# Characterization of the Ito Integral

## TL;DR

This paper characterizes the stochastic Ito integral with respect to Wiener processes, establishing existence, uniqueness, and key properties of the integral as a mapping from integrable processes to continuous adapted processes.

## Contribution

It provides a rigorous characterization of the Ito integral, including conditions for its existence, uniqueness, and convergence properties, enhancing theoretical understanding.

## Key findings

- Defines the Ito integral as a mapping from measurable, adapted processes to continuous adapted processes.
- Establishes conditions under which stochastic integrals of simple processes are calculated.
- Shows convergence in probability of the integrals when squared integrands' time integrals converge.

## Abstract

This paper provides an existence-and-uniqueness theorem characterizing the stochastic integral with respect to a Wiener process. The integral is represented as a mapping from the space of measurable and adapted pathwise locally integrable processes to the space of continuous adapted processes. It is characterized in terms of two properties: (1) how the stochastic integrals of simple processes are calculated and (2) how these integrals converge in probability when the time integrals of the squared integrands converge in probability.