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Parameter Estimation in Stochastic Differential Equations
Pacák, Daniel ; Maslowski, Bohdan (advisor) ; Hlubinka, Daniel (referee)
In the Thesis the problem of estimating an unknown parameter in a stochastic dif- ferential equation is studied. Linear equations with Volterra process as the source of noise are considered. Firstly, the properties of Volterra processes and the properties of stochastic integral with respect to a Volterra process are presented. Secondly, the prop- erties of the solution to the equation under consideration are discussed. This includes the existence of the strictly stationary solution, the properties of such solution and ergodic results. These results are then generalized to equations with a mixed noise. Ergodic results are used to derive strongly consistent estimators of the unknown parameter. 1
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Fractional geometric Brownian motion
Pacák, Daniel ; Šnupárková, Jana (advisor) ; Maslowski, Bohdan (referee)
The subject of this thesis is to study the geometric fracional Brownian motion. To do this, the necessary theory is presented. The first chapter summarizes the basic theory of stochastic processes. The second chapter deals with fractional Brownian motion. This is followed by the construction of Itô integral with respect to the Brownian motion. The main focus is the Itô's lemma. The formula for geometric Brownian motion is then derived using the Itô's lemma. In the last chapter deals with the geometric fractional Brownian motion. Its limit behaviour is studied. Some simulated examples are shown. 1
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Least Absolute Deviations
Pacák, Daniel ; Víšek, Jan Ámos (advisor) ; Červinka, Michal (referee)
This is a theoretical study of the Least Absolute Deviations (LAD) fits. In the first part, fundamental mathematical properties of LAD fits are established. Computational aspects of LAD fits are shown and the Barrodale-Roberts Al- gorithm for finding LAD fits is presented. In the second part, the statistical properties of LAD estimator are discussed in the concept of linear regression. It is shown that LAD estimator is a maximum likelihood estimator if the er- ror variables follow Laplace distribution. We state theorems establishing strong consistency and asymptotic normality of LAD estimator and we discuss the bias of LAD estimator. In the last section, we present the results of numerical experi- ments where we numerically showed consistency of LAD estimator, discussed its behaviour under different distributions of error variables with comparison to the Ordinary Least Squares (OLS) estimator. Lastly, we looked at the behaviour of LAD and OLS estimators in the presence of corrupted observations. 1
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