Entropy Coherent and Entropy Convex Measures of Risk - Eurandom

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

Distribution Invariant Representation [2] Theorem Suppose that ρ is γ-entropy convex. Then the following statements are equivalent: (i) ρ is distribution invariant. (ii) ρ(X) = sup ψ∈Ψ {eγ,ψ(X) − (ρ ∗ ) ′ (ψ)} with (ρ ∗ ) ′ (ψ) = sup X ∈L ∞ {eγ,ψ(X) − ρ(X)}. Entropy Coherent and Entropy Convex Measures of Risk Advances in Financial Mathematics, Eurandom, Eindhoven 36/40
6.1 Risk Sharing ◮ Suppose that there are two economic agents A and B measuring risk using a general entropy convex measure of risk ρ A and ρ B with γ A , γ B ∈ R + . ◮ Let ¯ρ = −ρ, ēγ,Q = −eγ,Q and ¯c = −c. ◮ Suppose that A owns a financial payoff X A and B owns a financial payoff X B . ◮ We solve explicitly the problem of optimal risk sharing given by R A,B (X A ,X B ) = sup F ∈L∞ {¯ρ A (X A − F + Π F ) + ¯ρ B (X B + F − Π F )} = sup ¯F ∈L∞ {¯ρ A (X A + X B − ¯ F) + ¯ρ B ( ¯ F)} =: ¯ρ A ¯ρ B (X A + X B ), where Π F is the agreed price of the financial derivative (risk transfer) F and where we have set ¯F := F + X B . ◮ In particular, under technical conditions (see paper), the optimal risk sharing is attained in the derivative F ∗ = γB γ A +γ B X A − γA γ A +γ B X B . Entropy Coherent and Entropy Convex Measures of Risk Advances in Financial Mathematics, Eurandom, Eindhoven 37/40
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 and 32: Characterization Results [3] Recall
Page 33 and 34: 4. Duality Results [1] Recall that
Page 35: 5. Distribution Invariant Represent
Page 39 and 40: Portfolio Optimization and Indiffer

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