Entropy Coherent and Entropy Convex Measures of Risk - Eurandom

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

Interpretation ◮ The function u, referred to as a utility function, represents the agent’s attitude towards wealth. ◮ The set Q represents the set of priors held by agents. ◮ Under multiple priors preferences, the degree of ambiguity is reflected by the multiplicity of the priors. ◮ Under general variational preferences, the degree of ambiguity is reflected by the multiplicity of the priors and the esteemed plausibility of the prior as reflected in the ambiguity index (or penalty function). Entropy Coherent and Entropy Convex Measures of Risk Advances in Financial Mathematics, Eurandom, Eindhoven 6/40
Homothetic Preferences ◮ Recently, Chateauneuf and Faro (2010) and, slightly more generally, Cerreia-Vioglio et al. (2008) axiomatized a multiplicative analog of variational preferences, referred to henceforth as homothetic preferences. ◮ It is represented as with β : Q → [0, ∞]. U(X) = inf Q∈Q {β(Q)EQ [u(X)]} , (1) ◮ It also includes multiple priors as a special case (β(Q) ≡ 1). Entropy Coherent and Entropy Convex Measures of Risk Advances in Financial Mathematics, Eurandom, Eindhoven 7/40
Page 1 and 2: Entropy Coherent and Entropy Convex
Page 3 and 4: Convex Measures of Risk ◮ Convex
Page 5: Multiple Priors Preferences ◮ A s
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 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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