Extensions of the Hitsuda-Skorokhod integral

TL;DR

This paper introduces new definitions of the Hitsuda-Skorokhod stochastic integral, extending its domain to $L^p$-spaces, and clarifies their relationships using the S-transform and exponential processes.

Contribution

It provides alternative, extended definitions of the Hitsuda-Skorokhod integral applicable to broader $L^p$-domains, based on S-transform characterization and exponential processes.

Findings

The new integral extends existing definitions to $L^p$-spaces.

Connections between different stochastic integrals are clarified.

The approach simplifies the understanding of the integral's domain extension.

Abstract

We present alternative definitions of the stochastic integral introduced by Ayew and Kuo and of the Hitsuda-Skorokhod integral extended to domains in $L^p$-spaces, $p \geq 1$. Our approach is motivated by the S-transform characterization of the Hitsuda-Skorokhod integral and based on simple processes of stochastic exponential type. We prove that the new stochastic integral extends the mentioned stochastic integrals above and we outline their connection.