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Stochastic Equations with Correlated Noise and Their Applications
Týbl, Ondřej ; Maslowski, Bohdan (advisor) ; Peszat, Szymon (referee) ; Hlubinka, Daniel (referee)
Stochastic Equations with Correlated Noise and Their Applications Ondřej Týbl Doctoral Thesis Abstract Properties of stochastic differential equations with jumps are stud- ied. Lyapunov-type methods are derived to assess long-time behavior of solu- tions and general results are applied in specific cases. In the first case, conditions in terms of the geometric properties of the coefficients for stability in terms of boundedness in probability in the mean are obtained. By means of Krylov Bogolyubov Theorem criterion for existence of invariant measures is given sub- sequentely. In the second case, the long-time behavior refers to existence of an almost sure single-point limit not depending on the initial condition. This result is then applied to get a continuous-time Robbins-Monro type stochastic approximation procedure for finding roots of a given function. 1
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Random Dynamical Systems and Their Applications
Iuzbashev, Artem ; Maslowski, Bohdan (advisor) ; Čoupek, Petr (referee)
This thesis extends the existing results in the theory of random dynamical systems driven by fractional noise in Hilbert space. In particular, it broadens the scope of ap- plicability of the results presented by Maria J. Garrido-Atienza, Bohdan Maslowski and Jana Snuparkova in Garrido-Atienza et al. [2016] for fractional noise whose sample paths have a Hölder exponent greater than 1/2. The main object of the research is the following stochastic equation: d u(t) = (A(t)u(t) + F(u(t)))d t + Bu(t)d ω(t), u(0) = u0 ∈ V, where (V, ∥ · ∥V ) is a separable Hilbert space, ω is a stochastic process and the stochastic integral is understood in the Zähle sense. This thesis contains the proof of a Fubini-type theorem for integration in the sense of Zähle. It is shown that the assumption about ergodicity for the underlying fractional noise in Garrido-Atienza et al. [2016] is redundant and the statements about random dynamical systems which are generated by the solution of the equation and its random attractor remain valid. The thesis also contains the proof of the existence and uniqueness of the solution to the equation above. 1
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Completeness of the Kantorovich-Rubinstein metric
Picek, Radovan ; Seidler, Jan (advisor) ; Maslowski, Bohdan (referee)
In the thesis the Kantorovich-Rubinstein metric in the space of Borel probability me- asures with a finite first moment on a separable complete metric space is studied. Its completeness is proved and convergence of sequences is characterized using elementary tools in Chapter 3. The proofs rely on results about the Dudley metric for weak conver- gence of probability measures, which are dealt with in Chapters 1 and 2. 1
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Feller's test for non-explosions
Rubín, Daniel ; Seidler, Jan (advisor) ; Maslowski, Bohdan (referee)
The main result of the work is a complete discussion of the solutions to stochastic differential equations on the half-line (0, ∞) with polynomial coefficients in terms of their lifetimes. To achieve this, Feller's test for non-explosion is utilized. The theorem is proven in detail, as existing proofs are too concise. 1
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Optimal control of Lévy-driven stochastic equations in Hilbert spaces
Kadlec, Karel ; Maslowski, Bohdan (advisor)
Controlled linear stochastic evolution equations driven by Lévy processes are studied in the Hilbert space setting. The control operator may be unbounded which makes the results obtained in the abstract setting applicable to parabolic SPDEs with boundary or point control. The first part contains some preliminary technical results, notably a version of Itô formula which is applicable to weak/mild solutions of controlled equations. In the second part, the ergodic control problem is solved: The feedback form of the optimal control and the formula for the optimal cost are found. The control problem is solved in the mean-value sense and, under selective conditions, in the pathwise sense. As examples, various parabolic type controlled SPDEs are studied. 1
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Random Dynamical Systems
Garaj, Tomáš ; Maslowski, Bohdan (advisor) ; Čoupek, Petr (referee)
Random differential equations are differential equations whose right-hand side con- tains a random noise. In most applications that noise is modelled by a stochastic process of certain properties or a metric dynamical system. In this thesis we examine random differential equations and find out under which conditions an equation through its so- lution generates a random dynamical system. To be able to consider a wider variety of functions on the right-hand side of the equation we employ the method of Lyapunov functions, obtaining less restrictive conditions than the ones normally presented. In the latter portion of the thesis we introduce the field of random attractors, present a theorem from literature regarding the conditions for the existence of a random attractor and for- mulate and prove our own version that is more closely related to the theory we concerned ourselves with before. 1
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Optimal risk sensitive control in radical chains
Picek, Radovan ; Dostál, Petr (advisor) ; Maslowski, Bohdan (referee)
In the first segment this thesis deal with Markov chains with discreet time and a finite set of states. Subsequently there is introduced valuation of transitions and a possibility of controlling these chains. Yields from valuation of transitions are then appointed to exponential utility funcion and discounted to the begining. Afterwards there is estab- lished Howard's iterative algorithm, which finds optimal control. The control is optimal amongst homogeneous and non-homogeneous controls. In the second segment, Markov chains are generalized to so called radical chains, again with discreet time and a finite set of states. The generalization is executed by adding an opportunity of choosing radical decisions, which take place out of real time. Howard's iterative algorithm is modified for this more general case. The control found by the algorithm is optimal amongst homoge- neous and non-homogeneous controls. 1
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Gebelein's inequality
Svoboda, Matěj ; Čoupek, Petr (advisor) ; Maslowski, Bohdan (referee)
In this thesis we deal with a correlation inequality for Gaussian random variables called Gebelein's inequality. In the first part of the thesis, we state the inequality, define Hermite polynomials, and derive several of their properties which we then use to prove the inequality. In the second part, we apply Gebelein's inequality in order to show that for normalized Gaussian sequences the Borel-Cantelli lemma and strong law of large numbers still hold when the assumption of independence is replaced by a sufficient fast decay of their correlation function. 1
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Non-smooth paths
Hendrych, František ; Čoupek, Petr (advisor) ; Maslowski, Bohdan (referee)
Sewing Lemmas are useful tools when one needs to give meaning to some abstract integral through given local approximations. This can be used in Rough Paths Theory, for example. Many versions of such lemmas are known. Stochastic Sewing is its particular generalization allowing us to relax the regularity requirements of the underlying objects. In the present thesis, we generalize the known Stochastic Sewing Lemma for stochastic processes viewed as Hölder functions with values in Lm (Ω) introduced by Lê [2020] and the Besov Sewing Lemma for functions of a Besov type introduced by Friz and Seeger [2021]. Their natural combination yields the Stochastic Besov Sewing Lemma for stochastic processes viewed as Besov-type functions with values in Lm (Ω). 1
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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.
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