A white noise approach to stochastic integration with respect to 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.