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
Duality Results [2] Theorem Suppose that ρ is γ-entropy convex with penalty function c. Then: (i) The dual conjugate of ρ, is given by the largest convex and lower-semicontinuous function α being dominated by infQ∈Q{γH(¯P|Q) + c(Q)}. (ii) If c is convex and lower-semicontinuous, then α is the largest lower-semicontinuous function being dominated by infQ∈Q{γH( ¯ P|Q) + c(Q)}. (iii) If c is convex and lower-semicontinuous, and satisfies additional integrability conditions (see paper), then the conjugate dual α( ¯ P) = min Q∈Q {γH(¯ P|Q) + c(Q)}. Entropy Coherent and Entropy Convex Measures of Risk Advances in Financial Mathematics, Eurandom, Eindhoven 34/40
5. Distribution Invariant Representation [1] ◮ Let Ψ = {ψ : [0, 1] → [0, 1] |ψ is concave, right-continuous at zero with ψ(0+) = 0 and ψ(1) = 1}. ◮ For ψ ∈ Ψ and X ∈ L ∞ we define Eψ [X] :=ÊXdψ(P). ◮ Furthermore, we define eγ,ψ(X) := γ logEψexp−X eγ,ψ(P)(X). γ=: Entropy Coherent and Entropy Convex Measures of Risk Advances in Financial Mathematics, Eurandom, Eindhoven 35/40
- 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
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- Page 25 and 26: A Basic Duality Result Define and T
- Page 27 and 28: 3. Characterization Results [1] Rec
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- Page 31 and 32: Characterization Results [3] Recall
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Duality Results [2]<br />
Theorem<br />
Suppose that ρ is γ-entropy convex with penalty function c. Then:<br />
(i) The dual conjugate <strong>of</strong> ρ, is given by the largest convex <strong>and</strong><br />
lower-semicontinuous function α being dominated by<br />
infQ∈Q{γH(¯P|Q) + c(Q)}.<br />
(ii) If c is convex <strong>and</strong> lower-semicontinuous, then α is the largest<br />
lower-semicontinuous function being dominated by<br />
infQ∈Q{γH( ¯ P|Q) + c(Q)}.<br />
(iii) If c is convex <strong>and</strong> lower-semicontinuous, <strong>and</strong> satisfies additional<br />
integrability conditions (see paper), then the conjugate dual<br />
α( ¯ P) = min<br />
Q∈Q {γH(¯ P|Q) + c(Q)}.<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 34/40
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