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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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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, Value-at-Risk and Conditio- nal Value-at-Risk, to the multivariate case. First, the theory of p-efficient points, possible generalization of a quantile, is presented. The Prékopa-Vizvári-Badics algorithm for finding p-efficient points in case of random vectors with finite support is presented and a generalization of the algorithm in special case is proposed. Multivariate Value-at-Risk and Multivariate Conditional Value-at-Risk are defined and some of the properties are discussed. A lot-sizing problem for different time horizons is solved. 1
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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
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Multi-Stage Stochastic Programming with CVaR: Modeling, Algorithms and Robustness
Kozmík, Václav ; Dupačová, Jitka (advisor) ; Morton, David (referee) ; Kaňková, Vlasta (referee)
Multi-Stage Stochastic Programming with CVaR: Modeling, Algorithms and Robustness RNDr. Václav Kozmík Abstract: We formulate a multi-stage 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 risk-averse setting are demonstrated. We present a new approach to evaluating policies in multi-stage risk-averse 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 mean-risk sampling with CVaR is provided. In order to evaluate robustness of the presented models we extend con- tamination technique to the case of large-scale programs, where a precise solution cannot be obtained. Our computational results are based on a simple multi-stage asset allocation model and confirm usefulness of the presented procedures, as well as give additional insights into the behavior of more complex models. Keywords: Multi-stage stochastic programming, stochastic dual dynamic programming, im- portance sampling, contamination, CVaR
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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 Value-at-Risk (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)
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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)
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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 so-called 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 well-known Flower-girl 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)
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Stochastic Programming Problems via Economic Problems
Kučera, Tomáš ; Kaňková, Vlasta (advisor) ; Dupačová, Jitka (referee)
This thesis' topic is stochastic programming, in particular with regard to portfolio optimization and heavy tailed data. The first part of the thesis mentions the most common types of problems associated with stochastic programming. The second part focuses on solving the stochastic programming problems via the SAA method, especially on the condition of data with heavy tailed distributions. In the final part, the theory is applied to the portfolio optimization problem and the thesis concludes with a numerical study programmed in R based on data collected from Google Finance.
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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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