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.