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Statistical Problems in Markov Chains
Adamová, Markéta ; Prášková, Zuzana (advisor) ; Branda, Martin (referee)
In this thesis we study basic statistical methods in Markov chains. In the case of discrete time, this thesis is focused on estimation of transition probability matrix and some basic tests (test for a specified transition probability matrix, test for homogeneity, test for independence, test for the order of Markov chain). In the case of continuous time we will concentrate on Poisson process and birth and death process. Estimation of parameters of these processes and tests for processes with specified parameters are mentioned. Developed estimates and test statistics are applied to real data in the final chapter.
Solving mixed-integer linear programming problems in GAMS
Škoda, Štěpán ; Branda, Martin (advisor) ; Kopa, Miloš (referee)
In the present work we study the problems of integer linear optimi- zation, at first from the theoretical point of view (Part I) and subsequently on the basis of empirical data (Part II). First section explains with what this field deals with and where is applied. Other sections contain annotated mathematical formulation of problems, definitions and theorems needed to understand the general methods for solving integer linear programs. In the last section of Part I, we introduce the two best known groups of algorithms that are used by commercial software. The second part provides more details on an Internet library that contains some practical problems, that has been needed to be solved in the past. Furthermore, there are sections dealing with solvers and the advanced options of GAMS. The last section presents data obtained in the course of solving problems using several codes (solvers) of software. 1
Probabilistic programs with discrete probability distributions
Murgaš, Karel ; Dupačová, Jitka (advisor) ; Branda, Martin (referee)
This thesis deals with stochastic programming problems with probabilistic constraits with discrete distribution. Finitness and corectness of algortithm for finding p-level efficient points is proved and I implement this algorithm in R. I relax the feasible set to get convex problem and I study properties of the relaxed set. Results for linear, integer and nonlinear problems are presented. In en example I compare discrete approach with the continuous one.
Nonconvex stochastic programming problems-formulations, sample approximations and stability
Branda, Martin ; Lachout, Petr (advisor) ; Kaňková, Vlasta (referee) ; H.van der Vlerk, Maarten (referee)
Title: Nonconvex stochastic programming problems - formulations, sample approximations and stability Author: RNDr. Martin Branda Author's e-mail address: branda@karlin.mff.cuni.cz Supervisor: Doc. RNDr. Petr Lachout, CSc. Supervisor's e-mail address: lachout@karlin.mff.cuni.cz Abstract: We deal with problems where integer variables may appear, hence no assumptions on convexity are made throughout this thesis. The goal of Chapter 2 is to introduce stochastic programming problems and to outline the most important tasks connected with solving the problems. In Chapter 3, we compare basic formulations of static stochastic programming problems with chance constraints, with integrated chance constraints and with penalties in the objective function. We show that the problems are asymptotically equivalent under mild conditions. We discuss solving the problems using sample approximation techniques and extend some results on rates of convergence. All the formulations and corresponding sample approximations are compared on an investment problem with real features with Value at Risk constraint, integer allocations and transaction costs. Then, stability of financial decision models where two-stage mixed-integer value function appears as a loss variable is studied. In Chapter 4, we study qualitative properties of the...
Risk measures - sensitivity and dynamics
Branda, Martin
Risk measures are subject to many scientific papers and monographs published on financial portfolio optimization problem within stochastic programming. Currently there are many functionals which measure risk of random future losses according to risk managers preferences. However, their sensitivity is studied less commonly, especially according to possible changes of input data or with respect to the portfolio allocation. This thesis deals with sensitivity of two frequently discussed measures - Value at Risk (VaR) and Conditional Value at Risk (CVaR). Explicit contamination bounds for relative VaR optimization problem are expressed using general results of parametric optimization valid for quadratic programming. A numerical study and a heuristic algorithm for correlation matrices stressing are involved. Sensitivity of VaR and CVaR is studied through their derivatives with respect to the portfolio allocation. Assumptions for the derivatives are formulated, Hessians introduced and convexity is discussed. At last, some dynamic risk measures for multi-period investory models are proposed.
Stable distribution and application to finance
Omelchenko, Vadym ; Branda, Martin (referee) ; Klebanov, Lev (advisor)
Title: Stable distributions and application to finance Author: Vadym Omelchenko Department: Department of Probability and Mathematical Statistics Supervisor: Prof. Lev Klebanov, DrSc. Supervisor's e-mail address: Lev.Klebanov@mff.cuni.cz Abstract: This work deals with the theory of the stable distributions, their parameter estimation, and their financial application. There arc given the methods of characteristic function and method of projections, which is rel- ative to ML-methodology, for estimation of the parameters of stable dis- tributions. We compare these methods with the conventional estimators. The quality of estimators is verified by the simulation of the sample having stable distribution with known parameters and comparing the estimates of these parameters with their real values. The aim of this work is estima- tion of parameters of the stable laws which iy applicable for modification of AHCH/GAHCH models with stable innovations. Keywords: stable distribution, ARGII/GARCII models, characteristic func- tion (CF) based estimators, maximum likelihood projection (MLP) estima- tors.

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