From forward integrals to Wick-It\^o integrals: the fractional Brownian motion and the Rosenblatt process cases

In this paper, we combine Hida distribution theory and Sobolev-Watanabe-Kree spaces in order to study finely the link between forward integrals obtained by regularization and Wick-It\^o integrals with respect to fractional Brownian motion and the Rosenblatt process. The new methodology developed in this paper allows to retrieve results for fractional Brownian motion and to obtain new results regarding the Rosenblatt process. In particular, an It\^o formula for functionals of the Rosenblatt process is obtained which holds in the space of square-integrable random variables.