In this thesis, the asymptotic behavior and oscillation of solutions of dynamic equations on time scales are studied. In the first part of the thesis, asymptotic equivalence and asymptotic equilibrium of dynamic systems are investigated. Sufficient conditions are established for the asymptotic equivalence of linear systems and linear and quasilinear systems, respectively, and for the asymptotic equilibrium of quasilinear systems by unifying and extending some known results for differential systems and difference systems to dynamic systems on arbitrary time scales. In particular, for the asymptotic equivalence of differential systems, the well-known theorems of Levinson and Yakubovich are improved and the well-known theorem of Wintner for the asymptotic equilibrium of linear differential systems is generalized to arbitrary time scales. Some of our results for asymptotic equilibrium are new even for difference systems. In the second part, the oscillation of solutions of a particular class of second order nonlinear delay dynamic equations and, more generally, two-dimensional nonlinear dynamic systems, including delay-dynamic systems, are discussed. Necessary and sufficient conditions are derived for the oscillation of solutions of nonlinear delay dynamic equations by extending some continuous results. Specifically, the classical theorems of Atkinson and Belohorec are generalized. Sufficient conditions are established for the oscillation of solutions of nonlinear dynamic systems by unifying and extending the corresponding continuous and discrete results. Particularly, the oscillation criteria of Atkinson, Belohorec, Waltman, and Hooker and Patula are generalized.

The submitted thesis studies penalty function methods for stochastic programming problems. The main objective of the paper is to examine penalty function methods for deterministic nonlinear programming, in particular exact penalty function methods, in order to enhance penalty function methods for stochastic programming. For this purpose, the equivalence of the original de- terministic nonlinear and the corresponding penalty function problem using arbi- trary vector norm as the penalty function is shown for convex and invex functions occurring in the problems, respectively. The obtained theorems are consequently applied to multiple chance constrained problems under finite discrete probability distribution to show the asymptotic equivalence of the probabilistic and the cor- responding penalty function problems. The practical use of the newly obtained methods is demonstrated on a numerical study, in which a comparison with other approaches is provided as well. 1

Diese Arbeit untersucht Inferenz für Streuungsparameter bedingter Gaußprozesse anhand diskreter verrauschter Beobachtungen in einem Hochfrequenz-Setting. Unser Ziel dabei ist es, eine asymptotische Charakterisierung von effizienter Schätzung in einem allgemeine Gaußschen Rahmen zu finden. Für ein parametrisches Fundamentalmodell wird ein Hájek-Le Cam-Faltungssatz hergeleitet, welcher eine exakte asymptotische untere Schranke für Schätzmethoden liefert. Dazu passende obere Schranken werden konstruiert und die Bedeutung des Satzes wird verdeutlicht anhand zahlreicher Beispiele wie der (fraktionellen) Brownschen Bewegung, dem Ornstein-Uhlenbeck-Prozess oder integrierten Prozessen. Die Herleitung der Effizienzresultate basiert auf asymptotischen Äquivalenzen und kann für verschiedene Verallgemeinerungen des parametrischen Fundamentalmodells verwendet werden. Als eine solche Erweiterung betrachten wir das Schätzen der quadrierten Kovariation eines stetigen Martingals anhand verrauschter asynchroner Beobachtungen, welches ein fundamentales Schätzproblem in der Öknometrie ist. Für dieses Modell erhalten wir einen semi-parametrischen Faltungssatz, welcher bisherige Resultate im Sinne von Multidimensionalität, Asynchronität und Annahmen verallgemeinert. Basierend auf den vorhergehenden Herleitungen entwickeln wir einen statistischen Test für den Hurst-Parameter einer fraktionellen Brownschen Bewegung. Ein Score- und ein Likelihood-Quotienten-Test werden implementiert sowie analysiert und erste empirische Eindrücke vermittelt. / This work studies inference on scaling parameters of a conditionally Gaussian process under discrete noisy observations in a high-frequency regime. Our aim is to find an asymptotic characterisation of efficient estimation for a general Gaussian framework. For a parametric basic case model a Hájek-Le Cam convolution theorem is derived, yielding an exact asymptotic lower bound for estimators. Matching upper bounds are constructed and the importance of the theorem is illustrated by various examples of interest such as the (fractional) Brownian motion, the Ornstein-Uhlenbeck process or integrated processes. The derivation of the efficiency result is based on asymptotic equivalences and can be employed for several generalisations of the parametric basic case model. As such an extension we consider estimation of the quadratic covariation of a continuous martingale from noisy asynchronous observations, which is a fundamental estimation problem in econometrics. For this model, a semi-parametric convolution theorem is obtained which generalises existing results in terms of multidimensionality, asynchronicity and assumptions. Based on the previous derivations, we develop statistical tests on the Hurst parameter of a fractional Brownian motion. A score test and a likelihood ratio type test are implemented as well as analysed and first empirical impressions are given.