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## Parametric inference for hypoelliptic ergodic diffusions with full observations

Anna Melnykova

#### 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. We build a consistent estimator of the drift and variance parameters with the help of a discretized log-likelihood of the continuous process when discrete time observations of both coordinates are available on an interval $T = n\Delta_n$, with $\Delta_n$ the time step between the observations. We discuss the difficulties generated by the hypoellipticity and provide a proof of the consistency and the asymptotic normality of the estimator in the asymptotic setting $T\to\infty$ as $\Delta_n\to 0$. We test our approach numerically on the hypoelliptic FitzHugh-Nagumo model, which describes the firing mechanism of a neuron.

### Dates and versions

hal-01704010 , version 1 (08-02-2018)
hal-01704010 , version 2 (11-01-2019)
hal-01704010 , version 3 (24-06-2020)
hal-01704010 , version 4 (15-07-2020)

### Identifiers

• HAL Id : hal-01704010 , version 3
• ARXIV :

### Cite

Anna Melnykova. Parametric inference for hypoelliptic ergodic diffusions with full observations. 2020. ⟨hal-01704010v3⟩

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