Stochastic integration based on simple, symmetric random walks

TL;DR

This paper introduces a new stochastic integration method using simple, symmetric random walks for pathwise approximation, aiming for greater transparency and simplicity compared to existing techniques.

Contribution

It presents a novel approach to stochastic integration based on a.s. uniform convergence of simple, symmetric random walks, which is more didactically advantageous and less technically demanding.

Findings

Achieves a.s. uniform convergence on compacts

Provides a.s. convergence for integrals of finite variation functions

Applicable even when the integrator is not cdlg

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.