Fractional processes, pathwise stochastic analysis and finance

Heikki Tikanmäki

Research output: ThesisDoctoral ThesisCollection of Articles


This thesis is about fractional processes, their pathwise stochastic analysis and financial applications. Firstly, we introduce a new definition for fractional Lévy processes using the integral representation of fractional Brownian motion on a compact interval. The properties of such processes, as well as connections to the earlier definitions of fractional Lévy processes, are studied. Secondly, the thesis contains integral representations for functionals depending on various averages of (geometric) fractional Brownian motion. These integral representations can be used for obtaining hedges for Asian options in fractional pricing models. Finally, we introduce the core financial contribution of the thesis that is to study the connections between pathwise functional calculus and robust hedging in so-called mixed models. In such models, the price of an asset is modeled as an exponential of the sum of Brownian motion and a fractional process.
Translated title of the contributionFraktionaaliset prosessit, poluttainen stokastinen analyysi ja rahoitusteoria
Original languageEnglish
QualificationDoctor's degree
Awarding Institution
  • Aalto University
  • Valkeila, Esko, Supervising Professor
  • Valkeila, Esko, Thesis Advisor
Print ISBNs978-952-60-4600-6
Electronic ISBNs978-952-60-4601-3
Publication statusPublished - 2012
MoE publication typeG5 Doctoral dissertation (article)


  • fractional Brownian motion
  • fractional Lévy process
  • functional Ito calculus
  • vertical derivative
  • Asian options
  • arithmetic average
  • robust hedging
  • nonsemimartingale models

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