Nonparametric Estimation in SDE Models Involving an Explanatory Process

This paper deals with the process [Formula: see text] defined by the stochastic differential equation (SDE) [Formula: see text], where [Formula: see text] is a Brownian motion and [Formula: see text] is an exogenous process. The first task—of probabilistic nature—is to properly define the model, to prove the existence and uniqueness of the solution of such an equation, and then to establish the existence and a suitable control of a density with respect to the Lebesgue measure of the distribution of [Formula: see text] ([Formula: see text]). In the second part of the paper, a risk bound and a rate of convergence in specific Sobolev spaces are established for a copies-based projection least squares estimator of the [Formula: see text]-valued function [Formula: see text]. Moreover, a model selection procedure making the adequate bias-variance compromise both in theory and practice is investigated.