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Ambiguity in Stochastic Optimization Problems with Nonlinear Dependence on a Probability Measure via Wasserstein Metric
Kaňková, Vlasta
Many economic and financial applications lead to deterministic optimization problems depending on a probability measure. It happens very often (in applications) that these problems have to be solved on the data base. Point estimates of an optimal value and estimates of an optimal solutionset can be obtained by this approach. A consistency, a rate of convergence and normal properties, of these estimates, have been discussed (many times) not only under assumptions of independent data corresponding to the distributions with light tails, but also for weak dependent data and the distributions with heavy tails. However, it is also possible to estimate (on the data base) a confidence intervals and bounds for the optimal value and the optimal solutions. To analyze this approach we focus on a special case of static problems depending nonlineary on the probability measure. Stability results based on the Wasserstein metric and the Valander approach will be employed for the above mentioned analysis.
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Bootstrap methods for dependent observations
Petrásek, Jakub ; Prášková, Zuzana (advisor) ; Kaňková, Vlasta (referee)
This Diploma thesis deals with principles, asymptotic properties and comparison of bootstrap methods for dependent observations. In the first chapter principal ideas and benefits of bootstrap method for independent data are introduced. Subsequently, these knowledge are applied to data exhibiting dependency. Block, frequency and sieve bootstrap methods are presented. Afterwards, principle of each method is described in broader context, asymptotic properties are presented and some of them are derived. Strong dependency of block bootstrap method on block length is discussed and algorithms for empirical choice of optimal block length are described. The main aim of this work is to compare discussed methods from theoretical point of view and via simulation study. Eventually, a few examples, which are based on real data sets, are presented. Discussed principles are implemented in software R and software Fortran.
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Multicriterial Optimization Problems with a Random Element and Stochastic Programming
Líkař, Jan ; Kaňková, Vlasta (advisor) ; Dupačová, Jitka (referee)
In practice we often have to solve optimization problems with several criteria. These problems are called multicriteria optimization problems. Such problems are presented in this thesis. It is important, whether parameters take unknown values at the moment of making decision. If these parameters are random variables, resulting problem is called stochastic multiobjective problem, otherwise it is called deterministic multiobjective problem. We describe how to choose some "good" solutions of deterministic problem. We investigate their relations as well. In the stochastic case we have to convert such problem to deterministic one. We introduce some possibilities how to do it. Then we are able to solve the problem. These concepts are demonstrated using examples. We present a numerical illustration as well (the Portfolio Selection problem).
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Multicriteria games
Tichá, Michaela ; Lachout, Petr (advisor) ; Kaňková, Vlasta (referee)
The concern of this thesis is to discuss different multicriteria games solution concepts. Multicriteria game is a special case from the game theory if the payoff function of at least one player is a vector and the player wants to maximize all the criteria at the same time. The thesis is divided into four chapters. In the first instance a few motivation examples are introduced. Subsequently the history of the multicriteria games is mentioned. The theoretical chapter follows. It contains five sections - introduction of new definitions; the structure of the set of equilibria for two person multicriteria games; searching equilibria points by the help of scalarization of the vector-valued function; introduction of ideal equilibria points and ways how to find them; the comparison of used methods. The last solution concept is demonstrated by the real example. Finally a theoretical chapter with new results is included. 1
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Problems of Stochastic Optimisation under Uncertainty, Quantitative Methods, Simulations, Applications in Gas Storage Valuation
Omelčenko, Vadim ; Kaňková, Vlasta (advisor) ; Ortobelli, Sergio (referee) ; Popela, Pavel (referee)
This dissertation deals with heavy-tailed distributions and the problematics of stochastic dominance for stable distributions. In terms of stochastic dominance in the setup of stable distributions, we prove novel results which are mostly based on the domain of attraction of stable distributions. We introduce a bivariate sub-family of stable distributions, which can easily be simulated and used for the joint modelling of dependent data (such as spot and forward prices). The marginals of these bivariate distributions are stable and can have a different tail index. We also present our approach for parameter estimation of stable distributions. The theoretical results achieved are used for the valuation of gas storage units. In this part of the dissertation, we use stochastic dynamic programming to address this problem, and we present several algorithms.
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New Trends in Stochastic Programming
Szabados, Viktor ; Kaňková, Vlasta (advisor) ; Lachout, Petr (referee)
Stochastic methods are present in our daily lives, especially when we need to make a decision based on uncertain events. In this thesis, we present basic approaches used in stochastic tasks. In the first chapter, we define the stochastic problem and introduce basic methods and tasks which are present in the literature. In the second chapter, we present various problems which are non-linearly dependent on the probability measure. Moreover, we introduce deterministic and non-deterministic multicriteria tasks. In the third chapter, we give an insight on the concept of stochastic dominance and we describe the methods that are used in tasks with multidimensional stochastic dominance. In the fourth chapter, we capitalize on the knowledge from chapters two and three and we try to solve the role of portfolio optimization on real data using different approaches. 1
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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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