|
|
Optimalization in allocation drivers at bus service
Sokol, Petr ; Bartušek, Bohumír (advisor) ; Štěpán, Josef (referee)
The thesis is focused on suggesting an algorithm for planning of optimal assignment drivers and vehicles to scheme of expeditions. When we make the assignment, we aspire to minimize the total transportation costs. By the same mail we push for well-balanced assignment, with respect to utilization of drivers and vehicles . The structure of the planning problem there is describing by making out of a wide linear model, whose parts are algebraic formulations of restrictions, which one must in traffic abide. In this model we can see binary and real variables that mean we draw up a model of mixed integer programming. We can say, that the integer programming is in general more complicated, then integer programming. We used optimization software named GAMS for solution this problem. The software uses Branch and Bound algorithm for the integer number problems. We draw up a program in GAMS, which is able to set an optimal pian for assignment drivers and vehicles to expeditions. A part of this work is couple of exemplary tasks, with their solution.
|
|
|
Proofs of the strong law of large numbers
Odintsov, Kirill ; Štěpán, Josef (advisor) ; Staněk, Jakub (referee)
This thesis concentrates on the Strong Law of Large Numbers. It features two proofs of this law. The first is less general, but simpler Borel's proof. The second one is more complex. It uses Kronecker's lemma and Kolmogorov-Khinchin's theorem, which is proven by Kolmogorov's inequality. The text includes all the necessary auxiliary theorems and lemmas along with their proofs. Since all the proofs are explored in a great detail this text is suitable for readers with only basic knowledge of probability theory and measure theory. Furthermore it contains numerous practical and mathematical examples thought out the whole text. Finally to demonstrate the importance of Strong Law of Large Numbers the text features four important applications of the law in mathematics.
|
|
|
Modifications of stochastic objects
Kadlec, Karel ; Štěpán, Josef (advisor) ; Dostál, Petr (referee)
In this thesis, we are concerned with the modifications of the stochastic processes and the random probability measures. First chapter is devoted to modifications of the stochastic process to the space of continuous functions, modifications of submartingale to the set of right-continuous with finite left-hand limits functions and separable modifications of stochastic process. In the second chapter is the attention on the regularization of random probability measure in Markov kernel focused. In particular, we work with random probability measures on the Borel subset of the Polish space, or Radon separable topological space.
|
|
|
Univariate difusion stochastic differential equations with applications to financial mathematics
Zahradník, Petr ; Štěpán, Josef (advisor)
In this thesis, the aim is to employ some of the advanced probability and calculus techniques to financial mathematics. In the first chapter some major facts from continuous - time probability theory are presented. In the second chapter, one - dimensional stochastic diferential equations are introduced, we touch upon the questions of existence and uniqueness of solutions in full generality, construct a weak solution to the Engelbert - Schmidt equation and thoroughly present a known procedure called a Feller's test for explosions. In chapter three, focus is directed to a brief presentation of the well known Dirichlet problem. The problem is also interpreted financially, applied to options valuation and related approximations are implemented. The fourth, final, chapter concentrates on the Cox - Ingersoll - Ross model. Techniques derived in the second and third chapters are employed to thoroughly study the model properties.
|
|
| |
|
|
Univariate difusion stochastic differential equations with applications to financial mathematics
Zahradník, Petr ; Štěpán, Josef (advisor)
In this thesis, the aim is to employ some of the advanced probability and calculus techniques to financial mathematics. In the first chapter some major facts from continuous - time probability theory are presented. In the second chapter, one - dimensional stochastic diferential equations are introduced, we touch upon the questions of existence and uniqueness of solutions in full generality, construct a weak solution to the Engelbert - Schmidt equation and thoroughly present a known procedure called a Feller's test for explosions. In chapter three, focus is directed to a brief presentation of the well known Dirichlet problem. The problem is also interpreted financially, applied to options valuation and related approximations are implemented. The fourth, final, chapter concentrates on the Cox - Ingersoll - Ross model. Techniques derived in the second and third chapters are employed to thoroughly study the model properties.
|
|
| |
|
|
Zero one laws in probabability and topology
Šimon, Prokop ; Štěpán, Josef (advisor) ; Maslowski, Bohdan (referee)
Práce se zabývá teorií funkcí typu PLIF, jejichž zavedení bylo motivo- váno matematickou statistikou. Je ukázána cesta vedoucí od statistického problému až k jeho zjednodušení pomocí PLIF, resp. SPLIF. Navazující pří- klady dávají odpově¤ na existenci těchto funkcí na vybraných prostorech, přirozeně je kladen d·raz na prostor všech nekonečných posloupností 0 a 1 {0, 1}N a jeho podprostory. Za použití silného zákona velkých čísel pro ná- hodnou procházku je uveden zajímavý příklad ukazující množinu 1. kategorie mající míru jedna. Dále je dokázán Oxtobyho 0-1 zákon. Celou práci uzavírá rozpracovaný d·kaz věty od D. Blackwella ukazující neexistenci borelovských SPLIF, ve kterém hraje klíčovou roli právě Oxtobyho 0-1 zákon. 1
|
|
|
Probability distributions on metric groups.
Ondřej, Josef ; Štěpán, Josef (advisor) ; Dostál, Petr (referee)
Title: Probability distributions on metric groups Author: Josef Ondřej Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Josef Štěpán, DrSc., Department of Probability and Mathematical Statistics Abstract: In this thesis we deal with the space of Borel probability measures at first on a metric space and later on a metric group. We define the notion of a weak convergence of Borel probability measures and in a special case we show this convergence is metrizable. Further we introduce operation of convolution of Borel probability measures on a metric group and we show that together with this operation the space of measures becomes a topological semigroup. We use the notion of convolution to define idempotent and Haar measure and we show a relation between them. Finally we use the mentioned results to describe all solutions of Choquet problem. At the end we demonstrate how the theory that we have developed applies to a group of complex units. Keywords: Metric group, weak convergence, Prokhorov's theorem, Choquet's theorem.
|
|
|
Three proofs of a limit theorem
Marcinčín, Martin ; Štěpán, Josef (advisor) ; Beneš, Viktor (referee)
We show three diferent proofs of the central limit theorem using elementary methods. The central limit theorem with the Feller - Lindeberg condition is proven using a convergence of charakteristic functions and Fejer theorem about uniform convergence of trigonometric polynoms on a bounded interval. The second proof is based on the fact that convergence in distribution is equivalent to convergence of means of functions with all derivatives bounded. The central limit theorem for sums of independent random variables with all moments finite is shown using convergence of all moments and determinacy of normal distribution by its moments.
|
guest ::
RSS feed.