Thi To Nhu Dang | Searcharxiv

arXiv2017

Estimation of the multifractional function and the stability index of linear multifractional stable processes

In this paper we are interested in multifractional stable processes where the self-similarity index $H$ is a function of time, in other words $H$ becomes time changing, and the stability index $α$ is a constant. Using $β$- negative power variations ($-1/2<β<0$), we propose estimators for the value of the multifractional function $H$ at a fixed time $t_0$ and for $α$ for two cases: multifractional Brownian motion ($α=2$) and linear multifractional stable motion ($0<α<2$). We get the consistency of our estimates for the underlying processes with the rate of convergence.

math.ST↗

arXiv2017

Estimation of the Hurst and the stability indices of a $H$-self-similar stable process

In this paper we estimate both the Hurst and the stable indices of a H-self-similar stable process. More precisely, let $X$ be a $H$-sssi (self-similar stationary increments) symmetric $α$-stable process. The process $X$ is observed at points $\frac{k}{n}$, $k=0,\ldots,n$. Our estimate is based on $β$-variations with $-\frac{1}{2}<β<0$. We obtain consistent estimators, with rate of convergence, for several classical $H$-sssi $α$-stable processes (fractional Brownian motion, well-balanced linear fractional stable motion, Takenaka's processes, Lévy motion). Moreover, we obtain asymptotic normality of our estimators for fractional Brownian motion and Lévy motion. Keywords: H-sssi processes; stable processes; self-similarity parameter estimator; stability parameter estimator.

math.ST↗