Nonparametric adaptive estimation for interacting particle systems

We consider a stochastic system of N$$ N $$ 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]$$ \left[0,T\right] $$ for fixed T$$ T $$ and large N$$ N $$. We define two collections of projection estimators belonging to finite‐dimensional subspaces of 𝕃2([0,T]). We study the 𝕃2‐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.