Integration in a Normal World: Fractional Brownian Motion and Beyond

Research output: ThesisDoctoral ThesisCollection of Articles

Abstract

This thesis is about stochastic integration with respect to Gaussian processes that are notsemimartingales. Firstly, we study approximations of integrals with respect to fractionalBrownian motion and derive an upper bound for an average approximation error. Secondly, westudy the existence of pathwise integrals with respect to a wide class of Gaussian processes andintegrands. 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 containsintegral representations for arbitrary random variables. Finally, we study a certain modelinvolving a Gaussian process and provide estimators for different parameters. We applyMalliavin calculus and divergence integrals to obtain central limit theorems for our estimators.
Translated title of the contributionIntegrointi normaalissa maailmassa: fraktionaalinen Brownin liike sekä laajennuksia
Original languageEnglish
QualificationDoctor's degree
Awarding Institution
  • Aalto University
Supervisors/Advisors
  • Nevanlinna, Olavi, Supervising Professor
  • Valkeila, Esko, Supervising Professor
  • Sottinen, Tommi, Thesis Advisor, External person
Publisher
Print ISBNs978-952-60-5548-0
Electronic ISBNs978-952-60-5549-7
Publication statusPublished - 2014
MoE publication typeG5 Doctoral dissertation (article)

Keywords

  • Gaussian process
  • fractional Brownian motion
  • approximation error
  • pathwise integrals
  • integral representation
  • parameter estimation
  • divergence integrals

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