Inference for ergodic McKean-Vlasov stochastic differential equations with polynomial interactions

We consider a family of one-dimensional McKean-Vlasov stochastic differential equations with no potential term and with interaction term modeled by an odd increasing polynomial. We assume that the observed process is in stationary regime and that the sample path is continuously observed on a time interval [0, 2T]. Due to the McKean-Vlasov structure, the drift function depends on the unknown marginal law of the process in addition to the unknown parameters present in the interaction function. This is why the exact likelihood function does not lead to computable estimators. We overcome this difficulty by a two-step approach: We use the observations on [0, T] to build empirical estimates of moments of the stationary distribution and on [T, 2T] to build an approximate likelihood. We derive explicit estimators of the interaction term parameters, which are proved to be consistent and asymptotically normal with rate T as T grows to infinity. Examples illustrating the theory are proposed.