|
|
Markov processes (analytic and probabilistic point of view)
Nováková, Eva ; Janák, Josef (advisor) ; Maslowski, Bohdan (referee)
This Bachelor Thesis tackles the basics of the Markov chains theory. The first four chapters describe fundamental definitions and theorems of the theory of Markov chains, both in continuous and discrete time and both with discrete and general state space. The last chapter contains examples of each type of Markov chains. The conclusion describes the relation between all four types of Markov chains.
|
|
|
Stochastic Differential Equations with Gaussian Noise
Janák, Josef ; Maslowski, Bohdan (advisor)
Title: Stochastic Differential Equations with Gaussian Noise Author: Josef Janák Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Bohdan Maslowski, DrSc., Department of Probability and Mathematical Statistics Abstract: Stochastic partial differential equations of second order with two un- known parameters are studied. The strongly continuous semigroup (S(t), t ≥ 0) for the hyperbolic system driven by Brownian motion is found as well as the formula for the covariance operator of the invariant measure Q (a,b) ∞ . Based on ergodicity, two suitable families of minimum contrast estimators are introduced and their strong consistency and asymptotic normality are proved. Moreover, another concept of estimation using "observation window" is studied, which leads to more families of strongly consistent estimators. Their properties and special cases are descibed as well as their asymptotic normality. The results are applied to the stochastic wave equation perturbed by Brownian noise and illustrated by several numerical simula- tions. Keywords: Stochastic hyperbolic equation, Ornstein-Uhlenbeck process, invariant measure, paramater estimation, strong consistency, asymptotic normality.
|
|
|
Analýza brzdových systémů motorových vozidel
Janák, Josef
Bachelor thesis deals with problems of motor vehicle braking systems and systems associated to them. The introduction deals with the division of brake systems and sys-tems according to various criteria. It continues with the composition and main parts of hydraulic brakes up to the main content of this work, namely the design and function of drum and disc brake systems. Then it follows a with a little description of the electro-hydraulic braking system and the end part of the thesis is focused with the anti-lock and stabilization systems associated with the brake system of the car and the length of the braking distance.
|
|
|
Stochastic Differential Equations with Gaussian Noise
Janák, Josef ; Maslowski, Bohdan (advisor)
Title: Stochastic Differential Equations with Gaussian Noise Author: Josef Janák Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Bohdan Maslowski, DrSc., Department of Probability and Mathematical Statistics Abstract: Stochastic partial differential equations of second order with two un- known parameters are studied. The strongly continuous semigroup (S(t), t ≥ 0) for the hyperbolic system driven by Brownian motion is found as well as the formula for the covariance operator of the invariant measure Q (a,b) ∞ . Based on ergodicity, two suitable families of minimum contrast estimators are introduced and their strong consistency and asymptotic normality are proved. Moreover, another concept of estimation using "observation window" is studied, which leads to more families of strongly consistent estimators. Their properties and special cases are descibed as well as their asymptotic normality. The results are applied to the stochastic wave equation perturbed by Brownian noise and illustrated by several numerical simula- tions. Keywords: Stochastic hyperbolic equation, Ornstein-Uhlenbeck process, invariant measure, paramater estimation, strong consistency, asymptotic normality.
|
|
|
Stochastic Differential Equations with Gaussian Noise
Janák, Josef ; Maslowski, Bohdan (advisor) ; Duncan, Tyrone E. (referee) ; Pawlas, Zbyněk (referee)
Title: Stochastic Differential Equations with Gaussian Noise Author: Josef Janák Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Bohdan Maslowski, DrSc., Department of Probability and Mathematical Statistics Abstract: Stochastic partial differential equations of second order with two un- known parameters are studied. The strongly continuous semigroup (S(t), t ≥ 0) for the hyperbolic system driven by Brownian motion is found as well as the formula for the covariance operator of the invariant measure Q (a,b) ∞ . Based on ergodicity, two suitable families of minimum contrast estimators are introduced and their strong consistency and asymptotic normality are proved. Moreover, another concept of estimation using "observation window" is studied, which leads to more families of strongly consistent estimators. Their properties and special cases are descibed as well as their asymptotic normality. The results are applied to the stochastic wave equation perturbed by Brownian noise and illustrated by several numerical simula- tions. Keywords: Stochastic hyperbolic equation, Ornstein-Uhlenbeck process, invariant measure, paramater estimation, strong consistency, asymptotic normality.
|
|
|
Heavy tailed distributions and their applications to finance
Korbel, Michal ; Klebanov, Lev (advisor) ; Janák, Josef (referee)
In this work we describe heavy tailed distributions. We show conditions necessary and sufficient for their existence. First we study the product of random number of random variables and their convergence to the Pareto distribution. We also show graphs that concur this theorem. Next we define stable distributions and we study their usefulness for approximating of sum of random number of random variables. We also define Gauss and infinitely divisible random variables and we show conditions for their existence. We also show that the only geometric stable distribution following the stable law are strictly geometric stable or improper geometric stable distributions. In the end we study applications of stable distributions in finance and we show example for their usage in computing VaR. Powered by TCPDF (www.tcpdf.org)
|
|
|
Conditional distributions and condeitional expectations
Čellár, Matúš ; Hlubinka, Daniel (advisor) ; Janák, Josef (referee)
This paper discusses conditional distributions and conditional expectations, their introduction and basic properties. We begin with the definition of conditional probability, show a few theorems and demonstrate their application on an example. From there we move on to the conditioning with respect to random events and discrete random variables. In the general case we help ourselves with the definition of conditional expectation as random variable, show its properties, ways of expression and the fact that the introduction in the discrete case does not lead to a contradiction with the general definition. Then we deduce the criteria that have to be met for the conditional distribution to exist and in the last part we solve a number of theoretical problems.
|
|
|
Radon-Nikodym Derivative in Probability Theory
Křepinská, Dana ; Dostál, Petr (advisor) ; Janák, Josef (referee)
This thesis concerns the Radon-Nikodym derivate, its properties, connection with measure derivative and its applications in the probability theory. The text defines the conditional probability distribution and solves the problem of unique- ness in the case of conditioning of an event which has zero probability of occuring. Next part of the text is about the conditional expactation, which is defined by the conditional distribution, and some of its properties. There are also few words about the Borel isomorphic spaces and the conditional variability and covariance. Last section of this work is about construction of the Brownian Bridge from the Wiener process and about its applications is the statistics.
|
|
|
Markov processes (analytic and probabilistic point of view)
Nováková, Eva ; Janák, Josef (advisor) ; Maslowski, Bohdan (referee)
This Bachelor Thesis tackles the basics of the Markov chains theory. The first four chapters describe fundamental definitions and theorems of the theory of Markov chains, both in continuous and discrete time and both with discrete and general state space. The last chapter contains examples of each type of Markov chains. The conclusion describes the relation between all four types of Markov chains.
|
|
|
Ornstein-Uhlenbeck bridge
Janák, Josef ; Dostál, Petr (referee) ; Maslowski, Bohdan (advisor)
In the Thesis we study the Ornstein-Uhlenbeck Bridges. First, we recall the notion of the fractional Brownian motion and introduce stochastic integral of a deterministic function with respect to (fBm). We summarize the results on existence and uniqueness of a solutions to the autonomic linear stochastic di erential equations that are called the Ornstein-Uhlenbeck processes. We introduce the concept of the Gaussian Bridge and we derive its representation, which we use for obtaining the formula for Ornstein-Uhlenbeck Bridge. The results are applied to some special examples. In the last part of the Thesis we mention a nonanticipative representation of the bridge.
|
guest ::
RSS feed.