# A note on stochastic integrals as $L^2$-curves

## TL;DR

This paper explores the representation of stochastic integrals as $L^2$-curves, completing the connection to the classical Itô-integral and discussing an application to stochastic PDEs.

## Contribution

It extends previous work by fully establishing the link between $L^2$-curves and the Itô-integral, and sketches an application to stochastic partial differential equations.

## Key findings

- Complete connection between $L^2$-curves and Itô-integral.
- Provides a framework for stochastic PDE applications.
- Builds on van Gaans (2005a) and Filipović and Tappe (2008).

## Abstract

In a work of van Gaans (2005a) stochastic integrals are regarded as $L^2$-curves. In Filipovi\'{c} and Tappe (2008) we have shown the connection to the usual It\^o-integral for c\`adl\`ag-integrands. The goal of this note is to complete this result and to provide the full connection to the It\^o-integral. We also sketch an application to stochastic partial differential equations.