Iguchi, Yuga; (2025) Statistical Inference for Elliptic and Hypo-elliptic Diffusions: Asymptotic Theory and Numerical Methodologies. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

The thesis aims to contribute to statistical inference for Stochastic Differential Equations (SDEs), a rich model class of continuous-time nonlinear Markov processes. The scope of this work includes an impor- tant model class characterised as hypo-elliptic SDEs that contain a degenerate diffusion matrix. The statistical inference for hypo-elliptic models has been recently highlighted in both theory and practice but was still not sufficiently understood analytically compared to the standard SDEs with non-degenerate diffusion matrix, i.e. elliptic SDEs. To close this gap, we establish theoretical and numerical foundations for parameter estimation of a broad class of SDEs, including the hypo-elliptic ones. This thesis first introduces model frameworks that cover such a wide class of SDEs. We then pro- pose time-discretisation schemes carefully designed for those model frameworks to construct tractable likelihood functions that are key to conducting statistical inference for SDEs. The use of the developed likelihood is justified via a standard asymptotic analysis in Statistics; precisely, we show that the max- imum likelihood type estimator based on the likelihood has the desirable asymptotic properties such as consistency and Central Limit Theorem (CLT) under the high-frequency observations regime, i.e. the scenario where the step size between consecutive data points is sufficiently small. In particular, the proposed estimators can achieve the CLT under a less restrictive condition on the data to make the parameter estimates reliable with larger step sizes. We also propose tractable higher order time discretisation schemes for SDEs, which can be used in parametric inference under the low-frequency observations regime, where one needs to carry out Data Augmentation (DA), i.e. by augmenting the missing data or latent variables between data points with simulation. We present analytical and numer- ical results showcasing that the use of developed schemes in DA effectively reduces the discretisation bias compared to the standard time-discretisation.

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