Nonparametric moment method for scalar McKean–Vlasov stochastic differential equations

We study the nonparametric estimation of both the potential and the interaction terms of a scalar McKean–Vlasov stochastic differential equation (SDE) in stationary regime from a continuous observation on a time interval [0, T], with asymptotic framework T → +∞. To estimate the two functions, we consider the observation of four i.i.d. sample paths. The observation of two sample paths could be enough at the cost of much more computations. Estimators of the potential and the interaction functions are built using a combination of a moment method and a projection method on sieves. The potential and the interaction term do not belong to 𝕃2(ℝ), so we define a specific risk fitted to this estimation problem and obtain a bound for it. A nonparametric estimator of the invariant density also is proposed. The method is implemented on simulated data for several examples of McKean–Vlasov SDEs and a model selection procedure is experimented.