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Slabá řešení stochastických diferenciálních rovnic
Hofmanová, Martina ; Seidler, Jan (advisor) ; Maslowski, Bohdan (referee)
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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Geometric Brownian motion in Hilbert space
Bártek, Jan ; Maslowski, Bohdan (advisor) ; Beneš, Viktor (referee)
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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Invariant measures for dissipative stochastic differential equations
Lavička, Karel ; Seidler, Jan (advisor) ; Maslowski, Bohdan (referee)
The main topic of this Thesis is a new simplified proof of the Sunyach theorem that provides suffici- ent conditions for existence and uniqueness of an invariant measure for a Markov kernel on a complete separable metric space equipped with its Borel σ-algebra. Weak convergence of measures following from Sunyach's theorem is strengthened to convergence in the total variation norm provided that the Markov kernel is strong Feller. Furthermore, sufficient conditions for geometric ergodicity are stated. Another topic treated is the strong Feller property: its characterization by absolute measurability and uniform integrability and derivation of some other sufficient conditions.
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Zero one laws in probabability and topology
Šimon, Prokop ; Štěpán, Josef (advisor) ; Maslowski, Bohdan (referee)
Práce se zabývá teorií funkcí typu PLIF, jejichž zavedení bylo motivo- váno matematickou statistikou. Je ukázána cesta vedoucí od statistického problému až k jeho zjednodušení pomocí PLIF, resp. SPLIF. Navazující pří- klady dávají odpově¤ na existenci těchto funkcí na vybraných prostorech, přirozeně je kladen d·raz na prostor všech nekonečných posloupností 0 a 1 {0, 1}N a jeho podprostory. Za použití silného zákona velkých čísel pro ná- hodnou procházku je uveden zajímavý příklad ukazující množinu 1. kategorie mající míru jedna. Dále je dokázán Oxtobyho 0-1 zákon. Celou práci uzavírá rozpracovaný d·kaz věty od D. Blackwella ukazující neexistenci borelovských SPLIF, ve kterém hraje klíčovou roli právě Oxtobyho 0-1 zákon. 1
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Stochastické modelování reakčně-difuzních procesů v biologii
Lipková, Jana ; Maslowski, Bohdan (advisor) ; Vejchodský, Tomáš (referee)
Many biological processes can be described in terms of chemical reactions and diffusion. In this thesis, reaction-diusion mechanisms related to the formation of Turing patterns are studied. Necessary and sufficient conditions under which Turing instability occur is presented. Behaviour of Turing patterns is investigated with a use of deterministic approach, compartment-based stochastic simulation algorithm and molecular-based stochastic simulation algorithm.
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Ito formula and its applications
Till, Alexander ; Haman, Jiří (advisor) ; Maslowski, Bohdan (referee)
Title: Itô formula and its applications Author: Alexander Till Department: Department of Probability and Mathematical Statistics Supervisor: Mgr. Jiří Haman Supervisor's e-mail address: j.haman@seznam.cz Abstract: The bachelor thesis contains basis and elementary findings of stochastic analysis. It includes definition and properties of stochastic integral with Wiener process as an integrator, definition of stochastic integral with Itô process as an integrator, Itô formula for functions of time and Wiener process, Itô formula for functions of time and Itô process. These conclusions are used to solve certain examples. Keywords: Wiener process, Stochastic integral, Itô formula 1
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Kalman-Bucy Filter in Continuous Time
Týbl, Ondřej ; Maslowski, Bohdan (advisor) ; Čoupek, Petr (referee)
In the Thesis we study the problem of linear filtration of Gaussian signals in finite-dimensional space. We use the Kalman-type equations for the filter to show that the filter depends continuously on the signal. Secondly, we show the same continuity property for the covariance of the error and verify existence and uniqueness of a solution to an integral equation that is satisfied by the filter even under more general assumptions. We present several examples of application of the continuity property that are based on the theory of stochastic differential equations driven by fractional Brownian motion. 1
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Markovské semigrupy
Žák, František ; Maslowski, Bohdan (advisor) ; Štěpán, Josef (referee)
In the presented work we study the existence of periodic solution to infinite dimensional stochastic equation with periodic coefficients driven by Cylindrical Wiener process. Used theory of infinite dimensional stochastic equations in Hilbert spaces and Markov processes is summarized in the first two chapters. In the third and last chapter we present the result itself. Necessary technical background mostly from operator theory is encapsulated in the Appendix. The proof of existence of periodic solution of corresponding equation is a combination of arguments by Khasminskii, which ensure under suitable conditions the existence of periodic Markov process, and the results of Da Prato, G¸atatrek and Zabczyk for the existence of invariant measure for homogeneous stochastic equation in Hilbert spaces. At the end we derive sufficient condition for the existence of periodic solution in the language of coefficients using the work of Ichikawa and illustrate the results by the example of Stochastic PDE. The work is written in English.
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