The classical Lebesgue±Stieltjes integral f dg of real or complex-valued functions on a ®nite interval Y 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 HoÈ lder 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 Itoà formula remains valid. Moreover, this approach enables us to derive calculation rules for pathwise de®ned stochastic integrals with respect to fractional Brownian motion.

No abstract

No abstract

Problems of intersections of processes of HAWSDORFF rectifiable closed sets in Rn are studied. The first part deals with integral geometric fundamentals and questions of measurability. In the second part the stochastic approach is given.for Y"-almost all z A n (Bz ) is Xm+P-*-naeasurabb and P+P-*-rectifiabb, and for yy( Vx-,cx,(")) Z r n ( d 4 V X ) = J h(r) Y " ( d r ) J s ( f ( X ) ) j qy( V x ( 4 ) X*(W @(dX)(by the property of f and the invariance of Zm) Vx-r(x)) s m ( d z ) ~( d x ) ~( d r ) x-r Friedrich-Schiller-Universitat Sektion Mathemntik DDR -6900 Jena Universitatshochhaus

For locally finite unions of sets with positive reach in R a, generalized unit normal bundles are introduced in support of a certain set additive index function. Given an appropriate orientation to the normal bundle, signed curvature measures may be defined by means of associated locally rectifiable currents (with index function as multiplicity) and specially chosen differential forms. In the case of 'regular' sets this is shown to be equivalent to well-known classical concepts via former results. The present approach leads to unified methods in proving integral-geometric relations. Some of them are stated in this paper, Geometriae Dedieata 23 (1987) 155-171