[Paper Review] Estimation of the Hurst and the stability indices of a $H$-self-similar stable process

Thi To Nhu Dang, Jacques Istas|arXiv (Cornell University)|Jun 18, 2015

Financial Risk and Volatility Modeling17 references17 citations

TL;DR

This paper proposes a novel estimation method for the Hurst parameter $ H $ and stability index $ \alpha $ of $ H $-self-similar symmetric $ \alpha $-stable processes using $ \beta $-negative power variations with $ -\frac{1}{2} < \beta < 0 $. The approach yields consistent estimators with established rates of convergence and asymptotic normality for fractional Brownian motion and $ S\alpha S $-stable Lévy motion, enabling separate estimation of $ H $ and $ \alpha $ without requiring prior knowledge of either parameter.

ABSTRACT

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 $\\alpha$-stable process. The process $X$ is observed at points $\\frac{k}{n}$, $k=0,\\ldots,n$. Our estimate is based on $\\beta$-variations with $-\\frac{1}{2}&lt;\\beta&lt;0$. We obtain consistent estimators, with rate of convergence, for several classical $H$-sssi $\\alpha$-stable processes (fractional Brownian motion, well-balanced linear fractional stable motion, Takenaka's processes, L\\'evy motion). Moreover, we obtain asymptotic normality of our estimators for fractional Brownian motion and L\\'evy motion. Keywords: H-sssi processes; stable processes; self-similarity parameter estimator; stability parameter estimator.

Motivation & Objective

• To develop consistent estimators for both the self-similarity parameter $ H $ and the stability index $ \alpha $ of $ H $-sssi symmetric $ \alpha $-stable processes.
• To enable separate estimation of $ H $ and $ \alpha $ without requiring prior knowledge of either parameter.
• To establish rates of convergence and asymptotic normality for the proposed estimators under general conditions on the covariance structure of negative power variations.
• To validate the method on classical processes such as fractional Brownian motion, $ S\alpha S $-stable Lévy motion, and others.

Proposed method

• Estimation is based on $ \beta $-negative power variations of the process $ X $ observed at discrete times $ \frac{k}{n} $, for $ -\frac{1}{2} < \beta < 0 $.
• The method relies on the existence of moments and covariances of $ |\Delta_{k,1}X|^\beta $, which are well-defined for $ \beta \in (-\frac{1}{2}, 0) $ due to the density of stable variables.
• Consistency and convergence rates are derived under a general assumption on the series of covariances of the negative power variations.
• Asymptotic normality is established for the estimators in the cases of fractional Brownian motion and $ S\alpha S $-stable Lévy motion.
• The approach avoids assumptions on the existence of higher-order moments of the underlying process.
• Theoretical results are validated through detailed analysis of four classical $ H $-sssi $ \alpha $-stable processes.

Experimental results

Research questions

• RQ1 Can consistent and asymptotically normal estimators be constructed for both $ H $ and $ \alpha $ in $ H $-sssi $ \alpha $-stable processes using a unified framework?
• RQ2 Can the estimation of $ H $ be performed without assuming knowledge of $ \alpha $, and vice versa?
• RQ3 What are the rates of convergence of the proposed estimators for different classes of $ H $-sssi $ \alpha $-stable processes?
• RQ4 Under what conditions does the asymptotic normality of the estimators hold?

Key findings

• The proposed estimators for $ H $ and $ \alpha $ are consistent under a general assumption on the covariance structure of $ \beta $-negative power variations.
• For fractional Brownian motion and $ S\alpha S $-stable Lévy motion, the estimators achieve asymptotic normality with convergence rate $ n^{-\frac{1}{2}} $.
• The method allows separate estimation of $ H $ and $ \alpha $, without requiring prior knowledge of either parameter.
• The rate of convergence for the estimators is $ O(n^{-1/2}) $ for fractional Brownian motion and $ S\alpha S $-stable Lévy motion.
• Theoretical convergence rates are validated through analysis of four classical processes: fractional Brownian motion, $ S\alpha S $-stable Lévy motion, well-balanced linear fractional stable motion, and Takenaka’s process.
• The method is robust to the absence of higher-order moments, as it relies only on the existence of negative power variations for $ \beta \in (-\frac{1}{2}, 0) $.