Normal Cones to Sublevel Sets: An Axiomatic Approach

AbstractCapital allocation is of central importance in portfolio management and risk-based performance measurement. Capital allocations for univariate risk measures have been extensively studied in the finance literature. In contrast to this situation, few papers dealt with capital allocations for multivariate risk measures. In this paper, we propose an axiom system for capital allocation with multivariate risk measures. We first recall the class of the positively homogeneous and subadditive multivariate risk measures, and provide the corresponding representation results. Then it is shown that for a given positively homogeneous and subadditive multivariate risk measure, there exists a capital allocation principle. Furthermore, the uniqueness of the capital allocation principe is characterized. Finally, examples are also given to derive the explicit capital allocation principles for the multivariate risk measures based on mean and standard deviation, including the multivariate mean-standard-deviation risk measures.

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Multivariate Lorenz Majorization and Heterogeneity Measures: An Axiomatic Approach

Analyse & Kritik ◽

10.1515/auk-2007-0206 ◽

2007 ◽

Vol 29 (2) ◽

Author(s):

Wolfgang Eichhorn ◽

Manfred Krtscha

Keyword(s):

Lorenz Curve ◽

Axiomatic Approach

AbstractThis work introduces two new curves that are multivariate generalizations of the “classical” Lorenz curve. All data of d-variate distributions can be visualized by drawing these curves in the plane, whereas Koshevoy’s and Mosler’s generalization by a lift zonoid in ℝ

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WEYL GEOMETRY AS CHARACTERIZATION OF SPACE-TIME

International Journal of Modern Physics Conference Series ◽

10.1142/s2010194511001176 ◽

2011 ◽

Vol 03 ◽

pp. 87-97 ◽

Cited By ~ 5

Author(s):

F. P. POULIS ◽

J. M. SALIM

Keyword(s):

General Relativity ◽

Scalar Field ◽

Riemannian Geometry ◽

Axiomatic Approach ◽

Space Time ◽

Weyl Geometry ◽

Test Particles ◽

Variational Formalism ◽

Physical Fields

Motivated by an axiomatic approach to characterize space-time it is investigated a reformulation of Einstein's gravity where the pseudo-riemannian geometry is substituted by a Weyl one. It is presented the main properties of the Weyl geometry and it is shown that it gives extra contributions to the trajectories of test particles, serving as one more motivation to study general relativity in Weyl geometry. It is introduced its variational formalism and it is established the coupling with other physical fields in such a way that the theory acquires a gauge symmetry for the geometrical fields. It is shown that this symmetry is still present for the red-shift and it is concluded that for cosmological models it opens the possibility that observations can be fully described by the new geometrical scalar field. It is concluded then that this reformulation, although representing a theoretical advance, still needs a complete description of their objects.

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Axiomatic approach to implication for approximate reasoning with fuzzy logic

Fuzzy Sets and Systems ◽

10.1016/0165-0114(80)90054-8 ◽

1980 ◽

Vol 3 (2) ◽

pp. 193-219 ◽

Cited By ~ 121

Author(s):

J.F. Baldwin ◽

B.W. Pilsworth

Keyword(s):

Fuzzy Logic ◽

Axiomatic Approach ◽

Approximate Reasoning

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