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.
Article PDF
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Received: 14 January 1998 / Revised version: 9 April 1998
Rights 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
Issue Date:
DOI: https://doi.org/10.1007/s004400050171