The Annals of Statistics

Spectral estimation of the fractional order of a Lévy process

Denis Belomestny

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We consider the problem of estimating the fractional order of a Lévy process from low frequency historical and options data. An estimation methodology is developed which allows us to treat both estimation and calibration problems in a unified way. The corresponding procedure consists of two steps: the estimation of a conditional characteristic function and the weighted least squares estimation of the fractional order in spectral domain. While the second step is identical for both calibration and estimation, the first one depends on the problem at hand. Minimax rates of convergence for the fractional order estimate are derived, the asymptotic normality is proved and a data-driven algorithm based on aggregation is proposed. The performance of the estimator in both estimation and calibration setups is illustrated by a simulation study.

Article information

Ann. Statist. Volume 38, Number 1 (2010), 317-351.

First available in Project Euclid: 31 December 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F10: Point estimation
Secondary: 62J12: Generalized linear models 62F25: Tolerance and confidence regions 62H12: Estimation

Regular Lévy processes Blumenthal–Getoor index semiparametric estimation


Belomestny, Denis. Spectral estimation of the fractional order of a Lévy process. Ann. Statist. 38 (2010), no. 1, 317--351. doi:10.1214/09-AOS715.

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