Nadaraya-Watson estimators for stochastic differential equations driven by fractional Brownian motion

Abstract

In this article, we investigate the nonparametric Nadaraya-Watson estimator for the drift function of stochastic differential equations driven by fractional Brownian motion of the Hurst parameter . The drift function is a one-sided dissipative Lipschitz that ensures the ergodic property for the stochastic differential equation. The explicit formula of the estimator is obtained by using the Wick product based on the discretely observed process, which is of the utmost importance for practical applications. With the proper bandwidth selectors, we derive the strong consistency of the proposed estimator, and the main tools are ergodic theory and Malliavin calculus.