Sur les fonctions d'ensemble additives et continues

Sierpiński, Wacław

doi:10.4064/fm-3-1-240-246

Cited by 50 publications

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“…Proof. Sierpiński's theorem on non-atomic measures tells us that for all 0 ≤ α ≤ 1 there is a measurable set E ⊆ Ω of measure α [24]. One deduces that there is a uniformly distributed random variable U on Ω.…”

Section: Discussionmentioning

confidence: 99%

Optimizing S-Shaped Utility and Implications for Risk Management

Armstrong

Brigo

2018

SSRN Journal

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We consider market players with tail-risk-seeking behaviour as exemplified by the S-shaped utility introduced by Kahneman and Tversky. We argue that risk measures such as value at risk (VaR) and expected shortfall (ES) are ineffective in constraining such players. We show that, in many standard market models, product design aimed at utility maximization is not constrained at all by VaR or ES bounds: the maximized utility corresponding to the optimal payoff is the same with or without ES constraints. By contrast we show that, in reasonable markets, risk management constraints based on a second more conventional concave utility function can reduce the maximum S-shaped utility that can be achieved by the investor, even if the constraining utility function is only rather modestly concave. It follows that product designs leading to unbounded S-shaped utilities will lead to unbounded negative expected constraining utilities when measured with such conventional utility functions. To prove these latter results we solve a general problem of optimizing an investor expected utility under risk management constraints where both investor and risk manager have conventional concave utility functions, but the investor has limited liability. We illustrate our results throughout with the example of the Black-Scholes option market. These results are particularly important given the historical role of VaR and that ES was endorsed by the Basel committee in 2012-2013.

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Section: Discussionmentioning

confidence: 99%

Optimizing S-Shaped Utility and Implications for Risk Management

Armstrong

Brigo

2018

SSRN Journal

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“…and for cos(2w) = −m, x = π/2 we get (22) in the case |m| ≤ 1. From (26) If f 0 gives the maximum, then z → f 0 (−z) gives the minimum, and vice versa.…”

Section: Lemma 6 Letmentioning

confidence: 96%

Bounded harmonic mappings

Szapiel

2010

JAMA

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Denote by B(τ) the class of all complex functions of the form f (z) ≡ τ + ∞ n=1 a n (f )z n + b n (f ) z n that are harmonic in the open unit disk D with f (D) ⊂ D.Both B(τ) and some of its closed convex subsets are strongly convex, e.g.,Simple extremal problems in B(τ) may have nontrivial solutions. To find them, we present various methods: the method of subordination, the Poisson integral, and calculus of variations. For instance, by analogy to the known result a n ( (τ)) = {w : |w| ≤ 1 − |τ| 2 }, we find the variability regions a n (B(τ)) = b n (B(τ)) = w :x + cos t √x 2 + 2x cos t + 1 dt .In the case n = 1, |τ| < 2/π (resp., |τ| = 2/π), the extremal functions realizing points of the circle w : |w| = ϕ 1/ϕ −1 (|τ|) are univalent self-mappings of the disk D (resp., univalent mappings of D onto a half unit disk). PreliminariesThe classes of all complex analytic functions and all complex harmonic functions in the disk D = {z ∈ C : |z| < 1} that are bounded there by one have a surprising property. Both of them are strongly convex, and hence their extreme and support points form dense subsets. The geometric nature of the sets considered requires some notions of abstract analysis.Let H be a separated locally convex linear space. Suppose that ∅ = F ⊂ H and write conv F, convF, ext F and spt F to denote the convex hull of F, the closed convex hull of F, the set of all extreme points of F, and the set of all (proper)

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“…Recall (cf. [39]) that, if σ ∈ P(X) is diffuse, then for every k ∈ N 1 and y ∈ int ∆ k−1 there exists X ∈ P k (X) such that σ X = y. , Fleming-Viot with parent-independent mutation [8]; see e.g. [36, §2] for an explicit construction) is the unique random probability over (X, B(X)) such that…”

Section: Recall the Following Representations Ofmentioning

confidence: 99%

Characteristic functionals of Dirichlet measures

Schiavo¹

2019

Electron. J. Probab.

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We compute characteristic functionals of Dirichlet-Ferguson measures over a locally compact Polish space and prove continuous dependence of the random measure on the parameter measure. In finite dimension, we identify the dynamical symmetry algebra of the characteristic functional of the Dirichlet distribution with a simple Lie algebra of type A. We study the lattice determined by characteristic functionals of categorical Dirichlet posteriors, showing that it has a natural structure of weight Lie algebra module and providing a probabilistic interpretation. A partial generalization to the case of the Dirichlet-Ferguson measure is also obtained.

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Sur les fonctions d'ensemble additives et continues

Cited by 50 publications

References 0 publications

Optimizing S-Shaped Utility and Implications for Risk Management

Optimizing S-Shaped Utility and Implications for Risk Management

Bounded harmonic mappings

Characteristic functionals of Dirichlet measures

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