Parametric inference for hypoelliptic ergodic diffusions with full observations

TL;DR

This paper develops a consistent and asymptotically normal parameter estimator for two-dimensional hypoelliptic diffusions observed discretely, addressing challenges posed by their degenerate structure, with applications to neuronal models.

Contribution

It introduces a novel estimation method for hypoelliptic diffusions using discretized likelihood, proving its consistency and asymptotic normality, and demonstrates its effectiveness on neuronal models.

Findings

Estimator is consistent and asymptotically normal.

Method successfully applied to FitzHugh-Nagumo neuronal model.

Addresses hypoellipticity challenges in parameter estimation.

Abstract

Multidimensional hypoelliptic diffusions arise naturally in different fields, for example to model neuronal activity. Estimation in those models is complex because of the degenerate structure of the diffusion coefficient. In this paper we consider hypoelliptic diffusions, given as a solution of two-dimensional stochastic differential equations (SDEs), with the discrete time observations of both coordinates being available on an interval $T = n\Delta_n$, with $\Delta_n$ the time step between the observations. The estimation is studied in the asymptotic setting, with $T\to\infty$ as $\Delta_n\to 0$. We build a consistent estimator of the drift and variance parameters with the help of a discretized log-likelihood of the continuous process. We discuss the difficulties generated by the hypoellipticity and provide a proof of the consistency and the asymptotic normality of the estimator. We test our approach numerically on the hypoelliptic FitzHugh-Nagumo model, which describes the firing mechanism of a neuron.