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Isomorphisms for spaces of predictable processes and an extension of the Itô integral

Barbara Rüdiger, Stefan Tappe

TL;DR

This work proves that spaces of Banach-space valued predictable, progressive, and adapted processes are isometrically isomorphic when the driving kernel is absolutely continuous, enabling a straightforward extension of the It\^{o} integral to Banach-space valued, adapted integrands. The authors construct the inverse embedding as the predictable projection and show that it reduces to the left-continuous version $\Phi_-$ for càdlàg integrands. They then develop extended It\^{o} integrals for three driving noises—martingales, infinite-dimensional Wiener processes, and compensated Poisson random measures—consistent with existing theory while preserving the martingale property. Finally, they illustrate the approach with stochastic partial differential equations, showing existence and uniqueness of mild solutions under Lipschitz conditions. The results provide a unified, operator-friendly framework for infinite-dimensional stochastic integration and SPDE analysis.

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 Itô integral in infinite dimensions. We also outline an application to stochastic partial differential equations.

Isomorphisms for spaces of predictable processes and an extension of the Itô integral

TL;DR

This work proves that spaces of Banach-space valued predictable, progressive, and adapted processes are isometrically isomorphic when the driving kernel is absolutely continuous, enabling a straightforward extension of the It\^{o} integral to Banach-space valued, adapted integrands. The authors construct the inverse embedding as the predictable projection and show that it reduces to the left-continuous version for càdlàg integrands. They then develop extended It\^{o} integrals for three driving noises—martingales, infinite-dimensional Wiener processes, and compensated Poisson random measures—consistent with existing theory while preserving the martingale property. Finally, they illustrate the approach with stochastic partial differential equations, showing existence and uniqueness of mild solutions under Lipschitz conditions. The results provide a unified, operator-friendly framework for infinite-dimensional stochastic integration and SPDE analysis.

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 Itô integral in infinite dimensions. We also outline an application to stochastic partial differential equations.

Paper Structure

This paper contains 7 sections, 2 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Isomorphisms for spaces of predictable processes
  3. The Itô integral for adapted, measurable processes
  4. The Itô integral with respect to martingales
  5. The Itô integral with respect to infinite dimensional Wiener processes
  6. The Itô integral with respect to compensated Poisson random measures
  7. Stochastic partial differential equations

Key Result

Theorem 2.1

Suppose there is a nonnegative, measurable function $f : \tilde{\Omega} \times [0,T] \rightarrow \mathbb{R}$ such that for each $\tilde{\omega} \in \tilde{\Omega}$ we have $K(\tilde{\omega},dt) = f(\tilde{\omega},t)dt$. Then we have

Theorems & Definitions (6)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4