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L. Petroczki:
"Stochastic Processes, Orthogonal Polynomials and Chaos Expansions";
Betreuer/in(nen): T. Levajković, M. Oberguggenberger; Fakultät für Mathematik, Informatik und Physik, Universität Innsbruck, 2019; Abschlussprüfung: 14.01.2019.

The main focus of this thesis is on the study of orthogonal polynomials and their con- nection to stochastic processes. One of the goals is to build families of orthogonal stochastic polynomials which serve as basis of Hilbert spaces of the square integrable random variables and stochastic processes. Each random variable is represented through formal power series in terms of stochastic polynomials that are built from deterministic orthogonal polynomials which fulfill the three-term orthogonality recurrence. The con- struction is based on the Askey scheme of hypergeometric functions. For the construction we followed the historical paper of Meixner. We study four families of Askey orthogo- nal polynomials: the Hermite, the Laguerre, the Charlier and the Meixner polynomials. The main requirement used to connect Askey polynomials with stochastic processes is the martingale property. We show the connection between families of polynomials and stochastic processes and introduce the Askey-Itô chaos expansion theorem. In the second part of the thesis we consider fractional calculus and chaos expansion representation of fractional stochastic processes. Particularly we first review the basics of the fractional calculus for Brownian motion and then present the results known in the literature for Lévy processes without Brownian part. As a particular example, in the last part of the thesis we represent the Gamma process through Askey-Itô chaos expansion representation, where the basis is built by the Laguerre polynomials. We focus on Gamma processes due to the fact that it is an important process appearing in mathematical finance and engineering. Finally we apply the method of chaos expansion to stochastic equations involving different type of noise processes.