Stochastic integration based on simple, symmetric random walks

Abstract

A new approach to stochastic integration is described, which is based on an a.s. pathwise approximation of the integrator by simple, symmetric random walks. Hopefully, this method is didactically more advantageous, more transparent, and technically less demanding than other existing ones. In a large part of the theory one has a.s. uniform convergence on compacts. In particular, it gives a.s. convergence for the stochastic integral of a finite variation function of the integrator, which is not c\`adl\`ag in general.