Nonparametric adaptive estimation for interacting particle systems

We consider a stochastic system of N interacting particles with constant diffusion coefficient 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 fixed T and large N. We define two collections of projection estimators belonging to finite-dimensional subspaces of L-2([0, T]). We study the L-2-risks of these estimators, where the risk is defined 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.