Nonparametric inference for fractional diffusion

TL;DR

This paper develops and analyzes non-parametric estimators for the drift function in a fractional diffusion model with fractional Brownian motion noise, providing theoretical guarantees and practical methods for both continuous and discrete data.

Contribution

It introduces two non-parametric estimators for the drift in fractional diffusion models and establishes their non-asymptotic properties under practical observation schemes.

Findings

Non-asymptotic deviation bounds for the estimators

Consistency of estimators under ergodic conditions

Effective estimation with discrete observations

Abstract

A non-parametric diffusion model with an additive fractional Brownian motion noise is considered in this work. The drift is a non-parametric function that will be estimated by two methods. On one hand, we propose a locally linear estimator based on the local approximation of the drift by a linear function. On the other hand, a Nadaraya-Watson kernel type estimator is studied. In both cases, some non-asymptotic results are proposed by means of deviation probability bound. The consistency property of the estimators are obtained under a one sided dissipative Lipschitz condition on the drift that insures the ergodic property for the stochastic differential equation. Our estimators are first constructed under continuous observations. The drift function is then estimated with discrete time observations that is of the most importance for practical applications.