On Pathwise Stochastic Integration of Processes with Unbounded Power Variation

Zhe Chen

Research output: ThesisDoctoral ThesisMonograph

Abstract

This dissertation concerns pathwise integrability of stochastic processes which are non-semimartingales with unbounded power variation. In this dissertation, a class of stochastic processes which can be represented as a composition of a Hölder continuous process with a nonrandom function of locally bounded variation is studied. Since the nonrandom function may contain discontinuities, stochastic processes in this class are usually of unbounded power variation. This kind of stochastic processes are of interest in many applications, for example in financial mathematics concerning option pricing. In this dissertation, new conditions are presented for the existence of generalized Lebesgue–Stieltjes integrals for the aforementioned one-dimensional stochastic processes with respect to general Hölder continuous processes. This dissertation also contains a new result on the existence of generalized Lebesgue–Stieltjes integrals for a certain class of multi-dimensional stochastic processes with respect to general Hölder continuous processes. Moreover, in this dissertation, a new proof is presented for a change of variables formula for sufficiently regular one-dimensional stochastic processes with unbounded power variation.
Translated title of the contributionOn Pathwise Stochastic Integration of Processes with Unbounded Power Variation
Original languageEnglish
QualificationDoctor's degree
Awarding Institution
  • Aalto University
Supervisors/Advisors
  • Leskelä, Lasse, Supervising Professor
  • Viitasaari, Lauri, Thesis Advisor
Publisher
Print ISBNs978-952-60-6671-4
Electronic ISBNs978-952-60-6672-1
Publication statusPublished - 2016
MoE publication typeG4 Doctoral dissertation (monograph)

Keywords

  • pathwise integration
  • Hölder process
  • unbounded p-variation
  • generalized Lebesgue-Stieltjes integration

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