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Hypoelliptic diffusions: filtering and inference from complete and partial observations

Abstract : The statistical problem of parameter estimation in partially observed hypoel-liptic diffusion processes is naturally occurring in many applications. However, due to the noise structure, where the noise components of the different coordinates of the multi-dimensional process operate on different time scales, standard inference tools are ill conditioned. In this paper, we propose to use a higher order scheme to discretize the process and approximate the likelihood, such that the different time scales are appropriately accounted for. We show consistency and asymptotic normality with non-typical convergence rates. When only partial observations are available, we embed the approximation into a filtering algorithm for the unobserved coordinates, and use this as a building block in a Stochastic Approximation Expectation Maximization algorithm. We illustrate on simulated data from three models; the Harmonic Oscillator, the FitzHugh-Nagumo model used to model the membrane potential evolution in neuroscience, and the Synaptic Inhibition and Excitation model used for determination of neuronal synaptic input.
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https://hal.archives-ouvertes.fr/hal-01627616
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Submitted on : Wednesday, December 26, 2018 - 4:57:35 PM
Last modification on : Sunday, March 29, 2020 - 1:08:14 AM
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Susanne Ditlevsen, Adeline Samson. Hypoelliptic diffusions: filtering and inference from complete and partial observations. Journal of the Royal Statistical Society: Series B, Royal Statistical Society, 2019, 81 (2), pp.361-384. ⟨10.1111/rssb.12307⟩. ⟨hal-01627616v3⟩

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