Two-dimensional (2-D) models of electrical resistivity are improved by jointly inverting radiomagnetotelluric (RMT) and direct-current resistivity (DCR) data or by allowing for displacement currents in the inversion of RMT data collected on highly resistive bedrock. Uniqueness and stability of the 2-D models are assessed with a model variance and resolution analysis that allows for the non-linearity of the inverse problem. Model variance and resolution are estimated with a truncated singular value decomposition (TSVD) of the sensitivity matrix. In the computation of model errors, inverse singular values are replaced by non-linear semi-axes and the number of included eigenvectors is increased until a given error threshold is reached. Non-linear error estimates are verified with most-squares inversions. For the obtained truncation levels, model resolution matrices are computed. For RMT data, non-linear error appraisals are smaller than linearized ones. Hence, the consideration of the non-linearity in RMT data leads to reduced model errors or enhanced model resolution. The dielectric effect on RMT data is investigated with a new 2-D forward and inverse code that allows for displacement currents. As compared to the quasi-static approximation, apparent resistivities and phases of the impedance tensor elements are found to be significantly smaller and the vertical magnetic transfer function exhibits more distinct sign reversals. More reliable models of electrical resistivity are obtained from areas with highly resistive bedrock, if displacement currents are allowed for. In contrast, inversions with a quasi-static scheme introduce artefactual structures with extremely low or high resistivities. A smoothness-constrained 2-D joint inversion of RMT and DCR data is presented. The non-linear model variance and resolution analysis is applied to single and joint inverse models. For DCR data, the errors estimated by most-squares inversions are consistently larger than those estimated by the non-linear semi-axes, indicating that DCR models are poorly resolved. Certain areas of the joint inverse models are better resolved than in the single inverse models.

The submitted work deals with option pricing. Mathematical approach is immediately followed by an economic interpretation. The main problem is to model the underlying uncertainities driving the stock price. Using two well-known valuation models, binomial model and Black-Scholes model, we explain basic principles, especially risk neutral pricing. Due to the empirical biases new models have been developped, based on pure jump process. Variance gamma process and its special symmetric case are presented.

The concept of outlier detection by statistical hypothesis testing in geodesy is briefly reviewed. The performance of such tests can only be measured or optimized with respect to a proper alternative hypothesis. Firstly, we discuss the important question whether gross errors should be treated as non-random quantities or as random variables. In the first case, the alternative hypothesis must be based on the common mean shift model, while in the second case, the variance inflation model is appropriate. Secondly, we review possible formulations of alternative hypotheses (inherent, deterministic, slippage, mixture) and discuss their implications. As measures of optimality of an outlier detection, we propose the premium and protection, which are briefly reviewed. Finally, we work out a practical example: the fit of a straight line. It demonstrates the impact of the choice of an alternative hypothesis for outlier detection. / Das Konzept der Ausreißererkennung durch statistische Hypothesentests in der Geodäsie wird kurz überblickt. Die Leistungsfähigkeit solch eines Tests kann nur gemessen oder optimiert werden in Bezug auf eine geeignete Alternativhypothese. Als erstes diskutieren wir die wichtige Frage, ob grobe Fehler als nicht-zufällige oder zufällige Größen behandelt werden sollten. Im ersten Fall muss die Alternativhypothese auf das Mean-Shift-Modell gegründet werden, im zweiten Fall ist das Variance-Inflation-Modell passend. Als zweites stellen wir mögliche Formulierungen von Alternativhypothesen zusammen und diskutieren ihre Implikationen. Als Optimalitätsmaß schlagen wir das Premium-Protection-Maß vor, welches kurz überblickt wird. Schließlich arbeiten wir ein praktisches Beispiel aus: Die Anpassung einer ausgleichenden Gerade. Es zeigt die Auswirkung der Wahl einer Alternativhypothese für die Ausreißererkennung.