Parametric inference for small variance and long time horizon McKean-Vlasov diffusion models

Let (Xt) be solution of a one-dimensional McKean-Vlasov stochastic differential equation with classical drift term V (α, x), self-stabilizing term Φ(β, x) and small noise amplitude e. Our aim is to study the estimation of the unknown parameters α, β from a continuous observation of (Xt, t ∈ [0, T ]) under the double asymptotic framework e tends to 0 and T tends to infinity. After centering and normalization of the process, uniform bounds for moments with respect to t ≥ 0 and e are derived. We then build an explicit approximate log-likelihood leading to consistent and asymptotically Gaussian estimators with original rates of convergence: the rate for the estimation of α is either e −1 or √ T e −1 , the rate for the estimation of β is √ T .