Self-similarity parameter estimation and reproduction property for non-Gaussian Hermite processes
By Alexandra Chronopoulou, Frederi G. Viens, Ciprian A. Tudor. 2008
49 citations recorded by the source corpus.
We consider the class of all the Hermite processes $(Z_{t}^{(q,H)})_{t\in \lbrack 0,1]}$ of order $q\in \mathbf{N}^{\ast}$ and with Hurst parameter $% H\in (\frac{1}{2},1)$. The process $Z^{(q,H)}$ is $H$-selfsimilar, it has stationary increments and it exhibits long-range dependence identical to that of fractional Brownian motion (fBm). For $q=1$, $Z^{(1,H)}$ is fBm, which is Gaussian; for $q=2$, $Z^{(2,H)}$ is the Rosenblatt process, which lives in the second Wiener chaos; for any $q>2$, $Z^{(q,H)}$ is a process in the $q$th Wiener chaos. We study the variations of $Z^{(q,H)}$ for any $q$, 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 their variations give rise to other Hermite processes of different orders and with different Hurst parameters. We apply our results to construct a strongly consistent estimator for the self-similarity parameter $H$ from discrete observations of $Z^{(q,H)}$; the asymptotics of this estimator, after appropriate normalization, are proved to be distributed like a Rosenblatt random variable (value at time $1$ of a Rosenblatt process).with self-similarity parameter $1+2(H-1)/q$.
Subfields: Economics and Econometrics, Finance
View Paper: https://www.semanticscholar.org/paper/f94b71c8d796100222bef5622878b49439b898f4
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In Economics and Econometrics
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- Ivan Nourdin
- T. T. Diu Tran
- Giovanni Peccati
- David Nualart
- Frederi G. Viens
- Gesine Reinert
- Khalifa Es-Sebaiy
- Anthony R'eveillac
Citations in the corpus
- Statistical inference for Vasicek-type model driven by Hermite processes (2017, background): “With the exception of the aforementioned papers [5, 6, 14, 27], all these papers, and the current paper, make no attempt to estimate the Hurst parameter H.”