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Love-Young Inequality and Its Consequences
Sýkora, Adam ; Čoupek, Petr (advisor) ; Hlubinka, Daniel (referee)
This thesis is focused on proving the Love-Young inequality and clarifying the manner in which it relates to a fractional Brownian motion. To begin with, several estimates alongside the concept of p-variation of a func- tion are presented. The connection between functions of finite p-variation and regulated functions is then highlighted and used to prove the aforementioned Love-Young inequality. Deficiency of the pathwise approach to stochastic in- tegration is recognised and later discussed amongst the properties of fractional Brownian motions. This constitutes the main application of the featured theory which is the integration with respect to irregular functions. 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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Central and Non-Central Limit Theorems
Kiška, Boris ; Čoupek, Petr (advisor) ; Beneš, Viktor (referee)
V této práci zkoumáme centrální limitní věty (CLT) a jejich různé varianty. Zpočátku je uvedena CLT pro nezávislé a stejně rozdělené náhodné veličiny. Dále studujeme případ nezávislých a nestejně rozdělených náhodných veličin, kde porovnáme různé verze a různé podmínky, za kterých CLT platí. Tyto klasické výsledky jsou prezentovány spolu s několika protipříklady, které porušují předpoklady CLT různými způsoby. V této práci je také uvažován případ závislých náhodných veličin. Zejména CLT pro a-mixující náhodné posloupnosti je dána společně s Rosenblattovým protipříkladem, který zahrnuje limitní ne- Gaussovské rozdělení, které se nyní nazývá Rosenblattovo rozdělení.
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Riemann zeta function
Čoupek, Petr ; Rokyta, Mirko (advisor) ; Zahradník, Miloš (referee)
Riemann zeta function represents an important tool in analytical number theory with various applications in quantum mechanics, probability theory and statistics. First introduced by Bernhard Riemann in 1859, zeta function is a central object of many outstanding problems. From previous results follows the importance of zeta function for further development in the field of number theory. This thesis provides basic properties of the Riemann zeta function. In particular, we prove theorems concerning the distribution of its roots outside and inside the critical strip which leads to the formulation of the Riemann hypothesis and theorems concerning the irrationality of selected values of the Riemann zeta function including the proof of the irrationality of ζ(3). 1
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Stochastic Evolution Equations
Čoupek, Petr ; Maslowski, Bohdan (advisor) ; Garrido-Atienza, María J. (referee) ; Hlubinka, Daniel (referee)
Stochastic Evolution Equations Petr Čoupek Doctoral Thesis Abstract Linear stochastic evolution equations with additive regular Volterra noise are studied in the thesis. Regular Volterra processes need not be Gaussian, Markov or semimartingales, but they admit a certain covariance structure instead. Particular examples cover the fractional Brownian motion of H > 1/2 and, in the non-Gaussian case, the Rosenblatt process. The solution is considered in the mild form, which is given by the variation of constants formula, and takes values either in a separable Hilbert space or the space Lp(D, µ) for large p. In the Hilbert-space setting, existence, space-time regularity and large-time behaviour of the solutions are studied. In the Lp setting, existence and regularity is studied, and in concrete cases of stochastic partial differential equations, the solution is shown to be a space-time continuous random field.
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Variation of Fractional Processes
Kiška, Boris ; Čoupek, Petr (advisor) ; Maslowski, Bohdan (referee)
In this thesis, we study various notions of variation of certain stochastic processes, namely $p$-variation, pathwise $p$-th variation along sequence of partitions and $p$-th variation along sequence of partitions. We study these concepts for fractional Brownian motions and Rosenblatt processes. A fractional Brownian motion is a Gaussian process and it has been intensively developed and studied over the last two decades because of its importance in modeling various phenomena. On the other hand, a Rosenblatt process, which is a non- Gaussian process that can be used for modeling non-Gaussian fluctuations, has not been getting as much attention as fractional Brownian motion. For that reason, we concentrate in this thesis on this process and we present some original results that deal with ergodicity, $p$-variation, pathwise $p$-th variation along sequence of partitions and $p$-th variation along sequence of partitions. Boris Kiška
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Kolmogorov-Chentsov Theorem
Lebeda, Matěj ; Čoupek, Petr (advisor) ; Kříž, Pavel (referee)
Is there a sufficient condition for continuity of sample paths of a random process? Or, is it at least possible to modify the process so that the paths would already be continuous? An affirmative answer is given by the Kolmogorov- Chentsov theorem, whose statement and proof are the subject of this thesis. First, we introduce the notion of a random process and briefly focus on the so-called Gaussian processes. The main focus of the second chapter is the Kolmogorov- Chentsov theorem, its proof and some auxiliary assertions are given. In the final third chapter, we deal with the applications of the theorem to some well-known Gaussian processes such as the Wiener process or the Brownian bridge. Finally, we look into the Poisson process, which on the contrary does not satisfy the condition of the theorem. 1
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Hájek-Renyi inequality
Bělohlávek, Ivan ; Prášková, Zuzana (advisor) ; Čoupek, Petr (referee)
Title: A Hájek-Renyi inequality Author: Ivan Bělohlávek Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Zuzana Prášková, CSc., Department of Probability and Mathematical Statistics Abstract: In this thesis we study the Hájek-Rényi inequality for mixingales and their special cases. First, we prove the Hájek-Rényi inequality for martingales. Then, we investigate the relationship between the Kolomogorov and Hájek-Rényi inequalities. After that, we prove the law of large numbers using the Hájek-Rényi inequality. We then provide a detailed proof of maximal inequality for mixingales, which we then use to derive the Hájek-Rényi inequality for mixingales. We then apply the inequality to a multitude of special cases of mixingales. Keywords: Hájek-Rényi inequality, martingales, mixingales, linear process 1
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Trajectories of Fractional Brownian Motions
Roubínová, Veronika ; Čoupek, Petr (advisor) ; Maslowski, Bohdan (referee)
This work concerns the fractional Brownian motion, in particular, the properties of its trajectories. Firstly some basic notions are defined and then the definiton of the fractional Brownian motion itself is given. Subsequently, its basic properties such as correlation of increments and self-similarity are derived. Continuity of its trajectories is shown using the Kolomogorov-Chentsov Theorem. The main chapter contains a thorough proof of the law of the iterated logarithm. It is complemented with simulations of limit behavior of trajectories and used to prove nondifferentiability. 1
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Semilinear stochastic evolution equations
Kršek, Daniel ; Maslowski, Bohdan (advisor) ; Čoupek, Petr (referee)
Stochastic partial differential equations have proven useful in many applied areas of mathematics, such as physics or mathematical finance. A major part of such equations consists of linear equations with additive noise. In certain cases, however, the drift part of the differential equation additionally contains a possibly problematic non-linear term, which makes it unsolvable by the standard methods and even a solution in the mild sense may be out of reach. In such situations, we may still find a solution in the weak sense by employing a suitable transformation of the probability space. This thesis deals with semilinear stochastic evolution equations in a separable Hilbert space, where the driving process is an element of a large class of processes - so called Volterra processes, which can be understood as a generalisation of the Wiener process and may be of use to model a wide range of phenomena. The weak solutions, however, have been studied so far only for equations with the cylindrical fractional Brownian motion as the driving process. In this thesis, we introduce a generalisation of the Girsanov theorem for cylindrical Gaussian Volterra processes and give, in full generality, sufficient conditions for the existence of a weak solution and the uniqueness of the equation in law. Further, we introduce...
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