Stochastic integration for a wide class of Gaussian stationary increment processes using an extension of the S-transform

TL;DR

This paper develops a method to define stochastic integrals for a broad class of Gaussian stationary increment processes using an extension of the S-transform within white noise space theory, including an Ito formula.

Contribution

It introduces an extension of the S-transform to define Wick-Ito integrals for Gaussian processes with stationary increments, expanding the applicability of white noise analysis.

Findings

Defines stochastic integrals for Gaussian stationary increment processes.

Establishes an Ito formula within this framework.

Connects spectral density with white noise space via Bochner-Minlos theorem.

Abstract

Given a Gaussian stationary increment processes with spectral density, we show that a Wick-Ito integral with respect to this process can be naturally obtained using Hida's white noise space theory. We use the Bochner-Minlos theorem to associate a probability space to the process, and define the counterpart of the S-transform in this space. We then use this transform to define the stochastic integral and prove an associated Ito formula.