a) Show that every coherent risk measure o: MR is a convex risk measure, that is g satisfies o(AL₁ + (1-X) L₂) ≤ Ag(L₁)+(1-X)g(L2), [0, 1], L₁, L₂ € M. b) Show that for positive-homogeneous risk measures convexity and coherence are equivalent. c) Show by means of a counterexample that without positive homogeneity the statement in b) is false.
a) Show that every coherent risk measure o: MR is a convex risk measure, that is g satisfies o(AL₁ + (1-X) L₂) ≤ Ag(L₁)+(1-X)g(L2), [0, 1], L₁, L₂ € M. b) Show that for positive-homogeneous risk measures convexity and coherence are equivalent. c) Show by means of a counterexample that without positive homogeneity the statement in b) is false.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![a) Show that every coherent risk measure o: M→ R is a convex risk measure, that is o
satisfies
o(AL₁ + (1-X) L₂) ≤ Ag(L₁) + (1-X)o(L₂), AE [0, 1], L₁, L2 € M.
b) Show that for positive-homogeneous risk measures convexity and coherence are equivalent.
c) Show by means of a counterexample that without positive homogeneity the statement in b)
is false.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6773fbd6-6acc-4514-b2e8-e0261e1e69e4%2F77a9b1af-3a3d-4fd4-85b9-92e2467c9a83%2Ffvhcrhc_processed.png&w=3840&q=75)
Transcribed Image Text:a) Show that every coherent risk measure o: M→ R is a convex risk measure, that is o
satisfies
o(AL₁ + (1-X) L₂) ≤ Ag(L₁) + (1-X)o(L₂), AE [0, 1], L₁, L2 € M.
b) Show that for positive-homogeneous risk measures convexity and coherence are equivalent.
c) Show by means of a counterexample that without positive homogeneity the statement in b)
is false.
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