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Random Processes in Reliability Analysis
Chovanec, Kamil ; Volf, Petr (advisor) ; Prokešová, Michaela (referee)
Title: Random Processes in Reliability Analysis Author: Kamil Chovanec Department: Department of Probability and Mathematical Statistics Supervisor: Doc. Petr Volf, CSc. Supervisor's e-mail address: volf@utia.cas.cz Abstract: The thesis is aimed at the reliability analysis with special em- phasis at the Aalen additive model. The result of hypothesis testing in the reliability analysis is often a process that converges to a Gaussian martingale under the null hypothesis. We can estimate the variance of the martingale using a uniformly consistent estimator. The result of this estimation is a new hypothesis about the process resulting from the original hypothesis. There are several ways to test for this hypothesis. The thesis presents some of these tests and compares their power for various models and sample sizes using Monte Carlo simulations. In a special case we derive a point that maximizes the asymptotic power of two of the tests. Keywords: Martingale, Aalen's additive model, hazard function 1
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Random walk
Baňasová, Barbora ; Omelka, Marek (advisor) ; Dostál, Petr (referee)
Random walk is a well-known mathematical model used in various scientific fields. The aim of this thesis is to explain and to show the relation between the basic characteristics of simple random walk. The paper summarizes theoretical knowledge concerning this mathematical model in terms of its symmetrical or asymmetrical version. It deals with the derivation of absorbing probabilities, probability of the first and repeated return to origin and clasification of simple random walk states. The final part presents random walk in a wider perspective as a martingale. The conditions under which a random walk equals a martingale are established as well. It is also shown how it is possible to apply this more general mathematical structure on the model of random walk.
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Optimal trading and pricing of financial derivatives
Samek, Daniel ; Dostál, Petr (advisor) ; Hlubinka, Daniel (referee)
In the text of this thesis we deal with the task of valuing financial derivatives. The theory is based on the Douglas theorem and its financial interpretation upon which we state replication theorem. These theorems connect martingale measures and existence of no-arbitrage price of derivative in both discrete and continuous time. Next part discusses trading strategies maximizing expected utility and their impact on existence of martingale measure. In the last chapter there are stated fundamental theorems of asset pricing which sum up main previous results. Powered by TCPDF (www.tcpdf.org)
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Random walk
Baňasová, Barbora ; Omelka, Marek (advisor) ; Dostál, Petr (referee)
Random walk is a well-known mathematical model used in various scientific fields. The aim of this thesis is to explain and to show the relation between the basic characteristics of simple random walk. The paper summarizes theoretical knowledge concerning this mathematical model in terms of its symmetrical or asymmetrical version. It deals with the derivation of absorbing probabilities, probability of the first and repeated return to origin and clasification of simple random walk states. The final part presents random walk in a wider perspective as a martingale. The conditions under which a random walk equals a martingale are established as well. It is also shown how it is possible to apply this more general mathematical structure on the model of random walk.
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Martingale measures and pricing of financial derivatives
Melicherčík, Martin ; Dostál, Petr (advisor) ; Haman, Jiří (referee)
Title: Martingale measures and pricing of financial derivatives Author: Martin Melicherčík Department: Department of Probability and Mathematical Statistics Supervisor: Mgr. Petr Dostál, Ph.D., Department of Probability and Mathema- tical Statistics Abstract: The theory written in this work explains basic tools for setting justified price of financial derivatives. Jusified pricing is based on principal of balance, which means, that in advance no side has bigger chance to profit than other. Because of this characteristic, the main pricing tool in the work are martingale measures, which respect the state of balance. From the point of view of martingale measures random processes keep their constant expected value, so we can never expect them to deflect to one side or another. The important part of the work, besides basics of martingales, is Douglas theorem, which answers the question of our ability to theoretically set the justified price of any financial derivative. In the last parts, there are also some manuals and examples how to determine the justified price. Keywords: martingale, martingale pricing, Douglas theorem, predictable process 1
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Random Processes in Reliability Analysis
Chovanec, Kamil ; Volf, Petr (advisor) ; Prokešová, Michaela (referee)
Title: Random Processes in Reliability Analysis Author: Kamil Chovanec Department: Department of Probability and Mathematical Statistics Supervisor: Doc. Petr Volf, CSc. Supervisor's e-mail address: volf@utia.cas.cz Abstract: The thesis is aimed at the reliability analysis with special em- phasis at the Aalen additive model. The result of hypothesis testing in the reliability analysis is often a process that converges to a Gaussian martingale under the null hypothesis. We can estimate the variance of the martingale using a uniformly consistent estimator. The result of this estimation is a new hypothesis about the process resulting from the original hypothesis. There are several ways to test for this hypothesis. The thesis presents some of these tests and compares their power for various models and sample sizes using Monte Carlo simulations. In a special case we derive a point that maximizes the asymptotic power of two of the tests. Keywords: Martingale, Aalen's additive model, hazard function 1
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