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Parametric inference for hypoelliptic ergodic diffusions with full observations

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  • Anna Melnykova

    (Université de Cergy-Pontoise, AGM UMR-CNRS 8088
    Université de Grenoble Alpes, LJK UMR-CNRS 5224)

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, with the discrete time observations of both coordinates being available on an interval $$T = n\varDelta _n$$ T = n Δ n , with $$\varDelta _n$$ Δ n the time step between the observations. The estimation is studied in the asymptotic setting, with $$T\rightarrow \infty $$ T → ∞ as $$\varDelta _n\rightarrow 0$$ Δ n → 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.

Suggested Citation

  • Anna Melnykova, 2020. "Parametric inference for hypoelliptic ergodic diffusions with full observations," Statistical Inference for Stochastic Processes, Springer, vol. 23(3), pages 595-635, October.
    • Handle: RePEc:spr:sistpr:v:23:y:2020:i:3:d:10.1007_s11203-020-09222-4
    DOI: 10.1007/s11203-020-09222-4
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