
MeanRisk Optimization Problem via Scalarization, Stochastic Dominance, Empirical Estimates
Kaňková, Vlasta
Many economic and financial situations depend simultaneously on a random element and on a decision parameter. Mostly it is possible to influence the above mentioned situation by an optimization model depending on a probability measure. We focus on a special case of onestage two objective stochastic “MeanRisk problem”. Of course to determine optimal solution simultaneously with respect to the both criteria is mostly impossible. Consequently, it is necessary to employ some approaches. A few of them are known (from the literature), however two of them are very important: first of them is based on a scalarizing technique and the second one is based on the stochastic dominance. First approach has been suggested (in special case) by Markowitz, the second approach is based on the second order stochastic dominance. The last approach corresponds (under some assumptions) to partial order in the set of the utility functions.\nThe aim of the contribution is to deal with the both main above mentioned approaches. First, we repeat their properties and further we try to suggest possibility to improve the both values simultaneously with respect to the both criteria. However, we focus mainly on the case when probability characteristics has to be estimated on the data base.


A Note on Optimal Value of Loans
Kaňková, Vlasta
People try to gain (in the last decades) own residence (a flat or a little house). Since young people do not posses necessary financial resources, bank sector offers them a mortgage. Of course, the aim of any bank is to profit from such a transaction. Therefore, according to their possibilities, the banks employ excellent experts to analyze the financial situation of potenitial clients. Consequently, the banks know what could be a maximal size of the loan (in dependence on the debtor's position, salary and age) and what is reasonable size of installments. The aim of this contribution is to analyze the situation from the second size. In particular, the aim is to investgate the possibilities of the debtors not only on the dependence on their present  day situation, but also on their future private and subjective decisions and on possible “unpleasant” events. Moreover, consequently according to these indexes, the aim of this contribution is to suggest a method for a recognition of a “safe” loan and simultaneously to offer tactics to state a suitable environment for future time.The stochastic programming theory will be employed to it.


MultiObjective Optimization Problems with Random Elements  Survey of Approaches
Kaňková, Vlasta
Many economic and financial situations depend simultaneously on a random element and a decision parameter. Mostly, it is possible to influence the above mentioned situation only by an optimization model depending on a probability measure. This optimization problem can be static (onestage), dynamic with finite or infinite horizon, singleobjective or multiobjective. We focus on onestage multiobjective problems corresponding to applications those are suitable to evaluate simultaneously by a few objectives. The aim of the contribution is to give a survey of different approaches (as they are known from the literature) of the above mentioned applications. To this end we start with wellknown meanrisk model and continue with other known approaches. Moreover, we try to complete every model by a suitable application. Except an analysis of a choice of the objective functions type we try to discuss suitable constraints set with respect to the problem base, possible investigation and relaxation. At the end we mention properties of the problem in the case when the theoretical „underlying“ probability measure is replaced by its „deterministic“ or „stochastic“ estimate.


Multicriteria and robust extension of newsboy problem
Šedina, Jaroslav ; Kopa, Miloš (advisor) ; Kaňková, Vlasta (referee)
This thesis studies a classic singleperiod stochastic optimization problem called the newsvendor problem. A newsboy must decide how many items to order un der the random demand. The simple model is extended in the following ways: endogenous demand in the additive and multiplicative manner, objective func tion composed of the expected value and Conditional Value at Risk (CVaR) of profit, multicriteria objective with pricedependent demand, multiproduct exten sion under dependent and independent demands, distributional robustness. In most cases, the optimal solution is provided. The thesis concludes with the nu merical study that compares results of two models after applying the Sample Average Approximation (SAA) method. This study is conducted on the real data. 1


Optimal Value of Loans via Stochastic Programming
Kaňková, Vlasta
A question of mortgage leads to serious and complicated problems of financial mathematics. On one side is a bank with an aim to have a “good” profit, on the other side is the client trying to invest money safely, with possible “small” risk.Let us suppose that a young married couple is in a position of client. Young people know that an expected and also unexpected unpleasant financial situation can happen. Many unpleasant financial situation can be caused by a random factor. Consequently stochastic methods are suitable to secure against them. The aim of the suggested model is not only to state a maximal reasonable value of loans, but also to endure unpleasant financial period. To this end we employ stochastic optimization theory. A few suitable models will be introduced. The choice of the model depends on environment of the young people. Models will be with “deterministic” constraints, probability constraints, but also with stochastic dominance constraints. The suggested models will be analyzed both from the numerical point of view and from possible method solution based on data. Except static oneobjective problem we suggest also multi–objective models.

 

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 nonlinearly dependent on the probability measure. Moreover, we introduce deterministic and nondeterministic 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


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 heavytailed 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 subfamily 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.


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)


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.
