Entropy Coherent and Entropy Convex Measures of Risk - Eurandom

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

1. Introduction Sign conventions used in this talk: ◮ Random variables represent payoffs of financial positions. Positive realizations represent gains. ◮ A risk measure represents a negative valuation. Entropy Coherent and Entropy Convex Measures of Risk Advances in Financial Mathematics, Eurandom, Eindhoven 2/40
Convex Measures of Risk ◮ Convex measures of risk (Föllmer and Schied, 2002, Fritelli and Rosazza Gianin, 2002, and Heath and Ku, 2004) are characterized by the axioms of monotonicity, translation invariance and convexity. ◮ They can (under additional assumptions on the space of random variables and on continuity properties of the risk measure) be represented in the form ρ(X) = sup {EQ [−X] − α(Q)}, Q∈Q where Q is a set of probability measures and α is a penalty function defined on Q. ◮ With α(Q) =0, if Q ∈ Q; ∞, otherwise; we obtain the subclass of coherent measures of risk, represented in the form ρ(X) = sup Q∈M⊂Q EQ [−X] . Entropy Coherent and Entropy Convex Measures of Risk Advances in Financial Mathematics, Eurandom, Eindhoven 3/40
Page 1: Entropy Coherent and Entropy 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 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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