Generalized ITO integral and Henstock-Young integral

Abstract

The Ito integral is an integral of adapted processes with respect to a Brownian motion. It is an integral of Stieltjes-type. Unfortunately, paths of a Brownian motion are of unbounded variation on a compact interval. Hence the classical measure and integration theory cannot be applied to the Ito integral. K. Ito defined his integral in 1944 by the L^2-limit of a Cauchy sequence of integrals of simple processes. This approach is less intuitive than that of the Riemann-Stieltjes approach. In this thesis, we shall use the Riemann-Stieltjes approach with nonuniform meshes to study integrals of processes with respect to a Brownian motion, without assuming adaptedness. Furthermore, we also use this approach to study integrals with integrators of unbounded variation.