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Stochastic Integration
Týbl, Ondřej ; Maslowski, Bohdan (advisor) ; Dostál, Petr (referee)
The object of this thesis is a theory of stochastic integration, i.e., an inte- gration of a stochastic process with respect to a stochastic process. First, the Ito integral with respect to processes with finite quadratic variation is presented. This integral is then used to define the Stratonovich integral and both integrals are subsequently compared in terms of a martingale property and so-called chain rule. The core of this work is then a comparison of these two integrals as limits of aproximating sums. A third variant of an integral, first introduced in Strato- novich (1966), is then defined as a limit of sums of a different type. The resulting integral is equivalent to the original Stratonovich integral when the integrand is the Wiener process, however, it may differ if even when integrating with respect to a continuous process (a counterexample Yor (1977) is provided). A sufficient condition for an equivalence of these two integrals from Protter (2004) is presen- ted. 1
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Stochastic Integrals
Karal, David ; Maslowski, Bohdan (advisor) ; Seidler, Jan (referee)
Stochastic Integrals David Karal Abstrakt In this thesis we study the Wiener process and stochastic integrals. The thesis defines the basic objects of stochastic analysis and the existence of the Wiener process and some of its properties are shown. This process is then used to con- struct the Itô stochastic integral, where the Wiener process acts as an integrator. The Itô stochastic integral is first defined for simple processes and subsequently extended to mathcalFt-progressively measurable processes. Then, the integral is generalized to stochastic integral driven by any continuous martingale. In the end of the thesis, the Stratonovich integral is defined and its relationship with the Itô integral is investigated. 1
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