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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

Wiener process-a random process with continuous time-plays an important role in mathematics, physics or economy. It is often good to know whether it contains any deterministic part, e.g. drift or scale. However, it is nearly impossible either observe the whole trajectory of the process or preserve its full history. This thesis deals with a statistical analysis based on partial observations, namely passage times through some given barriers. We propose several statistical methods for testing hypotheses about drift or scale using these observations. As supporting methods, we consider the maximum likelihood theory, non-parametric test against a trend, and binomial test. For testing the value of scale in the model with no drift and constant scale we recommend maximum likelihood theory. We derive the estimate and related tests in the case of observing only three barriers. The simulation study suggested observing more barriers for testing monotony of scale in a model with linear drift, or testing monotone and convex/concave drift in a model with constant scale. 1

The purpose of this work is to state the Donsker's invariance principle which is about the relation of a random walk and the Wiener process and to make its detailed proof. Then we will deal with the usage of the Donsker's invariance prin- ciple when simulating the trajectory of the Wiener process and we will simulate it with a few of distributions of steps of a random walk. In the next part we will focus on the first passage time of the random walk for which we will derive the distribution and compare to the first passage time of the Wiener process. 1

In this thesis we research and introduce several properties of paths of a Wiener process. At first we present a way to prove existence of a Wiener process and then we discuss its basic properties. The second chapter is devoted to analytical properties of Wiener's paths including monotonicity, differentiability, Hölder continuity and quadratic variation. In the third chapter we research the reflection principle and the distribution of maxima of paths in the case of a random walk and then also in the case of a Wiener process. The fourth chapter concentrates on the Skorohod embedding and its application in the proof of the classic central limit theorem. Finally, using the results of the first chapter we simulate a path of a Wiener process and illustrate some of the properties discussed earlier. To demonstrate the concepts, several problems were included in the relevant chapters together with an author's solution. Powered by TCPDF (www.tcpdf.org)