Entropy Coherent and Entropy Convex Measures of Risk - Eurandom

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

Again Two Interpretations [1] Suppose that the agent is only interested in downside tail risk and considers Tail-Value-at-Risk (TV@R) defined by It is well-known that TV@R α (X) = 1 αα 0 V@R λ (X)dλ, α ∈]0, 1]. TV@R α (X) = sup EQ [−X] , Q∈Mα where Mα is the set of all probability measures Q ≪ P such that dQ dP ≤ 1 α . Entropy Coherent and Entropy Convex Measures of Risk Advances in Financial Mathematics, Eurandom, Eindhoven 22/40
Again Two Interpretations [2] The economic agent may, however, not fully trust the probabilistic model of X under P, hence under Q. Therefore, for every fixed Q, the agent considers the supremum over all measures absolutely continuous with respect to Q, where measures that are ‘close’ to Q are esteemed more plausible than measures that are ‘distant’ from Q. This leads to a risk measure ρ given by ρ(X) = sup ¯P≪Q Q∈Mα sup {E¯P [−X] − γH( ¯ P|Q)} = sup ¯P≪P Q∈Mα = sup sup {E¯P [−X] − γH(¯P|Q)} = sup eγ,Q(X), Q∈Mα ¯P≪P Q∈Mα sup {E¯P [−X] − γH( ¯ P|Q)} where we have used in the second and last equalities that H(¯P|Q) = ∞ if ¯P is not absolutely continuous with respect to Q. Entropy Coherent and Entropy Convex Measures of Risk Advances in Financial Mathematics, Eurandom, Eindhoven 23/40
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: Entropy Coherence and Entropy Conve
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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