Distribution functions of linear combinations of lattice polynomials from the uniform distribution

We give the distribution functions, the expected values, and the moments of linear combinations of lattice polynomials from the uniform distribution. Linear combinations of lattice polynomials, which include weighted sums, linear combinations of order statistics, and lattice polynomials, are actually those continuous functions that reduce to linear functions on each simplex of the standard triangulation of the unit cube. They are mainly used in aggregation theory, combinatorial optimization, and game theory, where they are known as discrete Choquet integrals and Lovasz extensions.Comment: 11 page

Weighted lattice polynomials of independent random variables

We give the cumulative distribution functions, the expected values, and the moments of weighted lattice polynomials when regarded as real functions of independent random variables. Since weighted lattice polynomial functions include ordinary lattice polynomial functions and, particularly, order statistics, our results encompass the corresponding formulas for these particular functions. We also provide an application to the reliability analysis of coherent systems.Comment: 14 page

Finitely additive extensions of distribution functions and moment sequences: The coherent lower prevision approach

We study the information that a distribution function provides about the finitely additive probability measure inducing it. We show that in general there is an infinite number of finitely additive probabilities associated with the same distribution function. Secondly, we investigate the relationship between a distribution function and its given sequence of moments. We provide formulae for the sets of distribution functions, and finitely additive probabilities, associated with some moment sequence, and determine under which conditions the moments determine the distribution function uniquely. We show that all these problems can be addressed efficiently using the theory of coherent lower previsions

Multivariate integration of functions depending explicitly on the minimum and the maximum of the variables

By using some basic calculus of multiple integration, we provide an alternative expression of the integral $$\int_{]a,b[^n} f(\mathbf{x},\min x_i,\max x_i) d\mathbf{x},$$ in which the minimum and the maximum are replaced with two single variables. We demonstrate the usefulness of that expression in the computation of orness and andness average values of certain aggregation functions. By generalizing our result to Riemann-Stieltjes integrals, we also provide a method for the calculation of certain expected values and distribution functions.Comment: 15 page

On dynamic spectral risk measures, a limit theorem and optimal portfolio allocation

In this paper we propose the notion of continuous-time dynamic spectral risk-measure (DSR). Adopting a Poisson random measure setting, we define this class of dynamic coherent risk-measures in terms of certain backward stochastic differential equations. By establishing a functional limit theorem, we show that DSRs may be considered to be (strongly) time-consistent continuous-time extensions of iterated spectral risk-measures, which are obtained by iterating a given spectral risk-measure (such as Expected Shortfall) along a given time-grid. Specifically, we demonstrate that any DSR arises in the limit of a sequence of such iterated spectral risk-measures driven by lattice-random walks, under suitable scaling and vanishing time- and spatial-mesh sizes. To illustrate its use in financial optimisation problems, we analyse a dynamic portfolio optimisation problem under a DSR.Comment: To appear in Finance and Stochastic

An empirical study of statistical properties of Choquet and Sugeno integrals

This paper investigates the statistical properties of the Choquet and Sugeno integrals, used as multiattribute models. The investigation is done on an empirical basis, and focuses on two topics: the distribution of the output of these integrals when the input is corrupted with noise, and the robustness of these models, when they are identified using some set of learning data through some learning procedure.Choquet integral; Sugeno integral; output distribution

Pessimistic portfolio allocation and Choquet expected utility

Recent developments in the theory of choice under uncertainty and risk yield a pessimistic decision theory that replaces the classical expected utility criterion with a Choquet expectation that accentuates the likelihood of the least favorable outcomes. A parallel theory has recently emerged in the literature on risk assessment. It is shown that a general form of pessimistic portfolio optimization based on the Choquet approach may be formulated as a problem of linear quantile regression.

Isotropic cosmological singularities: other matter models

Isotropic cosmological singularities are singularities which can be removed by rescaling the metric. In some cases already studied (gr-qc/9903008, gr-qc/9903009, gr-qc/9903018) existence and uniqueness of cosmological models with data at the singularity has been established. These were cosmologies with, as source, either perfect fluids with linear equations of state or massless, collisionless particles. In this article we consider how to extend these results to a variety of other matter models. These are scalar fields, massive collisionless matter, the Yang-Mills plasma of Choquet-Bruhat, or matter satisfying the Einstein-Boltzmann equation.Comment: LaTeX, 19 pages, no figure

Modelling multi-scale microstructures with combined Boolean random sets: A practical contribution

Boolean random sets are versatile tools to match morphological and topological properties of real structures of materials and particulate systems. Moreover, they can be combined in any number of ways to produce an even wider range of structures that cover a range of scales of microstructures through intersection and union. Based on well-established theory of Boolean random sets, this work provides scientists and engineers with simple and readily applicable results for matching combinations of Boolean random sets to observed microstructures. Once calibrated, such models yield straightforward three-dimensional simulation of materials, a powerful aid for investigating microstructure property relationships. Application of the proposed results to a real case situation yield convincing realisations of the observed microstructure in two and three dimensions