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Degenerate Parabolic Stochastic Partial Differential Equations
Hofmanová, Martina ; Seidler, Jan (advisor) ; Perthame, Benoit (referee) ; Flandoli, Franco (referee)
In this thesis, we address several problems arising in the study of nondegenerate and degenerate parabolic SPDEs, stochastic hyper- bolic conservation laws and SDEs with continues coefficients. In the first part, we are interested in degenerate parabolic SPDEs, adapt the notion of kinetic formulation and kinetic solution and establish existence, uniqueness as well as continuous dependence on initial data. As a preliminary result we obtain regularity of solutions in the nondegenerate case under the hypothesis that all the coefficients are sufficiently smooth and have bounded derivatives. In the second part, we consider hyperbolic conservation laws with stochas- tic forcing and study their approximations in the sense of Bhatnagar-Gross- Krook. In particular, we describe the conservation laws as a hydrodynamic limit of the stochastic BGK model as the microscopic scale vanishes. In the last part, we provide a new and fairly elementary proof of Skorkohod's classical theorem on existence of weak solutions to SDEs with continuous coefficients satisfying a suitable Lyapunov condition. 1
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Kalman-Bucy Filter in Continuous Time
Týbl, Ondřej ; Maslowski, Bohdan (advisor) ; Čoupek, Petr (referee)
In the Thesis we study the problem of linear filtration of Gaussian signals in finite-dimensional space. We use the Kalman-type equations for the filter to show that the filter depends continuously on the signal. Secondly, we show the same continuity property for the covariance of the error and verify existence and uniqueness of a solution to an integral equation that is satisfied by the filter even under more general assumptions. We present several examples of application of the continuity property that are based on the theory of stochastic differential equations driven by fractional Brownian motion. 1
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Markovské semigrupy
Žák, František ; Maslowski, Bohdan (advisor) ; Štěpán, Josef (referee)
In the presented work we study the existence of periodic solution to infinite dimensional stochastic equation with periodic coefficients driven by Cylindrical Wiener process. Used theory of infinite dimensional stochastic equations in Hilbert spaces and Markov processes is summarized in the first two chapters. In the third and last chapter we present the result itself. Necessary technical background mostly from operator theory is encapsulated in the Appendix. The proof of existence of periodic solution of corresponding equation is a combination of arguments by Khasminskii, which ensure under suitable conditions the existence of periodic Markov process, and the results of Da Prato, G¸atatrek and Zabczyk for the existence of invariant measure for homogeneous stochastic equation in Hilbert spaces. At the end we derive sufficient condition for the existence of periodic solution in the language of coefficients using the work of Ichikawa and illustrate the results by the example of Stochastic PDE. The work is written in English.
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Degenerate Parabolic Stochastic Partial Differential Equations
Hofmanová, Martina
In this thesis, we address several problems arising in the study of nondegenerate and degenerate parabolic SPDEs, stochastic hyper- bolic conservation laws and SDEs with continues coefficients. In the first part, we are interested in degenerate parabolic SPDEs, adapt the notion of kinetic formulation and kinetic solution and establish existence, uniqueness as well as continuous dependence on initial data. As a preliminary result we obtain regularity of solutions in the nondegenerate case under the hypothesis that all the coefficients are sufficiently smooth and have bounded derivatives. In the second part, we consider hyperbolic conservation laws with stochas- tic forcing and study their approximations in the sense of Bhatnagar-Gross- Krook. In particular, we describe the conservation laws as a hydrodynamic limit of the stochastic BGK model as the microscopic scale vanishes. In the last part, we provide a new and fairly elementary proof of Skorkohod's classical theorem on existence of weak solutions to SDEs with continuous coefficients satisfying a suitable Lyapunov condition. 1
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Stochastic Differential Equations with Gaussian Noise and Their Applications
Camfrlová, Monika ; Čoupek, Petr (advisor) ; Večeř, Jan (referee)
In the thesis, multivariate fractional Brownian motions with possibly different Hurst indices in different coordinates are considered and a Girsanov-type theo- rem for these processes is shown. Two applications of this theorem to stochastic differential equations driven by multivariate fractional Brownian motions (SDEs) are given. Firstly, the existence of a weak solution to an SDE with a drift coeffi- cient that can be written as a sum of a regular and a singular part and a diffusion coefficient that is dependent on time and satisfies suitable conditions is shown. The results are applied for the proof of existence of a weak solution of an equation describing stochastic harmonic oscillator. Secondly, the Girsanov-type theorem is used to find the maximum likelihood scalar estimator that appears in the drift of an SDE with additive noise. 1
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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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Kalman-Bucy Filter in Continuous Time
Týbl, Ondřej ; Maslowski, Bohdan (advisor) ; Čoupek, Petr (referee)
In the Thesis we study the problem of linear filtration of Gaussian signals in finite-dimensional space. We use the Kalman-type equations for the filter to show that the filter depends continuously on the signal. Secondly, we show the same continuity property for the covariance of the error and verify existence and uniqueness of a solution to an integral equation that is satisfied by the filter even under more general assumptions. We present several examples of application of the continuity property that are based on the theory of stochastic differential equations driven by fractional Brownian motion. 1
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Degenerate Parabolic Stochastic Partial Differential Equations
Hofmanová, Martina
In this thesis, we address several problems arising in the study of nondegenerate and degenerate parabolic SPDEs, stochastic hyper- bolic conservation laws and SDEs with continues coefficients. In the first part, we are interested in degenerate parabolic SPDEs, adapt the notion of kinetic formulation and kinetic solution and establish existence, uniqueness as well as continuous dependence on initial data. As a preliminary result we obtain regularity of solutions in the nondegenerate case under the hypothesis that all the coefficients are sufficiently smooth and have bounded derivatives. In the second part, we consider hyperbolic conservation laws with stochas- tic forcing and study their approximations in the sense of Bhatnagar-Gross- Krook. In particular, we describe the conservation laws as a hydrodynamic limit of the stochastic BGK model as the microscopic scale vanishes. In the last part, we provide a new and fairly elementary proof of Skorkohod's classical theorem on existence of weak solutions to SDEs with continuous coefficients satisfying a suitable Lyapunov condition. 1
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Degenerate Parabolic Stochastic Partial Differential Equations
Hofmanová, Martina ; Seidler, Jan (advisor) ; Perthame, Benoit (referee) ; Flandoli, Franco (referee)
In this thesis, we address several problems arising in the study of nondegenerate and degenerate parabolic SPDEs, stochastic hyper- bolic conservation laws and SDEs with continues coefficients. In the first part, we are interested in degenerate parabolic SPDEs, adapt the notion of kinetic formulation and kinetic solution and establish existence, uniqueness as well as continuous dependence on initial data. As a preliminary result we obtain regularity of solutions in the nondegenerate case under the hypothesis that all the coefficients are sufficiently smooth and have bounded derivatives. In the second part, we consider hyperbolic conservation laws with stochas- tic forcing and study their approximations in the sense of Bhatnagar-Gross- Krook. In particular, we describe the conservation laws as a hydrodynamic limit of the stochastic BGK model as the microscopic scale vanishes. In the last part, we provide a new and fairly elementary proof of Skorkohod's classical theorem on existence of weak solutions to SDEs with continuous coefficients satisfying a suitable Lyapunov condition. 1
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Markovské semigrupy
Žák, František ; Maslowski, Bohdan (advisor) ; Štěpán, Josef (referee)
In the presented work we study the existence of periodic solution to infinite dimensional stochastic equation with periodic coefficients driven by Cylindrical Wiener process. Used theory of infinite dimensional stochastic equations in Hilbert spaces and Markov processes is summarized in the first two chapters. In the third and last chapter we present the result itself. Necessary technical background mostly from operator theory is encapsulated in the Appendix. The proof of existence of periodic solution of corresponding equation is a combination of arguments by Khasminskii, which ensure under suitable conditions the existence of periodic Markov process, and the results of Da Prato, G¸atatrek and Zabczyk for the existence of invariant measure for homogeneous stochastic equation in Hilbert spaces. At the end we derive sufficient condition for the existence of periodic solution in the language of coefficients using the work of Ichikawa and illustrate the results by the example of Stochastic PDE. The work is written in English.
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