Entropy Coherent and Entropy Convex Measures of Risk - Eurandom
Entropy Coherent and Entropy Convex Measures of Risk - Eurandom Entropy Coherent and Entropy Convex Measures of Risk - Eurandom
Characterization Results [2] Recall the question asked in the Introduction (slide 12). Answer: Theorem Suppose that the probability space is rich. Let φ be a strictly increasing and continuous function satisfying 0 ∈ closure(Image(φ)), φ(∞) = ∞ and φ ∈ C 3 (]φ −1 (0), ∞[). 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 γ-entropy convex with γ ∈ R + or ρ is ∞-entropy coherent, and the entropy subdifferential is always nonempty. Entropy Coherent and Entropy Convex Measures of Risk Advances in Financial Mathematics, Eurandom, Eindhoven 28/40
Remark 1 ◮ The case that ρ is entropy convex corresponds to ρ being the negative certainty equivalent of ¯ρ(X) = sup Q∈M β(Q)EQ [−X], where β : M → [0,1] can be viewed as a discount factor, and with φ being linear (implying β(Q) ≡ 1) or exponential. ◮ In this case, every model Q is discounted by a factor β(Q) corresponding to its esteemed plausibility. ◮ If β(Q) = 1 for all Q ∈ M, we are back in the framework of Gilboa-Schmeidler. ◮ However, if there exists a Q ∈ M such that β(Q) < 1, ρ is entropy convex with γ ∈ R + but not entropy coherent. Entropy Coherent and Entropy Convex Measures of Risk Advances in Financial Mathematics, Eurandom, Eindhoven 29/40
- 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: 3. Characterization Results [1] Rec
- 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
Remark 1<br />
◮ The case that ρ is entropy convex corresponds to ρ being the negative<br />
certainty equivalent <strong>of</strong> ¯ρ(X) = sup Q∈M β(Q)EQ [−X], where<br />
β : M → [0,1] can be viewed as a discount factor, <strong>and</strong> with φ being linear<br />
(implying β(Q) ≡ 1) or exponential.<br />
◮ In this case, every model Q is discounted by a factor β(Q) corresponding<br />
to its esteemed plausibility.<br />
◮ If β(Q) = 1 for all Q ∈ M, we are back in the framework <strong>of</strong><br />
Gilboa-Schmeidler.<br />
◮ However, if there exists a Q ∈ M such that β(Q) < 1, ρ is entropy convex<br />
with γ ∈ R + but not entropy coherent.<br />
<strong>Entropy</strong> <strong>Coherent</strong> <strong>and</strong> <strong>Entropy</strong> <strong>Convex</strong> <strong>Measures</strong> <strong>of</strong> <strong>Risk</strong> Advances in Financial Mathematics, Eur<strong>and</strong>om, Eindhoven 29/40
Change language