Nadaraya-Watson Estimator for I.I.D. Paths of Diffusion Processes

TL;DR

This paper introduces a nonparametric Nadaraya-Watson estimator for the drift function of diffusion processes, providing risk bounds, bandwidth selection methods, and numerical validation.

Contribution

It develops a novel estimator for diffusion process drifts with theoretical risk bounds and practical bandwidth selection techniques.

Findings

Risk bounds for the estimator and its approximation

Effective bandwidth selection methods demonstrated

Numerical experiments validate the approach

Abstract

This paper deals with a nonparametric Nadaraya-Watson estimator $\hat b$ of the drift function computed from independent continuous observations of a diffusion process. Risk bounds on $\hat b$ and its discrete-time approximation are established. The paper also deals with extensions of the PCO and leave-one-out cross validation bandwidth selection methods for $\hat b$. Finally, some numerical experiments are provided.