# Isomorphisms for spaces of predictable processes and an extension of the It\^{o} integral

## TL;DR

This paper demonstrates that spaces of predictable processes in Banach spaces are isomorphic to other process spaces, extending the Itô integral to infinite dimensions and applying it to stochastic PDEs.

## Contribution

It provides a simple proof of isomorphisms between predictable and adapted process spaces, extending the Itô integral to infinite-dimensional settings.

## Key findings

- Spaces of predictable processes are isomorphic to spaces of adapted processes.
- Extension of the Itô integral to infinite-dimensional Banach spaces.
- Application to stochastic partial differential equations.

## Abstract

Our goal of this note is to give an easy proof that spaces of predictable processes with values in a Banach space are isomorphic to spaces of progressive resp. adapted, measurable processes. This provides a straightforward extension of the It\^{o} integral in infinite dimensions. We also outline an application to stochastic partial differential equations.