Integration in a Normal World: Fractional Brownian Motion and Beyond
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چکیده
Aalto University, P.O. Box 11000, FI-00076 Aalto www.aalto.fi Author Lauri Viitasaari Name of the doctoral dissertation Integration in a Normal World: Fractional Brownian Motion and Beyond Publisher School of Science Unit Department of Mathematics and Systems Analysis Series Aalto University publication series DOCTORAL DISSERTATIONS 14/2014 Field of research Mathematics Manuscript submitted 12 November 2013 Date of the defence 28 February 2014 Permission to publish granted (date) 14 January 2014 Language English Monograph Article dissertation (summary + original articles) Abstract This thesis is about stochastic integration with respect to Gaussian processes that are not semimartingales. Firstly, we study approximations of integrals with respect to fractional Brownian motion and derive an upper bound for an average approximation error. Secondly, we study the existence of pathwise integrals with respect to a wide class of Gaussian processes and integrands. We prove the existence of two different notions of pathwise integrals. Moreover, these two different integrals coincide. As an application of these results, the thesis contains integral representations for arbitrary random variables. Finally, we study a certain model involving a Gaussian process and provide estimators for different parameters. We apply Malliavin calculus and divergence integrals to obtain central limit theorems for our estimators.
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تاریخ انتشار 2014