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Optimization problems of vertex coloring under uncertainty
Kučera, Petr ; Branda, Martin (advisor) ; Lachout, Petr (referee)
The goal of this paper is to compare two formulations of optimi- zation problems of vertex coloring under uncertainty. These two formulations are integer linear programming with constraints and integer quadratic without con- straints. First chapter introduces integer linear programming. In second chapter we learn about these two formulations. Third chapter deals with implementation of these two formulations in optimization program called GAMS. We randomly generate 20 optimization problems of vertex coloring under uncertainty and com- pare integer linear formulation with constraints and integer quadratic formulation without constraints. 1
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Optimal solutions and CLM sets
Semela, Ondřej ; Lachout, Petr (advisor) ; Branda, Martin (referee)
This thesis falls within the theory of optimization problems. In the first part, terms such as epi- convergence, lower and upper semicontinuous function, epi-continuity and CLM set are defined. For a better understanding, the definitions of the key terms are accompanied with illustrative examples and observations of their basic properties. The following part deals with searching of (local) minimizers of random or deterministic function. Using the knowledge from the first part it is showed that under a set of assumptions it is possible to transfer this search to a sequence of random functions of specific requirements. Powered by TCPDF (www.tcpdf.org)
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Exchange rate volatility, and central bank interventions
Kubů, Jan ; Lachout, Petr (advisor) ; Branda, Martin (referee)
The exchange rates of currencies of different countries show higher volatility than it could be explained by the volatility of the fundamental variables. There are introduced different models which try to describe the behavior of these exchange rates in this Diploma Thesis. Their comparison is made with respect to the ability to capture the volatility of the empirically observed data. The behavior of exchange rates may also be influenced by interventions of the state institutions and therefore we introduced models which allow the effect of such regulatory interventions. These models were applied on real data. The properties of the model predictions of exchange rates were compared and evaluated with respect to their ability to explain the volatility of the empirical data. At the summary of my work one of the models has been used to simulate the behavior of the exchange rate during the application of different intervention strategies of the Central Bank. Powered by TCPDF (www.tcpdf.org)
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Random operators for modeling time series of counts
Lahodová, Kateřina ; Prášková, Zuzana (advisor) ; Lachout, Petr (referee)
In the thesis the thinning operators used for modeling of time series of counts are studied. The main properties of binomial, generalised, random coefficient, hyper- geometric and generalised binomial thinning operators are listed and proved. The comparison of these operators is also described. The use of binomial thinning for mo- deling INAR(1), binomial AR(1) and semi INAR(1) models is shown. The parameters of these models are estimated and also tested on a few simulations.
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Nonlinear ARMA model
Šabata, Marek ; Lachout, Petr (advisor) ; Prášková, Zuzana (referee)
The thesis regards theory of nonlinear ARMA models and its application on financial mar- kets data. First of all, we present general framework of time series modeling. Afterwards the theory of linear ARMA models is layed out, since it plays a key role in the theory of nonlinear models as well. The nonlinear models presented are threshold autoregressive model (TAR), autoregressive conditional heteroscedastic model (ARCH) and generalized autoregressive conditional heteroscedastic model (GARCH). For each model, we derive a method for esti- mating the model's parameters, asymptotic properties of the estimators and consequently confidence regions and intervals for testing hypotheses about the parameters. The theory is then applied on financial data, speficically on the data from Standard and Poor's 500 index (S&P500). All models are implemented in statistical software R. 1
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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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Influence of errors to regression model
Poliačková, Vlasta ; Lachout, Petr (advisor) ; Hlávka, Zdeněk (referee)
Title: Influence of errors to regression model Author: Bc. Vlasta Poliačková Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Petr Lachout, CSc. Supervisor's e-mail address: Petr.Lachout@mff.cuni.cz Abstract: The submitted work deals with the regression model, and the influence of errors to regression. Thesis describes different types of violations of assumptions re- quired to the error term and their impact to the properties of the regression model. In the next part, there are discussed various statistical approaches applicable in the case of violation assumptions of regression model such as heteroscedasticity or autocor- relation of the residuals. In the application part, there is used mainly knowledge of Box - Jenkins methodology. In this section it is described in detail how to build a Box - Jenkins models and forecasts of future values for various real financial time series. In processing of the data are used models of ARMA, ARIMA and SARIMA. In an example, forecasts of the models are compared to real future values of the time series. Keywords: regression, violation of assumptions, error term, Box-Jenkins methodo- logy, time series
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Decision Problems and Empirical Data; Applications to New Types of Problems
Odintsov, Kirill ; Kaňková, Vlasta (advisor) ; Lachout, Petr (referee)
This thesis concentrates on different approaches of solving decision making problems with an aspect of randomness. The basic methodologies of converting stochastic optimization problems to deterministic optimization problems are described. The proximity of solution of a problem and its empirical counterpart is shown. The empirical counterpart is used when we don't know the distribution of the random elements of the former problem. The distribution with heavy tails, stable distribution and their relationship is described. The stochastic dominance and the possibility of defining problems with stochastic dominance is introduced. The proximity of solution of problem with second order stochastic dominance and the solution of its empirical counterpart is proven. A portfolio management problem with second order stochastic dominance is solved by solving the equivalent empirical problem. Powered by TCPDF (www.tcpdf.org)
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Modeling of risk aversion
Navrátil, František ; Lachout, Petr (advisor) ; Kopa, Miloš (referee)
of the master thesis Title: Modeling of risk aversion Author: František Navrátil Department: Department of Probability and Mathematical Statistics Supervisor: Doc. RNDr. Petr Lachout, CSc. Abstract: The thesis discusses various theories that are able to model investor's subjective attitude to risk. The goal of the thesis is to clearly recapitulate possible mathematical approaches and to apply them in a real situation. One of the ways to tackle the problem is to use expected utility theory and a specific shape of a utility function. Another way is to choose a suitable risk measure. Especially useful for the modelling of risk aversion is the class of spectral risk measures that enables investor to choose a risk spectrum that meets his perception of risk. The thesis contains basic definitions concerning stochastic programming - a theory essential to solve the related optimization problems. Keywords: Risk aversion, utility function, probability constraint.
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Selected risk parameters in IRB approach and their modeling
Malec, Jaromír ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
The determination of lending (credit) risk is one of the most important fields of bank activities. This thesis discusses the IRB approach under Basel II. This approach includes the LGD, EAD and PD parameters. All parameters are individually modelled by the bank using regulator approved models. Parameter PD is the most focused one in this thesis. Theory for this parameter is of interest in many papers. However, at present the need for modelling of PD parameter over more years is appearing. Parameter LGD is also discussed in this thesis. The parameter EAD is only briefly presented. The thesis begins with the IRB approach, regression models and evaluation indicators, and then it focuses on the above parameters.
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