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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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Statistical inference for random processes
Kvitkovičová, Andrea ; Hlubinka, Daniel (advisor) ; Štěpán, Josef (referee)
The thesis deals with testing hypotheses about the parameters of the Wiener process with a constant drift rate and instantaneous variance. The tests are based on the first time, when the process reaches a pre-specified boundary point. We consider a process with a non-negative drift rate, and we observe hitting a positive point. We focus on tests about the drift rate, in particular about the absence of any drift. We first study several basic properties of the Wiener process and its connection with the Wiener process with a drift. Using these, we derive distributional properties of the first hitting time. We also describe selected hypotheses testing techniques in the setting of exponential families. We construct uniformly most powerful unbiased tests of one parameter in the presence of a nuisance parameter. Further, we construct uniformly most powerful tests of hypotheses about the drift rate, while the variance is known, and we study this situation in more detail. Finally, we construct asymptotic simultaneous tests of both parameters based on the R'enyi divergences.
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Deterministic and Stochastic Epidemic Models
Staněk, Jakub ; Štěpán, Josef (advisor) ; Hlubinka, Daniel (referee) ; Dohnal, Gejza (referee)
Kermack-McKendrick model and its version with vaccination are presented. First, we introduce a model with vaccination and then a numerical study that includes comparison of di erent vaccination strategies and searching for optimal vaccination strategy is presented. We proceed to introduce a stochastic model with migration and consequently we suggest its generalization and prove the existence and uniqueness of a solution to the stochastic di erential equation (henceforth SDE) describing this model. Three stochastic versions of Kermack-McKendrick model with vaccination are suggested and compared. A procedure of nding the optimal vaccination strategy is presented. We also prove the theorem on the existence and uniqueness of a solution to the SDE that drives a model with multiple pathogens. Finally, the stochastic di erential equation describing the general model is presented. We study properties of a solution to this SDE and present sufficient conditions for the existence of a solution that is absorbed by the natural barrier of the model.
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Generalized Stable Models in Finance
Chovanec, Róbert ; Klebanov, Lev (advisor) ; Štěpán, Josef (referee)
In this contribution, a basic theoretical approach to stable laws is described. There are mentioned some definitions of the stable distributions, properties and behavior of stable distributed random variables. Next, conditional modeling under the stable laws are analyzed. One can find homoskedastic (ARMA) and heteroskedastic (GARCH) structures. The GARCH models are explained partly for the Gaussian case too. An empirical application of this paper is based on comparison between the models, established in theoretical part, under the normal, and stable distribution respectively, built on real data from energetics. One issues from unconditional, then continues with conditional ARMA and finally, there are mixed ARMA-GARCH models. The results of interpreted statistical analysis demonstrate that the models based on the stable distribution matched the empirical distribution better than the the models based on the Gaussian distribution.
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Obchodní strategie v neúplném trhu
Bunčák, Tomáš ; Karlova, Andrea (advisor) ; Štěpán, Josef (referee)
MASTER THESIS ABSTRACT TITLE: Trading Strategy in Incomplete Market AUTHOR: Tomáš Bunčák DEPARTMENT: Department of Probability and Mathematical Statistics, Charles University in Prague SUPERVISOR: Andrea Karlová We focus on the problem of finding optimal trading strategies (in a meaning corresponding to hedging of a contingent claim) in the realm of incomplete markets mainly. Although various ways of hedging and pricing of contingent claims are outlined, main subject of our study is the so-called mean-variance hedging (MVH). Sundry techniques used to treat this problem can be categorized into two approaches, namely a projection approach (PA) and a stochastic control approach (SCA). We review the methodologies used within PA in diversely general market models. In our research concerning SCA, we examine the possibility of using the methods of optimal stochastic control in MVH, and we study the problem of our interest in several settings of market models; involving cases of pure diffusion models and a jump- diffusion case. In order to reach an exemplary comparison, we provide solutions of the MVH problem in the setting of the Heston model via techniques of both of the approaches. Some parts of the thesis are accompanied with numerical illustrations.
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Ergodic Theory
Lisko, Adrian ; Hlubinka, Daniel (advisor) ; Štěpán, Josef (referee)
This Bachelor Thesis compiles basics of ergodic theory. Motivation for writing this text was interesting topic and linkeage between it and the mathematics already learned. The Thesis begins with defining measure-preserving transformations and continues with recurrence and Poincarré's recurrence theorem to ergodicity and Birkhoff's ergodic theorem and mixing. In the end, it is shown that Birkhoff's ergodic theorem generalizes Kolmogorov's strong law of large numbers for stationary random sequences. Theory is demonstrated on a handful of examples of basic transformations.
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