/J.1467-9965.2007.00332.X
Abstract: We consider the problem of optimal risk sharing of some given total risk between two economic agents characterized by law-invariant monetary utility functions or equivalently, law-invariant risk measures. We first prove existence of an optimal risk sharing allocation which is in addition increasing in terms of the total risk. We next provide an explicit characterization in the case where both agents’ utility functions are comonotone. The general form of the optimal contracts turns out to be given by a sum of options (stop-loss contracts, in the language of insurance) on the total risk. In order to show the robustness of this type of contracts to more general utility functions, we introduce a new notion of strict risk aversion conditionally on lower tail events, which is typically satisfied by the semi-deviation and the entropic risk measures. Then, in the context of an AV@R-agent facing an agent with strict monotone preferences and exhibiting strict risk aversion conditional on lower tail events, we prove that optimal contracts again are European options on the total risk. MSC 1991 subject classifications: Primary 91B06, 46A20; secondary 91B70.

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Topics: Coherent risk measure (64%), Spectral risk measure (62%), Dynamic risk measure (62%) ... read more

210 Citations


Book ChapterDOI: 10.1007/4-431-27233-X_2
01 Jan 2005-
Abstract: As a generalization of a result by Kusuoka (2001), we provide the representation of law invariant convex risk measures. Very particular cases of law invariant coherent and convex risk measures are also studied.

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Topics: Invariant measure (66%), Convex analysis (64%), Subderivative (63%) ... read more

153 Citations


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