Entropy Coherent and Entropy Convex Measures of Risk - Eurandom

More documents

Recommendations

Info

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

Remark 2 ◮ Recall the definition of entropy convexity (slide 21). ◮ As e∞,Q(X) = EQ [−X] , ρ is a convex risk measure if and only if it is ∞-entropy convex. ◮ As we will see later, however, with γ < ∞, not every convex risk measure is γ-entropy convex. ◮ This is important: we have seen that, under some technical conditions, negative certainty equivalents under homothetic preferences are translation invariant if and only if they are γ-entropy convex with γ ∈ R + or ∞-entropy coherent, ruling out the general ∞-entropy convex case. ◮ Translation invariant negative certainty equivalents under homothetic preferences do not span the class of convex risk measures. Entropy Coherent and Entropy Convex Measures of Risk Advances in Financial Mathematics, Eurandom, Eindhoven 30/40
Characterization Results [3] Recall the second question asked in the Introduction (slide 11). Answer: Theorem Suppose that the probability space is rich. Let φ be a strictly increasing and convex function with φ ∈ C 3 (R) and either φ(−∞) = −∞ or limx→∞ φ(x)/x = ∞. Then the following statements are equivalent: (i) ρ(X) = φ −1 (¯ρ(−φ(−X))) is translation invariant and the subdifferential of ¯ρ is always nonempty. (ii) ρ is a convex risk measure and the subdifferential is always nonempty. Entropy Coherent and Entropy Convex Measures of Risk Advances in Financial Mathematics, Eurandom, Eindhoven 31/40
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: Remark 1 ◮ The case that ρ is en
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

Create successful ePaper yourself

Delete template?

Save as template?