Integration With Respect To Fractal Functions And Stochastic Calculus

The classical Lebesgue--Stieltjes integral b R a f dg 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 Holder continuous functions f and g of summary 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 Slobodecki spaces and introduce a stochastic integral for which the classical Ito formula remains valid. Moreover, this approach enables us to derive calculation rules for pathwise defined stochastic integrals with respect to fractional Brownian motion. 0 Introduction In order to motivate our paper we recall some well--..