Efficient estimation for diffusions sampled at high frequency over a fixed time interval

TL;DR

This paper develops conditions for efficient, consistent, and rate-optimal parametric estimation of diffusion processes from high-frequency data over a fixed interval, with asymptotic distributions characterized by normal variance-mixtures.

Contribution

It introduces verifiable conditions on approximate martingale estimating functions ensuring estimator efficiency and normal convergence, extending high-frequency diffusion estimation theory.

Findings

Estimators are shown to be consistent and asymptotically normal.

Asymptotic distributions are normal variance-mixtures depending on the sample path.

Simulation studies compare efficient and non-efficient estimators in different models.

Abstract

Parametric estimation for diffusion processes is considered for high frequency observations over a fixed time interval. The processes solve stochastic differential equations with an unknown parameter in the diffusion coefficient. We find easily verified conditions on approximate martingale estimating functions under which estimators are consistent, rate optimal, and efficient under high frequency (in-fill) asymptotics. The asymptotic distributions of the estimators are shown to be normal variance-mixtures, where the mixing distribution generally depends on the full sample path of the diffusion process over the observation time interval. Utilising the concept of stable convergence, we also obtain the more easily applicable result that for a suitable data dependent normalisation, the estimators converge in distribution to a standard normal distribution. The theory is illustrated by a simulation study comparing an efficient and a non-efficient estimating function for an ergodic and a non-ergodic model.