Probabilistic properties and parametric inference of small variance nonlinear self-stabilizing stochastic differential equations

Abstract

We consider a process (Xt) solution of a one-dimensional nonlinear self-stabilizing stochastic differential equation, with classical drift term V(α,x) depending on an unknown parameter α, self-stabilizing term Φ(β,x) depending on another unknown parameter β and small noise amplitude ɛ. Self-Stabilization is the effect of including a mean-field interaction in addition to the state-dependent drift. Adding this term leads to a nonlinear or McKean-Vlasov Markov process with transitions depending on the distribution of Xt. We study the probabilistic properties of (Xt) as ɛ tends to 0 and exhibit a Gaussian approximating process for (Xt). Next, we study the estimation of (α,β) from a continuous observation of (Xt,t∈[0,T]). We build explicit estimators using an approximate log-likelihood function obtained from the exact log-likelihood function of a proxi-model. We prove that, for fixed T, as ɛ tends to 0, α can be consistently estimated with rate ɛ−1but notβ. Then, considering ni.i.d. sample paths (Xti,i=1,…,n), we build consistent and asymptotically Gaussian estimators of (α,β) with rates nɛ−1 for α and n for β. Finally, we prove that the statistical experiments generated by (Xt) and the proxi-model are asymptotically equivalent in the sense of the Le Cam Δ-distance both for the continuous observation of one path and for ni.i.d. paths under the condition nɛ→0, which justifies our statistical method.