Optimal risk sharing for law invariant monetary utility functions
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Citations
Optimal Capital and Risk Allocations for Law-and Cash-Invariant Convex Functions
Quantile-Based Risk Sharing
Valuations and dynamic convex risk measures
Quantile-Based Risk Sharing
To split or not to split: Capital allocation with convex risk measures
References
Coherent Measures of Risk
In search of homo economicus: Behavioral experiments in 15 small-scale societies
In Search of Homo Economicus: Behavioral Experiments in 15 Small- Scale Societies
Behavioral Game Theory
Convex measures of risk and trading constraints
Related Papers (5)
Frequently Asked Questions (11)
Q2. What is the value of a measure preserving transformation of (,F,P?
A measure preserving transformation of (Ω,F ,P) is a bi-measurable bijection τ : (Ω,F ,P) → (Ω,F ,P) leaving P invariant, i.e., τ(P) = P.
Q3. What is the definition of Pareto optimal allocations?
Pareto optimal allocations are only defined up to a constant, i.e. given a Pareto optimal allocation (ξ0, ξ1), the attainable allocation (ξ0 + c, ξ1− c) is also Pareto optimal for every c ∈ R, 2. Let (ξ0, ξ1) be an attainable allocation which is not Pareto optimal.
Q4. What is the proof for the comonotone law-invariant monetary utility?
Let U0 and U1 be two comonotone law-invariant monetary utility functions, and let ϕ̄0, ϕ̄1 ∈ C↘ be the associated functions in the representation (2.3).
Q5. What is the comonotonicity of the pairs?
By Lemma 4.2, the authors see thatqξ0 is constant on [β, 1] . (5.4)Since U1 is strictly risk-averse conditionally on lower tail events, it follows again from the comonotonicity of the pairs (−Z, ξi), i = 0, 1, together with Lemma 5.1 thatqξ1 is constant on [0, β] .
Q6. What is the proof for the monetary utility function U0 and V 1i?
By Remark 2.3, the functions Ui and V 1i = χCϕ̄i are 〈L ∞,L1(P)〉−conjugate, where χ is the indicator function in the sense of convex analysis.
Q7. What is the proof of equivalence with (iii)?
2The characterization (ii) in the above Theorem 3.1 has a well-known extension for general utility functions (see e.g. [26]), which involves two Lagrange multipliers (simply leaving out the last sentence in the above proof of (i)=⇒(ii)).
Q8. Why is the set of Pareto optimal allocations empty?
In this example due to F. Delbaen [20], the authors show that the set of Pareto optimal allocations might be empty in the context where all monetary utility functions U0, U1, U0 U1, are positively-homogeneous, and have the Fatou property.
Q9. What is the proof of the maximization problem in the definition of U0 U1?
The authors first start by proving that the maximization problem in the definition of the supconvolution U0 U1 can be restricted to pairs (ξ0, ξ1) ∈ A↗(X).
Q10. what is the subset of all functions in C which are continuous at the point t?
D↘ } and its pointwise closureC↘ = {ϕ : [0, 1] −→ [0, 1] non-decreasing, concave, ϕ(0) = 0 and ϕ(1) = 1} .Clearly, C↘ is the subset of all functions in C↘ which are continuous at the point t = 0.
Q11. What is the simplest example of a Pareto optimal?
This example was triggered by a question of N. El Karoui, and shows that, even in a finite-dimensional setting, it may happen that the set of Pareto optimal allocations is empty for any possible aggregate risk X ∈ L∞.