Distributional It\^o's Formula and Regularization of Generalized Wiener Functionals

TL;DR

This paper develops a distributional Itô's formula for diffusions and studies the conditions under which generalized Wiener functionals are Bochner integrable within certain Sobolev spaces.

Contribution

It introduces a new Itô formula in a distributional setting and analyzes Bochner integrability conditions for generalized Wiener functionals.

Findings

Formulated an Itô formula for diffusion processes in a distributional framework.

Characterized differentiability and integrability indices for Bochner integrals of Wiener functionals.

Provided conditions for the membership of generalized Wiener functionals in Sobolev spaces.

Abstract

We investigate Bochner integrabilities of generalized Wiener functionals. We further formulate an It\^o formula for a diffusion in a distributional setting, and apply to investigate differentiability-index $s$ and integrability-index $p \geqslant 2$ for which the Bochner integral belongs to $\mathbb{D}_{p}^{s}$.