General Stochastic Vector Integration - Three Approaches

The development of stochastic integration theory throughout the 20th century has led to several definitions and approaches of the stochastic integral, particularly for predictable integrands and semimartingale integrators. This survey provides an overview of the two prominent approaches in defining the stochastic integral: the classical approach attributed to Itô, Meyer and Jacod, and the more contemporary functional analytical approach mainly developed by Bichteler and Protter. It also delves into the historical milestones and achievements in this area and analyzes them from a modern perspective. Drawing inspiration from the similarities of existing approaches, this survey introduces a new topology-based approach to the general vector-valued stochastic integral for predictable integrands and semimartingale integrators. This new approach provides a faster, simpler way to define the general integral and offers a self-contained derivation of its key properties without depending on semimartingale decomposition theorems.