Entropy Coherent and Entropy Convex Measures of Risk - Eurandom

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

Again Two Interpretations [3] The definition of entropy convexity (whence the special case of entropy coherence as well) can also be motivated using microeconomic theory, as follows: ◮ An economic agent with a CARA (exponential) utility function u(x) = 1 − e − x γ computes the certainty equivalent to the payoff X with respect to the reference measure P. ◮ The agent is, however, uncertain about the probabilistic model under the reference measure, and therefore takes the infimum over all probability measures Q absolutely continuous with respect to P, where the penalty function c(Q) represents the esteemed plausibility of the probabilistic model under Q. ◮ The robust certainty equivalent thus computed is precisely −ρ(X). Entropy Coherent and Entropy Convex Measures of Risk Advances in Financial Mathematics, Eurandom, Eindhoven 24/40
A Basic Duality Result Define and Then the following result holds: ρ ∗ (Q) = sup X ∈L∞ {eγ,Q(X) − ρ(X)} ρ ∗∗ (X) = sup {eγ,Q(X) − ρ Q≪P ∗ (Q)}. Lemma A normalized mapping ρ is γ-entropy convex if and only if ρ ∗∗ = ρ. Furthermore, ρ ∗ is the minimal penalty function. Entropy Coherent and Entropy Convex Measures of Risk Advances in Financial Mathematics, Eurandom, Eindhoven 25/40
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: Again Two Interpretations [2] The e
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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