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Love-Young Inequality and Its Consequences
Sýkora, Adam ; Čoupek, Petr (advisor) ; Hlubinka, Daniel (referee)
This thesis is focused on proving the Love-Young inequality and clarifying the manner in which it relates to a fractional Brownian motion. To begin with, several estimates alongside the concept of p-variation of a func- tion are presented. The connection between functions of finite p-variation and regulated functions is then highlighted and used to prove the aforementioned Love-Young inequality. Deficiency of the pathwise approach to stochastic in- tegration is recognised and later discussed amongst the properties of fractional Brownian motions. This constitutes the main application of the featured theory which is the integration with respect to irregular functions. 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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Valuation of financial derivatives
Matušková, Radka ; Hurt, Jan (advisor) ; Zichová, Jitka (referee)
In the present thesis we deal with several possible approaches to financial de- rivatives pricing. In the first part, we introduce the basic types of derivatives and the methods of trading. Furthermore, we present several models for the valuati- on of specific financial derivative, i.e. options. Firstly we describe Black-Scholes model in detail, which considers that the development of the underlying asset price is governed by Wiener process. Following are the jumps diffusion models that are extension of the Black-Scholes model with jumps. Then we get to jump models, which are based on Lévy processes. Finally, we will deal with the model, which considers that the development of the underlying asset price is governed by fractional Brownian motion with Hurst's coefficient greater than 1/2. All models are suplemented with sample examples. 1
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Stochastic Evolution Equations
Čoupek, Petr ; Maslowski, Bohdan (advisor) ; Garrido-Atienza, María J. (referee) ; Hlubinka, Daniel (referee)
Stochastic Evolution Equations Petr Čoupek Doctoral Thesis Abstract Linear stochastic evolution equations with additive regular Volterra noise are studied in the thesis. Regular Volterra processes need not be Gaussian, Markov or semimartingales, but they admit a certain covariance structure instead. Particular examples cover the fractional Brownian motion of H > 1/2 and, in the non-Gaussian case, the Rosenblatt process. The solution is considered in the mild form, which is given by the variation of constants formula, and takes values either in a separable Hilbert space or the space Lp(D, µ) for large p. In the Hilbert-space setting, existence, space-time regularity and large-time behaviour of the solutions are studied. In the Lp setting, existence and regularity is studied, and in concrete cases of stochastic partial differential equations, the solution is shown to be a space-time continuous random field.
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Black-Scholes models of option pricing
Čekal, Martin
Title: Black-Scholes Models of Option Pricing Author: Martin Cekal Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Bohdan Maslowski, DrSc., Charles University in Prague, Faculty of Mathematics and Physics, Department of Probability and Mathematical Statistics. Abstract: In the present master thesis we study a generalization of Black-Scholes model using fractional Brownian motion and jump processes. The main goal is a derivation of the price of call option in a fractional jump market model. The first chapter introduces long memory and its modelling by discrete and continuous time models. In the second chapter fractional Brownian motion is defined, appropriate stochastic analysis is developed and we generalize the notion of Lévy and jump processes. The third chapter introduces fractional Black-Scholes model. In the fourth chapter, tools developed in the second chapter are used for the construction of jump fractional Black-Scholes model and derivation of explicit formula for the price of european call option. In the fifth chapter, we analyze long memory contained in simulated and empirical time series. Keywords: Black-Scholes model, fractional Brownian motion, fractional jump process, long- memory, options pricing.
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Stochastic Evolution Equations
Čoupek, Petr ; Maslowski, Bohdan (advisor)
Stochastic Evolution Equations Petr Čoupek Doctoral Thesis Abstract Linear stochastic evolution equations with additive regular Volterra noise are studied in the thesis. Regular Volterra processes need not be Gaussian, Markov or semimartingales, but they admit a certain covariance structure instead. Particular examples cover the fractional Brownian motion of H > 1/2 and, in the non-Gaussian case, the Rosenblatt process. The solution is considered in the mild form, which is given by the variation of constants formula, and takes values either in a separable Hilbert space or the space Lp(D, µ) for large p. In the Hilbert-space setting, existence, space-time regularity and large-time behaviour of the solutions are studied. In the Lp setting, existence and regularity is studied, and in concrete cases of stochastic partial differential equations, the solution is shown to be a space-time continuous random field.
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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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Love-Young Inequality and Its Consequences
Sýkora, Adam ; Čoupek, Petr (advisor) ; Hlubinka, Daniel (referee)
This thesis is focused on proving the Love-Young inequality and clarifying the manner in which it relates to a fractional Brownian motion. To begin with, several estimates alongside the concept of p-variation of a func- tion are presented. The connection between functions of finite p-variation and regulated functions is then highlighted and used to prove the aforementioned Love-Young inequality. Deficiency of the pathwise approach to stochastic in- tegration is recognised and later discussed amongst the properties of fractional Brownian motions. This constitutes the main application of the featured theory which is the integration with respect to irregular functions. 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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Stochastic Evolution Equations
Čoupek, Petr ; Maslowski, Bohdan (advisor)
Stochastic Evolution Equations Petr Čoupek Doctoral Thesis Abstract Linear stochastic evolution equations with additive regular Volterra noise are studied in the thesis. Regular Volterra processes need not be Gaussian, Markov or semimartingales, but they admit a certain covariance structure instead. Particular examples cover the fractional Brownian motion of H > 1/2 and, in the non-Gaussian case, the Rosenblatt process. The solution is considered in the mild form, which is given by the variation of constants formula, and takes values either in a separable Hilbert space or the space Lp(D, µ) for large p. In the Hilbert-space setting, existence, space-time regularity and large-time behaviour of the solutions are studied. In the Lp setting, existence and regularity is studied, and in concrete cases of stochastic partial differential equations, the solution is shown to be a space-time continuous random field.
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