Probability Theory and Related Fields

, Volume 111, Issue 3, pp 333–374 | Cite as

Integration with respect to fractal functions and stochastic calculus. I

  • M. Zähle


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.

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


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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

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

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