Spectral estimation of the fractional order of a L\'{e}vy process

TL;DR

This paper introduces a spectral method for estimating the fractional order of a Lévy process using low-frequency data, providing a unified approach for estimation and calibration with proven convergence rates.

Contribution

It develops a novel spectral estimation procedure for the fractional order of Lévy processes, including minimax convergence rates and a data-driven aggregation algorithm.

Findings

Estimator achieves minimax convergence rates.

Asymptotic normality of the estimator is established.

Simulation results demonstrate effectiveness in estimation and calibration.

Abstract

We consider the problem of estimating the fractional order of a L\'{e}vy 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.