Nonparametric estimation of the diffusion coefficient from S.D.E. paths

TL;DR

This paper introduces ridge estimators for the nonparametric estimation of the diffusion coefficient in stochastic differential equations, demonstrating their consistency and convergence rates through theoretical analysis and numerical simulations.

Contribution

It proposes a novel ridge estimator for the diffusion coefficient based on discrete high-frequency data, with proven consistency and convergence rates.

Findings

Estimators are consistent as sample size increases.

Convergence rates are derived for the estimators.

Numerical simulations validate theoretical results.

Abstract

Consider a diffusion process X=(X_t), with t in [0,1], observed at discrete times and high frequency, solution of a stochastic differential equation whose drift and diffusion coefficients are assumed to be unknown. In this article, we focus on the nonparametric esstimation of the diffusion coefficient. We propose ridge estimators of the square of the diffusion coefficient from discrete observations of X and that are obtained by minimization of the least squares contrast. We prove that the estimators are consistent and derive rates of convergence as the size of the sample paths tends to infinity, and the discretization step of the time interval [0,1] tend to zero. The theoretical results are completed with a numerical study over synthetic data.