Entropy Coherent and Entropy Convex Measures of Risk - Eurandom

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$Continuous-time random walks, fractional calculus and ... - Eurandom$

Question Rephrased [2] ◮ We consider in addition with with β : M → [0, 1]. ρ(X) = φ −1 (¯ρ(−φ(−X))), ¯ρ(X) = sup Q∈M⊂Q β(Q)EQ [−X] , ◮ Preferences of Chateauneuf and Faro (2010). ◮ With ¯ρ as given above, negative certainty equivalents are invariant under positive multiplication of u (or φ); complementary case. ◮ β : M → [0,1] can be viewed as a discount factor; ¯ρ seems natural. ◮ Includes multiple priors preferences as a special case. ◮ Recall question: find sufficient and necessary conditions under which ρ (not ¯ρ) is a convex risk measure. Entropy Coherent and Entropy Convex Measures of Risk Advances in Financial Mathematics, Eurandom, Eindhoven 12/40
Results [1] The contribution of this paper is twofold. ◮ First we derive precise connections between risk measurement under the theories of variational, homothetic and multiple priors preferences on the one hand and risk measurement using convex measures of risk on the other. ◮ This is, despite the vast literature on both paradigms, a hitherto open problem. ◮ In particular, we identify two subclasses of convex risk measures that we call entropy coherent and entropy convex measures of risk, and that include all coherent risk measures. ◮ We show that, under technical conditions, negative certainty equivalents under variational, homothetic, and multiple priors preferences are translation invariant if and only if they are convex, entropy convex, and entropy coherent measures of risk, respectively. Entropy Coherent and Entropy Convex Measures of Risk Advances in Financial Mathematics, Eurandom, Eindhoven 13/40
Page 1 and 2: Entropy Coherent and Entropy Convex
Page 3 and 4: Convex Measures of Risk ◮ Convex
Page 5 and 6: Multiple Priors Preferences ◮ A s
Page 7 and 8: Homothetic Preferences ◮ Recently
Page 9 and 10: Variational and Homothetic Preferen
Page 11: Question Rephrased [1] ◮ In other
Page 15 and 16: Results [3] ◮ These connections s
Page 17 and 18: 2. Entropic Measures of Risk [1] We
Page 19 and 20: Two Interpretations 1. Kullback-Lei
Page 21 and 22: Entropy Coherence and Entropy Conve
Page 23 and 24: Again Two Interpretations [2] The e
Page 25 and 26: A Basic Duality Result Define and T
Page 27 and 28: 3. Characterization Results [1] Rec
Page 29 and 30: Remark 1 ◮ The case that ρ is en
Page 31 and 32: Characterization Results [3] Recall
Page 33 and 34: 4. Duality Results [1] Recall that
Page 35 and 36: 5. Distribution Invariant Represent
Page 37 and 38: 6.1 Risk Sharing ◮ Suppose that t
Page 39 and 40: Portfolio Optimization and Indiffer

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