INFERENCE FOR ERGODIC MCKEAN-VLASOV STOCHASTIC DIFFERENTIAL EQUATIONS WITH POLYNOMIAL INTERACTIONS

We consider a specific 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 leading to an approximate likelihood function. We then derive explicit estimators of the coefficients of the interaction term and prove their consistency and asymptotic normality with rate √ T as T grows to infinity. Examples illustrating the theory are proposed