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Ito formula and its applications
Till, Alexander ; Haman, Jiří (advisor) ; Maslowski, Bohdan (referee)
Title: Itô formula and its applications Author: Alexander Till Department: Department of Probability and Mathematical Statistics Supervisor: Mgr. Jiří Haman Supervisor's e-mail address: j.haman@seznam.cz Abstract: The bachelor thesis contains basis and elementary findings of stochastic analysis. It includes definition and properties of stochastic integral with Wiener process as an integrator, definition of stochastic integral with Itô process as an integrator, Itô formula for functions of time and Wiener process, Itô formula for functions of time and Itô process. These conclusions are used to solve certain examples. Keywords: Wiener process, Stochastic integral, Itô formula 1
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Stochastic Integrals Driven by Isonormal Gaussian Processes and Applications
Čoupek, Petr ; Maslowski, Bohdan (advisor) ; Dostál, Petr (referee)
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.
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Ito formula and its applications
Till, Alexander ; Haman, Jiří (advisor) ; Maslowski, Bohdan (referee)
Title: Itô formula and its applications Author: Alexander Till Department: Department of Probability and Mathematical Statistics Supervisor: Mgr. Jiří Haman Supervisor's e-mail address: j.haman@seznam.cz Abstract: The bachelor thesis contains basis and elementary findings of stochastic analysis. It includes definition and properties of stochastic integral with Wiener process as an integrator, definition of stochastic integral with Itô process as an integrator, Itô formula for functions of time and Wiener process, Itô formula for functions of time and Itô process. These conclusions are used to solve certain examples. Keywords: Wiener process, Stochastic integral, Itô formula 1
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