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Testing time-series characteristics of prices of financial derivatives
Vdovičenko, Martin ; Kadavý, Matěj (advisor) ; Šnupárková, Jana (referee)
This work discusses Brownian motion and its basic transformations. The work describes basic properties of its trajectories and shows that Brownian motion is a martingale and a self-similar process. Next, we discuss time series analysis. We introduce graphical tools for analyzing data and we describe theoretical basics of some normality and independence tests. Finally, we consider the hypothesis that in the short run the price of financial assets can be modelled by Brownian motion. We conduct basic statistical tests on real data using the R progam and we talk through our results.
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Stochastic evolution equations with multiplicative fractional noise
Šnupárková, Jana ; Maslowski, Bohdan (advisor)
Title: Stochastic evolution equations with multiplicative fractional noise Author: Jana Šnupárková Departement: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Bohdan Maslowski, DrSc. Supervisor's e-mail address: maslow@karlin.mff.cuni.cz Abstract: The fractional Gaussian noise is a formal derivative of a fractional Brownian motion with Hurst parameter H ∈ (0, 1). An explicit formula for a solution to stochastic differential equations with a multiplicative fractional Gaussian noise in a separable Hilbert space is given. The large time behaviour of the solution is studied. In addition, equations of this type with a nonlinear perturbation of a drift part are investigated in the case H > 1/2. Keywords: Fractional Brownian Motion, Stochastic Differential Equations in Hilbert Space, Explicit Formula for Solution
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Stochastic evolution equations with multiplicative fractional noise
Šnupárková, Jana ; Maslowski, Bohdan (advisor)
Title: Stochastic evolution equations with multiplicative fractional noise Author: Jana Šnupárková Departement: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Bohdan Maslowski, DrSc. Supervisor's e-mail address: maslow@karlin.mff.cuni.cz Abstract: The fractional Gaussian noise is a formal derivative of a fractional Brownian motion with Hurst parameter H ∈ (0, 1). An explicit formula for a solution to stochastic differential equations with a multiplicative fractional Gaussian noise in a separable Hilbert space is given. The large time behaviour of the solution is studied. In addition, equations of this type with a nonlinear perturbation of a drift part are investigated in the case H > 1/2. Keywords: Fractional Brownian Motion, Stochastic Differential Equations in Hilbert Space, Explicit Formula for Solution
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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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Girsanov Theorem
Navrátil, Robert ; Šnupárková, Jana (advisor) ; Maslowski, Bohdan (referee)
Girsanov Theorem Bachelor's thesis - Robert Navrátil Abstract Modern theory of probability and financial mathematics require the theory of stochastic calculus. Its foundations contain Wiener process (Brownian motion) and the integral of stochastic process with respect to another stochastic process. This thesis deals with building the mathematical theory needed to construct the stochastic integral, with the construction itself, the Girsanov Theorem and its applications. The Girsanov Theorem uses equivalent probability measure to transform Wiener process with drift to Wiener process without drift. Using the Girsanov Theorem, we change our measure to the equivalent risk neutral measure and we deduce Black-Scholes formula which estimates the prize of European call option with underlying stock asset. The stock prize is modelled using the geometric Brownian motion. Finally, we demonstrate, on real life data, how this model works and what are its outcomes. 1
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Testing time-series characteristics of prices of financial derivatives
Vdovičenko, Martin ; Kadavý, Matěj (advisor) ; Šnupárková, Jana (referee)
This work discusses Brownian motion and its basic transformations. The work describes basic properties of its trajectories and shows that Brownian motion is a martingale and a self-similar process. Next, we discuss time series analysis. We introduce graphical tools for analyzing data and we describe theoretical basics of some normality and independence tests. Finally, we consider the hypothesis that in the short run the price of financial assets can be modelled by Brownian motion. We conduct basic statistical tests on real data using the R progam and we talk through our results.
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Stochastic evolution equations with multiplicative fractional noise
Šnupárková, Jana ; Maslowski, Bohdan (advisor) ; Hlubinka, Daniel (referee) ; Seidler, Jan (referee)
Title: Stochastic evolution equations with multiplicative fractional noise Author: Jana Šnupárková Departement: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Bohdan Maslowski, DrSc. Supervisor's e-mail address: maslow@karlin.mff.cuni.cz Abstract: The fractional Gaussian noise is a formal derivative of a fractional Brownian motion with Hurst parameter H ∈ (0, 1). An explicit formula for a solution to stochastic differential equations with a multiplicative fractional Gaussian noise in a separable Hilbert space is given. The large time behaviour of the solution is studied. In addition, equations of this type with a nonlinear perturbation of a drift part are investigated in the case H > 1/2. Keywords: Fractional Brownian Motion, Stochastic Differential Equations in Hilbert Space, Explicit Formula for Solution
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