Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Integration with respect to fractal functions and stochastic calculus. I
Download PDF
Download PDF
  • Article
  • Published: July 1998

Integration with respect to fractal functions and stochastic calculus. I

  • M. Zähle1 

Probability Theory and Related Fields volume 111, pages 333–374 (1998)Cite this article

  • 849 Accesses

  • 355 Citations

  • Metrics details

Abstract.

The classical Lebesgue–Stieltjes integral ∫b a fdg of real or complex-valued functions on a finite interval (a,b) is extended to a large class of integrands f and integrators g of unbounded variation. The key is to use composition formulas and integration-by-part rules for fractional integrals and Weyl derivatives. In the special case of Hölder continuous functions f and g of summed order greater than 1 convergence of the corresponding Riemann–Stieltjes sums is proved.

The results are applied to stochastic integrals where g is replaced by the Wiener process and f by adapted as well as anticipating random functions. In the anticipating case we work within Slobodeckij spaces and introduce a stochastic integral for which the classical Itô formula remains valid. Moreover, this approach enables us to derive calculation rules for pathwise defined stochastic integrals with respect to fractional Brownian motion.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

Author information

Authors and Affiliations

  1. Mathematical Institute, University of Jena, Ernst-Abbe-Platz 1-4, D-07740 Jena, Germany. e-mail: zaehle@minet.uni-jena.de, , , , , , DE

    M. Zähle

Authors
  1. M. Zähle
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Received: 14 January 1998 / Revised version: 9 April 1998

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Zähle, M. Integration with respect to fractal functions and stochastic calculus. I. Probab Theory Relat Fields 111, 333–374 (1998). https://doi.org/10.1007/s004400050171

Download citation

  • Issue Date: July 1998

  • DOI: https://doi.org/10.1007/s004400050171

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

  • Mathematical Subject Classification(1991): Primary 60H05; Secondary 26A33
  • 26A42
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature