A white noise approach to stochastic integration with respect to the Rosenblatt process

TL;DR

This paper develops a white noise-based stochastic calculus for the Rosenblatt process, including explicit formulas, Itô's formula, and comparisons with other methods, expanding tools for non-Gaussian stochastic analysis.

Contribution

It introduces a novel white noise distribution framework for stochastic integration with respect to the Rosenblatt process, including explicit variance formulas and Itô's formula.

Findings

Derived the translated characteristic function of the Rosenblatt process.

Defined stochastic integrals using white noise derivatives and Wick multiplication.

Provided explicit variance formulas and Itô's formula for functionals of the Rosenblatt process.

Abstract

In this paper, we define a stochastic calculus with respect to the Rosenblatt process by means of white noise distribution theory. For this purpose, we compute the translated characteristic function of the Rosenblatt process at time $t>0$ in any direction $\xi\in S(\mathbb{R})$ and the derivative of the Rosenblatt process in the white noise sense. Using Wick multiplication by the former derivative and Pettis integration, we define our stochastic integral with respect to the Rosenblatt process for a wide class of distribution processes. We obtain an explicit formula for the variance of such a stochastic integral and It\^o's formulae for a certain class of functionals of the Rosenblatt process. Finally, we compare our stochastic integral to other approaches.