Entropy Coherent and Entropy Convex Measures of Risk - Eurandom

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

Another Characterization Result Reconsider the question asked in the Introduction (slide 12). Another answer: Theorem Suppose that the probability space is rich. Let φ be a strictly increasing and continuous function satisfying 0 ∈ closure(Image(φ)), φ(∞) = ∞ and φ ∈ C 3 (]φ −1 (0), ∞[). Then the following statements are equivalent: (i) ρ(X) = φ −1 (¯ρ(−φ(−X))) is convex, ρ(m) = −m for all m ∈ R and the subdifferential of ¯ρ is always nonempty. (ii) ρ is γ-entropy convex with γ ∈ R + or ρ is ∞-entropy coherent, and the entropy subdifferential is always nonempty. Entropy Coherent and Entropy Convex Measures of Risk Advances in Financial Mathematics, Eurandom, Eindhoven 32/40
4. Duality Results [1] Recall that if ρ is a convex risk measure then (under additional continuity assumptions) there exists a unique α : Q → R ∪ {∞}, referred to as the dual conjugate of ρ, such that the following dual representation holds: with ρ(X) = sup Q∈QÒEQ [−X] − α(Q)Ó, α(Q) = sup X ∈L ∞ÒEQ [−X] − ρ(X)Ó. Entropy Coherent and Entropy Convex Measures of Risk Advances in Financial Mathematics, Eurandom, Eindhoven 33/40
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 and 12: Question Rephrased [1] ◮ In other
Page 13 and 14: Results [1] The contribution of thi
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: Characterization Results [3] Recall
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

Entropy Coherent and Entropy Convex Measures of Risk - Eurandom

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