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Risk Process with Random Income
Ringlerová, Anna ; Klebanov, Lev (advisor) ; Mazurová, Lucie (referee)
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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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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Stable distribution and application to finance
Omelchenko, Vadym ; Klebanov, Lev (advisor) ; Branda, Martin (referee)
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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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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Characterization of probability distributions.
Pacáková, Andrea ; Klebanov, Lev (advisor) ; Bubelíny, Peter (referee)
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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Extremal measures in probability
Kešelj, Sonja ; Dostál, Petr (advisor) ; Klebanov, Lev (referee)
Pólya urn scheme is a parametric probability model with interesting characteristics, which we shall look into within the scope of this thesis. Furthermore, using Bayesian approach we will show that, under certain conditions, the aforementioned model is equivalent to the Bernoulli scheme of independent alternative trials with random parameter that has beta distribution. Another subject of the thesis is ergodic theory of stationary sequances, as well as extremal analysis of probability measures that are invariant under some measurable transformation. This is illustrated on an example of homogegenous Markov chain with stationary distribution. The final segment of the thesis focuses on the theory of financial derivatives pricing - more specifically, finding arbitrage-free price using martingale measures. To this we add examples of application on binomial pricing trees. Keywords: extremal measure, Pólya urn scheme, ergodic and stationary sequences, financial derivatives pricing Powered by TCPDF (www.tcpdf.org)
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Statistical methods for determination of the weight of evidence in the identification process
Slovák, Dalibor ; Zvárová, Jana (advisor) ; Klebanov, Lev (referee)
In the present work we study an identification of culprit and assesment of evidence against him. At first we define a simple model called the island problem and we derive the weight-of-evidence formula in its basic form. In the next chapters we analyse several modifications of island problem and related issues. We find how we can deal with uncertainty about basic parametres of model, like size of population. We investigate possibility of inclusion of different errors or influence of relatedness and subpopulation structure into model. At the close we enlarge mixtures of DNA, including deriving and programming of appropriate formulas.
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