Pathwise Stieltjes integrals of discontinuously evaluated stochastic processes

Zhe Chen, Lasse Leskelä*, Lauri Viitasaari

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

1 Citation (Scopus)

Abstract

In this article we study the existence of pathwise Stieltjes integrals of the form ∫f(Xt)dYt for nonrandom, possibly discontinuous, evaluation functions f and Hölder continuous random processes X and Y. We discuss a notion of sufficient variability for the process X which ensures that the paths of the composite process t↦f(Xt) are almost surely regular enough to be integrable. We show that the pathwise integral can be defined as a limit of Riemann–Stieltjes sums for a large class of discontinuous evaluation functions of locally finite variation, and provide new estimates on the accuracy of numerical approximations of such integrals, together with a change of variables formula for integrals of the form ∫f(Xt)dXt.

Original languageEnglish
Pages (from-to)2723-2757
Number of pages35
JournalStochastic Processes and their Applications
Volume129
Issue number8
Early online date1 Jan 2018
DOIs
Publication statusPublished - 1 Aug 2019
MoE publication typeA1 Journal article-refereed

Keywords

  • Bounded p-variation
  • Composite stochastic process
  • Fractional calculus
  • Fractional Sobolev space
  • Fractional Sobolev–Slobodeckij space
  • Gagliardo–Slobodeckij seminorm
  • Generalised Stieltjes integral
  • Riemann–Liouville integral

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