On a Universal Strictly Decreasing Nonparametric Estimator Applied to the Drift Function of a Recurrent Diffusion Process Estimation

Abstract

This paper deals with a copies-based continuously differentiable and strictly decreasing estimator of the drift function for stochastic differential equations defining recurrent diffusion processes. The first part of our paper deals with non-asymptotic $\mathbb L^1$-risk bounds and a bandwidths selection procedure for a universal monotone estimator. These results are tailor-made to our framework, and then applied to the estimation of the drift function of recurrent diffusion processes in the second part of the paper.