Nonparametric Estimation in Fractional SDE

TL;DR

This paper investigates the consistency and convergence rate of a nonparametric estimator for the drift in fractional SDEs driven by fractional Brownian motion, using advanced stochastic analysis techniques.

Contribution

It introduces a new analysis of the Nadaraya-Watson estimator's properties for fractional SDEs, leveraging long-time behavior and Skorokhod integral properties.

Findings

Estimator is consistent for fractional SDEs.

Derived explicit convergence rates.

Illustrated on fractional Ornstein-Uhlenbeck process.

Abstract

This paper deals with the consistency and a rate of convergence for a Nadaraya-Watson estimator of the drift function of a stochastic differential equation driven by an additive fractional noise. The results of this paper are obtained via both some long-time behavior properties of Hairer and some properties of the Skorokhod integral with respect to the fractional Brownian motion. These results are illustrated on the fractional Ornstein-Uhlenbeck process.