Thesis (Master').

The theoretical study of nonlinear expectations is the focus of attention for applications in a variety of different fields — often with the objective to model systems under incomplete information. Especially in mathematical finance, advances in the theory of sublinear expectations (also referred to as coherent risk measures) lay the theoretical foundation for modern approaches to evaluations under the presence of Knightian uncertainty. In this book, we introduce and study a large class of jump-type processes for sublinear expectations, which can be interpreted as Lévy-type processes under uncertainty in their characteristics. Moreover, we establish an existence and uniqueness theory for related nonlinear, nonlocal Hamilton-Jacobi-Bellman equations with non-dominated jump terms.

This thesis uses innovative approaches to analyse energy policy interventions aimed at enhancing the environmental sustainability of energy use as well as its consequential welfare implications. First, we examine the relationship between energy efficiency improvement and CO2 emissions at the macro level. We use the Index Decomposition Analysis to derive energy efficiency by separating out the impact of shifts in economic activity on energy intensity. We then employ econometric models to relate energy efficiency and CO2 emissions accounting for non-economic factors such as consumers lifestyle and attitudes. The applications for 13 OPEC and 30 OECD countries show that at the country-group and individual country level, increase in energy intensity for OPEC is associated with both deteriorations in energy efficiency and shifts towards energy-intensive activities. The model results suggest that the reduction in energy efficiency in general go in tandem with substantial increases in CO2 emissions. The decline in energy intensity for OECD can be attributed mainly to improvements in energy efficiency which is found to compensate for the impact on CO2 emissions of income changes. The results confirm the empirical relevance of energy efficiency improvements for the mitigation of CO2 emissions. The method developed in this chapter further enables the separate assessment of non-economic behavioural factors which according to the results exert a non-trivial influence on CO2 emissions. Secondly, having empirically confirmed the relationship between energy efficiency improvements and CO2 emission at the macro level in Chapter 2, we investigate potential underlying drivers of energy efficiency improvements taking into account potential asymmetric effects of energy price change in Chapter 3. This is crucial for designing effective and efficient policy measures that can promote energy efficiency. In addition to the Index Decomposition Analysis used to estimate the economy-wide energy efficiency in Chapter 2, we also use Stochastic Frontier Analysis and Data Envelop Analysis as alternative methods. The driving factors are examined using static and dynamic panel model methods that account for both observed and unobserved country heterogeneity. The application for 32 OECD countries shows that none of the three methods leads to correspondence in term of ranking between energy efficiency estimates and energy intensity at the country level corroborating the criticism that energy intensity is a poor proxy for energy efficiency. The panel-data regression results using the results of the three methods show similarities in the impacts of the determinants on the energy efficiency levels. Also, we find insignificant evidence of asymmetric effects of total energy price but there is proof of asymmetry using energy specific prices. Thirdly, in Chapter 4 we offer an improved understanding of the impacts to expect of abolishing fuel price subsidy on fuel consumption, and also of the welfare and distributional impacts at the household level. We develop a two-step approach for this purpose. Key aspect of the first step is a two-stage budgeting model to estimate various fuel types elasticities using micro-data. Relying on these estimates and the information on households expenditure shares for different commodities, the second step estimates the welfare (direct and indirect) and distributional impacts. The application for Nigeria emphasises the relevance of this approach. We find heterogeneous elasticities of fuel demand among household groups. The distributional impact of abolishing the kerosene subsidy shows a regressive welfare loss. Although we find a progressive loss for petrol, the loss gap between the low- and high-income groups is small relative to the loss gap from stopping kerosene subsidy, making the low-income groups to suffer a higher total welfare loss. Finally, from the highlighted results, we draw the following concluding remarks in chapter 5. Energy efficiency appears a key option to mitigate CO2 emissions but there is also a need for additional policies aiming for behavioural change; energy specific prices and allowing for asymmetry in analysing the changes in energy efficiency is more appropriate and informative in formulating reliable energy policies; the hypothesis that only the rich would be worse-off from fuel subsidy removal is rejected and the results further suggest that timing of the fuel subsidy removal would be crucial as a higher international oil price will lead to higher deregulated fuel price and consequently, larger welfare loss.

My thesis is focused on the construction of automated trading system and evaluation of its trading with selected commodities.

The presented paper deals with a theoretical analysis of the ultimate limit state. The results of experimental research carried out at the Institute of Metal and Timber Structures headed by Assoc. Prof. Karmazinová and Professor Melcher were applied. The geometrically and materially nonlinear solution based on the Timošenko’s solution is verified by the FEM model in the computer programme system ANSYS. The random influence of initial imperfections is taken into consideration. The FEM model also includes the influence of residual stress. In the parametric study, the influence of residual stress on the cross-section plastification is researched into, its influence on the load carrying capacity limit is, together with the influence of other imperfections, the subject of the stochastic analysis, applying the Latin Hypercube Sampling (LHS). Further on, the study proves a direct effect of the concrete part of the cross-section (combination of materials steel-concrete) on the decrease of load carrying capacity limit of the beam caused by influence of the residual stress of steel. With regard to the importance of time dependent phenomena of the concrete creep for the load carrying capacity, the studies given in the Ph.D. thesis are oriented in this respect. The parametric studies of the influence of the concrete creep having applied the Standard Eurocode 2 provide both a comparison of load carrying capacity limits when using common and high-strength concrete types, and also the variability of load carrying capacities. It follows from the comparison of the statistical analysis outputs according to the design reliability conditions of the Standard EN1990 and of the approach of Eurocode 4 that the Eurocode 4 can be recommended for dimensioning of this member type. According to the studies which we carried out, the design in compliance with Eurocode 4 can be evaluated as the reliable one. A larger set of experimental data is necessary to determine the economy.

Motivation and background The enormous amount of capabilities that every human learns throughout his life, is probably among the most remarkable and fascinating aspects of life. Learning has therefore drawn lots of interest from scientists working in very different fields like philosophy, biology, sociology, educational sciences, computer sciences and mathematics. This thesis focuses on the information theoretical and mathematical aspects of learning. We are interested in the learning process of an agent (which can be for example a human, an animal, a robot, an economical institution or a state) that interacts with its environment. Common models for this interaction are Markov decision processes (MDPs) and partially observable Markov decision processes (POMDPs). Learning is then considered to be the maximization of the expectation of a predefined reward function. In order to formulate general principles (like a formal definition of curiosity-driven learning or avoidance of unpleasant situation) in a rigorous way, it might be desirable to have a theoretical framework for the optimization of more complex functionals of the underlying process law. This might include the entropy of certain sensor values or their mutual information. An optimization of the latter quantity (also known as predictive information) has been investigated intensively both theoretically and experimentally using computer simulations by N. Ay, R. Der, K Zahedi and G. Martius. In this thesis, we develop a mathematical theory for learning in the sensorimotor loop beyond expected reward maximization. Approaches and results This thesis covers four different topics related to the theory of learning in the sensorimotor loop. First of all, we need to specify the model of an agent interacting with the environment, either with learning or without learning. This interaction naturally results in complex causal dependencies. Since we are interested in asymptotic properties of learning algorithms, it is necessary to consider infinite time horizons. It turns out that the well-understood theory of causal networks known from the machine learning literature is not powerful enough for our purpose. Therefore we extend important theorems on causal networks to infinite graphs and general state spaces using analytical methods from measure theoretic probability theory and the theory of discrete time stochastic processes. Furthermore, we prove a generalization of the strong Markov property from Markov processes to infinite causal networks. Secondly, we develop a new idea for a projected stochastic constraint optimization algorithm. Generally a discrete gradient ascent algorithm can be used to generate an iterative sequence that converges to the stationary points of a given optimization problem. Whenever the optimization takes place over a compact subset of a vector space, it is possible that the iterative sequence leaves the constraint set. One possibility to cope with this problem is to project all points to the constraint set using Euclidean best-approximation. The latter is sometimes difficult to calculate. A concrete example is an optimization over the unit ball in a matrix space equipped with operator norm. Our idea consists of a back-projection using quasi-projectors different from the Euclidean best-approximation. In the matrix example, there is another canonical way to force the iterative sequence to stay in the constraint set: Whenever a point leaves the unit ball, it is divided by its norm. For a given target function, this procedure might introduce spurious stationary points on the boundary. We show that this problem can be circumvented by using a gradient that is tailored to the quasi-projector used for back-projection. We state a general technical compatibility condition between a quasi-projector and a metric used for gradient ascent, prove convergence of stochastic iterative sequences and provide an appropriate metric for the unit-ball example. Thirdly, a class of learning problems in the sensorimotor loop is defined and motivated. This class of problems is more general than the usual expected reward maximization and is illustrated by numerous examples (like expected reward maximization, maximization of the predictive information, maximization of the entropy and minimization of the variance of a given reward function). We also provide stationarity conditions together with appropriate gradient formulas. Last but not least, we prove convergence of a stochastic optimization algorithm (as considered in the second topic) applied to a general learning problem (as considered in the third topic). It is shown that the learning algorithm converges to the set of stationary points. Among others, the proof covers the convergence of an improved version of an algorithm for the maximization of the predictive information as proposed by N. Ay, R. Der and K. Zahedi. We also investigate an application to a linear Gaussian dynamic, where the policies are encoded by the unit-ball in a space of matrices equipped with operator norm.