Diverse probabilistic results are used in the design of random univariate generators. General methods based on these are classified and relevant theoretical properties derived. This is followed by a comparative review of specific algorithms currently available for continuous and discrete univariate distributions. A need for a Zeta generator is established, and two new methods, based on inversion and rejection with a truncated Pareto envelope respectively are developed and compared. The paucity of algorithms for multivariate generation motivates a classification of general methods, and in particular, a new method involving envelope rejection with a novel target distribution is proposed. A new method for generating first passage times in a Wiener Process is constructed. This is based on the ratio of two random numbers, and its performance is compared to an existing method for generating inverse Gaussian variates. New "hybrid" algorithms for Poisson and Negative Binomial distributions are constructed, using an Alias implementation, together with a Geometric tail procedure. These are shown to be robust, exact and fast for a wide range of parameter values. Significant modifications are made to Atkinson's Poisson generator (PA), and the resulting algorithm shown to be complementary to the hybrid method. A new method for Von Mises generation via a comparison of random numbers follows, and its performance compared to that of Best and Fisher's Wrapped Cauchy rejection method. Finally new methods are proposed for sampling from distribution tails, using optimally designed Exponential envelopes. Timings are given for Gamma and Normal tails, and in the latter case the performance is shown to be significantly better than Marsaglia's tail generation procedure.

In this paper we will investigate the connection between a random walk and a continuous time stochastic process. Donsker's Theorem states that a random walk under certain conditions will converge to a Wiener process. We will provide a detailed proof of this theorem which will be used to prove that a geometric random walk converges to a geometric Brownian motion.

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Wiener process-a random process with continuous time-plays an important role in mathematics, physics or economy. It is often good to know whether it contains any deterministic part, e.g. drift or scale. However, it is nearly impossible either observe the whole trajectory of the process or preserve its full history. This thesis deals with a statistical analysis based on partial observations, namely passage times through some given barriers. We propose several statistical methods for testing hypotheses about drift or scale using these observations. As supporting methods, we consider the maximum likelihood theory, non-parametric test against a trend, and binomial test. For testing the value of scale in the model with no drift and constant scale we recommend maximum likelihood theory. We derive the estimate and related tests in the case of observing only three barriers. The simulation study suggested observing more barriers for testing monotony of scale in a model with linear drift, or testing monotone and convex/concave drift in a model with constant scale. 1

The purpose of the thesis is to provide a useful concept in the framework of stochastic analysis applicable in finance. The thesis offers proof for Doob- Meyer theorem for boundend martingales which is then extended for local martingales. It also proves the strong Markov theorem for Wiener process and some of its significant consequences. The built framework is then used for creating a method for solution of different tasks in applied finance. 1

Dynamic Linear Models are a state space model framework based on the Kalman filter. We use this framework to do seasonal adjustments of empirical and artificial data. A simple model and an extended model based on Gibbs sampling are used and the results are compared with the results of a standard seasonal adjustment method. The state space approach is then extended to discuss direct and indirect seasonal adjustments. This is achieved by applying a seasonal level model with no trend and some specific input variances that render different signal-to-noise ratios. This is illustrated for a system consisting of two artificial time series. Relative efficiencies between direct, indirect and multivariate, i.e. optimal, variances are then analyzed. In practice, standard seasonal adjustment packages do not support optimal/multivariate seasonal adjustments, so a univariate approach to simultaneous estimation is presented by specifying a Holt-Winters exponential smoothing method. This is applied to two sets of time series systems by defining a total loss function that is specified with a trade-off weight between the individual series’ loss functions and their aggregate loss function. The loss function is based on either the more conventional squared errors loss or on a robust Huber loss. The exponential decay parameters are then estimated by minimizing the total loss function for different trade-off weights. It is then concluded what approach, direct or indirect seasonal adjustment, is to be preferred for the two time series systems. The dynamic linear modeling approach is also applied to Swedish political opinion polls to assert the true underlying political opinion when there are several polls, with potential design effects and bias, observed at non-equidistant time points. A Wiener process model is used to model the change in the proportion of voters supporting either a specific party or a party block. Similar to stock market models, all available (political) information is assumed to be capitalized in the poll results and is incorporated in the model by assimilating opinion poll results with the model through Bayesian updating of the posterior distribution. Based on the results, we are able to assess the true underlying voter proportion and additionally predict the elections. / <p>At the time of doctoral defence the following papers were unpublished and had a status as follows: Paper 3: Manuscript; Paper 4: Manuscripts</p>

In this dissertation, we first prove that for a random differential equation with the multiplicative driving noise constructed from a Q-Wiener process and the Wiener shift, which is an approximation to a stochastic evolution equation, there exists a unique solution that generates a local dynamical system. There also exist a local center, unstable, stable, centerunstable, center-stable manifold, and a local stable foliation, an unstable foliation on the center-unstable manifold, and a stable foliation on the center-stable manifold, the smoothness of which depend on the vector fields of the equation. In the second half of the dissertation, we show that any two arbitrary local center manifolds constructed as above are conjugate. We also show the same conjugacy result holds for a stochastic evolution equation with the multiplicative Stratonovich noise term as u â—¦ dW

V předložené práci je definována stochastická diferenciální rovnice a jsou uvedeny její základní vlastnosti. Stochastické diferenciální rovnice se používají k popisu fyzikálních jevů, které jsou ovlivněny i náhodnými vlivy. Řešením stochastického modelu je náhodný proces. Cílem analýzy náhodných procesů je konstrukce vhodného modelu, který umožní porozumět mechanismům, na jejichž základech jsou generována sledovaná data. Znalost modelu také umožňuje předvídání budoucnosti a je tak možné kontrolovat a optimalizovat činnost daného systému. V práci je nejdříve definován pravděpodobnostní prostor a Wienerův proces. Na tomto základě je definována stochastická diferenciální rovnice a jsou uvedeny její základní vlastnosti. Závěrečná část práce obsahuje příklad ilustrující použití stochastických diferenciálních rovnic v praxi.

>Magister Scientiae - MSc / The case of pricing options under constant volatility has been common practise for decades. Yet market data proves that the volatility is a stochastic phenomenon, this is evident in longer duration instruments in which the volatility of underlying asset is dynamic and unpredictable. The methods of valuing options under stochastic volatility that have been extensively published focus mainly on stock markets and on options written on a single reference asset. This work probes the effect of valuing European call option written on a basket of currencies, under constant volatility and under stochastic volatility models. We apply a family of the stochastic models to investigate the relative performance of option prices. For the valuation of option under constant volatility, we derive a closed form analytic solution which relaxes some of the assumptions in the Black-Scholes model. The problem of two-dimensional random diffusion of exchange rates and volatilities is treated with present value scheme, mean reversion and non-mean reversion stochastic volatility models. A multi-factor Gaussian distribution function is applied on lognormal asset dynamics sampled from a normal distribution which we generate by the Box-Muller method and make inter dependent by Cholesky factor matrix decomposition. Furthermore, a Monte Carlo simulation method is adopted to approximate a general form of numeric solution The historic data considered dates from 31 December 1997 to 30 June 2008. The basket contains ZAR as base currency, USD, GBP, EUR and JPY are foreign currencies.

Scientific Workflows bieten flexible Möglichkeiten für die Modellierung und den Austausch komplexer Arbeitsabläufe zur Analyse wissenschaftlicher Daten. In den letzten Jahrzehnten sind verschiedene Systeme entstanden, die den Entwurf, die Ausführung und die Verwaltung solcher Scientific Workflows unterstützen und erleichtern. In mehreren wissenschaftlichen Disziplinen wachsen die Mengen zu verarbeitender Daten inzwischen jedoch schneller als die Rechenleistung und der Speicherplatz verfügbarer Rechner. Parallelisierung und verteilte Ausführung werden häufig angewendet, um mit wachsenden Datenmengen Schritt zu halten. Allerdings sind die durch verteilte Infrastrukturen bereitgestellten Ressourcen häufig heterogen, instabil und unzuverlässig. Um die Skalierbarkeit solcher Infrastrukturen nutzen zu können, müssen daher mehrere Anforderungen erfüllt sein: Scientific Workflows müssen parallelisiert werden. Simulations-Frameworks zur Evaluation von Planungsalgorithmen müssen die Instabilität verteilter Infrastrukturen berücksichtigen. Adaptive Planungsalgorithmen müssen eingesetzt werden, um die Nutzung instabiler Ressourcen zu optimieren. Hadoop oder ähnliche Systeme zur skalierbaren Verwaltung verteilter Ressourcen müssen verwendet werden. Diese Dissertation präsentiert neue Lösungen für diese Anforderungen. Zunächst stellen wir DynamicCloudSim vor, ein Simulations-Framework für Cloud-Infrastrukturen, welches verschiedene Aspekte der Variabilität adäquat modelliert. Im Anschluss beschreiben wir ERA, einen adaptiven Planungsalgorithmus, der die Ausführungszeit eines Scientific Workflows optimiert, indem er Heterogenität ausnutzt, kritische Teile des Workflows repliziert und sich an Veränderungen in der Infrastruktur anpasst. Schließlich präsentieren wir Hi-WAY, eine Ausführungsumgebung die ERA integriert und die hochgradig skalierbare Ausführungen in verschiedenen Sprachen beschriebener Scientific Workflows auf Hadoop ermöglicht. / Scientific workflows provide a means to model, execute, and exchange the increasingly complex analysis pipelines necessary for today's data-driven science. Over the last decades, scientific workflow management systems have emerged to facilitate the design, execution, and monitoring of such workflows. At the same time, the amounts of data generated in various areas of science outpaced hardware advancements. Parallelization and distributed execution are generally proposed to deal with increasing amounts of data. However, the resources provided by distributed infrastructures are subject to heterogeneity, dynamic performance changes at runtime, and occasional failures. To leverage the scalability provided by these infrastructures despite the observed aspects of performance variability, workflow management systems have to progress: Parallelization potentials in scientific workflows have to be detected and exploited. Simulation frameworks, which are commonly employed for the evaluation of scheduling mechanisms, have to consider the instability encountered on the infrastructures they emulate. Adaptive scheduling mechanisms have to be employed to optimize resource utilization in the face of instability. State-of-the-art systems for scalable distributed resource management and storage, such as Apache Hadoop, have to be supported. This dissertation presents novel solutions for these aspirations. First, we introduce DynamicCloudSim, a cloud computing simulation framework that is able to adequately model the various aspects of variability encountered in computational clouds. Secondly, we outline ERA, an adaptive scheduling policy that optimizes workflow makespan by exploiting heterogeneity, replicating bottlenecks in workflow execution, and adapting to changes in the underlying infrastructure. Finally, we present Hi-WAY, an execution engine that integrates ERA and enables the highly scalable execution of scientific workflows written in a number of languages on Hadoop.