Nonparametric Estimation for Independent and Identically Distributed Stochastic Differential Equations with Space-Time Dependent Coefficients

Abstract

SIAM/ASA Journal on Uncertainty Quantification, Volume 12, Issue 2, Page 377-410, June 2024.
Abstract. We consider [math] independent and identically distributed one-dimensional inhomogeneous diffusion processes [math] with drift [math] and diffusion coefficient [math], where [math] and the functions [math] and [math] are known. Our concern is the nonparametric estimation of the [math]-dimensional unknown function [math] from the continuous observation of the sample paths [math] throughout a fixed time interval [math]. A collection of projection estimators belonging to a product of finite-dimensional subspaces of [math] is built. The [math]-risk is defined by the expectation of either an empirical norm or a deterministic norm fitted to the problem. Rates of convergence for large [math] are discussed. A data-driven choice of the dimensions of the projection spaces is proposed. The theoretical results are illustrated by numerical experiments on simulated data.