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

Abstract

Let $(X_t)$ be solution of a one-dimensional McKean-Vlasov stochastic differential equation with classical drift term $V(\mathit{\alpha },x)$, self-stabilizing term $\mathrm{\Phi }(\mathit{\beta },x)$ and small noise amplitude ε. Our aim is to study the estimation of the unknown parameters $\mathit{\alpha },\mathit{\beta }$ from a continuous observation of $(X_t,t\in [0,T])$ 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 $t\ge 0$ and ε are derived. We then build an explicit approximate log-likelihood leading to consistent and asymptotically Gaussian estimators, under the condition that $\mathit{\epsilon }\sqrt{T}$ tends to 0, with original rates of convergence: the rate for the estimation of α is either ${\mathit{\epsilon }}^{-1}$ or $\sqrt{T}{\mathit{\epsilon }}^{-1}$, the rate for the estimation of β is $\sqrt{T}$. Moreover, the estimators are asymptotically efficient.