
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 onedimensional 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 Cvine 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)


Stochastic DEA and dominance
Majerová, Michaela ; Kopa, Miloš (advisor) ; Dupačová, Jitka (referee)
At the beginning of this thesis we discuss DEA methods, which measure efficiency of Decision Making Units by comparing weighted inputs and outputs. First we describe basic DEA models without random inputs and outputs then stochastic DEA models which are derived from the deterministic ones. We describe more approaches to stochastic DEA models, for example using scenario approach or chance constrained programming problems. Another approach for measuring efficiency employs stochastic dominance. Stochastic dominance is a relation that allows to compare two random variables. We describe the first and second order stochastic dominance. First we consider pairwise stochastic efficiency, then we discuss the first and second order stochastic dominance portfolio efficiency. We describe different tests to measure this type of efficiency. At the end of this thesis we study efficiency of US stock portfolios using real historical data and we compare results obtained when using stochastic DEA models and stochastic dominance. Powered by TCPDF (www.tcpdf.org)


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  mdependence, 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.


Multivariate risk measures in stochastic optimization
Rauš, Jaroslav ; Branda, Martin (advisor) ; Dupačová, Jitka (referee)
The thesis deals with possible generalization of widely used risk measures, ValueatRisk and Conditio nal ValueatRisk, to the multivariate case. First, the theory of pefficient points, possible generalization of a quantile, is presented. The PrékopaVizváriBadics algorithm for finding pefficient points in case of random vectors with finite support is presented and a generalization of the algorithm in special case is proposed. Multivariate ValueatRisk and Multivariate Conditional ValueatRisk are defined and some of the properties are discussed. A lotsizing problem for different time horizons is solved. 1


Robustness of the Markowitz portfolios
Petráš, Tomáš ; Dupačová, Jitka (advisor) ; Kopa, Miloš (referee)
This diploma thesis deals with the problem of portfolio optimization in relation to the mean vector and the variance matrix of yields. The emphasis is put on Mar kowitz model. In the thesis there are explored some possibilities of robustification based on the used parametric set. Beside the classic formulation of the task our focus is also devoted to the cases in which short sales are not allowed. The core of the thesis constitutes of a simulation study that models the impact of errors in the estimation of the input parameters of Markowitz model. It takes into account different types of risk aversions and different approaches to modelling parameter perturbations . Therefore it specifies the hypothesis of the dominating influence of the mean vector estimate which is valid only for a risk lover. 1


MultiStage Stochastic Programming with CVaR: Modeling, Algorithms and Robustness
Kozmík, Václav ; Dupačová, Jitka (advisor) ; Morton, David (referee) ; Kaňková, Vlasta (referee)
MultiStage Stochastic Programming with CVaR: Modeling, Algorithms and Robustness RNDr. Václav Kozmík Abstract: We formulate a multistage stochastic linear program with three different risk measures based on CVaR and discuss their properties, such as time consistency. The stochastic dual dynamic programming algorithm is described and its draw backs in the riskaverse setting are demonstrated. We present a new approach to evaluating policies in multistage riskaverse programs, which aims to elimi nate the biggest drawback  lack of a reasonable upper bound estimator. Our approach is based on an importance sampling scheme, which is thoroughly ana lyzed. A general variance reduction scheme for meanrisk sampling with CVaR is provided. In order to evaluate robustness of the presented models we extend con tamination technique to the case of largescale programs, where a precise solution cannot be obtained. Our computational results are based on a simple multistage asset allocation model and confirm usefulness of the presented procedures, as well as give additional insights into the behavior of more complex models. Keywords: Multistage stochastic programming, stochastic dual dynamic programming, im portance sampling, contamination, CVaR

 

Newsboy problem
Šedina, Jaroslav ; Dupačová, Jitka (advisor) ; Lachout, Petr (referee)
This thesis deals with the newsboy problem and its various modifications. The first part of the thesis mentions definitions and theorems that are essential for investigation of the optimal solution of the problem. In the second part, various formulations of newsboy problem are discussed and their solutions are presented. For instance, we use Sample Average Approximation method. In the final part, the results are applied to calculate Conditional ValueatRisk (CVaR) and the thesis concludes with a numerical study programmed in R which compares parametric and nonparametric approach to the problem. The text is consecutively supplemented with graphs. Powered by TCPDF (www.tcpdf.org)


Scenario generation by the moment fitting method
Koláčková, Hana ; Dupačová, Jitka (advisor) ; Branda, Martin (referee)
The thesis presents four methods for scenario generating leading to the resulting discrete probability distribution that replicates given values of the moments. The first method uses heuristic algorithm, the second method generates by symmetrically distributing values around the mean value, the third one is based on solving the system of nonlinear equations and finally the last method is based on goal programming. Next section describes the nature of problems solved by the goal programming. It also details possible ways of parameter specification to allow control of the computational complexity. In the last part of the thesis the results of several suitable methods for chosen types of problem are compared. Powered by TCPDF (www.tcpdf.org)


Robust optimization for solution of uncertain optimization programs
Komora, Antonín ; Dupačová, Jitka (advisor) ; Kopa, Miloš (referee)
Robust optimization is a valuable alternative to stochastic programming, where all underlying probabilistic structures are replaced by the socalled uncertainty sets and all related conditions must be satisfied under all circumstances. This thesis reviews the fundamental aspects of robust optimization and discusses the most common types of problems as well as different choices of uncertainty sets. It focuses mainly on polyhedral and elliptical uncertainty and for the latter, in the case of linear, quadratic, semidefinite or discrete programming, computationally tractable equivalents are formulated. The final part of this thesis then deals with the wellknown Flowergirl problem. First, using the principles of robust methodology, a basis for the construction of the robust counterpart is provided, then multiple versions of computationally tractable equivalents are formulated, tested and compared. Powered by TCPDF (www.tcpdf.org)
