Boundedness and convergence of singular integrals on fractal type sets

The topic of this dissertation lies in the intersection of harmonic analysis and fractal geometry. We particulary consider singular integrals in Euclidean spaces with respect to general measures, and we study how the geometric structure of the measures affects certain analytic properties of the operators. The thesis consists of three research articles and an overview. In the first article we construct singular integral operators on lower dimensional Sierpinski gaskets associated with homogeneous Calderón-Zygmund kernels. While these operators are bounded their principal values fail to exist almost everywhere. Conformal iterated function systems generate a broad range of fractal sets. In the second article we prove that many of these limit sets are porous in a very strong sense, by showing that they contain holes spread in every direction. In the following we connect these results with singular integrals. We exploit the fractal structure of these limit sets, in order to establish that singular integrals associated with very general kernels converge weakly. Boundedness questions consist a central topic of investigation in the theory of singular integrals. In the third article we study singular integrals of different measures. We prove a very general boundedness result in the case where the two underlying measures are separated by a Lipshitz graph. As a consequence we show that a certain weak convergence holds for a large class of singular integrals.

Stochastic Filtering

The stochastic filtering has been in general an estimation of indirectly observed states given observed data. This means that one is discussing conditional expected values as being one of the most accurate estimation, given the observations in the context of probability space. In my thesis, I have presented the theory of filtering using two different kind of observation process: the first one is a diffusion process which is discussed in the first chapter, while the third chapter introduces the latter which is a counting process. The majority of the fundamental results of the stochastic filtering is stated in form of interesting equations, such the unnormalized Zakai equation that leads to the Kushner-Stratonovich equation. The latter one which is known also by the normalized Zakai equation or equally by Fujisaki-Kallianpur-Kunita (FKK) equation, shows the divergence between the estimate using a diffusion process and a counting process. I have also introduced an example for the linear gaussian case, which is mainly the concept to build the so-called Kalman-Bucy filter. As the unnormalized and the normalized Zakai equations are in terms of the conditional distribution, a density of these distributions will be developed through these equations and stated by Kushner Theorem. However, Kushner Theorem has a form of a stochastic partial differential equation that needs to be verify in the sense of the existence and uniqueness of its solution, which is covered in the second chapter.

Computing the Stochastic Complexity of Simple Probabilistic Graphical Models

Minimum Description Length (MDL) is an information-theoretic principle that can be used for model selection and other statistical inference tasks. There are various ways to use the principle in practice. One theoretically valid way is to use the normalized maximum likelihood (NML) criterion. Due to computational difficulties, this approach has not been used very often. This thesis presents efficient floating-point algorithms that make it possible to compute the NML for multinomial, Naive Bayes and Bayesian forest models. None of the presented algorithms rely on asymptotic analysis and with the first two model classes we also discuss how to compute exact rational number solutions.

Growth and Convergence: The case of China

Growth and Convergence: The Case of China Since the initiation of economic reforms in 1978, China has become one of the world’s fast-growing economies. The rapid growth, however, has not been shared equally across the different regions in China. The prominent feature of substantial differences in incomes and growth rates across the different Chinese regions has attracted the attention of many researchers. This book focuses on issues related to economic growth and convergence across the Chinese regions over the past three decades. The book has eight chapters. Apart from an introduction chapter and a concluding chapter, all the other chapters each deal with some certain aspects of the central issue of regional growth and convergence across China over the past three decades. The whole book is organized as follows. Chapter 1 provides an introduction to the basic issues involved in this book. Chapter 2 tests economic growth and convergence across 31 Chinese provinces during 1981-2005, based on the theoretical framework of the Solow growth model. Chapter 3 investigates the relationship between openness to foreign economic activities, such as foreign trade and foreign direct investment, and the regional economic growth in the case of China during 1981-2005. Chapter 4, based on data of 31 Chinese provinces over the period 1980-2004, presents new evidence on the effects of structural shocks and structural transformation on growth and convergence among the Chinese regions. Chapter 5, by building up an empirical model that takes account of different potential effects of foreign direct investment, focuses on the impacts of foreign direct investment on China’s regional economic performance and growth. Chapter 6 reconsiders the growth and convergence problem of the Chinese regions in an alternative theoretical framework with endogenous saving behavior and capital mobility across regions. Chapter 7, by building up a theoretical model concerning comparative advantage and transaction efficiency, focuses on one of the potential mechanisms through which China achieves its fast economic growth over the past few decades. Chapter 8 concludes the book by summarizing the results from the previous chapters and suggesting directions for further studies.

Pricing Index Options with Stochastic Volatility

The objective of this paper is to investigate the pricing accuracy under stochastic volatility where the volatility follows a square root process. The theoretical prices are compared with market price data (the German DAX index options market) by using two different techniques of parameter estimation, the method of moments and implicit estimation by inversion. Standard Black & Scholes pricing is used as a benchmark. The results indicate that the stochastic volatility model with parameters estimated by inversion using the available prices on the preceding day, is the most accurate pricing method of the three in this study and can be considered satisfactory. However, as the same model with parameters estimated using a rolling window (the method of moments) proved to be inferior to the benchmark, the importance of stable and correct estimation of the parameters is evident.

Producing interactivity : Does media convergence promote interactivity and audience participation?

This study investigates the process of producing interactivity in a converged media environment. The study asks whether more media convergence equals more interactivity. The research object is approached through semi-structured interviews of prominent decision makers within the Finnish media. The main focus of the study are the three big ones of the traditional media, radio, television and the printing press, and their ability to adapt to the changing environment. The study develops theoretical models for the analysis of interactive features and convergence. Case-studies are formed from the interview data and they are evaluated against the models. As a result the cases arc plotted and compared on a four-fold table. The cases are Radio Rock, NRJ, Biu Brother, Television Chat, Olivia and Sanoma News. It is found out that the theoretical models can accurately forecast the results of the case studies. The models are also able to distinguish different aspects of both interactivity and convergence so that a case, which at a first glance seems not to be very interactive is in the end found out to receive second highest scores on the analysis. The highest scores are received by Big Brother and Sanoma News. Through the theory and the analysis of the research data it is found out that the concepts of interactivity and convergence arc intimately intertwined and very hard in many cases to separate from each other. Hence the answer to the main question of this study is yes, convergence does promote interactivity and audience participation. The main theoretical background for the analysis of interactivity follows the work of Came Fleeter, Spiro Kiousis and Sally McMillan. Heeler's six-dimensional definition of interactivity is used as the basis for operationalizing interactivity. The actor-network theory is used as the main theoretical framework to analyze convergence. The definition and operationalization of the actor-network theory into a model of convergence follows the work of Michel Callon. Bruno Latour and especially John Law and Felix Stalder.