Self-similarity parameter estimation and reproduction property for non-Gaussian Hermite processes

Abstract

Let (Z (q;H) t )t2(0;1) be a Hermite processes of order q and with Hurst parameter H 2 ( 1 ;1). This process is H-self-similar, it has stationary increments and it exhibits long-range dependence. This class contains the fractional Brownian motion (for q = 1) and the Rosenblatt process (for q = 2). We study in this paper the variations of Z (q;H) by using multiple Wiener -It^o stochastic integrals and Malliavin calculus. We prove a reproduction property for this class of processes in the sense that the terms appearing in the chaotic decomposition of the their variations give birth to other Hermite processes of dierent orders and with dierent Hurst parameters. We apply our results to construct a consistent estimator for the self-similarity parameter from discrete observations of a Hermite process.