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Empiciral Estimates in Stochastic Programming; Dependent Data
Kolafa, Ondřej ; Kaňková, Vlasta (advisor) ; Dupačová, Jitka (referee)
This thesis concentrates on stochastic programming problems based on empirical and theoretical distributions and their relationship. Firstly, it focuses on the case where the empirical distribution is an independent random sample. The basic properties are shown followed by the convergence between the problem based on the empirical distribution and the same problem applied to the theoretical distribution. The thesis continues with an overview of some types of dependence - m-dependence, mixing, and also more general weak dependence. For sequences with some of these types of dependence, properties are shown to be similar to those holding for independent sequences. In the last section, the theory is demonstrated using numerical examples, and dependent and independent sequences, including sequences with different types of dependence, are compared.
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Stochastic activity networks
Sůva, Pavel ; Dupačová, Jitka (advisor) ; Kaňková, Vlasta (referee)
In the present work, stochastic network models representing a project as a set of activities are studied, as well as different approaches to these models. The critical path method, stochastic network models with probability constraints, finding a reference project duration, worst-case analysis in stochastic networks and optimization of the parameters of the probability distributions of the activity durations are studied. Use of stochastic network models in telecommunications networks is also briefly presented. In a numerical study, some of these models are implemented and the~related numerical results are analyzed.
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Probabilistic programs with discrete probability distributions
Murgaš, Karel ; Dupačová, Jitka (advisor) ; Branda, Martin (referee)
This thesis deals with stochastic programming problems with probabilistic constraits with discrete distribution. Finitness and corectness of algortithm for finding p-level efficient points is proved and I implement this algorithm in R. I relax the feasible set to get convex problem and I study properties of the relaxed set. Results for linear, integer and nonlinear problems are presented. In en example I compare discrete approach with the continuous one.
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Stochastic dominance portfolio efficiency measures
Jakubcová, Monika ; Kopa, Miloš (advisor) ; Dupačová, Jitka (referee)
In the present work we study the stochastic dominance portfolio e ciency measures. The investor's risk attitude is given by the type of an utility function. If this information is unknown or a general investor is assumed, it is possible to use the stochastic dominance principle, in which the portfolio is only classi ed as e cient or ine cient. We build on the works of Post, Kuosmanen and Kopa, who formulated the criteria of portfolio e ciency for nonsatiate and risk averse investors. On the basis of these criteria, we de ne the second-order stochastic dominance (SSD) portfolio e ciency measures. We examine the properties of SSD ine ciency measures, which allow to compare SSD ine cient portfolios. We prove mutual relationships for the de ned SSD ine ciency measures. Eventually, we test the SSD e ciency of a US market portfolio on real-world US Stock Exchange data.
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Scenario generation for multidimensional distributions
Olos, Marek ; Dupačová, Jitka (advisor) ; Kaňková, Vlasta (referee)
Some methods for generating scenarios from multidimensional distribution assume we are able to generate scenarios from the one-dimensional distribution. We dedicate chapter 3 to this problem. At the end of the chapter, we provide references for applicable algorithms. Chapter 4 is focused on selected methods for generating scenarios from multidimensional distributions. In chapter 4.3, we introduce an algorithm for generating scenarios, which do not use any assumption about the distribution, except the first four moments and correlations to be specified. A method of generating scenarios based on approximation of multivariate normal distribution by the binomial distribution is described in chapter 4.5. Dimension reduction technique using principal components is presented in chapter 4.4. The algorithm is presented under the assumption of normal distribution. In chapter 4.6, we introduce the basics of the copula theory and a method for generating scenarios by C-vine copula. In chapter 5, we implement selected methods for generating scenarios for the estimation of daily value at risk for selected indexes and we discuss the results. Powered by TCPDF (www.tcpdf.org)
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Optimization and stress tests
Fašungová, Diana ; Dupačová, Jitka (advisor) ; Kozmík, Václav (referee)
Title: Optimization and stress tests Author: Diana Fašungová Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Jitka Dupačová, DrSc., Department of Probability and Mathematical Statistics Abstract: In the thesis we apply contamination technique on a portfolio optimiza- tion problem using minimization of risk measure CVaR. The problem is considered from a risk manager point of view. We stress correlation structure of data and of revenues using appropriately chosen data for this kind of problem and for ge- nerated stress scenarios. From behaviour of CVaR with regard to contamination bounds, we formulate recommendations for the risk manager optimizing his port- folio. The recommendations are interpreted for both types of stress scenarios. In the end, limitations of the model and possible ways of improvement are discussed. Keywords: contamination bounds, stress tests, portfolio optimization, risk mana- gement
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Portfolio efficiency with continuous probability distribution of returns
Kozmík, Václav ; Kopa, Miloš (advisor) ; Dupačová, Jitka (referee)
Present work deals with the portfolio selection problem using mean-risk models. The main goal of this work is to investigate the convergence of approxi mate solutions using generated scenarios to the analytic solution and its sensitivity to chosen risk measure and probability distribution. The considered risk measures are: variance, VaR, cVaR, absolute deviation and semivariance. We present analytical solutions for all risk measures under the assumption of normal or Student distribution. For log-normal distribution, we use the approximate assumption that the sum of log-normal random variables has log-normal distribution. Optimization models for discrete scenarios are derived for all risk measures and compared with analytical solution. In case of approximate solution with scenarios, we repeat the procedure multiple times and present our own approach to nding the optimal solution using the cluster analysis. All optimization models are written in GAMS language. Testing and estimating are realized using an application developed in C++ language.
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Scenario generation for multidimensional distributions
Olos, Marek ; Dupačová, Jitka (advisor) ; Kaňková, Vlasta (referee)
Some methods for generating scenarios from multidimensional distribution assume we are able to generate scenarios from the one-dimensional distribution. We dedicate chapter 3 to this problem. At the end of the chapter, we provide references for applicable algorithms. Chapter 4 is focused on selected methods for generating scenarios from multidimensional distributions. In chapter 4.3, we introduce an algorithm for generating scenarios, which do not use any assumption about the distribution, except the first four moments and correlations to be specified. A method of generating scenarios based on approximation of multivariate normal distribution by the binomial distribution is described in chapter 4.5. Dimension reduction technique using principal components is presented in chapter 4.4. The algorithm is presented under the assumption of normal distribution. In chapter 4.6, we introduce the basics of the copula theory and a method for generating scenarios by C-vine copula. In chapter 5, we implement selected methods for generating scenarios for the estimation of daily value at risk for selected indexes and we discuss the results. Powered by TCPDF (www.tcpdf.org)
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