Let be solution of a one-dimensional McKean-Vlasov stochastic differential equation with classical drift term , self-stabilizing term and small noise amplitude ε. Our aim is to study the estimation of the unknown parameters from a continuous observation of under the double asymptotic framework ε tends to 0 and T tends to infinity. After centering and normalization of the process, uniform bounds for moments with respect to and ε are derived. We then build an explicit approximate log-likelihood leading to consistent and asymptotically Gaussian estimators, under the condition that tends to 0, with original rates of convergence: the rate for the estimation of α is either or , the rate for the estimation of β is . Moreover, the estimators are asymptotically efficient.
We thank the two referees for their helpful and detailed comments.
"Parametric inference for small variance and long time horizon McKean-Vlasov diffusion models." Electron. J. Statist. 15 (2) 5811 - 5854, 2021. https://doi.org/10.1214/21-EJS1922