Weak characterizations of stochastic integrability and Dudley's theorem in infinite dimensions

TL;DR

This paper explores weak characterizations of stochastic integrability in infinite dimensions, providing counterexamples and extending Dudley's representation theorem to infinite-dimensional spaces.

Contribution

It introduces new weak characterizations of stochastic integrability, demonstrates their limitations with counterexamples, and extends Dudley's representation theorem to infinite-dimensional settings.

Findings

Counterexample shows limitations of weak characterizations

Extended Dudley's theorem to infinite dimensions

Provided new representation results for infinite-dimensional martingales

Abstract

In this paper we consider stochastic integration with respect to cylindrical Brownian motion in infinite dimensional spaces. We study weak characterizations of stochastic integrability and present a natural continuation of results of van Neerven, Weis and the second named author. The limitation of weak characterizations will be demonstrated with a nontrivial counterexample. The second subject treated in the paper addresses representation theory for random variables in terms of stochastic integrals. In particular, we provide an infinite dimensional version of Dudley's representation theorem for random variables and an extension of Doob's representation for martingales.