Nonparametric volatility estimation in scalar diffusions: Optimality across observation frequencies

TL;DR

This paper introduces a new nonparametric estimator for scalar diffusion volatility that achieves optimal convergence rates across different observation frequencies, supported by theoretical proofs and numerical validation.

Contribution

It presents the first estimator that attains minimax optimal rates for both high and low-frequency data in scalar diffusion models.

Findings

Estimator achieves minimax optimal convergence rates.

Theoretical proofs validate estimator's optimality.

Numerical example demonstrates practical performance.

Abstract

The nonparametric volatility estimation problem of a scalar diffusion process observed at equidistant time points is addressed. Using the spectral representation of the volatility in terms of the invariant density and an eigenpair of the infinitesimal generator the first known estimator that attains the minimax optimal convergence rates for both high and low-frequency observations is constructed. The proofs are based on a posteriori error bounds for generalized eigenvalue problems as well as the path properties of scalar diffusions and stochastic analysis. The finite sample performance is illustrated by a numerical example.