Parametric inference for ergodic McKean-Vlasov stochastic differential equations

We consider a one-dimensional McKean-Vlasov stochastic differential equation with potential and interaction terms depending on unknown parameters. The sample path is continuously observed on a time interval 0,2T. We assume that the process is in the stationary regime. As this distribution is not explicit, the exact likelihood does not lead to computable estimators. To overcome this difficulty, we consider a kernel estimator of the invariant density based on the sample path on 0,T and obtain new properties for this estimator. Then, we derive an explicit approximate likelihood using the sample path on T,2T, including the kernel estimator of the invariant density and study the associated estimators of the unknown parameters. We prove their consistency and asymptotic normality with rate T as T grows to infinity. Several classes of models illustrate the theory.