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1	Stochastické integrály řízené isonormálními gaussovskými procesy a aplikace / Stochastic Integrals Driven by Isonormal Gaussian Processes and Applications Čoupek, Petr January 2013 (has links) Stochastic Integrals Driven by Isonormal Gaussian Processes and Applications Master Thesis - Petr Čoupek Abstract In this thesis, we introduce a stochastic integral of deterministic Hilbert space valued functions driven by a Gaussian process of the Volterra form βt = t 0 K(t, s)dWs, where W is a Brownian motion and K is a square integrable kernel. Such processes generalize the fractional Brownian motion BH of Hurst parameter H ∈ (0, 1). Two sets of conditions on the kernel K are introduced, the singular case and the regular case, and, in particular, the regular case is studied. The main result is that the space H of β-integrable functions can be, in the strictly regular case, embedded in L 2 1+2α ([0, T]; V ) which corresponds to the space L 1 H ([0, T]) for the fractional Brownian mo- tion. Further, the cylindrical Gaussian Volterra process is introduced and a stochastic integral of deterministic operator-valued functions, driven by this process, is defined. These results are used in the theory of stochastic differential equations (SDE), in particular, measurability of a mild solution of a given SDE is proven.
2	Stochastické evoluční rovnice / Stochastic Evolution Equations Čoupek, Petr January 2017 (has links) 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.
3	Nonparametric Methods in Spot Volatility Estimation / Nichtparametrische Methoden für das Schätzen der Spot-Volatilität Schmidt-Hieber, Anselm Johannes 26 October 2010 (has links) No description available. Adaptives Schätzen Fourierreihen hochfrequente Daten Volterra Prozess Minimax Mikrostrukturrauschen nichtparametrisches Schätzen Semimartingal Spot-Volatilität Wavelets Adaptive estimation Fourier series high-frequency data Volterra process minimax microstructure noise nonparametric estimation semimartingale spot volatility wavelets

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