Spectral estimation for diffusions with random sampling times

TL;DR

This paper develops an adaptive spectral estimation method for scalar diffusions observed at random times, achieving minimax optimality and extending previous spectral techniques to more general sampling schemes.

Contribution

It generalizes the spectral estimation approach to handle random sampling times and proves its optimality and adaptivity in a nonparametric setting.

Findings

Estimator is minimax optimal.

Method adapts to sampling time distribution.

Numerical example demonstrates effectiveness.

Abstract

The nonparametric estimation of the volatility and the drift coefficient of a scalar diffusion is studied when the process is observed at random time points. The constructed estimator generalizes the spectral method by Gobet, Hoffmann and Rei{\ss} [Ann. Statist. 32 (2006), 2223-2253]. The estimation procedure is optimal in the minimax sense and adaptive with respect to the sampling time distribution and the regularity of the coefficients. The proofs are based on the eigenvalue problem for the generalized transition operator. The finite sample performance is illustrated in a numerical example.