Penalized nonparametric mean square estimation of the coefficients of diffusion processes

TL;DR

This paper introduces a nonparametric, penalized least squares method for estimating the drift and diffusion coefficients of stationary diffusion processes from discrete observations, achieving optimal convergence rates.

Contribution

It develops a data-driven, finite-dimensional estimator with non-asymptotic risk bounds for stationary diffusion processes, reaching minimax optimal rates as sampling becomes finer.

Findings

Estimators achieve minimax optimal convergence rates.

Numerical simulations demonstrate estimator effectiveness.

Method adapts to unknown function complexity.

Abstract

We consider a one-dimensional diffusion process $(X_t)$ which is observed at $n+1$ discrete times with regular sampling interval $\Delta$. Assuming that $(X_t)$ is strictly stationary, we propose nonparametric estimators of the drift and diffusion coefficients obtained by a penalized least squares approach. Our estimators belong to a finite-dimensional function space whose dimension is selected by a data-driven method. We provide non-asymptotic risk bounds for the estimators. When the sampling interval tends to zero while the number of observations and the length of the observation time interval tend to infinity, we show that our estimators reach the minimax optimal rates of convergence. Numerical results based on exact simulations of diffusion processes are given for several examples of models and illustrate the qualities of our estimation algorithms.