Notes on the Ogawa integrability and a condition for convergence in the multidimensional case

TL;DR

This paper reviews the Ogawa stochastic integral within abstract Wiener spaces, investigates the conditions for its universal integrability in multiple dimensions, and demonstrates the necessity of a renormalization term through explicit examples.

Contribution

It introduces a necessary renormalization term for universal Ogawa integrability in multidimensional settings and provides explicit examples illustrating this requirement.

Findings

Universal Ogawa integrability generally requires a renormalization term in multiple dimensions.

Explicit examples show cases where integrability fails without renormalization.

The paper clarifies conditions under which the Ogawa integral converges in the multidimensional case.

Abstract

The Ogawa stochastic integral is shortly reviewed and formulated in the framework of abstract Wiener spaces. The condition of universal Ogawa integrability in the multidimensional case is investigated, proving that it cannot hold in general without the introduction of a "renormalization term". Explicit examples are provided.