Projection Estimators of the Stationary Density of a Differential Equation Driven by the Fractional Brownian Motion

TL;DR

This paper introduces projection estimators for the stationary density of solutions to differential equations driven by fractional Brownian motion, providing a model selection method and an oracle inequality for adaptive estimation.

Contribution

It presents a novel approach to density estimation for fractional Brownian motion-driven systems, including a model selection technique and theoretical guarantees.

Findings

Established an oracle inequality for the adaptive estimator.

Provided a concentration inequality for Lipschitz functionals of discrete samples.

Demonstrated the effectiveness of the projection estimators in this context.

Abstract

The paper deals with projection estimators of the density of the stationary solution $X$ to a differential equation driven by the fractional Brownian motion under a dissipativity condition on the drift function. A model selection method is provided and, thanks to the concentration inequality for Lipschitz functionals of discrete samples of $X$ proved in Bertin et al. (2020), an oracle inequality is established for the adaptive estimator.