Nonparametric adaptive estimation for Interacting particle systems

Abstract

. We consider a stochastic system of N interacting particles with constant di(cid:27)usion coe(cid:30)cient and drift linear in space, time-depending on two unknown deterministic functions. Our concern here is the nonparametric estimation of these functions from a continuous observation of the process on [0 , T ] for (cid:28)xed T and large N . We de(cid:28)ne two collections of projection estimators belonging to (cid:28)nite-dimensional subspaces of L 2 ([0 , T ]) . We study the L 2 -risks of these estimators, where the risk is de(cid:28)ned either by the expectation of an empirical norm or by the expectation of a deterministic norm. Afterwards, we propose a data-driven choice of the dimensions and study the risk of the adaptive estimators. The results are illustrated by numerical experiments on simulated data.