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Application for Management of Arbitration Procedures
Maslowski, Bohdan ; Poch, Tomáš (referee) ; Ježek, Pavel (advisor)
The aim of this work is to create an application that would simplify the implementation of arbitration procedures. The result is an application written in C# with the use of .Net Framework 3.5 and WPF technology. The application facilitates management of arbitration procedures data, generation of MS Word documents, obtaining law-suit participants data from public web services and arbitration fee calculation. There was a strong emphasis on the simplicity of form modifications and the possibility of adding new law-suit types. Praetor Forms library has been developed, facilitating complex handling of forms defined in xml format.
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Ornstein-Uhlenbeck bridge
Janák, Josef ; Dostál, Petr (referee) ; Maslowski, Bohdan (advisor)
In the Thesis we study the Ornstein-Uhlenbeck Bridges. First, we recall the notion of the fractional Brownian motion and introduce stochastic integral of a deterministic function with respect to (fBm). We summarize the results on existence and uniqueness of a solutions to the autonomic linear stochastic di erential equations that are called the Ornstein-Uhlenbeck processes. We introduce the concept of the Gaussian Bridge and we derive its representation, which we use for obtaining the formula for Ornstein-Uhlenbeck Bridge. The results are applied to some special examples. In the last part of the Thesis we mention a nonanticipative representation of the bridge.
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Geometric Brownian motion in Hilbert space
Bártek, Jan ; Beneš, Viktor (referee) ; Maslowski, Bohdan (advisor)
The present work describes the relation between solutions of a special kind of nonlinear stochastic partial differential equation with multiplicative noise, driven by fractional Brownian motion (fBm), and the solutions of deterministic version of this equation. Solution of the stochastic equation is given explicitly by means of solution to the deterministic equation and trajectories of fBm. The geometric fractional Brownian motion plays an important role here. The solutions are considered both in strong and weak sense. Stochastic integral wrt. fBm with Hurst index H can be defined in various ways. Here we consider a Stratonovich type integral for H > 1/2. The results obtained are used for the study of properties of solution of stochastic porous media equation - the expected value of total mass of the solution and the long-time behaviour of the solution.
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Modelování úrokových sazeb s využitím Lévyho procesů
Slámová, Lenka ; Maslowski, Bohdan (referee) ; Beneš, Viktor (advisor)
In this work we study the HJM model of the term structure of interest rates driven by a Lévy process. We study the no-arbitrage dynamics of the discounted bond prices and obtain a risk-neutral dynamics of the short rate as a consequence. We study in particular the short rate process and formulate a criteria for mean reversion. The theory gives us a machinery producing short rate processes associated with a general Lévy driver and general volatility structure and we show the non-emptiness of the theory by demonstrating the previous on an example of an OU type process associated with the generalized inverse Gaussian distribution. The upshot is an explicitely given short rate process that generalizes the Vašíček model, and moreover stays positive. Finally we study numerical methods for thus constructed short rate process such as simulations and lattice approximations.
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Slabá řešení stochastických diferenciálních rovnic
Hofmanová, Martina ; Maslowski, Bohdan (referee) ; Seidler, Jan (advisor)
In the present work we study a stochastic di fferential equation with coefficients continuous in x having in this variable linear growth. As a main result we show that there exists a weak solution to this equation by a new, more elementary method. Standard methods are based either on the concept of the weak solution or equivalently on solving a martingale problem. However, both approaches employ the integral representation theorem for martingales, whose proof becomes rather complicated in dimension greater than one. By a simple modi cation of the usual procedure, one can identify the weak solution elementary, with no need to apply the above mentioned theorem. In the preliminaries we summarize some auxiliary results: namely, some properties of the space of continuous functions as the space of trajectories are established and an important theorem which allows us to approximate continuous function by functions Lipschitz continuous is proved.
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Stochastické rovnice a stochastické metody v parciálních diferenciálních rovnicích
Maslowski, Bohdan
In the first part of the paper, basic notions of stochastic analysis and probability theory are recalled, including the concept of stochastic integration, and some basic results of the theory of partial differential equations are reviewed. The second part of the paper is a review of applications of stochastic analysis techniques in deterministic PDE theory. The last part contains a brief introduction to stochastic PDEs and other stochastic infinite dimensional systems.
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