Multivariate risk measures in the non-convex setting

The family of admissible positions in a transaction costs model is a random closed set, which is convex in case of proportional transaction costs. However, the convexity fails, e.g., in case of fixed transaction costs or when only a finite number of transfers are possible. The paper presents an approach to measure risks of such positions based on the idea of considering all selections of the portfolio and checking if one of them is acceptable. Properties and basic examples of risk measures of non-convex portfolios are presented.

ON THE SPECIFIC AREA OF INHOMOGENEOUS BOOLEAN MODELS. EXISTENCE RESULTS AND APPLICATIONS

The problem of the evaluation of the so-called specific area of a random closed set, in connection with its mean boundary measure, is mentioned in the classical book by Matheron on random closed sets (Matheron, 1975, p. 50); it is still an open problem, in general. We offer here an overview of some recent results concerning the existence of the specific area of inhomogeneous Boolean models, unifying results from geometric measure theory and from stochastic geometry. A discussion of possible applications to image analysis concerning the estimation of the mean surface density of random closed sets, and, in particular, to material science concerning birth-and-growth processes, is also provided.

3D IMAGE ANALYSIS OF OPEN FOAMS USING RANDOM TESSELLATIONS

Volume image analysis provides a number of methods for the characterization of the microstructure of open foams. Mean values of characteristics of the edge system are measured directly from the volume image. Further characteristics like the intensity and mean size of the cells are obtained using model assumptions where the edge system of the foam is interpreted as a realization of a random closed set. Macroscopically homogeneous random tessellations provide a suitable model for foam structures. However, their cells often lack the degree of regularity observed in real data. In this respect some deterministic models seem to be closer to realistic structures, although they do not capture the microscopic heterogeneity of real foams. In this paper, the influence of the model choice on the obtained mean values is studied. Moreover, a method for reconstruction of the cells of an open foam from its edge system is described and tested for the tessellations under consideration.

RANDOM CLOSED SET MODELS: ESTIMATING AND SIMULATING BINARY IMAGES

In this paper we show the use of the Boolean model and a class of RACS models that is a generalization of it to obtain simulations of random binary images able to imitate natural textures such as marble or wood. The different tasks required, parameter estimation, goodness-of-fit test and simulation, are reviewed. In addition to a brief review of the theory, simulation studies of each model are included.

MESH FREE ESTIMATION OF THE STRUCTURE MODEL INDEX

The structure model index (SMI) is a means of subsuming the topology of a homogeneous random closed set under just one number, similar to the isoperimetric shape factors used for compact sets. Originally, the SMI is defined as a function of volume fraction, specific surface area and first derivative of the specific surface area, where the derivative is defined and computed using a surface meshing. The generalised Steiner formula yields however a derivative of the specific surface area that is – up to a constant – the density of the integral of mean curvature. Consequently, an SMI can be defined without referring to a discretisation and it can be estimated from 3D image data without need to mesh the surface but using the number of occurrences of 2×2×2 pixel configurations, only. Obviously, it is impossible to completely describe a random closed set by one number. In this paper, Boolean models of balls and infinite straight cylinders serve as cautionary examples pointing out the limitations of the SMI. Nevertheless, shape factors like the SMI can be valuable tools for comparing similar structures. This is illustrated on real microstructures of ice, foams, and paper.

Spectral theory for random closed sets and estimating the covariance via frequency space

A spectral theory for stationary random closed sets is developed and provided with a sound mathematical basis. The definition and a proof of the existence of the Bartlett spectrum of a stationary random closed set as well as the proof of a Wiener-Khinchin theorem for the power spectrum are used to two ends. First, well-known second-order characteristics like the covariance can be estimated faster than usual via frequency space. Second, the Bartlett spectrum and the power spectrum can be used as second-order characteristics in frequency space. Examples show that in some cases information about the random closed set is easier to obtain from these characteristics in frequency space than from their real-world counterparts.