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Sample approximation technique in stochastic programming
Vörös, Eszter ; Branda, Martin (advisor) ; Kozmík, Václav (referee)
Title: Sample approximation technique in stochastic programming Author: Eszter V¨or¨os Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Martin Branda, Ph.D., Department of Probability and Mathematical Statistics Abstract: This thesis deals with the problem of stochastic programming. Sto- chastic problems are usually applied for optimalization problems involving uncer- tain parameters. The problem, which we are aimed to solve, is approximated with the so-called sample average approximation method. The sample used to estimate the true problem is generated by the Monte Carlo method. This technique allows us to use standard algorithms for the further treatment of the problem. The aim of this thesis is to discuss the convergence properites of the optimal value and the optimal solution of the approximed problem to the optimal value and the optimal solution of the real problem. The thesis ends with a practical demonstration of the theoretical results on a portfolio optimization problem. Keywords: stochastic programming, sample average approximation, Monte Carlo method, portfolio optimization 1
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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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Importance Sampling methods in solving optimization problems
Zavřel, Lukáš ; Kozmík, Václav (advisor) ; Kopa, Miloš (referee)
Present work deals with the portfolio selection problem using mean-risk models where analysed risk measures include variance, VaR and CVaR. The main goal is to approximate solution of optimization problems using simulation techniques like Monte Carlo and Importance Sampling. For both simulation techniques we present a numerical study of their variance and efficiency with respect to optimal solution. For normal distribution with particular expected value and variance the values of parameters for sampling using Importance Sampling method are empirically deduced and they are consequently used for solving a practical problem of choice of optimal portfolio from ten stocks, when their weekly historical prices are available. All optimization problems are solved in Wolfram Mathematica program. Powered by TCPDF (www.tcpdf.org)
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Portfolio efficiency with continuous probability distribution of returns
Kozmík, Václav
Present work deals with the portfolio selection problem using mean-risk models. The main goal of this work is to investigate the convergence of approximate 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 finding 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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Solution of Emission Management Problem
Šmíd, Martin ; Kozmík, Václav
Optimal covering of emissions stemming from random production is a multistage stochastic programming problem. Solving it in a usual way - by means of deterministic equivalent - is possible only given an unrealistic approximation of random parameters. There exists an efficient way of solving multistage problems - stochastic dual dynamic programming (SDDP), however, it requires the inter-stage independence of random parameters, which is not the case which our problem. In the paper, we discuss a modified version of SDDP, allowing for some form of interstage dependence.
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Two Algorithms for Risk-averse Reformulation of Multi-stage Stochastic Programming Problems
Šmíd, Martin ; Kozmík, Václav
Many real-life applications lead to risk-averse multi-stage stochastic problems, therefore effective solution of these problems is of great importance. Many tools can be used to their solution (GAMS, Coin-OR, APML or, for smaller problems, Excel), it is, however, mostly up to researcher to reformulate the problem into its deterministic equivalent. Moreover, such solutions are usually one-time, not easy to modify for different applications. We overcome these problems by providing a front-end software package, written in C++, which enables to enter problem definitions in a way close to their mathematical definition. Creating of a deterministic equivalent (and its solution) is up to the computer. In particular, our code is able to solve linear multi-stage with Multi-period Mean-CVaR or Nested Mean-CVaR criteria. In the present paper, we describe the algorithms, transforming these problems into their deterministic equivalents.
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Importance Sampling methods in solving optimization problems
Zavřel, Lukáš ; Kozmík, Václav (advisor) ; Kopa, Miloš (referee)
Present work deals with the portfolio selection problem using mean-risk models where analysed risk measures include variance, VaR and CVaR. The main goal is to approximate solution of optimization problems using simulation techniques like Monte Carlo and Importance Sampling. For both simulation techniques we present a numerical study of their variance and efficiency with respect to optimal solution. For normal distribution with particular expected value and variance the values of parameters for sampling using Importance Sampling method are empirically deduced and they are consequently used for solving a practical problem of choice of optimal portfolio from ten stocks, when their weekly historical prices are available. All optimization problems are solved in Wolfram Mathematica program. Powered by TCPDF (www.tcpdf.org)
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Characterization of convex sets
Lžičař, Jiří ; Lachout, Petr (advisor) ; Kozmík, Václav (referee)
The idea of convexity is very important especially for probability theory, optimization and stochastic optimization. Convexity is a unique set pro- perty in many ways, which is worth to be studied. Various properties of convex sets are generally known, such as the ones related to separability. It however becomes apparent that the definition of convexity is very interesting, since it is possible to replace the definition by various collections of properties which are equivalent to it. There also exist set operations preserving convexity and another ones which preserve it when supported by another requirements. 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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