Volker Krätschmer

A unified approach to fuzzy random variables

The concept of fuzzy random variable was introduced as an analogous notion to random variables in order to extend statistical analysis to situations when the outcomes of some random experiment are fuzzy sets. But in contrary to the classical statistical methods no unique definition has been established yet. In this paper a set-theoretical concept of fuzzy random variable will be presented. This notion provides a useful framework to compare different concepts of fuzzy random variables, using methods of general topology and drawing on results from topological measure theory and the theory of analytic spaces. As the main result, it will be shown that the introduced concept of fuzzy random variable is a unification of the already known ones.

Comparative and qualitative robustness for law-invariant risk measures

When estimating the risk of a P&L from historical data or Monte Carlo simulation, the robustness of the estimate is important. We argue here that Hampel’s classical notion of qualitative robustness is not suitable for risk measurement, and we propose and analyze a refined notion of robustness that applies to tail-dependent law-invariant convex risk measures on Orlicz spaces. This concept captures the tradeoff between robustness and sensitivity and can be quantified by an index of qualitative robustness. By means of this index, we can compare various risk measures, such as distortion risk measures, in regard to their degree of robustness. Our analysis also yields results of independent interest such as continuity properties and consistency of estimators for risk measures, or a Skorohod representation theorem for ψ-weak convergence.

Limit theorems for fuzzy-random variables

This paper deals with limit theorems for fuzzy-valued measurable mappings which provide, as a whole, a foundation of statistical analysis with fuzzy data. A strong law of large numbers, a central limit theorem and a Gliwenko-Cantelli theorem are proved. The results are formulated simultaneously with respect to the Lp-metrics on the fuzzy sample spaces, investigated by Diamond and Kloeden. In particular, these versions of the limit theorems are related to identical, compatible concepts of convergence and measurability in the fuzzy sample spaces. The proofs of the theorems are based heavily on isomorphic isometric embeddings of the fuzzy sample spaces, endowed with Lp-metrics, into respective Lp-spaces, which are Banach spaces of type 2. These embeddings provide the application of convergence results in Banach spaces.

Qualitative and infinitesimal robustness of tail-dependent statistical functionals

The main goal of this article is to introduce a new notion of qualitative robustness that applies also to tail-dependent statistical functionals and that allows us to compare statistical functionals in regards to their degree of robustness. By means of new versions of the celebrated Hampel theorem, we show that this degree of robustness can be characterized in terms of certain continuity properties of the statistical functional. The proofs of these results rely on strong uniform Glivenko-Cantelli theorems in fine topologies, which are of independent interest. We also investigate the sensitivity of tail-dependent statistical functionals w.r.t. infinitesimal contaminations, and we introduce a new notion of infinitesimal robustness. The theoretical results are illustrated by means of several examples including general L- and V-functionals.

Some complete metrics on spaces of fuzzy subsets

The classes of the Lp,∞- and Lp-metrics play an important role to develop a probability theory in fuzzy sample spaces. All of these metrics are known to be separable, but not complete. The classes are closely related as for each Lp,∞-metric there exists some Lp-metric which induces the same topology. This paper deals with the completion of the Lp,∞- and Lp-metrics. We can also show that the relationship between the classes of Lp,∞- and Lp-metrics still holds for the obtained respective classes of their completions.

When fuzzy measures are upper envelopes of probability measures

Abstract Fuzzy measures which are upper envelopes of probability measures may play an important role to develop a general theory of Bayesian statistics. Especially from a technical point of view, a widely accepted generalized Bayes rule would be applicable for those kind of fuzzy measures. We give sufficient general conditions to ensure that fuzzy measures are upper envelopes of probability measures. They are applied to some special classes of important types of fuzzy measures, namely Sugeno fuzzy, plausibility and possibility measures. The proof of the main result is based on recently systemized inner extension procedures within abstract measure theory.

Least-squares estimation in linear regression models with vague concepts

The paper is a contribution to parameter estimation in fuzzy regression models with random fuzzy sets. Here models with crisp parameters and fuzzy observations of the variables are investigated. This type of regression models may be understood as an extension of the ordinary single equation linear regression models by integrating additionally the physical vagueness of the involved items. So the significance of these regression models is to improve the empirical meaningfulness of the relationship between the items by a more sensitive attention to the fundamental adequacy problem of measurement. Concerning the parameter estimation the ordinary least-squares method is extended. The existence of estimators by the suggested method is shown, and some of their stochastic properties are surveyed.

On s-additive robust representation of convex risk measures for unbounded financial positions in the presence of uncertainty about the market model

Recently, Frittelli and Scandolo ([9]) extend the notion of risk measures, originally introduced by Artzner, Delbaen, Eber and Heath ([1]), to the risk assessment of abstract financial positions, including pay offs spread over different dates, where liquid derivatives are admitted to serve as financial instruments. The paper deals with s-additive robust representations of convex risk measures in the extended sense, dropping the assumption of an existing market model, and allowing also unbounded financial positions. The results may be applied for the case that a market model is available, and they encompass as well as improve criteria obtained for robust representations of the original convex risk measures for bounded positions ([4], [7], [16]).

Integrals of random fuzzy sets

This paper tries to give a systematic investigation of integration of random fuzzy sets. Besides the widely used Aumann-integral adaptions of Pettis- and Bochner-integration for random elements in Banach spaces are introduced. The mutual relationships of these competing concepts will be explored comprehensively, completing and improving former results from literature. As a by product dominated convergence theorems, strong laws of large numbers and central limit theorems for random fuzzy sets can be derived. They are based on weaker assumptions than previous versions from literature.

Parametric estimation of risk neutral density functions

This chapter deals with the estimation of risk neutral distributions for pricing index options resulting from the hypothesis of the risk neutral valuation principle. After justifying this hypothesis, we shall focus on parametric estimation methods for the risk neutral density functions determining the risk neutral distributions. We we shall differentiate between the direct and the indirect way. Following the direct way, parameter vectors are estimated which characterize the distributions from selected statistical families to model the risk neutral distributions. The idea of the indirect approach is to calibrate characteristic parameter vectors for stochastic models of the asset price processes, and then to extract the risk neutral density function via Fourier methods. For every of the reviewed methods the calculation of option prices under hypothetically true risk neutral distributions is a building block. We shall give explicit formula for call and put prices w.r.t. reviewed parametric statistical families used for direct estimation. Additionally, we shall introduce the Fast Fourier Transform method of call option pricing developed in Carr and Madan [J. Comput. Finance 2(4):61–73, 1999]. It is intended to compare the reviewed estimation methods empirically.