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Options under Stable Laws
Karlová, Andrea ; Volf, Petr (advisor) ; Klebanov, Lev (referee) ; Witzany, Jiří (referee)
Title: Options under Stable Laws. Author: Andrea Karlová Department: Department of Probability and Mathematical Statistics Supervisor: Doc. Petr Volf, CSc. Abstract: Stable laws play a central role in the convergence problems of sums of independent random variables. In general, densities of stable laws are represented by special functions, and expressions via elementary functions are known only for a very few special cases. The convenient tool for investigating the properties of stable laws is provided by integral transformations. In particular, the Fourier transform and Mellin transform are greatly useful methods. We first discuss the Fourier transform and we give overview on the known results. Next we consider the Mellin transform and its applicability on the problem of the product of two independent random variables. We establish the density of the product of two independent stable random variables, discuss the properties of this product den- sity and give its representation in terms of power series and Fox's H-functions. The fourth chapter of this thesis is focused on the application of stable laws into option pricing. In particular, we generalize the model introduced by Louise Bachelier into stable laws. We establish the option pricing formulas under this model, which we refer to as the Lévy Flight...
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Normality and its testing
Hájek, Štěpán ; Bašta, Milan (advisor) ; Klebanov, Lev (referee)
This thesis is concerned with normality and its testing. We often encounter with this topic when using statistical tests and models. Among others, examples such as t tests, analysis of variance and linear regression might be given. In this thesis these tests and models are overviewed and the consequences of the violation of the normality assumption are briefly mentioned. The following section describes statistical tests of normality. For example Shapiro-Wilk test or Anderson-Darling test are explored. For each test of normality is given test statistic and conditions for rejection of the null hypothesis. The last section provides a simulation study. The first part of this study is devoted to exploring whether the empirical relative frequency of Type I error corresponds to the nominal significance level of the test. The second part of the simulation study explores the power of normality tests against various alternatives. The results are summarized and discussed. 1
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Introduction to Order Statistics Theory
Hanuš, Antonín ; Kulich, Michal (advisor) ; Klebanov, Lev (referee)
This thesis deals with the theory of order statistics. Its aim is to summarize the basic knowledge concerning the distribution of the order statistics of random variables that are absolutely continuous with respect to the Lebesgue Measure and afterwards use those order statistics for some specific distributions. The first chapter describes the derivation of the density and distribution function of order statistics in several ways as well as dealing with some functions of order statistics and their conditional distribution. The second chapter is devoted to the moments of order statistics and formulae for their calculation and to the relations between them. In the conclusion the previous theoretical findings are applied to the uniform, exponential and normal distributions. 1
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Deterministické a stochastické modely v molekulární a buněčné biologii
Krasnovský, Pavol ; Vejchodský, Tomáš (advisor) ; Klebanov, Lev (referee)
This thesis presents the main methods that are used to model the time evolution of the number of molecules in a cell. Two of the main aims in cell biology are to compute first the transi- tion probability function and second the density of the invariant measure. These two problems imply a number of conditions and hence we also include the ergodic theory and theory of the invariant measure. We use two illustrative examples of the application of the previously mentioned theories. We verify the necessary and sufficient conditions for the computation of the transition probability function and the density of the invariant measure in case of two types of a chemical system. The probability function and the density are then given by a numerical solution to the Fokker-Planck equation in both the dynamic and the stationary case. Furthermore, we compare the obtained solu- tions to the results from the Monte Carlo simulation. We find that the solutions give almost identical results as the Monte Carlo simulation. At the end of this thesis, we formulate and analyze a chemical system represented by a human cell infected by an influenza virus. Given the complexity of the sys- tem, we compute the results using the Monte Carlo method. In addition we define this problem by a stochastic differential equation with random...
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Large deviations and their applications in insurance mathematics
Fuchsová, Lucia ; Pawlas, Zbyněk (advisor) ; Klebanov, Lev (referee)
Title: Large deviations and their applications in insurance mathematics Author: Lucia Fuchsová Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Zbyněk Pawlas, Ph.D. Supervisor's e-mail address: Zbynek.Pawlas@mff.cuni.cz Abstract: In the present work we study large deviations theory. We discuss heavy-tailed distributions, which describe the probability of large claim oc- curence. We are interested in the use of large deviations theory in insurance. We simulate claim sizes and their arrival times for Cramér-Lundberg model and first we analyze the probability that ruin happens in dependence on the parameters of our model for Pareto distributed claim size, next we compare ruin probability for other claim size distributions. For real life data we model the probability of large claim size occurence by generalized Pareto distribu- tion. 1
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Characterization of probability distributions.
Pacáková, Andrea ; Bubelíny, Peter (referee) ; Klebanov, Lev (advisor)
Naxev pracc: Charakterizace pravdepodohnoslnich rozdekni Autor: Andrea Pacdkovd Katedra:K(//f dm pravdepodobnosti a matematicke statist iky Vedouci bakalafskc praee: Prof. Lev Kk'banov, DrSc. e-mail vcdouciho prace: kk'banov(alkarlin.mff.cuni.cz Abstrakt: Tato pracc se zabyvd charakterizaci normalniho a slabilniho rozdeleni. Vime. ze rozdeleni souciti nezavislych normalnich nahodnych velicin je normalni, a prave studium jisteho opaku lohoto tvrzeni je hlavnim client teto prace. Podstatna cast nasledncho te\ln je venovana sludiu vlastnosti intenzivne monotomuch operatorit a silne E-pozitivnkh rotiin funkci, ponioci nichz jsoit dokazany zajiniavc skutccnosti, jako je nask'dujici: Mitzeme-li predpoklddat shodn rozdclcm jeihw ndhodnc vcliciny a lineami jormy z nczavislych nahodnych velicin, pak za priddni dalskh predpokladit dokazetnc jiz pfesne urcil jcjich rozdekni. Posledni kapitola je vOnovana Bernsk'inove vt'tc ajejinnt diikazit zalozcnem pravc na vetdch o charaktcrizaci normahuho rozdekni. Klicova slova: intenzivne monotonni operator, siltie E-pozitivni rodina, lineurniforma, normalni a stahilni rozdekni Title: Characterization of probability distributions Author: Andrea Pacakova Department: Department of Probability and Mathematical Statistics Supervisor: Prof. Lev Kk'banov, DrSc. Supervisor's e-mail...
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Risk Process with Random Income
Ringlerová, Anna ; Mazurová, Lucie (referee) ; Klebanov, Lev (advisor)
This diploma thesis deals with risk processes. It describes a classical risk process and mentions the ruin probability. A convolution formula and the Beekman convolution formula for calculating the ruin probability are deduced for the classical risk process. The following part of the thesis provides the investigation of the Cram¶er-Lundberg, the Beekman-Bowers and the De Vylder approximation to the ruin probability. The accuracy of approximations is illustrated in two examples. Afterwards, a risk process with random income is studied and a convolution formula for such a process is derived. In an example, the classical risk process is taken as a specic type of the risk process with random income. For such a process, the ruin probability computed by the convolution formula for classical risk process is compared to the ruin probability computed by the convolution formula for the risk process with random income. It is shown that sometimes the ruin probability is undervalued when computed by the convolution formula for classical risk process.
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Stable distribution and application to finance
Omelchenko, Vadym ; Branda, Martin (referee) ; Klebanov, Lev (advisor)
Title: Stable distributions and application to finance Author: Vadym Omelchenko Department: Department of Probability and Mathematical Statistics Supervisor: Prof. Lev Klebanov, DrSc. Supervisor's e-mail address: Lev.Klebanov@mff.cuni.cz Abstract: This work deals with the theory of the stable distributions, their parameter estimation, and their financial application. There arc given the methods of characteristic function and method of projections, which is rel- ative to ML-methodology, for estimation of the parameters of stable dis- tributions. We compare these methods with the conventional estimators. The quality of estimators is verified by the simulation of the sample having stable distribution with known parameters and comparing the estimates of these parameters with their real values. The aim of this work is estima- tion of parameters of the stable laws which iy applicable for modification of AHCH/GAHCH models with stable innovations. Keywords: stable distribution, ARGII/GARCII models, characteristic func- tion (CF) based estimators, maximum likelihood projection (MLP) estima- tors.
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