Asymptotic behavior of weighted cubic variation of sub-fractional Brownian motion. (English) Zbl 1357.60058
Summary: In this article, we investigate the convergence of renormalized weighted cubic variation of a sub-fractional Brownian motion \(S^{H}\) with Hurst index \(H\). When \(0<H<\frac{1}{6}\), we prove by means of Malliavin calculus that the convergence holds in \(L^{2}\) toward an explicit limit which only depends on \(S^{H}\). We also numerically simulate the sample paths of such a type of sub-fractional Brownian motion regulated by different Hurst index \(H\).
MSC:
| 60H07 | Stochastic calculus of variations and the Malliavin calculus |
| 60G15 | Gaussian processes |
| 60G22 | Fractional processes, including fractional Brownian motion |
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