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Selected problems of random walks
Pavčová, Eva ; Hlubinka, Daniel (advisor) ; Pawlas, Zbyněk (referee)
Title: Selected problems of random walks Author: Eva Pavčová Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Daniel Hlubinka, Ph.D., Department of Probability and Mathematical Statistics Abstract: This thesis deals with simple random walks and solutions of theoretical selected problems. We define the path which can be interpreted as the realization of a random walk. We bring forward examples of paths with illustrations and basic properties such as ballot theorem and reflection principle. Random walk is defined and also the probability of its is brought forward. Our attention is concentrated on the main lemma. We derive from it other interesting assertions such as arcsin law. The aim of this thesis is to solve the selected problems using theoretical knowledge. The problems are concerned with probabilities and numbers of paths with certain restrictions. The specific problem of positive paths proves geometrically the equality of numbers of two types of paths. Specially, we are interested in the proof of reformulation of main lemma. Keywords: path, reflection principle, main lemma, arcsin law
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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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The Analysis of the Relative Efficiency of the Czech and Polish Financial Market.
Džmuráňová, Hana ; Rippel, Milan (advisor) ; Todica, Doina (referee)
of bachelor thesis Author: Hana Džmuráňová The topic of this bachelor thesis is the Theory of efficient markets. The thesis is split into two related parts. The first part aims to introduce the Theory of efficient markets and behavioral finance. It focuses on several anomalies and limitations in the Theory of efficient markets that have been found as a result of behavioral finance research. The second part of the thesis is an empirical text dedicated to the relative weak form efficiency analysis of the two Central and Eastern Europe Markets - the Prague and the Warsaw Stock Exchange. Relative efficiency is tested by the random walk properties of market index returns and by the OLS method for autoregressive process for market index returns. It has been found that the Warsaw Stock Exchange is relatively more efficient in the weak form efficiency than the Prague Stock Exchange.
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Application of Monte Carlo simulations in banking
Boruta, Matěj ; Teplý, Petr (advisor) ; Fučík, Vojtěch (referee)
Currently, banking is exposed to huge market risks. One of those risks is occurrence of negative interest rates in the EU. Nowadays, it is important to use sophisticated and modern measurement tools and approaches to measure and manage banking risks. One of those methods is Monte Carlo simulation. This bachelor thesis is aimed at analysis and prediction of 3-month maturity Prague Interest Offer Rate (PRIBOR) for 3, 6 and 12 months with using Monte Carlo simulations. It was found that this method is suitable for prediction market variables with low volatility. If anybody uses this method, it is necessity to have in mind all pitfalls and assumptions, that this method includes, as an adequate random generated number of scenarios, approximation of correct probability distribution, independence of dataset and not least, as far as possible, to focus on factors generating randomness of market variable and not the prices, that express rather consequences of randomness than its cause. Further, the Monte Carlo prediction was compared with prognosis of the Czech Nation Bank and it was found that Monte Carlo prediction is more accurate for short term predictions. 12-month prediction of Monte Carlo simulation discovered also possible occurrence of negative interest rate at 0,05% level of probability in compare to the Czech National Bank prognosis, where was no negative interest rate predicted.
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Determining the size of the state space
Rybanský, Marian ; Krček, Petr (referee) ; Roupec, Jan (advisor)
Objective of the thesis is to describe and evaluate the appropriate methods to determine the size of the state space. State space will be created by using a program (generator) which generates a state space, from the input values and regulations (model). To objectively assess the state space we classifies according to their parameters. Classified state space will be added to a class, in which the demarcation perform evaluation methods will be done.
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Determining the size of the state space
Rybanský, Marian ; Krček, Petr (referee) ; Roupec, Jan (advisor)
Objective of the thesis is to describe and evaluate the appropriate methods to determine the size of the state space. State space will be created by using a program (generator) which generates a state space, from the input values and regulations (model). To objectively assess the state space we classifies according to their parameters. Classified state space will be added to a class, in which the demarcation perform evaluation methods will be done.
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Random walk and game theory
Vondrušková, Ivana ; Malá, Ivana (advisor) ; Novák, Ilja (referee)
A random walk is a mathematical formalization of a trajectory that consists of taking succesive random steps. The results od random walk analysis have been applied to computer science, economics, biology tec. This thesis is concentrated on connection between random walk and games theory especially with thinglering simulation. The application part of this thesis is focused on game strategies, that determine the amount of bet in next game.
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